Research Article Designing a Robust Nonlinear Dynamic...
Transcript of Research Article Designing a Robust Nonlinear Dynamic...
Research ArticleDesigning a Robust Nonlinear Dynamic Inversion Controller forSpacecraft Formation Flying
Inseok Yang1 Dongik Lee2 and Dong Seog Han2
1 Center for ICT and Automobile Convergence Kyungpook National University Daegu 702-701 Republic of Korea2 School of Electronics Engineering Kyungpook National University Daegu 702-701 Republic of Korea
Correspondence should be addressed to Dong Seog Han dshanknuackr
Received 18 April 2014 Accepted 1 July 2014 Published 17 July 2014
Academic Editor Vivian Martins Gomes
Copyright copy 2014 Inseok Yang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
The robust nonlinear dynamic inversion (RNDI) control technique is proposed to keep the relative position of spacecrafts whileformation flying The proposed RNDI control method is based on nonlinear dynamic inversion (NDI) NDI is nonlinear controlmethod that replaces the original dynamics into the user-selected desired dynamics Because NDI removes nonlinearities in themodel by inverting the original dynamics directly it also eliminates the need of designing suitable controllers for each equilibriumpoint that is NDI works as self-scheduled controller Removing the original model also provides advantages of ease to satisfythe specific requirements by simply handling desired dynamics Therefore NDI is simple and has many similarities to classicalcontrol In real applications however it is difficult to achieve perfect cancellation of the original dynamics due to uncertaintiesthat lead to performance degradation and even make the system unstable This paper proposes robustness assurance method forNDI The proposed RNDI is designed by combining NDI and sliding mode control (SMC) SMC is inherently robust using high-speed switching inputs This paper verifies similarities of NDI and SMC firstly And then RNDI control method is proposed Theperformance of the proposed method is evaluated by simulations applied to spacecraft formation flying problem
1 Introduction
Spacecraft formation flying (SFF) problem is a cooperativecontrol problem that distributes the task of a single spacecraftinto a group of spacecrafts to improve the robustness of aspace mission by decreasing the possibility of a single failurethat can lead to total mission loss [1ndash3] For this reason SFFhas attracted considerable interest owing to its advantagesof not only increased mission success probability but alsoincreased feasibility flexibility and so forth According to thedistribution strategies the control methods for SFF can becategorized into two ways centralized and decentralized [1]In decentralized control each spacecraft in a formation groupcan communicate with each other In particular by transmit-ting the condition of each vehicle they can avoid the fault-sensitive problem that can lead to serious problems suchas collision The main drawbacks of decentralized formationcontrol are the difficulties in analyzing the global SFF stabilityand the increase of system complexity significantly if forma-tion member is big In contrast centralized formation flying
such as the leader-follower approach is easy to implementbecause formation can be achieved by controlling the relativeposition or velocity The objective of SFF problem in thispaper is to evaluate the formation keeping performance usingthe proposed controller Therefore the leader-follower for-mation problem of two spacecrafts is considered This paperconsiders the full nonlinear dynamics describing the relativeposition to design a suitable nonlinear controller for SFF
In this paper spacecraft formation flying using the robustnonlinear dynamic inversion (RNDI) control method isproposed The proposed RNDI control method is basedon well-known nonlinear dynamic inversion (NDI) NDIis a nonlinear control synthesis technique that steers thesystem states to track a user-designed desired trajectory [4ndash6] By inverting the original dynamics to remove the systemnonlinearities directly NDI does not require linearizingand designing gain-scheduled controllers for each equilib-rium point Therefore NDI provides a solution for thedifficulties in ensuring stabilities and performances betweenvarious operational points in gain-scheduled controllersThe
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 471352 12 pageshttpdxdoiorg1011552014471352
2 Mathematical Problems in Engineering
Dynamic inversion
PI-typedesired
dynamics
Dynamicinversion Plant
(a) PI-type of desired dynamics based method
Dynamic inversion
Robustlinear
controller
Dynamicinversion Plant
Desireddynamics
(b) Robust linear controller based method
Figure 1 Block diagrams that have been proposed to achieve robustness
advantages of NDI control are that it is conceptually simpleand has many similarities to classical control methods Inaddition it naturally handles nonlinear systemswithout gain-scheduling because it works as a self-scheduled controllerFor this reason NDI has become a popular flight controllerand been applied to various high-performance aircrafts suchas X-38 [4 5] F-18 HARV [7 8] and F-16 [8 9] On theother hand accurate knowledge of the nonlinear systemdynamics is needed to obtain a perfect cancellation of theoriginal dynamics Actually in real applications such anassumption of taking all accurate information about systemnonlinearities is rarely met due to uncertainties such asmodel mismatches disturbances and measurement noises[4 9ndash13] As these uncertainties lead to the closed-loopsystem control performance degradation and make the sys-tem unstable robustness issues against uncertainties must beconsidered when designing NDI controller
A variety of methods have been proposed to achieverobustness inNDI controllerMany studies have reported thatthe system controlled by NDI with a proportional-integral-(PI-) type of desired dynamics can improve the robustness[10 13 14] As shown in Figure 1(a) the PI controller isdesigned as the desired dynamics satisfying the specificrequirements that the states track after removing the originaldynamics by NDI The PI controller takes advantages of sim-ple design and similaritieswith classical controlHowever thedesired dynamics types are selected by considering the sys-tem requirements so there are some systems where the PI-type of desired dynamics is unsuitable One of the mostwidely used methods for solving the robustness issue is toemploy an additional linear controller such as the structuredsingular value (120583-analysis) and 119867
infinsynthesis [4 5 8 9 11]
In these methods NDI works as an inner-loop controllerwhile an additional linear robust controller is employed asan outer-loop controller as shown in Figure 1(b) Thereforethe linear controller attempts improve the robustness of theoverall control system These methods however are basedon linearized system equations and lead to an increase of theorder of the control system For example the controller orderincreases to 14while designing the119867
infincontroller for the X-38
[5] Recently Yang et al proposed a robust dynamic inversion(RDI) controlmethod using slidingmode control (SMC) [15]SMC is a robust nonlinear control method that takes distinctvalues as control inputs to steer the system states into a user-designed sliding surface and to maintain the states on it [1617] By combining dynamic inversion (DI) and SMC the RDIcontroller guarantees stability against uncertainties withoutusing any additional outer-loop controller and moreoverit takes the advantages of both controllers (Figure 2) [15]However the proposed RDI is designed as a linear controllerso it cannot cover the advantages of self-scheduling of NDIcontrollerThis paper extends the results of [15] for nonlinearsystems and verifies the feasibility of the proposed RNDIcontroller with application to the SFF problem
2 Mathematical Model ofSpacecraft Formation Flying
This section presents the derivation of a nonlinear SFFmodelFormation flying considered in this paper is assumed tobe composed of two spacecrafts leader and follower Theleader spacecraft provides the reference trajectory assumedto be a circular orbit with constant velocity 120596
0while the
follower spacecraft navigates the neighborhood of the leader
Mathematical Problems in Engineering 3
Robust dynamic inversion
Desireddynamics
Robustdynamicinversion
Plant
Figure 2 Block diagram of the proposed robust nonlinear dynamic inversion
p
Leader
Follower
Ez
Ex
Ey
eh
rf
er
e120579
rl
Figure 3 Reference coordinate frame [1]
according to the desired reference trajectory Figure 3 showsthe coordinate reference frames used throughout this paperTwo main coordinate frames are the Earth centered inertiaframe E
119909E119910E119911 whose origin is located in the center of the
Earth and the leader orbit frame e119903 e120579 eℎ whose origin is
attached to the center of mass of the leader spacecraft Thenonlinear dynamics of the leader and follower with respectto the e
119903 e120579 eℎ frame can be represented as follows [1 3]
r119897= minus
120583
1199033
119897
r119897+F119889119897
119898119897
+u119897
119898119897
(1)
r119891= minus
120583
1199033
119891
r119891+F119889119891
119898119891
+u119891
119898119891
(2)
where u119897isin R3 and u
119891isin R3 are control input vectors of the
leader and follower spacecrafts respectively And F119889119897isin R3
and F119889119891
isin R3 are the disturbance force vectors acting onthe leader and follower spacecrafts respectively It is assumedthat the masses of spacecrafts are small relative to the massof the Earth that is 119872 ≫ 119898
119897 119898119891 so 119866(119872 + 119898
119897) asymp 119866119872
and 119866(119872 + 119898119891) asymp 119866119872 Because the relative position of the
follower spacecraft p = r119891minus r119897 the second derivative of the
relative position can be represented as follows
p = r119891minus r119897= (minus
120583
1199033
119891
(r119897+ p) +
F119889119891
119898119891
+u119891
119898119891
)
minus (minus120583
1199033
119897
r119897+F119889119897
119898119897
+u119897
119898119897
)
= minus119872119866(r119897+ p
1003817100381710038171003817r119897 + p10038171003817100381710038173minus
r119897
1199033
119897
) minusF119889
119898119891
+u119905
119898119891
(3)
where F119889= F119889119891minus (119898119891119898119897)F119889119897and u
119905= u119891minus (119898119891119898119897)u119897
In this paper it is assumed that the leader spacecraft is freeflying that is u
119897= 0 then it satisfies that u
119905= u119891
If the relative position p in the leader orbit frame isgiven as p = 119909e
119903+ 119910e120579+ 119911eℎfor the unit vectors e
119903 e120579
and eℎalong with the E
119909E119910E119911-frame then the relative
acceleration vector p can be obtained as follows
p = ( minus 21205960119910 minus 1205962
0119909) e119903+ ( 119910 + 2120596
0 minus 1205962
0119910) e120579+ eℎ (4)
In the moving leader orbit frame the position vector r119897is
constant that is r119897= 119903119897e119903 By substituting (4) into (3) the
nonlinear dynamics of the follower spacecraft relative to theleader can be obtained as
( minus 21205960119910 minus 1205962
0119909) e119903+ ( 119910 + 2120596
0 minus 1205962
0119910) e120579+ eℎ
= minus119872119866((119903119897+ 119909) e
119903+ 119910e120579+ 119911eℎ
1003817100381710038171003817r119897 + p10038171003817100381710038173
minus119903119897e119903
1199033
119897
)
minus119865119889119909e119903+ 119865119889119910e120579+ 119865119889119911eℎ
119898119891
+119906119909e119903+ 119906119910e120579+ 119906119911eℎ
119898119891
997904rArr [
[
119910
]
]
= minus[
[
0 minus212059600
21205960
0 0
0 0 0
]
]
[
[
119910
]
]
+
[[[[[[[[
[
1205962
0minus
119872119866
1003817100381710038171003817r119897 + p10038171003817100381710038173
0 0
0 1205962
0minus
119872119866
1003817100381710038171003817r119897 + p10038171003817100381710038173
0
0 0 minus119872119866
1003817100381710038171003817r119897 + p10038171003817100381710038173
]]]]]]]]
]
[
[
119909
119910
119911
]
]
4 Mathematical Problems in Engineering
minus119872119866[[[
[
119903119897
1003817100381710038171003817r119897 + p10038171003817100381710038173minus1
1199032119897
0
0
]]]
]
minus1
119898119891
[
[
119865119889119909
119865119889119910
119865119889119911
]
]
+1
119898119891
[
[
119906119909
119906119910
119906119911
]
]
(5)
Denote the relative position and velocity vectors as p119897=
(119909 119910 119911) and v119897= (119889119909119889119905 119889119910119889119905 119889119911119889119905) in the leader orbit
frame respectively Then (5) can be represented as follows
p119897= k119897
k119897= minus C (120596
0) k119897minusD (p
119897 r119897 1205960) p119897minus N (p
119897 r119897) minus
F119889
119898119891
+u119905
119898119891
(6)
where
C (1205960) = [
[
0 minus212059600
21205960
0 0
0 0 0
]
]
N (p119897 r119897) = 119872119866
[[[
[
119903119897
1003817100381710038171003817r119897 + p10038171003817100381710038173minus1
1199032119897
0
0
]]]
]
F119889= [
[
119865119889119909
119865119889119910
119865119889119911
]
]
D (p119897 r119897 1205960)
=
[[[[[[[[
[
1205962
0minus
119872119866
1003817100381710038171003817r119897 + p10038171003817100381710038173
0 0
0 1205962
0minus
119872119866
1003817100381710038171003817r119897 + p10038171003817100381710038173
0
0 0 minus119872119866
1003817100381710038171003817r119897 + p10038171003817100381710038173
]]]]]]]]
]
(7)
3 The Proposed Robust NonlinearDynamic Inversion
In this section the mathematical backgrounds of slidingmode control and nonlinear dynamic inversion are intro-duced And then the robust nonlinear dynamic inversionmethod is proposed
31 Sliding Mode Controller Sliding mode control (SMC)also called variable structure control (VSC) is a high-speedswitching nonlinear control method that forces the systemstates into a user-designed sliding surface and maintains thestates on the surface [17 18] SMC consists of two phases(Figure 4) structuring the sliding surface and constructingthe switching feedback gains
Consider the following system
x (119905) = f (x (119905)) + g (x (119905)) u (119905) (8)
Sliding surface
Sliding phase
Reaching phase
120590(x) = 0
x2
x1
x(t1)
x(t0)
Figure 4The behavior of the system states in sliding mode control
where x(119905) isin R119899 is a state vector u(119905) isin R119898 is an input vectorwith 119899 ge 119898 and f R119899 rarr R119899 and g R119898 rarr R119899times119898are smooth functions Let a set Σ
119894be the regular (119899 minus 1)
dimensional submanifold in R119899 defined in [16] such as
Σ119894≜ x isin R119899 | 120590
119894(x) = 0 119894 = 1 2 119898 (9)
where 120590119894 R119899 rarr R (119894 = 1 2 119898) is a smooth function
Define120590119894(x) = 0 as the individual sliding surface or individual
switching surface Then surface constructed by intersecting119898 individual sliding surfaces is defined as the sliding surface[16]
Σ ≜
119898
⋂
119894=1
Σ119894=
119898
⋂
119894=1
x isin R119899 | 120590119894(x) = 0 ≜ x isin R119899 | 120590 (x) = 0
(10)
where 120590(x) = [1205901(x) 1205902(x) 120590
119898(x)]119879 If the states x(119905) of
the system are in Σ that is 120590(x(119905)) = 0 then the behavior ofthe states is called sliding motion or sliding mode Moreoverif sliding mode exists the tangent vectors of the states alwaysare forced to track the sliding surface as shown in Figure 4Then each entry 119906
119894(119905) (119894 = 1 2 119898) of the sliding mode
control law can be obtained by taking one of the followingvalues [16]
119906119894(119905) =
119906+
119894 if 120590
119894(x (119905)) gt 0
119906minus
119894 if 120590
119894(x (119905)) lt 0
(11)
where 119906+119894= 119906minus
119894
As shown in (11) designing the sliding surface is crucialbecause the sliding surface determines the control input andmoreover the performance of the closed-loop system Forthis reason various designing methods such as Fillipovrsquosmethod and equivalent control method have been proposed[18 19] In this paper the equivalent control method is
Mathematical Problems in Engineering 5
considered to analyze the similar characteristics betweenSMC and NDI If the system states intercept the slidingsurface at 119905 = 119905
0and a sliding mode exists for 119905 ge 119905
0 then
(119889119889119905)120590(x(119905)) = 0 and 120590(x(119905)) = 0 for all 119905 ge 1199050 Then the
equivalent input ueq that forces the state trajectory to stay onthe sliding surface can be analyzed as follows [19]
(x) = 120597120590120597x(f (x) + g (x)ueq) = 0
997904rArr ueq = minus(120597120590
120597xg(x))
minus1
(120597120590
120597xf (x))
(12)
Substituting (12) into (1) the system dynamics on the slidingsurface for 119905 ge 119905
0is governed by
x = f (x) + g (x) ueq = [I minus g (x) (120597120590120597x
g(x))minus1
(120597120590
120597x)] f (x)
(13)
The dynamics represented in (13) is defined as the idealsliding dynamics [16] Consequently the closed-loop systemdynamics in sliding mode are specified by the sliding surface
One of the widely used designing methods of the slidingmode controller combines switching inputs and the equiv-alent inputs as represented in (12) that is if the slidingsurface is determined then the sliding mode controller canbe obtained by adding switching input (119906sw119894) of which entry119906119894(119905) yields [18]
119906119894= 119906eq119894 + 119906sw119894 (119894 = 1 2 119898) (14)
where 119906eq119894 is the 119894th entry of the equivalent input analyzedin (12) And 119906sw119894 is the switching input that forces the systemstates tomove towards the sliding surface In [18] the possiblecandidates for designing the switching input are introducedfor example
(i) relay with constant gains 119906sw119894(x) = 120572119894 sgn(120590119894(x))with120572119894lt 0
(ii) relay with state dependent gains 119906sw119894(x) =
120572119894(x) sgn(120590
119894(x)) with 120572
119894(sdot) lt 0
(iii) univector nonlinearity with scale factorusw(x)=120588120590(x)120590(x) with 120588 lt 0
32 Similar Characteristics between Sliding Mode Control andNonlinear Dynamic Inversion Before analyzing the similarcharacteristics between SMC andNDI the conventional NDIcontroller is firstly introduced NDI controller consists oftwo blocks (Figure 5) desired dynamic block and dynamicinversion (DI) block [4ndash6] In the desired dynamic blockthe control variables are defined and the rate commandsof the selected control variables are generated And in theDI block proper control inputs are generated by invertingthe plant dynamics in order to make the inner closed-loop transfer function as an integrator [20] Hence thesystem states controlled by the dynamic inversion block willfollow the user-selected control variables For this reason thedesired dynamics are usually designed to satisfy the specific
dynamicsDynamicinversion
x x
xPlant
xcmd asymp1
s
Figure 5 Structure of dynamic inversion [4]
performancesrequirements so that the performance of thecontrolled system can achieve the system requirements In[4] some forms of desired dynamics are introduced propor-tional proportional integral flying quality ride quality andso forth
For the nonlinear system shown in (8) NDI control inputuNDI(119905) can be given by [4]
uNDI (119905) = [g (x (119905))]minus1
[xdes (119905) minus f (x (119905))] (15)
where xdes(119905) represents the desired dynamicsIt is worth noting that the requirement for existing NDI
control law in (15) is that [g(x)]minus1 must exist Howeverthe matrix g(x) is not generally a full rank for nonflatsystems These systems have a larger number of states thanthe number of control inputs Hence the number of statesthat can be inverted is less than or equal to the numberof inputs One way of achieving [g(x)]minus1 is to formulatethe existence problem as a two-time scale problem A two-time scale problem makes the system states separate into fastand slow dynamics The fast dynamics comprise a group ofstates affected directly by the control inputs On the otherhand the slow dynamics consists of states influenced by thefast dynamics Separating the system states into two groupsreduces the system order enough to provide a chance toexist more number of inputs than the number of the fastdynamic states Consequently [g(x)]minus1 can be obtained [4ndash6] In contrast for an overactuated system that adopts a largernumber of inputs than the number of states the inverse canbe obtained by reducing the number of inputs to be equal tothe number of states using control allocation [8 9]
By substituting (15) into (8) the inner-loop dynamics inFigure 5 controlled by conventional NDI yields
xdes = x (16)
Hence NDI controller replaces the original dynamics intothe user-selected desired dynamics by inverting the originaldynamics Consequently the control system can guaranteethe stability without gain-scheduling However it is impos-sible to achieve perfect cancellation of the original dynamicsdue to uncertainties such asmodelmismatches disturbancesand measurement noises which leads to poor robustnessof NDI controller Hence robustness issues against uncer-tainties must be considered when designing NDI control-ler
To achieve the robustness of NDI the equivalent controlmethod used to design sliding surface is considered Fordesired trajectory xdes isin R119899 let xlowast = xdes minus x isin R119899
6 Mathematical Problems in Engineering
Select a set of nonlinear smooth functions 120590(xlowast) = [1205901(xlowast)
1205902(xlowast) 120590
119898(xlowast)]119879 = 0 as a sliding surface If the system
states intercept the sliding surface at 119905 = 1199050and the sliding
mode exists for 119905 ge 1199050 then (119889119889119905)120590(x(119905)) = 0 and 120590(x(119905)) = 0
for all 119905 ge 1199050 Then the equivalent input ueq that the state
trajectory stays on the sliding surface can be analyzed asfollows [13ndash15]
(xlowast) = 120597120590
120597xlowastxlowast = 120597120590
120597xlowast(xdes minus x) = 0
997904rArr120597120590
120597xlowastxdes minus
120597120590
120597xlowast(f (x) + g (x) ueq) = 0
997904rArr ueq = (120597120590
120597xlowastg (x))
minus1
(120597120590
120597xlowastxdes minus
120597120590
120597xlowastf (x))
(17)
If 119898 = 119899 and 120597120590120597xlowast = I119899 then (17) can be represented as a
form of NDI input shown in (15)
ueq = g(x)minus1 [xdes minus f (x)] = uNDI (18)
Hence if the sliding surface is designed by an identity matrixthen the equivalent input of SMC can be represented as NDIinput
Definition 1 Let 120590NDI(x) be a set of smooth functions120590NDI119894(x) (119894 = 1 2 119899) that is 120590NDI(x)=[120590NDI1(x)120590NDI2(x) 120590NDI119899(x)]
119879 Then 120590NDI(x) = x is defined as theNDI surface that is the NDI surface is the sliding surfacedesigned by the identity matrix
It is worth noting that the ideal sliding dynamics shownin (13) yields the desired dynamics as follows
x = f (x) + g (x)ueq
= [f (x) + g (x) ( 120597120590120597xlowast
g (x))minus1
(120597120590
120597xlowastxdes minus
120597120590
120597xlowastf (x))]
= xdes(19)
In (19) the states on the identity sliding surface are governedby the desired dynamics Moreover the performances ofthe control system in sliding mode are determined only bythe desired dynamics This suggests that the difficulties indesigning a sliding mode controller to satisfy the specificsystem performances are converted to the desired dynamicsdesigning problem
33 Design of the Robust Nonlinear Dynamic Inversion Simi-lar to the designmethodof SMC each entry119906
119894(119905) (119894 = 1 2
119898) of the RNDI control law can be obtained by introducingthe following values
119906119894(119905) =
119906+
119894 if 120590
119894(119909des119894 minus 119909119894) = 119909des119894 minus 119909119894 gt 0
119906minus
119894 if 120590
119894(119909des119894 minus 119909119894) = 119909des119894 minus 119909119894 lt 0
(20)
where 119906+119894= 119906minus
119894satisfies the following for 119891
119894(x) the 119894th entry
of f(x) and for 119892119894119895(x) the (119894 119895)-entry of g(x)
lim119909119894rarr119909
+
des119894
119891119894(119909) +
119899
sum
119895=1
119895 = 119894
119892119894119895(119909) 119906119895+ 119892119894119894(119909) 119906minus
119894lt des119894
lim119909119894rarr119909
minus
des119894
119891119894(119909) +
119899
sum
119895=1
119895 = 119894
119892119894119895(119909) 119906119895+ 119892119894119894(119909) 119906+
119894gt des119894
(21)
where the superscripts + and minus denote the right- and left-hand limit respectivelyThe following theorem and corollarypropose the method for designing the RNDI controller andprovide the stability analytically
Theorem 2 Consider the following input
u = u119873119863119868
+ gminus1 (x) u119904119908 (22)
where u119873119863119868
is the conventional NDI input and u119904119908
isthe switching input that satisfies the following for u
119904119908=
[1199061199041199081 1199061199041199082 119906
119904119908119899]119879
119906119904119908119894
gt 0 119894119891 120590119894(x119889119890119904minus x) = 119909
119889119890119904119894minus 119909119894gt 0
119906119904119908119894
lt 0 119894119891 120590119894(x119889119890119904minus x) = 119909
119889119890119904119894minus 119909119894lt 0
(23)
Then the closed-loop system controlled by (22) is globally stable
Proof Select a Lyapunov candidate as
119881 =1
2120590(xdes minus x)119879120590 (xdes minus x) (24)
From (22) and (23)
= 120590(xdes minus x)119879 (xdes minus x)
= 120590(xdes minus x)119879 (xdes minus x)
= 120590(xdes minus x)119879 [xdes minus f (x) + g (x) u]
= 120590(xdes minus x)119879
times [xdes minus f (x) + g (x) (uNDI + gminus1 (x) usw)]
= 120590(xdes minus x)119879
times [xdes minus f (x) + g (x) uNDI minus usw]
= minus120590(xdes minus x)119879usw
= minus
119899
sum
119894=1
120590119894(xdes minus x) 119906sw119894
(25)
From (25) if 120590119894(xdes minus x) gt 0 then 119906sw119894 gt 0 and lt 0
Similarly if 120590119894(xdes minus x) lt 0 then 119906sw119894 lt 0 and lt 0 Hence
the closed-loop system controlled by (22) is globally stable
Mathematical Problems in Engineering 7
NDI surface
fi(x) +n
sumj=1jnei
gij(x)uj + gii(x)u+i
fi(x) +n
sumj=1jnei
gij(x)uj + gii(x)uminusi
120590i( minus ) gt 0x xdes
120590i( minus x) lt 0xdes
120590i( minus ) = 0xdes x
Figure 6 State trajectory on the NDI surface
Since the system controlled by (22) is globally stable fromTheorem 2 the states are forced to the NDI surface as shownin Figure 6
Corollary 3 If the switching input denoted as u119904119908= [1199061199041199081
1199061199041199082 119906
119904119908119899]119879 is designed with 119906
119904119908119894= 119896119894sgn(120590
119894(x119889119890119904minus x))
for a positive 119896119894 then the control system is globally stable
Proof If 120590119894(xdes minus x) gt 0 then 119906sw119894 = 119896119894 gt 0 and conversely
if 120590119894(xdes minus x) lt 0 then 119906sw119894 = minus119896119894 lt 0 Hence according to
Theorem 2 the control system is globally stable
Definition 4 The control law proposed inTheorem 2 is calledthe robust nonlinear dynamic inversion (RNDI) law that is fora diagonal matrix K = diag(119896
119894) (119894 = 1 2 119899) with 119896
119894gt 0
the RNDI control law yields
uRNDI = uNDI + usw
= g(x)minus1 [xdes minus f (x) + K sgn (xdes minus x)] (26)
It is worth noting that the form of the switching inputrepresented in Corollary 3 is a relay with constant gains formin SMC as mentioned in Section 31 Actually the switchingfunction satisfying (23) can be obtained directly from SMCdesign method as follows
(i) relay with constant gains 119906sw119894 = 119896119894 sgn(120590119894(xdes minus x))with 119896
119894gt 0
(ii) relay with state dependent gains 119906sw119894 = 119896119894(xdes minus
x) sgn(120590119894(xdes minus x)) with 119896
119894(sdot) gt 0
(iii) univector nonlinearity with scale factor usw =
119896119894120590(xdes minus x)120590(xdes minus x) with 119896
119894gt 0
It is also remarkable that using the sgn function whiledesigning the switching input leads to unfavorable resultsin control systems such as the chattering problem Thechattering problem makes the system have unmodeled highfrequencies or actuator saturation In sliding mode controltheory several methods have been proposed to overcomethis problem by designing a continuous control input insteadof a discontinuous switching input This method forces the
controlled system to follow the approximated sliding motionin some boundaries instead of the ideal sliding motionFollowing the similar manner the chattering problem in theRNDI control can be solved using the continuous controlinput For example replace sgn(xdes minus x) into sat(xdes minus x) =[sat1(119909des1 minus 1199091) sat2(119909des2 minus 1199092) sat119899(119909des119899 minus 119909119899)]
119879 inorder to design a continuous switching input as follows for apositive 120575
119894
sat119894(119909des119894 minus 119909119894
120575119894
) =
1 if (119909des119894 minus 119909119894) gt 120575119909des119894 minus 119909119894
120575119894
if 1003816100381610038161003816119909des119894 minus 1199091198941003816100381610038161003816 lt 120575
minus1 if (119909des119894 minus 119909119894) lt minus120575(27)
Then 119906sw119894 = 119896119894sat119894((119909des119894minus119909119894)120575119894) In (27) plusmn120575 is the boundarylayer in which the states are governed by the followingdynamics for sat((xdes minus x)120575) = [sat
1((119909des1 minus 1199091)1205751)
sat2((119909des2 minus 1199092)1205752) sat119899((119909des119899 minus 119909119899)120575119899)]
119879 and K =
diag(119896119894) (119894 = 1 2 119899) with 119896
119894gt 0
x = f (x) + g (x)u
= f (x) + g (x)
times [(g (x))minus1 (xdes minus f (x))
+(g (x))minus1Ksat((xdes minus x)
120575)]
= xdes + Ksat((xdes minus x)
120575)
= xdes
+ [1198961(119909des1 minus 1199091)
1205751
1198962(119909des2 minus 1199092)
1205752
sdot sdot sdot
119896119899(119909des119899 minus 119909119899)
120575119899
]
119879
(28)
8 Mathematical Problems in Engineering
uf
RefTime
Angular
Position of Referencetrajectorygenerator
the leader spacecraft
Reffcn
PositionControl
Dynamics of the Robust nonlinear dynamic inversion follower spacecraft
PositionScope
DisturbancesFd
++
t
ww
plplinputs
velocity
Figure 7 Block diagram using MATLAB Simulink
34 Stability of the Proposed RNDI Controller In this paperthe stability of the closed-loop system controlled by theproposed RNDI is analyzed by a Lyapunov stability criterionConsider the following nonlinear system including boundedmodel uncertainties Δf(x(119905)) isin R119899 and Δg(x(119905)) isin R119899 andbounded disturbance d(119905) isin R119899
x (119905) = [f (x (119905)) + Δf (x (119905))]
+ [g (x (119905)) + Δg (x (119905))] u (119905) + d (119905) (29)
It is assumed that 120585(119905 x(119905) u(119905)) = Δf(x(119905)) + Δg(x(119905)) + d(119905)satisfying 120585(119905 x(119905) u(119905))
2lt 120582min(K) where 120582min(K) is the
minimum eigenvalue of K For the desired state xdes isin 119877119899denote xlowast(119905) = xdesminusx To analyze the stability of the proposedRNDI controller select a Lyapunov candidate as follows
119881 (xlowast (119905)) = 12xlowast(119905)119879xlowast (119905) (30)
Then the derivative of 119881(xlowast(119905)) yields
(xlowast (119905))
= xlowast(119905)119879xlowast (119905)
= xlowast(119905)119879 [xdes (119905) minus [f (x (119905)) + Δf (x (119905))]
+ [g (x (119905)) + Δg (x (119905))] u (119905) + d (119905) ]
= xlowast(119905)119879 [xdes (119905) minus (f (x (119905)) + g (x (119905)) u (119905))
minus (Δf (x (119905)) + Δg (x (119905)) u (119905) + d (119905)) ]
= xlowast(119905)119879 [minus120585 (119905 x (119905) u (119905)) minus K sgn (xlowast (119905))]
le minusxlowast(119905)119879120585 (119905 x (119905) u (119905)) minus 120582min (K)1003817100381710038171003817xlowast
(119905)10038171003817100381710038171
le 120585 (119905 x (119905) u (119905))1
1003817100381710038171003817119909lowast
(119905)10038171003817100381710038171 minus 120582min (K)
1003817100381710038171003817xlowast
(119905)10038171003817100381710038171
le [120585 (119905 x (119905) u (119905))2minus 120582min (K)]
1003817100381710038171003817xlowast
(119905)10038171003817100381710038171
(31)
By the hypothesis 120585(119905 x(119905) u(119905))2lt 120582min(K) the derivative
of 119881(xlowast(119905)) is always negative Hence the system controlledby the proposed RNDI controller is globally stable againstdisturbances noises and model mismatches
Table 1 Parameters for spacecraft formation flying [1 3]
Symbol Value Unit119866 6673 times 10minus11 m3kgsdots2
119872 5974 times 1024 kg119898119891
410 kg119898119897
1550 kgr119897
[4224 times 1024 0 0]119879 m1205960
7272 times 10minus5 rads
4 Simulation Results
In this section numerical simulations are conducted to eva-luate the performance of the proposed robust nonlineardynamic inversion controller
41 Simulation Description The aim of SFF is to design thefeasible control input u
119905(119905) such that p
119897rarr p119889as 119905 rarr infin
for a given reference relative position trajectory p119889isin R3
of the follower spacecraft with respect to the leader space-craft To evaluate the performance of the proposed RNDIcontroller MATLAB Simulink is selected as a simulationtool Figure 7 shows the block diagram of the overall SFFstructure using MATLAB Simulink In this paper it isassumed that the reference trajectory is set to p
119889=
[100 sin(41205960119905) 100 cos(4120596
0119905) 0]119879 that is the follower space-
craft tracks a circular orbit centered at the leader spacecraftwith a radius of 100 meters on a plane generated by e
119903and e120579
with an angular velocity 41205960 The initial relative position and
velocity are assumed to be respectively as follows
p119897(0) = [10 90 minus20]
119879
k119897(0) = [0 0 0]
119879
(32)
Table 1 lists some parameters and their values used in thissimulation Some uncertainties such as disturbances and sen-sor noises of velocity and position are induced in this simu-lation Disturbances acting on the follower spacecraft areassumed such that F
119889= [290532 31775 minus112298]
119879 (N)And it is also assumed that maximally 20 of random velo-city sensor noises in three axes create difficulties in allowingthe follower spacecraft to obtain accurate velocity informa-tion Moreover maximally 20 and 40 of random positionsensor noises in e
119903and e
ℎaxes are also considered in this
simulation
Mathematical Problems in Engineering 9
0 10 20 30 40 50 60 70 80 90 100
0
5
10
Time (s)
No disturbanceDisturbance
minus5
minus10
minus15
e r-d
irect
ion
(m)
(a) Position tracking error in e119903-direction
0 10 20 30 40 50 60 70 80 90 100
0
2
Time (s)
No disturbanceDisturbance
minus2
minus4
minus6
minus8
minus10
e 120579-d
irect
ion
(m)
(b) Position tracking error in e120579-direction
0 10 20 30 40 50 60 70 80 90 100
0
5
10
Time (s)
No disturbanceDisturbance
minus5
minus10
minus15
minus20
e h-d
irect
ion
(m)
(c) Position tracking error in eℎ-direction
Figure 8 Relative position errors of no disturbance case and disturbance injected case The black thin line represents the trajectories of thespacecraft without considering any disturbances The blue thick line shows the results of the trajectories of the disturbed spacecraft
42 Simulation Results Figure 8 describes the position track-ing error results of the no disturbance case and disturbancesinjected case The objective of this simulation is to verifythe performance degradation of the disturbances injectedspacecraft In both cases NDI with the proportional type ofthe desired dynamics is designed as a primary controller forSFF As shown in Figure 8 if no disturbance is acting on thespacecraft then the position errors between the referencedposition and the follower position are zero in 30 secHoweverthe lack of robustness in NDI controller leads a significantperformance degradation of the disturbed spacecraft so ittracks the references with a tracking error of almost 8m 1mand 4m in e
119903 e120579 and e
ℎaxes respectively
Figures 9 and 10 show the trajectories of the followerspacecraft controlled by NDI and RNDI controllers Asmentioned in Figure 8 the position errors between thereferenced position and follower position controlled by NDIcannot be zero due to the induced uncertainties However bycombining the switching input into conventional NDI con-troller the RNDI control law attempts to reduce the effect ofthe uncertainties Consequently the state can track the refer-ence signal within 1m error as shown in Figure 9The controlinputs generated by NDI and RNDI controllers are described
in Figure 10 To compensate the induced uncertainties thespacecraft controlled by RNDI generates a large input force ineℎaxis around 27 sec (Figure 10(b)) Figure 11 shows the tra-
jectory of the spacecraft controlled by the RNDI controllerThe spacecraft maneuvers around a circle with a 100-meterradius Although the spacecraft is vibrated due to the induceduncertainties the error distance between the reference trajec-tory and the position of the follower is less than 1m as shownin Figure 9 Hence the follower spacecraft controlled by theproposed RNDI can track the reference trajectory
5 Conclusion
In this paper the robust nonlinear dynamic inversion controlstrategy that improves the robustness of NDI has been pro-posed NDI is a nonlinear control technique that eliminatesthe need of linearizing and designing gain-scheduled con-trollers for each equilibrium point by canceling the originaldynamics Because NDI removes the system nonlinearitiesdirectly it provides an opportunity to overcome the difficul-ties of ensuring stabilities and performances between variousoperational points while gain-scheduling However the maindrawback of designing NDI controller is the weak robustness
10 Mathematical Problems in Engineering
0 10 20 30 40 50 60 70 80 90 100
0
5
10
Time (m)
NDIRNDI
e r-d
irect
ion
(m)
minus5
minus15
minus10
(a) Position tracking error in e119903-direction
0
2
0 10 20 30 40 50 60 70 80 90 100Time (s)
NDIRNDI
e 120579-d
irect
ion
(m)
minus6
minus2
minus4
minus8
minus10
(b) Position tracking error in e120579-direction
0 10 20 30 40 50 60 70 80 90 100
0
5
10
Time (s)
NDIRNDI
e h-d
irect
ion
(m)
minus20
minus15
minus10
minus5
(c) Position tracking error in eℎ-direction
Figure 9 Simulation results of the position tracking error of the spacecrafts controlled by NDI and RNDIThe black thin line represents thetrajectories of the spacecraft controlled by NDI while the blue thick line represents the trajectories of the spacecraft controlled by RNDI
20 30 40 50 60 70 80 90 100
0
100
200
300
400
Time (s)
Inpu
t for
ces (
N)
minus200
minus100
ux along with er-axis
uz along with eh-axis
uy along with e120579-axis
(a) Control inputs generated by NDI
20 30 40 50 60 70 80 90 100
0
100
200
300
400
Time (s)
Inpu
t for
ces (
N)
minus200
minus100
ux along with er-axis
uz along with eh-axis
uy along with e120579-axis
(b) Control inputs generated by RNDI
Figure 10 Control inputs generated by NDI and RNDI
due to difficulties in obtaining accurate information of thenonlinear system dynamics In contrast to NDI SMC isconsidered a robust nonlinear control technique that usesswitching values as inputs to force the system states intoa sliding surface and to maintain the states on it Theproposed RNDI controller is designed by combining the
conventional NDI controller and switching inputs analyzedin SMC Hence RNDI takes the advantages of both NDI andSMC such that it
(i) is easy to design and implement(ii) eliminates the need of gain-scheduling
Mathematical Problems in Engineering 11
0 50 100 150050100150
0
10
e120579 axis er axis
e hax
is
minus50minus50minus100minus100 minus150minus150
minus10
minus20
Figure 11 Trajectory of the follower spacecraft controlled by theproposed RNDI controller
(iii) is easy to achieve the system required performancedue to the similar concept to classical control meth-ods and
(iv) is inherently robust
However the drawback of SMC such as the chatteringproblem still remains as a problem of RNDI control It isexpected that various chattering-free methods can be usedto reduce the effect of chattering phenomenon for examplereplacing the sign function into the saturation function
Moreover the equivalence of NDI input and equivalentinput designed by the identity sliding surface is also analyzedin this paper The reduced order dynamics on the slidingsurface can be converted to the user-selected desired dynam-ics As a result the difficulties in designing a sliding modecontroller for satisfying the specific system performances areconverted to a designing problem of the desired dynamicsHence RNDI provides an opportunity to solve the difficultiesin selecting the sliding surface of SMC The effectiveness ofthe proposed RNDI controller is confirmed by a set of simu-lations with application to the spacecraft formation flying
Nomenclature
G Universal gravity constantM Mass of the Earth119898119891 Mass of the follower spacecraft
119898119897 Mass of the leader spacecraft
r119891 Distance from the center of the Earth to
the center of the follower spacecraftr119897 Distance from the center of the Earth to
the center of the leader spacecraftxdes Desired dynamics
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research was supported by the MSIP (Ministry ofScience ICT amp Future Planning) Korea under the C-ITRC(Convergence Information Technology Research Center)
support program (NIPA-2014-H0401-14-1004) supervised bythe NIPA (National IT Industry Promotion Agency)
References
[1] Y Lv Q Hu G Ma and J Zhou ldquo6 DOF synchronized con-trol for spacecraft formation flying with input constraint andparameter uncertaintiesrdquo ISA Transactions vol 50 no 4 pp573ndash580 2011
[2] D Forta F Bordi and C Scolese ldquoLow-cost minimum sizesatellites for demonstration of formation flying modes at smallkilometer-size distancerdquo in Proceedings of the 13th AIAAUSUConference on Small Satellite SSC99-VI-3 pp 1ndash15 1999
[3] M S De Queiroz V Kapila and Q Yan ldquoAdaptive nonlinearcontrol ofmultiple spacecraft formation flyingrdquo Journal of Guid-ance Control and Dynamics vol 23 no 3 pp 385ndash390 2000
[4] J Georgie and J Valasek ldquoEvaluation of longitudinal desireddynamics for dynamic-inversion controlled generic reentryvehiclesrdquo Journal of Guidance Control and Dynamics vol 26no 5 pp 811ndash819 2003
[5] D Ito J Georgie J Valasek and D T Ward Reentry VehicleFlight Controls Design Guidelines Dynamic Inversion NASATPmdash2002ndash210771 2002
[6] A J Ostroff and B J Bacon ldquoForce and moment approachfor achievable dynamics using nonlinear dynamic inversionrdquo inProceedings of theAIAAGuidance Navigation andControl Con-ference and Exhibit pp 424ndash434 Reston Va USA 1999
[7] D Enns D Bugajski R Hendrick and G Stein ldquoDynamicinversion an evolving methodology for flight control designrdquoInternational Journal of Control vol 59 no 1 pp 71ndash91 1994
[8] J A Richard J M Buffington A G Sparks and S S BandaRobust Multivariable Flight Control Springer New York NYUSA 1994
[9] R J Adams and S S Banda ldquoRobust flight control design usingdynamic inversion and structured singular value synthesisrdquoIEEE Transactions on Control Systems Technology vol 1 no 2pp 80ndash92 1993
[10] M B McFarland and S M Hoque ldquoRobustness of a nonlinearmissile autopilot designed using dynamic inversionrdquo in Pro-ceedings of the AIAA Guidance Navigation and Control Confer-ence and Exhibit AIAA-2000-3970 Denver Colo USA August2000
[11] G Looye and H-D Joos ldquoDesign of robust dynamic inversioncontrol laws usingmulti-objective optimizationrdquo in Proceedingsof the AIAA Guidance Navigation and Control Conference andExhibit AIAA-2001-4285 2001
[12] P B Acquatella W Falkena E van Kampen and Q P ChuldquoRobust nonlinear spacecraft attitude control using incrementalnonlinear dynamic inversionrdquo in Proceedings of the AIAAGuid-ance Navigation and Control Conference and Exhibit Min-neapolis Minn USA August 2012
[13] P K Menon V R Iragavarapu and E J Ohlmeyer ldquoNonlinearMissile Autopilot Design using Time-Scale Separationrdquo in Pro-ceedings of the AIAA Guidance Navigation and Control Confer-ence and Exhibit AIAA-1997-3765 1997
[14] A Steinicke and G Michalka ldquoImproving transient perfor-mance of dynamic inversion missile autopilot by use of back-steppingrdquo in Proceedings of the AIAAGuidance Navigation andControl Conference and Exhibit AIAA-2002-4658 MontereyCalif USA August 2002
12 Mathematical Problems in Engineering
[15] I Yang D Kim and D Lee ldquoA flight control strategy usingrobust dynamic inversion based on sliding mode controlrdquo inproceeding of the AIAA Guidance Navigation and ControlConference August 2012
[16] H Sira-Ramirez ldquoDifferential geometric methods in variable-structure controlrdquo International Journal of Control vol 48 no4 pp 1359ndash1390 1988
[17] V I Utkin ldquoVariable structure systems with sliding modesrdquoIEEE Transactions on Automatic Control vol 22 no 2 pp 212ndash222 1977
[18] R A DeCarlo S H Zak and G P Matthews ldquoVariable struc-ture control of nonlinear multivariable systems a tutorialrdquo Pro-ceedings of the IEEE vol 76 no 3 pp 212ndash232 1988
[19] J Y HungWGao and J CHung ldquoVariable structure control asurveyrdquo IEEE Transactions on Industrial Electronics vol 40 no1 pp 2ndash22 1993
[20] G Papageorgiou andM Polansky ldquoTuning a dynamic inversionpitch axis autopilot using McFarlane-Glover loop shapingrdquoOptimal Control Applications ampMethods vol 30 no 3 pp 287ndash308 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Mathematical Problems in Engineering
Dynamic inversion
PI-typedesired
dynamics
Dynamicinversion Plant
(a) PI-type of desired dynamics based method
Dynamic inversion
Robustlinear
controller
Dynamicinversion Plant
Desireddynamics
(b) Robust linear controller based method
Figure 1 Block diagrams that have been proposed to achieve robustness
advantages of NDI control are that it is conceptually simpleand has many similarities to classical control methods Inaddition it naturally handles nonlinear systemswithout gain-scheduling because it works as a self-scheduled controllerFor this reason NDI has become a popular flight controllerand been applied to various high-performance aircrafts suchas X-38 [4 5] F-18 HARV [7 8] and F-16 [8 9] On theother hand accurate knowledge of the nonlinear systemdynamics is needed to obtain a perfect cancellation of theoriginal dynamics Actually in real applications such anassumption of taking all accurate information about systemnonlinearities is rarely met due to uncertainties such asmodel mismatches disturbances and measurement noises[4 9ndash13] As these uncertainties lead to the closed-loopsystem control performance degradation and make the sys-tem unstable robustness issues against uncertainties must beconsidered when designing NDI controller
A variety of methods have been proposed to achieverobustness inNDI controllerMany studies have reported thatthe system controlled by NDI with a proportional-integral-(PI-) type of desired dynamics can improve the robustness[10 13 14] As shown in Figure 1(a) the PI controller isdesigned as the desired dynamics satisfying the specificrequirements that the states track after removing the originaldynamics by NDI The PI controller takes advantages of sim-ple design and similaritieswith classical controlHowever thedesired dynamics types are selected by considering the sys-tem requirements so there are some systems where the PI-type of desired dynamics is unsuitable One of the mostwidely used methods for solving the robustness issue is toemploy an additional linear controller such as the structuredsingular value (120583-analysis) and 119867
infinsynthesis [4 5 8 9 11]
In these methods NDI works as an inner-loop controllerwhile an additional linear robust controller is employed asan outer-loop controller as shown in Figure 1(b) Thereforethe linear controller attempts improve the robustness of theoverall control system These methods however are basedon linearized system equations and lead to an increase of theorder of the control system For example the controller orderincreases to 14while designing the119867
infincontroller for the X-38
[5] Recently Yang et al proposed a robust dynamic inversion(RDI) controlmethod using slidingmode control (SMC) [15]SMC is a robust nonlinear control method that takes distinctvalues as control inputs to steer the system states into a user-designed sliding surface and to maintain the states on it [1617] By combining dynamic inversion (DI) and SMC the RDIcontroller guarantees stability against uncertainties withoutusing any additional outer-loop controller and moreoverit takes the advantages of both controllers (Figure 2) [15]However the proposed RDI is designed as a linear controllerso it cannot cover the advantages of self-scheduling of NDIcontrollerThis paper extends the results of [15] for nonlinearsystems and verifies the feasibility of the proposed RNDIcontroller with application to the SFF problem
2 Mathematical Model ofSpacecraft Formation Flying
This section presents the derivation of a nonlinear SFFmodelFormation flying considered in this paper is assumed tobe composed of two spacecrafts leader and follower Theleader spacecraft provides the reference trajectory assumedto be a circular orbit with constant velocity 120596
0while the
follower spacecraft navigates the neighborhood of the leader
Mathematical Problems in Engineering 3
Robust dynamic inversion
Desireddynamics
Robustdynamicinversion
Plant
Figure 2 Block diagram of the proposed robust nonlinear dynamic inversion
p
Leader
Follower
Ez
Ex
Ey
eh
rf
er
e120579
rl
Figure 3 Reference coordinate frame [1]
according to the desired reference trajectory Figure 3 showsthe coordinate reference frames used throughout this paperTwo main coordinate frames are the Earth centered inertiaframe E
119909E119910E119911 whose origin is located in the center of the
Earth and the leader orbit frame e119903 e120579 eℎ whose origin is
attached to the center of mass of the leader spacecraft Thenonlinear dynamics of the leader and follower with respectto the e
119903 e120579 eℎ frame can be represented as follows [1 3]
r119897= minus
120583
1199033
119897
r119897+F119889119897
119898119897
+u119897
119898119897
(1)
r119891= minus
120583
1199033
119891
r119891+F119889119891
119898119891
+u119891
119898119891
(2)
where u119897isin R3 and u
119891isin R3 are control input vectors of the
leader and follower spacecrafts respectively And F119889119897isin R3
and F119889119891
isin R3 are the disturbance force vectors acting onthe leader and follower spacecrafts respectively It is assumedthat the masses of spacecrafts are small relative to the massof the Earth that is 119872 ≫ 119898
119897 119898119891 so 119866(119872 + 119898
119897) asymp 119866119872
and 119866(119872 + 119898119891) asymp 119866119872 Because the relative position of the
follower spacecraft p = r119891minus r119897 the second derivative of the
relative position can be represented as follows
p = r119891minus r119897= (minus
120583
1199033
119891
(r119897+ p) +
F119889119891
119898119891
+u119891
119898119891
)
minus (minus120583
1199033
119897
r119897+F119889119897
119898119897
+u119897
119898119897
)
= minus119872119866(r119897+ p
1003817100381710038171003817r119897 + p10038171003817100381710038173minus
r119897
1199033
119897
) minusF119889
119898119891
+u119905
119898119891
(3)
where F119889= F119889119891minus (119898119891119898119897)F119889119897and u
119905= u119891minus (119898119891119898119897)u119897
In this paper it is assumed that the leader spacecraft is freeflying that is u
119897= 0 then it satisfies that u
119905= u119891
If the relative position p in the leader orbit frame isgiven as p = 119909e
119903+ 119910e120579+ 119911eℎfor the unit vectors e
119903 e120579
and eℎalong with the E
119909E119910E119911-frame then the relative
acceleration vector p can be obtained as follows
p = ( minus 21205960119910 minus 1205962
0119909) e119903+ ( 119910 + 2120596
0 minus 1205962
0119910) e120579+ eℎ (4)
In the moving leader orbit frame the position vector r119897is
constant that is r119897= 119903119897e119903 By substituting (4) into (3) the
nonlinear dynamics of the follower spacecraft relative to theleader can be obtained as
( minus 21205960119910 minus 1205962
0119909) e119903+ ( 119910 + 2120596
0 minus 1205962
0119910) e120579+ eℎ
= minus119872119866((119903119897+ 119909) e
119903+ 119910e120579+ 119911eℎ
1003817100381710038171003817r119897 + p10038171003817100381710038173
minus119903119897e119903
1199033
119897
)
minus119865119889119909e119903+ 119865119889119910e120579+ 119865119889119911eℎ
119898119891
+119906119909e119903+ 119906119910e120579+ 119906119911eℎ
119898119891
997904rArr [
[
119910
]
]
= minus[
[
0 minus212059600
21205960
0 0
0 0 0
]
]
[
[
119910
]
]
+
[[[[[[[[
[
1205962
0minus
119872119866
1003817100381710038171003817r119897 + p10038171003817100381710038173
0 0
0 1205962
0minus
119872119866
1003817100381710038171003817r119897 + p10038171003817100381710038173
0
0 0 minus119872119866
1003817100381710038171003817r119897 + p10038171003817100381710038173
]]]]]]]]
]
[
[
119909
119910
119911
]
]
4 Mathematical Problems in Engineering
minus119872119866[[[
[
119903119897
1003817100381710038171003817r119897 + p10038171003817100381710038173minus1
1199032119897
0
0
]]]
]
minus1
119898119891
[
[
119865119889119909
119865119889119910
119865119889119911
]
]
+1
119898119891
[
[
119906119909
119906119910
119906119911
]
]
(5)
Denote the relative position and velocity vectors as p119897=
(119909 119910 119911) and v119897= (119889119909119889119905 119889119910119889119905 119889119911119889119905) in the leader orbit
frame respectively Then (5) can be represented as follows
p119897= k119897
k119897= minus C (120596
0) k119897minusD (p
119897 r119897 1205960) p119897minus N (p
119897 r119897) minus
F119889
119898119891
+u119905
119898119891
(6)
where
C (1205960) = [
[
0 minus212059600
21205960
0 0
0 0 0
]
]
N (p119897 r119897) = 119872119866
[[[
[
119903119897
1003817100381710038171003817r119897 + p10038171003817100381710038173minus1
1199032119897
0
0
]]]
]
F119889= [
[
119865119889119909
119865119889119910
119865119889119911
]
]
D (p119897 r119897 1205960)
=
[[[[[[[[
[
1205962
0minus
119872119866
1003817100381710038171003817r119897 + p10038171003817100381710038173
0 0
0 1205962
0minus
119872119866
1003817100381710038171003817r119897 + p10038171003817100381710038173
0
0 0 minus119872119866
1003817100381710038171003817r119897 + p10038171003817100381710038173
]]]]]]]]
]
(7)
3 The Proposed Robust NonlinearDynamic Inversion
In this section the mathematical backgrounds of slidingmode control and nonlinear dynamic inversion are intro-duced And then the robust nonlinear dynamic inversionmethod is proposed
31 Sliding Mode Controller Sliding mode control (SMC)also called variable structure control (VSC) is a high-speedswitching nonlinear control method that forces the systemstates into a user-designed sliding surface and maintains thestates on the surface [17 18] SMC consists of two phases(Figure 4) structuring the sliding surface and constructingthe switching feedback gains
Consider the following system
x (119905) = f (x (119905)) + g (x (119905)) u (119905) (8)
Sliding surface
Sliding phase
Reaching phase
120590(x) = 0
x2
x1
x(t1)
x(t0)
Figure 4The behavior of the system states in sliding mode control
where x(119905) isin R119899 is a state vector u(119905) isin R119898 is an input vectorwith 119899 ge 119898 and f R119899 rarr R119899 and g R119898 rarr R119899times119898are smooth functions Let a set Σ
119894be the regular (119899 minus 1)
dimensional submanifold in R119899 defined in [16] such as
Σ119894≜ x isin R119899 | 120590
119894(x) = 0 119894 = 1 2 119898 (9)
where 120590119894 R119899 rarr R (119894 = 1 2 119898) is a smooth function
Define120590119894(x) = 0 as the individual sliding surface or individual
switching surface Then surface constructed by intersecting119898 individual sliding surfaces is defined as the sliding surface[16]
Σ ≜
119898
⋂
119894=1
Σ119894=
119898
⋂
119894=1
x isin R119899 | 120590119894(x) = 0 ≜ x isin R119899 | 120590 (x) = 0
(10)
where 120590(x) = [1205901(x) 1205902(x) 120590
119898(x)]119879 If the states x(119905) of
the system are in Σ that is 120590(x(119905)) = 0 then the behavior ofthe states is called sliding motion or sliding mode Moreoverif sliding mode exists the tangent vectors of the states alwaysare forced to track the sliding surface as shown in Figure 4Then each entry 119906
119894(119905) (119894 = 1 2 119898) of the sliding mode
control law can be obtained by taking one of the followingvalues [16]
119906119894(119905) =
119906+
119894 if 120590
119894(x (119905)) gt 0
119906minus
119894 if 120590
119894(x (119905)) lt 0
(11)
where 119906+119894= 119906minus
119894
As shown in (11) designing the sliding surface is crucialbecause the sliding surface determines the control input andmoreover the performance of the closed-loop system Forthis reason various designing methods such as Fillipovrsquosmethod and equivalent control method have been proposed[18 19] In this paper the equivalent control method is
Mathematical Problems in Engineering 5
considered to analyze the similar characteristics betweenSMC and NDI If the system states intercept the slidingsurface at 119905 = 119905
0and a sliding mode exists for 119905 ge 119905
0 then
(119889119889119905)120590(x(119905)) = 0 and 120590(x(119905)) = 0 for all 119905 ge 1199050 Then the
equivalent input ueq that forces the state trajectory to stay onthe sliding surface can be analyzed as follows [19]
(x) = 120597120590120597x(f (x) + g (x)ueq) = 0
997904rArr ueq = minus(120597120590
120597xg(x))
minus1
(120597120590
120597xf (x))
(12)
Substituting (12) into (1) the system dynamics on the slidingsurface for 119905 ge 119905
0is governed by
x = f (x) + g (x) ueq = [I minus g (x) (120597120590120597x
g(x))minus1
(120597120590
120597x)] f (x)
(13)
The dynamics represented in (13) is defined as the idealsliding dynamics [16] Consequently the closed-loop systemdynamics in sliding mode are specified by the sliding surface
One of the widely used designing methods of the slidingmode controller combines switching inputs and the equiv-alent inputs as represented in (12) that is if the slidingsurface is determined then the sliding mode controller canbe obtained by adding switching input (119906sw119894) of which entry119906119894(119905) yields [18]
119906119894= 119906eq119894 + 119906sw119894 (119894 = 1 2 119898) (14)
where 119906eq119894 is the 119894th entry of the equivalent input analyzedin (12) And 119906sw119894 is the switching input that forces the systemstates tomove towards the sliding surface In [18] the possiblecandidates for designing the switching input are introducedfor example
(i) relay with constant gains 119906sw119894(x) = 120572119894 sgn(120590119894(x))with120572119894lt 0
(ii) relay with state dependent gains 119906sw119894(x) =
120572119894(x) sgn(120590
119894(x)) with 120572
119894(sdot) lt 0
(iii) univector nonlinearity with scale factorusw(x)=120588120590(x)120590(x) with 120588 lt 0
32 Similar Characteristics between Sliding Mode Control andNonlinear Dynamic Inversion Before analyzing the similarcharacteristics between SMC andNDI the conventional NDIcontroller is firstly introduced NDI controller consists oftwo blocks (Figure 5) desired dynamic block and dynamicinversion (DI) block [4ndash6] In the desired dynamic blockthe control variables are defined and the rate commandsof the selected control variables are generated And in theDI block proper control inputs are generated by invertingthe plant dynamics in order to make the inner closed-loop transfer function as an integrator [20] Hence thesystem states controlled by the dynamic inversion block willfollow the user-selected control variables For this reason thedesired dynamics are usually designed to satisfy the specific
dynamicsDynamicinversion
x x
xPlant
xcmd asymp1
s
Figure 5 Structure of dynamic inversion [4]
performancesrequirements so that the performance of thecontrolled system can achieve the system requirements In[4] some forms of desired dynamics are introduced propor-tional proportional integral flying quality ride quality andso forth
For the nonlinear system shown in (8) NDI control inputuNDI(119905) can be given by [4]
uNDI (119905) = [g (x (119905))]minus1
[xdes (119905) minus f (x (119905))] (15)
where xdes(119905) represents the desired dynamicsIt is worth noting that the requirement for existing NDI
control law in (15) is that [g(x)]minus1 must exist Howeverthe matrix g(x) is not generally a full rank for nonflatsystems These systems have a larger number of states thanthe number of control inputs Hence the number of statesthat can be inverted is less than or equal to the numberof inputs One way of achieving [g(x)]minus1 is to formulatethe existence problem as a two-time scale problem A two-time scale problem makes the system states separate into fastand slow dynamics The fast dynamics comprise a group ofstates affected directly by the control inputs On the otherhand the slow dynamics consists of states influenced by thefast dynamics Separating the system states into two groupsreduces the system order enough to provide a chance toexist more number of inputs than the number of the fastdynamic states Consequently [g(x)]minus1 can be obtained [4ndash6] In contrast for an overactuated system that adopts a largernumber of inputs than the number of states the inverse canbe obtained by reducing the number of inputs to be equal tothe number of states using control allocation [8 9]
By substituting (15) into (8) the inner-loop dynamics inFigure 5 controlled by conventional NDI yields
xdes = x (16)
Hence NDI controller replaces the original dynamics intothe user-selected desired dynamics by inverting the originaldynamics Consequently the control system can guaranteethe stability without gain-scheduling However it is impos-sible to achieve perfect cancellation of the original dynamicsdue to uncertainties such asmodelmismatches disturbancesand measurement noises which leads to poor robustnessof NDI controller Hence robustness issues against uncer-tainties must be considered when designing NDI control-ler
To achieve the robustness of NDI the equivalent controlmethod used to design sliding surface is considered Fordesired trajectory xdes isin R119899 let xlowast = xdes minus x isin R119899
6 Mathematical Problems in Engineering
Select a set of nonlinear smooth functions 120590(xlowast) = [1205901(xlowast)
1205902(xlowast) 120590
119898(xlowast)]119879 = 0 as a sliding surface If the system
states intercept the sliding surface at 119905 = 1199050and the sliding
mode exists for 119905 ge 1199050 then (119889119889119905)120590(x(119905)) = 0 and 120590(x(119905)) = 0
for all 119905 ge 1199050 Then the equivalent input ueq that the state
trajectory stays on the sliding surface can be analyzed asfollows [13ndash15]
(xlowast) = 120597120590
120597xlowastxlowast = 120597120590
120597xlowast(xdes minus x) = 0
997904rArr120597120590
120597xlowastxdes minus
120597120590
120597xlowast(f (x) + g (x) ueq) = 0
997904rArr ueq = (120597120590
120597xlowastg (x))
minus1
(120597120590
120597xlowastxdes minus
120597120590
120597xlowastf (x))
(17)
If 119898 = 119899 and 120597120590120597xlowast = I119899 then (17) can be represented as a
form of NDI input shown in (15)
ueq = g(x)minus1 [xdes minus f (x)] = uNDI (18)
Hence if the sliding surface is designed by an identity matrixthen the equivalent input of SMC can be represented as NDIinput
Definition 1 Let 120590NDI(x) be a set of smooth functions120590NDI119894(x) (119894 = 1 2 119899) that is 120590NDI(x)=[120590NDI1(x)120590NDI2(x) 120590NDI119899(x)]
119879 Then 120590NDI(x) = x is defined as theNDI surface that is the NDI surface is the sliding surfacedesigned by the identity matrix
It is worth noting that the ideal sliding dynamics shownin (13) yields the desired dynamics as follows
x = f (x) + g (x)ueq
= [f (x) + g (x) ( 120597120590120597xlowast
g (x))minus1
(120597120590
120597xlowastxdes minus
120597120590
120597xlowastf (x))]
= xdes(19)
In (19) the states on the identity sliding surface are governedby the desired dynamics Moreover the performances ofthe control system in sliding mode are determined only bythe desired dynamics This suggests that the difficulties indesigning a sliding mode controller to satisfy the specificsystem performances are converted to the desired dynamicsdesigning problem
33 Design of the Robust Nonlinear Dynamic Inversion Simi-lar to the designmethodof SMC each entry119906
119894(119905) (119894 = 1 2
119898) of the RNDI control law can be obtained by introducingthe following values
119906119894(119905) =
119906+
119894 if 120590
119894(119909des119894 minus 119909119894) = 119909des119894 minus 119909119894 gt 0
119906minus
119894 if 120590
119894(119909des119894 minus 119909119894) = 119909des119894 minus 119909119894 lt 0
(20)
where 119906+119894= 119906minus
119894satisfies the following for 119891
119894(x) the 119894th entry
of f(x) and for 119892119894119895(x) the (119894 119895)-entry of g(x)
lim119909119894rarr119909
+
des119894
119891119894(119909) +
119899
sum
119895=1
119895 = 119894
119892119894119895(119909) 119906119895+ 119892119894119894(119909) 119906minus
119894lt des119894
lim119909119894rarr119909
minus
des119894
119891119894(119909) +
119899
sum
119895=1
119895 = 119894
119892119894119895(119909) 119906119895+ 119892119894119894(119909) 119906+
119894gt des119894
(21)
where the superscripts + and minus denote the right- and left-hand limit respectivelyThe following theorem and corollarypropose the method for designing the RNDI controller andprovide the stability analytically
Theorem 2 Consider the following input
u = u119873119863119868
+ gminus1 (x) u119904119908 (22)
where u119873119863119868
is the conventional NDI input and u119904119908
isthe switching input that satisfies the following for u
119904119908=
[1199061199041199081 1199061199041199082 119906
119904119908119899]119879
119906119904119908119894
gt 0 119894119891 120590119894(x119889119890119904minus x) = 119909
119889119890119904119894minus 119909119894gt 0
119906119904119908119894
lt 0 119894119891 120590119894(x119889119890119904minus x) = 119909
119889119890119904119894minus 119909119894lt 0
(23)
Then the closed-loop system controlled by (22) is globally stable
Proof Select a Lyapunov candidate as
119881 =1
2120590(xdes minus x)119879120590 (xdes minus x) (24)
From (22) and (23)
= 120590(xdes minus x)119879 (xdes minus x)
= 120590(xdes minus x)119879 (xdes minus x)
= 120590(xdes minus x)119879 [xdes minus f (x) + g (x) u]
= 120590(xdes minus x)119879
times [xdes minus f (x) + g (x) (uNDI + gminus1 (x) usw)]
= 120590(xdes minus x)119879
times [xdes minus f (x) + g (x) uNDI minus usw]
= minus120590(xdes minus x)119879usw
= minus
119899
sum
119894=1
120590119894(xdes minus x) 119906sw119894
(25)
From (25) if 120590119894(xdes minus x) gt 0 then 119906sw119894 gt 0 and lt 0
Similarly if 120590119894(xdes minus x) lt 0 then 119906sw119894 lt 0 and lt 0 Hence
the closed-loop system controlled by (22) is globally stable
Mathematical Problems in Engineering 7
NDI surface
fi(x) +n
sumj=1jnei
gij(x)uj + gii(x)u+i
fi(x) +n
sumj=1jnei
gij(x)uj + gii(x)uminusi
120590i( minus ) gt 0x xdes
120590i( minus x) lt 0xdes
120590i( minus ) = 0xdes x
Figure 6 State trajectory on the NDI surface
Since the system controlled by (22) is globally stable fromTheorem 2 the states are forced to the NDI surface as shownin Figure 6
Corollary 3 If the switching input denoted as u119904119908= [1199061199041199081
1199061199041199082 119906
119904119908119899]119879 is designed with 119906
119904119908119894= 119896119894sgn(120590
119894(x119889119890119904minus x))
for a positive 119896119894 then the control system is globally stable
Proof If 120590119894(xdes minus x) gt 0 then 119906sw119894 = 119896119894 gt 0 and conversely
if 120590119894(xdes minus x) lt 0 then 119906sw119894 = minus119896119894 lt 0 Hence according to
Theorem 2 the control system is globally stable
Definition 4 The control law proposed inTheorem 2 is calledthe robust nonlinear dynamic inversion (RNDI) law that is fora diagonal matrix K = diag(119896
119894) (119894 = 1 2 119899) with 119896
119894gt 0
the RNDI control law yields
uRNDI = uNDI + usw
= g(x)minus1 [xdes minus f (x) + K sgn (xdes minus x)] (26)
It is worth noting that the form of the switching inputrepresented in Corollary 3 is a relay with constant gains formin SMC as mentioned in Section 31 Actually the switchingfunction satisfying (23) can be obtained directly from SMCdesign method as follows
(i) relay with constant gains 119906sw119894 = 119896119894 sgn(120590119894(xdes minus x))with 119896
119894gt 0
(ii) relay with state dependent gains 119906sw119894 = 119896119894(xdes minus
x) sgn(120590119894(xdes minus x)) with 119896
119894(sdot) gt 0
(iii) univector nonlinearity with scale factor usw =
119896119894120590(xdes minus x)120590(xdes minus x) with 119896
119894gt 0
It is also remarkable that using the sgn function whiledesigning the switching input leads to unfavorable resultsin control systems such as the chattering problem Thechattering problem makes the system have unmodeled highfrequencies or actuator saturation In sliding mode controltheory several methods have been proposed to overcomethis problem by designing a continuous control input insteadof a discontinuous switching input This method forces the
controlled system to follow the approximated sliding motionin some boundaries instead of the ideal sliding motionFollowing the similar manner the chattering problem in theRNDI control can be solved using the continuous controlinput For example replace sgn(xdes minus x) into sat(xdes minus x) =[sat1(119909des1 minus 1199091) sat2(119909des2 minus 1199092) sat119899(119909des119899 minus 119909119899)]
119879 inorder to design a continuous switching input as follows for apositive 120575
119894
sat119894(119909des119894 minus 119909119894
120575119894
) =
1 if (119909des119894 minus 119909119894) gt 120575119909des119894 minus 119909119894
120575119894
if 1003816100381610038161003816119909des119894 minus 1199091198941003816100381610038161003816 lt 120575
minus1 if (119909des119894 minus 119909119894) lt minus120575(27)
Then 119906sw119894 = 119896119894sat119894((119909des119894minus119909119894)120575119894) In (27) plusmn120575 is the boundarylayer in which the states are governed by the followingdynamics for sat((xdes minus x)120575) = [sat
1((119909des1 minus 1199091)1205751)
sat2((119909des2 minus 1199092)1205752) sat119899((119909des119899 minus 119909119899)120575119899)]
119879 and K =
diag(119896119894) (119894 = 1 2 119899) with 119896
119894gt 0
x = f (x) + g (x)u
= f (x) + g (x)
times [(g (x))minus1 (xdes minus f (x))
+(g (x))minus1Ksat((xdes minus x)
120575)]
= xdes + Ksat((xdes minus x)
120575)
= xdes
+ [1198961(119909des1 minus 1199091)
1205751
1198962(119909des2 minus 1199092)
1205752
sdot sdot sdot
119896119899(119909des119899 minus 119909119899)
120575119899
]
119879
(28)
8 Mathematical Problems in Engineering
uf
RefTime
Angular
Position of Referencetrajectorygenerator
the leader spacecraft
Reffcn
PositionControl
Dynamics of the Robust nonlinear dynamic inversion follower spacecraft
PositionScope
DisturbancesFd
++
t
ww
plplinputs
velocity
Figure 7 Block diagram using MATLAB Simulink
34 Stability of the Proposed RNDI Controller In this paperthe stability of the closed-loop system controlled by theproposed RNDI is analyzed by a Lyapunov stability criterionConsider the following nonlinear system including boundedmodel uncertainties Δf(x(119905)) isin R119899 and Δg(x(119905)) isin R119899 andbounded disturbance d(119905) isin R119899
x (119905) = [f (x (119905)) + Δf (x (119905))]
+ [g (x (119905)) + Δg (x (119905))] u (119905) + d (119905) (29)
It is assumed that 120585(119905 x(119905) u(119905)) = Δf(x(119905)) + Δg(x(119905)) + d(119905)satisfying 120585(119905 x(119905) u(119905))
2lt 120582min(K) where 120582min(K) is the
minimum eigenvalue of K For the desired state xdes isin 119877119899denote xlowast(119905) = xdesminusx To analyze the stability of the proposedRNDI controller select a Lyapunov candidate as follows
119881 (xlowast (119905)) = 12xlowast(119905)119879xlowast (119905) (30)
Then the derivative of 119881(xlowast(119905)) yields
(xlowast (119905))
= xlowast(119905)119879xlowast (119905)
= xlowast(119905)119879 [xdes (119905) minus [f (x (119905)) + Δf (x (119905))]
+ [g (x (119905)) + Δg (x (119905))] u (119905) + d (119905) ]
= xlowast(119905)119879 [xdes (119905) minus (f (x (119905)) + g (x (119905)) u (119905))
minus (Δf (x (119905)) + Δg (x (119905)) u (119905) + d (119905)) ]
= xlowast(119905)119879 [minus120585 (119905 x (119905) u (119905)) minus K sgn (xlowast (119905))]
le minusxlowast(119905)119879120585 (119905 x (119905) u (119905)) minus 120582min (K)1003817100381710038171003817xlowast
(119905)10038171003817100381710038171
le 120585 (119905 x (119905) u (119905))1
1003817100381710038171003817119909lowast
(119905)10038171003817100381710038171 minus 120582min (K)
1003817100381710038171003817xlowast
(119905)10038171003817100381710038171
le [120585 (119905 x (119905) u (119905))2minus 120582min (K)]
1003817100381710038171003817xlowast
(119905)10038171003817100381710038171
(31)
By the hypothesis 120585(119905 x(119905) u(119905))2lt 120582min(K) the derivative
of 119881(xlowast(119905)) is always negative Hence the system controlledby the proposed RNDI controller is globally stable againstdisturbances noises and model mismatches
Table 1 Parameters for spacecraft formation flying [1 3]
Symbol Value Unit119866 6673 times 10minus11 m3kgsdots2
119872 5974 times 1024 kg119898119891
410 kg119898119897
1550 kgr119897
[4224 times 1024 0 0]119879 m1205960
7272 times 10minus5 rads
4 Simulation Results
In this section numerical simulations are conducted to eva-luate the performance of the proposed robust nonlineardynamic inversion controller
41 Simulation Description The aim of SFF is to design thefeasible control input u
119905(119905) such that p
119897rarr p119889as 119905 rarr infin
for a given reference relative position trajectory p119889isin R3
of the follower spacecraft with respect to the leader space-craft To evaluate the performance of the proposed RNDIcontroller MATLAB Simulink is selected as a simulationtool Figure 7 shows the block diagram of the overall SFFstructure using MATLAB Simulink In this paper it isassumed that the reference trajectory is set to p
119889=
[100 sin(41205960119905) 100 cos(4120596
0119905) 0]119879 that is the follower space-
craft tracks a circular orbit centered at the leader spacecraftwith a radius of 100 meters on a plane generated by e
119903and e120579
with an angular velocity 41205960 The initial relative position and
velocity are assumed to be respectively as follows
p119897(0) = [10 90 minus20]
119879
k119897(0) = [0 0 0]
119879
(32)
Table 1 lists some parameters and their values used in thissimulation Some uncertainties such as disturbances and sen-sor noises of velocity and position are induced in this simu-lation Disturbances acting on the follower spacecraft areassumed such that F
119889= [290532 31775 minus112298]
119879 (N)And it is also assumed that maximally 20 of random velo-city sensor noises in three axes create difficulties in allowingthe follower spacecraft to obtain accurate velocity informa-tion Moreover maximally 20 and 40 of random positionsensor noises in e
119903and e
ℎaxes are also considered in this
simulation
Mathematical Problems in Engineering 9
0 10 20 30 40 50 60 70 80 90 100
0
5
10
Time (s)
No disturbanceDisturbance
minus5
minus10
minus15
e r-d
irect
ion
(m)
(a) Position tracking error in e119903-direction
0 10 20 30 40 50 60 70 80 90 100
0
2
Time (s)
No disturbanceDisturbance
minus2
minus4
minus6
minus8
minus10
e 120579-d
irect
ion
(m)
(b) Position tracking error in e120579-direction
0 10 20 30 40 50 60 70 80 90 100
0
5
10
Time (s)
No disturbanceDisturbance
minus5
minus10
minus15
minus20
e h-d
irect
ion
(m)
(c) Position tracking error in eℎ-direction
Figure 8 Relative position errors of no disturbance case and disturbance injected case The black thin line represents the trajectories of thespacecraft without considering any disturbances The blue thick line shows the results of the trajectories of the disturbed spacecraft
42 Simulation Results Figure 8 describes the position track-ing error results of the no disturbance case and disturbancesinjected case The objective of this simulation is to verifythe performance degradation of the disturbances injectedspacecraft In both cases NDI with the proportional type ofthe desired dynamics is designed as a primary controller forSFF As shown in Figure 8 if no disturbance is acting on thespacecraft then the position errors between the referencedposition and the follower position are zero in 30 secHoweverthe lack of robustness in NDI controller leads a significantperformance degradation of the disturbed spacecraft so ittracks the references with a tracking error of almost 8m 1mand 4m in e
119903 e120579 and e
ℎaxes respectively
Figures 9 and 10 show the trajectories of the followerspacecraft controlled by NDI and RNDI controllers Asmentioned in Figure 8 the position errors between thereferenced position and follower position controlled by NDIcannot be zero due to the induced uncertainties However bycombining the switching input into conventional NDI con-troller the RNDI control law attempts to reduce the effect ofthe uncertainties Consequently the state can track the refer-ence signal within 1m error as shown in Figure 9The controlinputs generated by NDI and RNDI controllers are described
in Figure 10 To compensate the induced uncertainties thespacecraft controlled by RNDI generates a large input force ineℎaxis around 27 sec (Figure 10(b)) Figure 11 shows the tra-
jectory of the spacecraft controlled by the RNDI controllerThe spacecraft maneuvers around a circle with a 100-meterradius Although the spacecraft is vibrated due to the induceduncertainties the error distance between the reference trajec-tory and the position of the follower is less than 1m as shownin Figure 9 Hence the follower spacecraft controlled by theproposed RNDI can track the reference trajectory
5 Conclusion
In this paper the robust nonlinear dynamic inversion controlstrategy that improves the robustness of NDI has been pro-posed NDI is a nonlinear control technique that eliminatesthe need of linearizing and designing gain-scheduled con-trollers for each equilibrium point by canceling the originaldynamics Because NDI removes the system nonlinearitiesdirectly it provides an opportunity to overcome the difficul-ties of ensuring stabilities and performances between variousoperational points while gain-scheduling However the maindrawback of designing NDI controller is the weak robustness
10 Mathematical Problems in Engineering
0 10 20 30 40 50 60 70 80 90 100
0
5
10
Time (m)
NDIRNDI
e r-d
irect
ion
(m)
minus5
minus15
minus10
(a) Position tracking error in e119903-direction
0
2
0 10 20 30 40 50 60 70 80 90 100Time (s)
NDIRNDI
e 120579-d
irect
ion
(m)
minus6
minus2
minus4
minus8
minus10
(b) Position tracking error in e120579-direction
0 10 20 30 40 50 60 70 80 90 100
0
5
10
Time (s)
NDIRNDI
e h-d
irect
ion
(m)
minus20
minus15
minus10
minus5
(c) Position tracking error in eℎ-direction
Figure 9 Simulation results of the position tracking error of the spacecrafts controlled by NDI and RNDIThe black thin line represents thetrajectories of the spacecraft controlled by NDI while the blue thick line represents the trajectories of the spacecraft controlled by RNDI
20 30 40 50 60 70 80 90 100
0
100
200
300
400
Time (s)
Inpu
t for
ces (
N)
minus200
minus100
ux along with er-axis
uz along with eh-axis
uy along with e120579-axis
(a) Control inputs generated by NDI
20 30 40 50 60 70 80 90 100
0
100
200
300
400
Time (s)
Inpu
t for
ces (
N)
minus200
minus100
ux along with er-axis
uz along with eh-axis
uy along with e120579-axis
(b) Control inputs generated by RNDI
Figure 10 Control inputs generated by NDI and RNDI
due to difficulties in obtaining accurate information of thenonlinear system dynamics In contrast to NDI SMC isconsidered a robust nonlinear control technique that usesswitching values as inputs to force the system states intoa sliding surface and to maintain the states on it Theproposed RNDI controller is designed by combining the
conventional NDI controller and switching inputs analyzedin SMC Hence RNDI takes the advantages of both NDI andSMC such that it
(i) is easy to design and implement(ii) eliminates the need of gain-scheduling
Mathematical Problems in Engineering 11
0 50 100 150050100150
0
10
e120579 axis er axis
e hax
is
minus50minus50minus100minus100 minus150minus150
minus10
minus20
Figure 11 Trajectory of the follower spacecraft controlled by theproposed RNDI controller
(iii) is easy to achieve the system required performancedue to the similar concept to classical control meth-ods and
(iv) is inherently robust
However the drawback of SMC such as the chatteringproblem still remains as a problem of RNDI control It isexpected that various chattering-free methods can be usedto reduce the effect of chattering phenomenon for examplereplacing the sign function into the saturation function
Moreover the equivalence of NDI input and equivalentinput designed by the identity sliding surface is also analyzedin this paper The reduced order dynamics on the slidingsurface can be converted to the user-selected desired dynam-ics As a result the difficulties in designing a sliding modecontroller for satisfying the specific system performances areconverted to a designing problem of the desired dynamicsHence RNDI provides an opportunity to solve the difficultiesin selecting the sliding surface of SMC The effectiveness ofthe proposed RNDI controller is confirmed by a set of simu-lations with application to the spacecraft formation flying
Nomenclature
G Universal gravity constantM Mass of the Earth119898119891 Mass of the follower spacecraft
119898119897 Mass of the leader spacecraft
r119891 Distance from the center of the Earth to
the center of the follower spacecraftr119897 Distance from the center of the Earth to
the center of the leader spacecraftxdes Desired dynamics
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research was supported by the MSIP (Ministry ofScience ICT amp Future Planning) Korea under the C-ITRC(Convergence Information Technology Research Center)
support program (NIPA-2014-H0401-14-1004) supervised bythe NIPA (National IT Industry Promotion Agency)
References
[1] Y Lv Q Hu G Ma and J Zhou ldquo6 DOF synchronized con-trol for spacecraft formation flying with input constraint andparameter uncertaintiesrdquo ISA Transactions vol 50 no 4 pp573ndash580 2011
[2] D Forta F Bordi and C Scolese ldquoLow-cost minimum sizesatellites for demonstration of formation flying modes at smallkilometer-size distancerdquo in Proceedings of the 13th AIAAUSUConference on Small Satellite SSC99-VI-3 pp 1ndash15 1999
[3] M S De Queiroz V Kapila and Q Yan ldquoAdaptive nonlinearcontrol ofmultiple spacecraft formation flyingrdquo Journal of Guid-ance Control and Dynamics vol 23 no 3 pp 385ndash390 2000
[4] J Georgie and J Valasek ldquoEvaluation of longitudinal desireddynamics for dynamic-inversion controlled generic reentryvehiclesrdquo Journal of Guidance Control and Dynamics vol 26no 5 pp 811ndash819 2003
[5] D Ito J Georgie J Valasek and D T Ward Reentry VehicleFlight Controls Design Guidelines Dynamic Inversion NASATPmdash2002ndash210771 2002
[6] A J Ostroff and B J Bacon ldquoForce and moment approachfor achievable dynamics using nonlinear dynamic inversionrdquo inProceedings of theAIAAGuidance Navigation andControl Con-ference and Exhibit pp 424ndash434 Reston Va USA 1999
[7] D Enns D Bugajski R Hendrick and G Stein ldquoDynamicinversion an evolving methodology for flight control designrdquoInternational Journal of Control vol 59 no 1 pp 71ndash91 1994
[8] J A Richard J M Buffington A G Sparks and S S BandaRobust Multivariable Flight Control Springer New York NYUSA 1994
[9] R J Adams and S S Banda ldquoRobust flight control design usingdynamic inversion and structured singular value synthesisrdquoIEEE Transactions on Control Systems Technology vol 1 no 2pp 80ndash92 1993
[10] M B McFarland and S M Hoque ldquoRobustness of a nonlinearmissile autopilot designed using dynamic inversionrdquo in Pro-ceedings of the AIAA Guidance Navigation and Control Confer-ence and Exhibit AIAA-2000-3970 Denver Colo USA August2000
[11] G Looye and H-D Joos ldquoDesign of robust dynamic inversioncontrol laws usingmulti-objective optimizationrdquo in Proceedingsof the AIAA Guidance Navigation and Control Conference andExhibit AIAA-2001-4285 2001
[12] P B Acquatella W Falkena E van Kampen and Q P ChuldquoRobust nonlinear spacecraft attitude control using incrementalnonlinear dynamic inversionrdquo in Proceedings of the AIAAGuid-ance Navigation and Control Conference and Exhibit Min-neapolis Minn USA August 2012
[13] P K Menon V R Iragavarapu and E J Ohlmeyer ldquoNonlinearMissile Autopilot Design using Time-Scale Separationrdquo in Pro-ceedings of the AIAA Guidance Navigation and Control Confer-ence and Exhibit AIAA-1997-3765 1997
[14] A Steinicke and G Michalka ldquoImproving transient perfor-mance of dynamic inversion missile autopilot by use of back-steppingrdquo in Proceedings of the AIAAGuidance Navigation andControl Conference and Exhibit AIAA-2002-4658 MontereyCalif USA August 2002
12 Mathematical Problems in Engineering
[15] I Yang D Kim and D Lee ldquoA flight control strategy usingrobust dynamic inversion based on sliding mode controlrdquo inproceeding of the AIAA Guidance Navigation and ControlConference August 2012
[16] H Sira-Ramirez ldquoDifferential geometric methods in variable-structure controlrdquo International Journal of Control vol 48 no4 pp 1359ndash1390 1988
[17] V I Utkin ldquoVariable structure systems with sliding modesrdquoIEEE Transactions on Automatic Control vol 22 no 2 pp 212ndash222 1977
[18] R A DeCarlo S H Zak and G P Matthews ldquoVariable struc-ture control of nonlinear multivariable systems a tutorialrdquo Pro-ceedings of the IEEE vol 76 no 3 pp 212ndash232 1988
[19] J Y HungWGao and J CHung ldquoVariable structure control asurveyrdquo IEEE Transactions on Industrial Electronics vol 40 no1 pp 2ndash22 1993
[20] G Papageorgiou andM Polansky ldquoTuning a dynamic inversionpitch axis autopilot using McFarlane-Glover loop shapingrdquoOptimal Control Applications ampMethods vol 30 no 3 pp 287ndash308 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 3
Robust dynamic inversion
Desireddynamics
Robustdynamicinversion
Plant
Figure 2 Block diagram of the proposed robust nonlinear dynamic inversion
p
Leader
Follower
Ez
Ex
Ey
eh
rf
er
e120579
rl
Figure 3 Reference coordinate frame [1]
according to the desired reference trajectory Figure 3 showsthe coordinate reference frames used throughout this paperTwo main coordinate frames are the Earth centered inertiaframe E
119909E119910E119911 whose origin is located in the center of the
Earth and the leader orbit frame e119903 e120579 eℎ whose origin is
attached to the center of mass of the leader spacecraft Thenonlinear dynamics of the leader and follower with respectto the e
119903 e120579 eℎ frame can be represented as follows [1 3]
r119897= minus
120583
1199033
119897
r119897+F119889119897
119898119897
+u119897
119898119897
(1)
r119891= minus
120583
1199033
119891
r119891+F119889119891
119898119891
+u119891
119898119891
(2)
where u119897isin R3 and u
119891isin R3 are control input vectors of the
leader and follower spacecrafts respectively And F119889119897isin R3
and F119889119891
isin R3 are the disturbance force vectors acting onthe leader and follower spacecrafts respectively It is assumedthat the masses of spacecrafts are small relative to the massof the Earth that is 119872 ≫ 119898
119897 119898119891 so 119866(119872 + 119898
119897) asymp 119866119872
and 119866(119872 + 119898119891) asymp 119866119872 Because the relative position of the
follower spacecraft p = r119891minus r119897 the second derivative of the
relative position can be represented as follows
p = r119891minus r119897= (minus
120583
1199033
119891
(r119897+ p) +
F119889119891
119898119891
+u119891
119898119891
)
minus (minus120583
1199033
119897
r119897+F119889119897
119898119897
+u119897
119898119897
)
= minus119872119866(r119897+ p
1003817100381710038171003817r119897 + p10038171003817100381710038173minus
r119897
1199033
119897
) minusF119889
119898119891
+u119905
119898119891
(3)
where F119889= F119889119891minus (119898119891119898119897)F119889119897and u
119905= u119891minus (119898119891119898119897)u119897
In this paper it is assumed that the leader spacecraft is freeflying that is u
119897= 0 then it satisfies that u
119905= u119891
If the relative position p in the leader orbit frame isgiven as p = 119909e
119903+ 119910e120579+ 119911eℎfor the unit vectors e
119903 e120579
and eℎalong with the E
119909E119910E119911-frame then the relative
acceleration vector p can be obtained as follows
p = ( minus 21205960119910 minus 1205962
0119909) e119903+ ( 119910 + 2120596
0 minus 1205962
0119910) e120579+ eℎ (4)
In the moving leader orbit frame the position vector r119897is
constant that is r119897= 119903119897e119903 By substituting (4) into (3) the
nonlinear dynamics of the follower spacecraft relative to theleader can be obtained as
( minus 21205960119910 minus 1205962
0119909) e119903+ ( 119910 + 2120596
0 minus 1205962
0119910) e120579+ eℎ
= minus119872119866((119903119897+ 119909) e
119903+ 119910e120579+ 119911eℎ
1003817100381710038171003817r119897 + p10038171003817100381710038173
minus119903119897e119903
1199033
119897
)
minus119865119889119909e119903+ 119865119889119910e120579+ 119865119889119911eℎ
119898119891
+119906119909e119903+ 119906119910e120579+ 119906119911eℎ
119898119891
997904rArr [
[
119910
]
]
= minus[
[
0 minus212059600
21205960
0 0
0 0 0
]
]
[
[
119910
]
]
+
[[[[[[[[
[
1205962
0minus
119872119866
1003817100381710038171003817r119897 + p10038171003817100381710038173
0 0
0 1205962
0minus
119872119866
1003817100381710038171003817r119897 + p10038171003817100381710038173
0
0 0 minus119872119866
1003817100381710038171003817r119897 + p10038171003817100381710038173
]]]]]]]]
]
[
[
119909
119910
119911
]
]
4 Mathematical Problems in Engineering
minus119872119866[[[
[
119903119897
1003817100381710038171003817r119897 + p10038171003817100381710038173minus1
1199032119897
0
0
]]]
]
minus1
119898119891
[
[
119865119889119909
119865119889119910
119865119889119911
]
]
+1
119898119891
[
[
119906119909
119906119910
119906119911
]
]
(5)
Denote the relative position and velocity vectors as p119897=
(119909 119910 119911) and v119897= (119889119909119889119905 119889119910119889119905 119889119911119889119905) in the leader orbit
frame respectively Then (5) can be represented as follows
p119897= k119897
k119897= minus C (120596
0) k119897minusD (p
119897 r119897 1205960) p119897minus N (p
119897 r119897) minus
F119889
119898119891
+u119905
119898119891
(6)
where
C (1205960) = [
[
0 minus212059600
21205960
0 0
0 0 0
]
]
N (p119897 r119897) = 119872119866
[[[
[
119903119897
1003817100381710038171003817r119897 + p10038171003817100381710038173minus1
1199032119897
0
0
]]]
]
F119889= [
[
119865119889119909
119865119889119910
119865119889119911
]
]
D (p119897 r119897 1205960)
=
[[[[[[[[
[
1205962
0minus
119872119866
1003817100381710038171003817r119897 + p10038171003817100381710038173
0 0
0 1205962
0minus
119872119866
1003817100381710038171003817r119897 + p10038171003817100381710038173
0
0 0 minus119872119866
1003817100381710038171003817r119897 + p10038171003817100381710038173
]]]]]]]]
]
(7)
3 The Proposed Robust NonlinearDynamic Inversion
In this section the mathematical backgrounds of slidingmode control and nonlinear dynamic inversion are intro-duced And then the robust nonlinear dynamic inversionmethod is proposed
31 Sliding Mode Controller Sliding mode control (SMC)also called variable structure control (VSC) is a high-speedswitching nonlinear control method that forces the systemstates into a user-designed sliding surface and maintains thestates on the surface [17 18] SMC consists of two phases(Figure 4) structuring the sliding surface and constructingthe switching feedback gains
Consider the following system
x (119905) = f (x (119905)) + g (x (119905)) u (119905) (8)
Sliding surface
Sliding phase
Reaching phase
120590(x) = 0
x2
x1
x(t1)
x(t0)
Figure 4The behavior of the system states in sliding mode control
where x(119905) isin R119899 is a state vector u(119905) isin R119898 is an input vectorwith 119899 ge 119898 and f R119899 rarr R119899 and g R119898 rarr R119899times119898are smooth functions Let a set Σ
119894be the regular (119899 minus 1)
dimensional submanifold in R119899 defined in [16] such as
Σ119894≜ x isin R119899 | 120590
119894(x) = 0 119894 = 1 2 119898 (9)
where 120590119894 R119899 rarr R (119894 = 1 2 119898) is a smooth function
Define120590119894(x) = 0 as the individual sliding surface or individual
switching surface Then surface constructed by intersecting119898 individual sliding surfaces is defined as the sliding surface[16]
Σ ≜
119898
⋂
119894=1
Σ119894=
119898
⋂
119894=1
x isin R119899 | 120590119894(x) = 0 ≜ x isin R119899 | 120590 (x) = 0
(10)
where 120590(x) = [1205901(x) 1205902(x) 120590
119898(x)]119879 If the states x(119905) of
the system are in Σ that is 120590(x(119905)) = 0 then the behavior ofthe states is called sliding motion or sliding mode Moreoverif sliding mode exists the tangent vectors of the states alwaysare forced to track the sliding surface as shown in Figure 4Then each entry 119906
119894(119905) (119894 = 1 2 119898) of the sliding mode
control law can be obtained by taking one of the followingvalues [16]
119906119894(119905) =
119906+
119894 if 120590
119894(x (119905)) gt 0
119906minus
119894 if 120590
119894(x (119905)) lt 0
(11)
where 119906+119894= 119906minus
119894
As shown in (11) designing the sliding surface is crucialbecause the sliding surface determines the control input andmoreover the performance of the closed-loop system Forthis reason various designing methods such as Fillipovrsquosmethod and equivalent control method have been proposed[18 19] In this paper the equivalent control method is
Mathematical Problems in Engineering 5
considered to analyze the similar characteristics betweenSMC and NDI If the system states intercept the slidingsurface at 119905 = 119905
0and a sliding mode exists for 119905 ge 119905
0 then
(119889119889119905)120590(x(119905)) = 0 and 120590(x(119905)) = 0 for all 119905 ge 1199050 Then the
equivalent input ueq that forces the state trajectory to stay onthe sliding surface can be analyzed as follows [19]
(x) = 120597120590120597x(f (x) + g (x)ueq) = 0
997904rArr ueq = minus(120597120590
120597xg(x))
minus1
(120597120590
120597xf (x))
(12)
Substituting (12) into (1) the system dynamics on the slidingsurface for 119905 ge 119905
0is governed by
x = f (x) + g (x) ueq = [I minus g (x) (120597120590120597x
g(x))minus1
(120597120590
120597x)] f (x)
(13)
The dynamics represented in (13) is defined as the idealsliding dynamics [16] Consequently the closed-loop systemdynamics in sliding mode are specified by the sliding surface
One of the widely used designing methods of the slidingmode controller combines switching inputs and the equiv-alent inputs as represented in (12) that is if the slidingsurface is determined then the sliding mode controller canbe obtained by adding switching input (119906sw119894) of which entry119906119894(119905) yields [18]
119906119894= 119906eq119894 + 119906sw119894 (119894 = 1 2 119898) (14)
where 119906eq119894 is the 119894th entry of the equivalent input analyzedin (12) And 119906sw119894 is the switching input that forces the systemstates tomove towards the sliding surface In [18] the possiblecandidates for designing the switching input are introducedfor example
(i) relay with constant gains 119906sw119894(x) = 120572119894 sgn(120590119894(x))with120572119894lt 0
(ii) relay with state dependent gains 119906sw119894(x) =
120572119894(x) sgn(120590
119894(x)) with 120572
119894(sdot) lt 0
(iii) univector nonlinearity with scale factorusw(x)=120588120590(x)120590(x) with 120588 lt 0
32 Similar Characteristics between Sliding Mode Control andNonlinear Dynamic Inversion Before analyzing the similarcharacteristics between SMC andNDI the conventional NDIcontroller is firstly introduced NDI controller consists oftwo blocks (Figure 5) desired dynamic block and dynamicinversion (DI) block [4ndash6] In the desired dynamic blockthe control variables are defined and the rate commandsof the selected control variables are generated And in theDI block proper control inputs are generated by invertingthe plant dynamics in order to make the inner closed-loop transfer function as an integrator [20] Hence thesystem states controlled by the dynamic inversion block willfollow the user-selected control variables For this reason thedesired dynamics are usually designed to satisfy the specific
dynamicsDynamicinversion
x x
xPlant
xcmd asymp1
s
Figure 5 Structure of dynamic inversion [4]
performancesrequirements so that the performance of thecontrolled system can achieve the system requirements In[4] some forms of desired dynamics are introduced propor-tional proportional integral flying quality ride quality andso forth
For the nonlinear system shown in (8) NDI control inputuNDI(119905) can be given by [4]
uNDI (119905) = [g (x (119905))]minus1
[xdes (119905) minus f (x (119905))] (15)
where xdes(119905) represents the desired dynamicsIt is worth noting that the requirement for existing NDI
control law in (15) is that [g(x)]minus1 must exist Howeverthe matrix g(x) is not generally a full rank for nonflatsystems These systems have a larger number of states thanthe number of control inputs Hence the number of statesthat can be inverted is less than or equal to the numberof inputs One way of achieving [g(x)]minus1 is to formulatethe existence problem as a two-time scale problem A two-time scale problem makes the system states separate into fastand slow dynamics The fast dynamics comprise a group ofstates affected directly by the control inputs On the otherhand the slow dynamics consists of states influenced by thefast dynamics Separating the system states into two groupsreduces the system order enough to provide a chance toexist more number of inputs than the number of the fastdynamic states Consequently [g(x)]minus1 can be obtained [4ndash6] In contrast for an overactuated system that adopts a largernumber of inputs than the number of states the inverse canbe obtained by reducing the number of inputs to be equal tothe number of states using control allocation [8 9]
By substituting (15) into (8) the inner-loop dynamics inFigure 5 controlled by conventional NDI yields
xdes = x (16)
Hence NDI controller replaces the original dynamics intothe user-selected desired dynamics by inverting the originaldynamics Consequently the control system can guaranteethe stability without gain-scheduling However it is impos-sible to achieve perfect cancellation of the original dynamicsdue to uncertainties such asmodelmismatches disturbancesand measurement noises which leads to poor robustnessof NDI controller Hence robustness issues against uncer-tainties must be considered when designing NDI control-ler
To achieve the robustness of NDI the equivalent controlmethod used to design sliding surface is considered Fordesired trajectory xdes isin R119899 let xlowast = xdes minus x isin R119899
6 Mathematical Problems in Engineering
Select a set of nonlinear smooth functions 120590(xlowast) = [1205901(xlowast)
1205902(xlowast) 120590
119898(xlowast)]119879 = 0 as a sliding surface If the system
states intercept the sliding surface at 119905 = 1199050and the sliding
mode exists for 119905 ge 1199050 then (119889119889119905)120590(x(119905)) = 0 and 120590(x(119905)) = 0
for all 119905 ge 1199050 Then the equivalent input ueq that the state
trajectory stays on the sliding surface can be analyzed asfollows [13ndash15]
(xlowast) = 120597120590
120597xlowastxlowast = 120597120590
120597xlowast(xdes minus x) = 0
997904rArr120597120590
120597xlowastxdes minus
120597120590
120597xlowast(f (x) + g (x) ueq) = 0
997904rArr ueq = (120597120590
120597xlowastg (x))
minus1
(120597120590
120597xlowastxdes minus
120597120590
120597xlowastf (x))
(17)
If 119898 = 119899 and 120597120590120597xlowast = I119899 then (17) can be represented as a
form of NDI input shown in (15)
ueq = g(x)minus1 [xdes minus f (x)] = uNDI (18)
Hence if the sliding surface is designed by an identity matrixthen the equivalent input of SMC can be represented as NDIinput
Definition 1 Let 120590NDI(x) be a set of smooth functions120590NDI119894(x) (119894 = 1 2 119899) that is 120590NDI(x)=[120590NDI1(x)120590NDI2(x) 120590NDI119899(x)]
119879 Then 120590NDI(x) = x is defined as theNDI surface that is the NDI surface is the sliding surfacedesigned by the identity matrix
It is worth noting that the ideal sliding dynamics shownin (13) yields the desired dynamics as follows
x = f (x) + g (x)ueq
= [f (x) + g (x) ( 120597120590120597xlowast
g (x))minus1
(120597120590
120597xlowastxdes minus
120597120590
120597xlowastf (x))]
= xdes(19)
In (19) the states on the identity sliding surface are governedby the desired dynamics Moreover the performances ofthe control system in sliding mode are determined only bythe desired dynamics This suggests that the difficulties indesigning a sliding mode controller to satisfy the specificsystem performances are converted to the desired dynamicsdesigning problem
33 Design of the Robust Nonlinear Dynamic Inversion Simi-lar to the designmethodof SMC each entry119906
119894(119905) (119894 = 1 2
119898) of the RNDI control law can be obtained by introducingthe following values
119906119894(119905) =
119906+
119894 if 120590
119894(119909des119894 minus 119909119894) = 119909des119894 minus 119909119894 gt 0
119906minus
119894 if 120590
119894(119909des119894 minus 119909119894) = 119909des119894 minus 119909119894 lt 0
(20)
where 119906+119894= 119906minus
119894satisfies the following for 119891
119894(x) the 119894th entry
of f(x) and for 119892119894119895(x) the (119894 119895)-entry of g(x)
lim119909119894rarr119909
+
des119894
119891119894(119909) +
119899
sum
119895=1
119895 = 119894
119892119894119895(119909) 119906119895+ 119892119894119894(119909) 119906minus
119894lt des119894
lim119909119894rarr119909
minus
des119894
119891119894(119909) +
119899
sum
119895=1
119895 = 119894
119892119894119895(119909) 119906119895+ 119892119894119894(119909) 119906+
119894gt des119894
(21)
where the superscripts + and minus denote the right- and left-hand limit respectivelyThe following theorem and corollarypropose the method for designing the RNDI controller andprovide the stability analytically
Theorem 2 Consider the following input
u = u119873119863119868
+ gminus1 (x) u119904119908 (22)
where u119873119863119868
is the conventional NDI input and u119904119908
isthe switching input that satisfies the following for u
119904119908=
[1199061199041199081 1199061199041199082 119906
119904119908119899]119879
119906119904119908119894
gt 0 119894119891 120590119894(x119889119890119904minus x) = 119909
119889119890119904119894minus 119909119894gt 0
119906119904119908119894
lt 0 119894119891 120590119894(x119889119890119904minus x) = 119909
119889119890119904119894minus 119909119894lt 0
(23)
Then the closed-loop system controlled by (22) is globally stable
Proof Select a Lyapunov candidate as
119881 =1
2120590(xdes minus x)119879120590 (xdes minus x) (24)
From (22) and (23)
= 120590(xdes minus x)119879 (xdes minus x)
= 120590(xdes minus x)119879 (xdes minus x)
= 120590(xdes minus x)119879 [xdes minus f (x) + g (x) u]
= 120590(xdes minus x)119879
times [xdes minus f (x) + g (x) (uNDI + gminus1 (x) usw)]
= 120590(xdes minus x)119879
times [xdes minus f (x) + g (x) uNDI minus usw]
= minus120590(xdes minus x)119879usw
= minus
119899
sum
119894=1
120590119894(xdes minus x) 119906sw119894
(25)
From (25) if 120590119894(xdes minus x) gt 0 then 119906sw119894 gt 0 and lt 0
Similarly if 120590119894(xdes minus x) lt 0 then 119906sw119894 lt 0 and lt 0 Hence
the closed-loop system controlled by (22) is globally stable
Mathematical Problems in Engineering 7
NDI surface
fi(x) +n
sumj=1jnei
gij(x)uj + gii(x)u+i
fi(x) +n
sumj=1jnei
gij(x)uj + gii(x)uminusi
120590i( minus ) gt 0x xdes
120590i( minus x) lt 0xdes
120590i( minus ) = 0xdes x
Figure 6 State trajectory on the NDI surface
Since the system controlled by (22) is globally stable fromTheorem 2 the states are forced to the NDI surface as shownin Figure 6
Corollary 3 If the switching input denoted as u119904119908= [1199061199041199081
1199061199041199082 119906
119904119908119899]119879 is designed with 119906
119904119908119894= 119896119894sgn(120590
119894(x119889119890119904minus x))
for a positive 119896119894 then the control system is globally stable
Proof If 120590119894(xdes minus x) gt 0 then 119906sw119894 = 119896119894 gt 0 and conversely
if 120590119894(xdes minus x) lt 0 then 119906sw119894 = minus119896119894 lt 0 Hence according to
Theorem 2 the control system is globally stable
Definition 4 The control law proposed inTheorem 2 is calledthe robust nonlinear dynamic inversion (RNDI) law that is fora diagonal matrix K = diag(119896
119894) (119894 = 1 2 119899) with 119896
119894gt 0
the RNDI control law yields
uRNDI = uNDI + usw
= g(x)minus1 [xdes minus f (x) + K sgn (xdes minus x)] (26)
It is worth noting that the form of the switching inputrepresented in Corollary 3 is a relay with constant gains formin SMC as mentioned in Section 31 Actually the switchingfunction satisfying (23) can be obtained directly from SMCdesign method as follows
(i) relay with constant gains 119906sw119894 = 119896119894 sgn(120590119894(xdes minus x))with 119896
119894gt 0
(ii) relay with state dependent gains 119906sw119894 = 119896119894(xdes minus
x) sgn(120590119894(xdes minus x)) with 119896
119894(sdot) gt 0
(iii) univector nonlinearity with scale factor usw =
119896119894120590(xdes minus x)120590(xdes minus x) with 119896
119894gt 0
It is also remarkable that using the sgn function whiledesigning the switching input leads to unfavorable resultsin control systems such as the chattering problem Thechattering problem makes the system have unmodeled highfrequencies or actuator saturation In sliding mode controltheory several methods have been proposed to overcomethis problem by designing a continuous control input insteadof a discontinuous switching input This method forces the
controlled system to follow the approximated sliding motionin some boundaries instead of the ideal sliding motionFollowing the similar manner the chattering problem in theRNDI control can be solved using the continuous controlinput For example replace sgn(xdes minus x) into sat(xdes minus x) =[sat1(119909des1 minus 1199091) sat2(119909des2 minus 1199092) sat119899(119909des119899 minus 119909119899)]
119879 inorder to design a continuous switching input as follows for apositive 120575
119894
sat119894(119909des119894 minus 119909119894
120575119894
) =
1 if (119909des119894 minus 119909119894) gt 120575119909des119894 minus 119909119894
120575119894
if 1003816100381610038161003816119909des119894 minus 1199091198941003816100381610038161003816 lt 120575
minus1 if (119909des119894 minus 119909119894) lt minus120575(27)
Then 119906sw119894 = 119896119894sat119894((119909des119894minus119909119894)120575119894) In (27) plusmn120575 is the boundarylayer in which the states are governed by the followingdynamics for sat((xdes minus x)120575) = [sat
1((119909des1 minus 1199091)1205751)
sat2((119909des2 minus 1199092)1205752) sat119899((119909des119899 minus 119909119899)120575119899)]
119879 and K =
diag(119896119894) (119894 = 1 2 119899) with 119896
119894gt 0
x = f (x) + g (x)u
= f (x) + g (x)
times [(g (x))minus1 (xdes minus f (x))
+(g (x))minus1Ksat((xdes minus x)
120575)]
= xdes + Ksat((xdes minus x)
120575)
= xdes
+ [1198961(119909des1 minus 1199091)
1205751
1198962(119909des2 minus 1199092)
1205752
sdot sdot sdot
119896119899(119909des119899 minus 119909119899)
120575119899
]
119879
(28)
8 Mathematical Problems in Engineering
uf
RefTime
Angular
Position of Referencetrajectorygenerator
the leader spacecraft
Reffcn
PositionControl
Dynamics of the Robust nonlinear dynamic inversion follower spacecraft
PositionScope
DisturbancesFd
++
t
ww
plplinputs
velocity
Figure 7 Block diagram using MATLAB Simulink
34 Stability of the Proposed RNDI Controller In this paperthe stability of the closed-loop system controlled by theproposed RNDI is analyzed by a Lyapunov stability criterionConsider the following nonlinear system including boundedmodel uncertainties Δf(x(119905)) isin R119899 and Δg(x(119905)) isin R119899 andbounded disturbance d(119905) isin R119899
x (119905) = [f (x (119905)) + Δf (x (119905))]
+ [g (x (119905)) + Δg (x (119905))] u (119905) + d (119905) (29)
It is assumed that 120585(119905 x(119905) u(119905)) = Δf(x(119905)) + Δg(x(119905)) + d(119905)satisfying 120585(119905 x(119905) u(119905))
2lt 120582min(K) where 120582min(K) is the
minimum eigenvalue of K For the desired state xdes isin 119877119899denote xlowast(119905) = xdesminusx To analyze the stability of the proposedRNDI controller select a Lyapunov candidate as follows
119881 (xlowast (119905)) = 12xlowast(119905)119879xlowast (119905) (30)
Then the derivative of 119881(xlowast(119905)) yields
(xlowast (119905))
= xlowast(119905)119879xlowast (119905)
= xlowast(119905)119879 [xdes (119905) minus [f (x (119905)) + Δf (x (119905))]
+ [g (x (119905)) + Δg (x (119905))] u (119905) + d (119905) ]
= xlowast(119905)119879 [xdes (119905) minus (f (x (119905)) + g (x (119905)) u (119905))
minus (Δf (x (119905)) + Δg (x (119905)) u (119905) + d (119905)) ]
= xlowast(119905)119879 [minus120585 (119905 x (119905) u (119905)) minus K sgn (xlowast (119905))]
le minusxlowast(119905)119879120585 (119905 x (119905) u (119905)) minus 120582min (K)1003817100381710038171003817xlowast
(119905)10038171003817100381710038171
le 120585 (119905 x (119905) u (119905))1
1003817100381710038171003817119909lowast
(119905)10038171003817100381710038171 minus 120582min (K)
1003817100381710038171003817xlowast
(119905)10038171003817100381710038171
le [120585 (119905 x (119905) u (119905))2minus 120582min (K)]
1003817100381710038171003817xlowast
(119905)10038171003817100381710038171
(31)
By the hypothesis 120585(119905 x(119905) u(119905))2lt 120582min(K) the derivative
of 119881(xlowast(119905)) is always negative Hence the system controlledby the proposed RNDI controller is globally stable againstdisturbances noises and model mismatches
Table 1 Parameters for spacecraft formation flying [1 3]
Symbol Value Unit119866 6673 times 10minus11 m3kgsdots2
119872 5974 times 1024 kg119898119891
410 kg119898119897
1550 kgr119897
[4224 times 1024 0 0]119879 m1205960
7272 times 10minus5 rads
4 Simulation Results
In this section numerical simulations are conducted to eva-luate the performance of the proposed robust nonlineardynamic inversion controller
41 Simulation Description The aim of SFF is to design thefeasible control input u
119905(119905) such that p
119897rarr p119889as 119905 rarr infin
for a given reference relative position trajectory p119889isin R3
of the follower spacecraft with respect to the leader space-craft To evaluate the performance of the proposed RNDIcontroller MATLAB Simulink is selected as a simulationtool Figure 7 shows the block diagram of the overall SFFstructure using MATLAB Simulink In this paper it isassumed that the reference trajectory is set to p
119889=
[100 sin(41205960119905) 100 cos(4120596
0119905) 0]119879 that is the follower space-
craft tracks a circular orbit centered at the leader spacecraftwith a radius of 100 meters on a plane generated by e
119903and e120579
with an angular velocity 41205960 The initial relative position and
velocity are assumed to be respectively as follows
p119897(0) = [10 90 minus20]
119879
k119897(0) = [0 0 0]
119879
(32)
Table 1 lists some parameters and their values used in thissimulation Some uncertainties such as disturbances and sen-sor noises of velocity and position are induced in this simu-lation Disturbances acting on the follower spacecraft areassumed such that F
119889= [290532 31775 minus112298]
119879 (N)And it is also assumed that maximally 20 of random velo-city sensor noises in three axes create difficulties in allowingthe follower spacecraft to obtain accurate velocity informa-tion Moreover maximally 20 and 40 of random positionsensor noises in e
119903and e
ℎaxes are also considered in this
simulation
Mathematical Problems in Engineering 9
0 10 20 30 40 50 60 70 80 90 100
0
5
10
Time (s)
No disturbanceDisturbance
minus5
minus10
minus15
e r-d
irect
ion
(m)
(a) Position tracking error in e119903-direction
0 10 20 30 40 50 60 70 80 90 100
0
2
Time (s)
No disturbanceDisturbance
minus2
minus4
minus6
minus8
minus10
e 120579-d
irect
ion
(m)
(b) Position tracking error in e120579-direction
0 10 20 30 40 50 60 70 80 90 100
0
5
10
Time (s)
No disturbanceDisturbance
minus5
minus10
minus15
minus20
e h-d
irect
ion
(m)
(c) Position tracking error in eℎ-direction
Figure 8 Relative position errors of no disturbance case and disturbance injected case The black thin line represents the trajectories of thespacecraft without considering any disturbances The blue thick line shows the results of the trajectories of the disturbed spacecraft
42 Simulation Results Figure 8 describes the position track-ing error results of the no disturbance case and disturbancesinjected case The objective of this simulation is to verifythe performance degradation of the disturbances injectedspacecraft In both cases NDI with the proportional type ofthe desired dynamics is designed as a primary controller forSFF As shown in Figure 8 if no disturbance is acting on thespacecraft then the position errors between the referencedposition and the follower position are zero in 30 secHoweverthe lack of robustness in NDI controller leads a significantperformance degradation of the disturbed spacecraft so ittracks the references with a tracking error of almost 8m 1mand 4m in e
119903 e120579 and e
ℎaxes respectively
Figures 9 and 10 show the trajectories of the followerspacecraft controlled by NDI and RNDI controllers Asmentioned in Figure 8 the position errors between thereferenced position and follower position controlled by NDIcannot be zero due to the induced uncertainties However bycombining the switching input into conventional NDI con-troller the RNDI control law attempts to reduce the effect ofthe uncertainties Consequently the state can track the refer-ence signal within 1m error as shown in Figure 9The controlinputs generated by NDI and RNDI controllers are described
in Figure 10 To compensate the induced uncertainties thespacecraft controlled by RNDI generates a large input force ineℎaxis around 27 sec (Figure 10(b)) Figure 11 shows the tra-
jectory of the spacecraft controlled by the RNDI controllerThe spacecraft maneuvers around a circle with a 100-meterradius Although the spacecraft is vibrated due to the induceduncertainties the error distance between the reference trajec-tory and the position of the follower is less than 1m as shownin Figure 9 Hence the follower spacecraft controlled by theproposed RNDI can track the reference trajectory
5 Conclusion
In this paper the robust nonlinear dynamic inversion controlstrategy that improves the robustness of NDI has been pro-posed NDI is a nonlinear control technique that eliminatesthe need of linearizing and designing gain-scheduled con-trollers for each equilibrium point by canceling the originaldynamics Because NDI removes the system nonlinearitiesdirectly it provides an opportunity to overcome the difficul-ties of ensuring stabilities and performances between variousoperational points while gain-scheduling However the maindrawback of designing NDI controller is the weak robustness
10 Mathematical Problems in Engineering
0 10 20 30 40 50 60 70 80 90 100
0
5
10
Time (m)
NDIRNDI
e r-d
irect
ion
(m)
minus5
minus15
minus10
(a) Position tracking error in e119903-direction
0
2
0 10 20 30 40 50 60 70 80 90 100Time (s)
NDIRNDI
e 120579-d
irect
ion
(m)
minus6
minus2
minus4
minus8
minus10
(b) Position tracking error in e120579-direction
0 10 20 30 40 50 60 70 80 90 100
0
5
10
Time (s)
NDIRNDI
e h-d
irect
ion
(m)
minus20
minus15
minus10
minus5
(c) Position tracking error in eℎ-direction
Figure 9 Simulation results of the position tracking error of the spacecrafts controlled by NDI and RNDIThe black thin line represents thetrajectories of the spacecraft controlled by NDI while the blue thick line represents the trajectories of the spacecraft controlled by RNDI
20 30 40 50 60 70 80 90 100
0
100
200
300
400
Time (s)
Inpu
t for
ces (
N)
minus200
minus100
ux along with er-axis
uz along with eh-axis
uy along with e120579-axis
(a) Control inputs generated by NDI
20 30 40 50 60 70 80 90 100
0
100
200
300
400
Time (s)
Inpu
t for
ces (
N)
minus200
minus100
ux along with er-axis
uz along with eh-axis
uy along with e120579-axis
(b) Control inputs generated by RNDI
Figure 10 Control inputs generated by NDI and RNDI
due to difficulties in obtaining accurate information of thenonlinear system dynamics In contrast to NDI SMC isconsidered a robust nonlinear control technique that usesswitching values as inputs to force the system states intoa sliding surface and to maintain the states on it Theproposed RNDI controller is designed by combining the
conventional NDI controller and switching inputs analyzedin SMC Hence RNDI takes the advantages of both NDI andSMC such that it
(i) is easy to design and implement(ii) eliminates the need of gain-scheduling
Mathematical Problems in Engineering 11
0 50 100 150050100150
0
10
e120579 axis er axis
e hax
is
minus50minus50minus100minus100 minus150minus150
minus10
minus20
Figure 11 Trajectory of the follower spacecraft controlled by theproposed RNDI controller
(iii) is easy to achieve the system required performancedue to the similar concept to classical control meth-ods and
(iv) is inherently robust
However the drawback of SMC such as the chatteringproblem still remains as a problem of RNDI control It isexpected that various chattering-free methods can be usedto reduce the effect of chattering phenomenon for examplereplacing the sign function into the saturation function
Moreover the equivalence of NDI input and equivalentinput designed by the identity sliding surface is also analyzedin this paper The reduced order dynamics on the slidingsurface can be converted to the user-selected desired dynam-ics As a result the difficulties in designing a sliding modecontroller for satisfying the specific system performances areconverted to a designing problem of the desired dynamicsHence RNDI provides an opportunity to solve the difficultiesin selecting the sliding surface of SMC The effectiveness ofthe proposed RNDI controller is confirmed by a set of simu-lations with application to the spacecraft formation flying
Nomenclature
G Universal gravity constantM Mass of the Earth119898119891 Mass of the follower spacecraft
119898119897 Mass of the leader spacecraft
r119891 Distance from the center of the Earth to
the center of the follower spacecraftr119897 Distance from the center of the Earth to
the center of the leader spacecraftxdes Desired dynamics
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research was supported by the MSIP (Ministry ofScience ICT amp Future Planning) Korea under the C-ITRC(Convergence Information Technology Research Center)
support program (NIPA-2014-H0401-14-1004) supervised bythe NIPA (National IT Industry Promotion Agency)
References
[1] Y Lv Q Hu G Ma and J Zhou ldquo6 DOF synchronized con-trol for spacecraft formation flying with input constraint andparameter uncertaintiesrdquo ISA Transactions vol 50 no 4 pp573ndash580 2011
[2] D Forta F Bordi and C Scolese ldquoLow-cost minimum sizesatellites for demonstration of formation flying modes at smallkilometer-size distancerdquo in Proceedings of the 13th AIAAUSUConference on Small Satellite SSC99-VI-3 pp 1ndash15 1999
[3] M S De Queiroz V Kapila and Q Yan ldquoAdaptive nonlinearcontrol ofmultiple spacecraft formation flyingrdquo Journal of Guid-ance Control and Dynamics vol 23 no 3 pp 385ndash390 2000
[4] J Georgie and J Valasek ldquoEvaluation of longitudinal desireddynamics for dynamic-inversion controlled generic reentryvehiclesrdquo Journal of Guidance Control and Dynamics vol 26no 5 pp 811ndash819 2003
[5] D Ito J Georgie J Valasek and D T Ward Reentry VehicleFlight Controls Design Guidelines Dynamic Inversion NASATPmdash2002ndash210771 2002
[6] A J Ostroff and B J Bacon ldquoForce and moment approachfor achievable dynamics using nonlinear dynamic inversionrdquo inProceedings of theAIAAGuidance Navigation andControl Con-ference and Exhibit pp 424ndash434 Reston Va USA 1999
[7] D Enns D Bugajski R Hendrick and G Stein ldquoDynamicinversion an evolving methodology for flight control designrdquoInternational Journal of Control vol 59 no 1 pp 71ndash91 1994
[8] J A Richard J M Buffington A G Sparks and S S BandaRobust Multivariable Flight Control Springer New York NYUSA 1994
[9] R J Adams and S S Banda ldquoRobust flight control design usingdynamic inversion and structured singular value synthesisrdquoIEEE Transactions on Control Systems Technology vol 1 no 2pp 80ndash92 1993
[10] M B McFarland and S M Hoque ldquoRobustness of a nonlinearmissile autopilot designed using dynamic inversionrdquo in Pro-ceedings of the AIAA Guidance Navigation and Control Confer-ence and Exhibit AIAA-2000-3970 Denver Colo USA August2000
[11] G Looye and H-D Joos ldquoDesign of robust dynamic inversioncontrol laws usingmulti-objective optimizationrdquo in Proceedingsof the AIAA Guidance Navigation and Control Conference andExhibit AIAA-2001-4285 2001
[12] P B Acquatella W Falkena E van Kampen and Q P ChuldquoRobust nonlinear spacecraft attitude control using incrementalnonlinear dynamic inversionrdquo in Proceedings of the AIAAGuid-ance Navigation and Control Conference and Exhibit Min-neapolis Minn USA August 2012
[13] P K Menon V R Iragavarapu and E J Ohlmeyer ldquoNonlinearMissile Autopilot Design using Time-Scale Separationrdquo in Pro-ceedings of the AIAA Guidance Navigation and Control Confer-ence and Exhibit AIAA-1997-3765 1997
[14] A Steinicke and G Michalka ldquoImproving transient perfor-mance of dynamic inversion missile autopilot by use of back-steppingrdquo in Proceedings of the AIAAGuidance Navigation andControl Conference and Exhibit AIAA-2002-4658 MontereyCalif USA August 2002
12 Mathematical Problems in Engineering
[15] I Yang D Kim and D Lee ldquoA flight control strategy usingrobust dynamic inversion based on sliding mode controlrdquo inproceeding of the AIAA Guidance Navigation and ControlConference August 2012
[16] H Sira-Ramirez ldquoDifferential geometric methods in variable-structure controlrdquo International Journal of Control vol 48 no4 pp 1359ndash1390 1988
[17] V I Utkin ldquoVariable structure systems with sliding modesrdquoIEEE Transactions on Automatic Control vol 22 no 2 pp 212ndash222 1977
[18] R A DeCarlo S H Zak and G P Matthews ldquoVariable struc-ture control of nonlinear multivariable systems a tutorialrdquo Pro-ceedings of the IEEE vol 76 no 3 pp 212ndash232 1988
[19] J Y HungWGao and J CHung ldquoVariable structure control asurveyrdquo IEEE Transactions on Industrial Electronics vol 40 no1 pp 2ndash22 1993
[20] G Papageorgiou andM Polansky ldquoTuning a dynamic inversionpitch axis autopilot using McFarlane-Glover loop shapingrdquoOptimal Control Applications ampMethods vol 30 no 3 pp 287ndash308 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Mathematical Problems in Engineering
minus119872119866[[[
[
119903119897
1003817100381710038171003817r119897 + p10038171003817100381710038173minus1
1199032119897
0
0
]]]
]
minus1
119898119891
[
[
119865119889119909
119865119889119910
119865119889119911
]
]
+1
119898119891
[
[
119906119909
119906119910
119906119911
]
]
(5)
Denote the relative position and velocity vectors as p119897=
(119909 119910 119911) and v119897= (119889119909119889119905 119889119910119889119905 119889119911119889119905) in the leader orbit
frame respectively Then (5) can be represented as follows
p119897= k119897
k119897= minus C (120596
0) k119897minusD (p
119897 r119897 1205960) p119897minus N (p
119897 r119897) minus
F119889
119898119891
+u119905
119898119891
(6)
where
C (1205960) = [
[
0 minus212059600
21205960
0 0
0 0 0
]
]
N (p119897 r119897) = 119872119866
[[[
[
119903119897
1003817100381710038171003817r119897 + p10038171003817100381710038173minus1
1199032119897
0
0
]]]
]
F119889= [
[
119865119889119909
119865119889119910
119865119889119911
]
]
D (p119897 r119897 1205960)
=
[[[[[[[[
[
1205962
0minus
119872119866
1003817100381710038171003817r119897 + p10038171003817100381710038173
0 0
0 1205962
0minus
119872119866
1003817100381710038171003817r119897 + p10038171003817100381710038173
0
0 0 minus119872119866
1003817100381710038171003817r119897 + p10038171003817100381710038173
]]]]]]]]
]
(7)
3 The Proposed Robust NonlinearDynamic Inversion
In this section the mathematical backgrounds of slidingmode control and nonlinear dynamic inversion are intro-duced And then the robust nonlinear dynamic inversionmethod is proposed
31 Sliding Mode Controller Sliding mode control (SMC)also called variable structure control (VSC) is a high-speedswitching nonlinear control method that forces the systemstates into a user-designed sliding surface and maintains thestates on the surface [17 18] SMC consists of two phases(Figure 4) structuring the sliding surface and constructingthe switching feedback gains
Consider the following system
x (119905) = f (x (119905)) + g (x (119905)) u (119905) (8)
Sliding surface
Sliding phase
Reaching phase
120590(x) = 0
x2
x1
x(t1)
x(t0)
Figure 4The behavior of the system states in sliding mode control
where x(119905) isin R119899 is a state vector u(119905) isin R119898 is an input vectorwith 119899 ge 119898 and f R119899 rarr R119899 and g R119898 rarr R119899times119898are smooth functions Let a set Σ
119894be the regular (119899 minus 1)
dimensional submanifold in R119899 defined in [16] such as
Σ119894≜ x isin R119899 | 120590
119894(x) = 0 119894 = 1 2 119898 (9)
where 120590119894 R119899 rarr R (119894 = 1 2 119898) is a smooth function
Define120590119894(x) = 0 as the individual sliding surface or individual
switching surface Then surface constructed by intersecting119898 individual sliding surfaces is defined as the sliding surface[16]
Σ ≜
119898
⋂
119894=1
Σ119894=
119898
⋂
119894=1
x isin R119899 | 120590119894(x) = 0 ≜ x isin R119899 | 120590 (x) = 0
(10)
where 120590(x) = [1205901(x) 1205902(x) 120590
119898(x)]119879 If the states x(119905) of
the system are in Σ that is 120590(x(119905)) = 0 then the behavior ofthe states is called sliding motion or sliding mode Moreoverif sliding mode exists the tangent vectors of the states alwaysare forced to track the sliding surface as shown in Figure 4Then each entry 119906
119894(119905) (119894 = 1 2 119898) of the sliding mode
control law can be obtained by taking one of the followingvalues [16]
119906119894(119905) =
119906+
119894 if 120590
119894(x (119905)) gt 0
119906minus
119894 if 120590
119894(x (119905)) lt 0
(11)
where 119906+119894= 119906minus
119894
As shown in (11) designing the sliding surface is crucialbecause the sliding surface determines the control input andmoreover the performance of the closed-loop system Forthis reason various designing methods such as Fillipovrsquosmethod and equivalent control method have been proposed[18 19] In this paper the equivalent control method is
Mathematical Problems in Engineering 5
considered to analyze the similar characteristics betweenSMC and NDI If the system states intercept the slidingsurface at 119905 = 119905
0and a sliding mode exists for 119905 ge 119905
0 then
(119889119889119905)120590(x(119905)) = 0 and 120590(x(119905)) = 0 for all 119905 ge 1199050 Then the
equivalent input ueq that forces the state trajectory to stay onthe sliding surface can be analyzed as follows [19]
(x) = 120597120590120597x(f (x) + g (x)ueq) = 0
997904rArr ueq = minus(120597120590
120597xg(x))
minus1
(120597120590
120597xf (x))
(12)
Substituting (12) into (1) the system dynamics on the slidingsurface for 119905 ge 119905
0is governed by
x = f (x) + g (x) ueq = [I minus g (x) (120597120590120597x
g(x))minus1
(120597120590
120597x)] f (x)
(13)
The dynamics represented in (13) is defined as the idealsliding dynamics [16] Consequently the closed-loop systemdynamics in sliding mode are specified by the sliding surface
One of the widely used designing methods of the slidingmode controller combines switching inputs and the equiv-alent inputs as represented in (12) that is if the slidingsurface is determined then the sliding mode controller canbe obtained by adding switching input (119906sw119894) of which entry119906119894(119905) yields [18]
119906119894= 119906eq119894 + 119906sw119894 (119894 = 1 2 119898) (14)
where 119906eq119894 is the 119894th entry of the equivalent input analyzedin (12) And 119906sw119894 is the switching input that forces the systemstates tomove towards the sliding surface In [18] the possiblecandidates for designing the switching input are introducedfor example
(i) relay with constant gains 119906sw119894(x) = 120572119894 sgn(120590119894(x))with120572119894lt 0
(ii) relay with state dependent gains 119906sw119894(x) =
120572119894(x) sgn(120590
119894(x)) with 120572
119894(sdot) lt 0
(iii) univector nonlinearity with scale factorusw(x)=120588120590(x)120590(x) with 120588 lt 0
32 Similar Characteristics between Sliding Mode Control andNonlinear Dynamic Inversion Before analyzing the similarcharacteristics between SMC andNDI the conventional NDIcontroller is firstly introduced NDI controller consists oftwo blocks (Figure 5) desired dynamic block and dynamicinversion (DI) block [4ndash6] In the desired dynamic blockthe control variables are defined and the rate commandsof the selected control variables are generated And in theDI block proper control inputs are generated by invertingthe plant dynamics in order to make the inner closed-loop transfer function as an integrator [20] Hence thesystem states controlled by the dynamic inversion block willfollow the user-selected control variables For this reason thedesired dynamics are usually designed to satisfy the specific
dynamicsDynamicinversion
x x
xPlant
xcmd asymp1
s
Figure 5 Structure of dynamic inversion [4]
performancesrequirements so that the performance of thecontrolled system can achieve the system requirements In[4] some forms of desired dynamics are introduced propor-tional proportional integral flying quality ride quality andso forth
For the nonlinear system shown in (8) NDI control inputuNDI(119905) can be given by [4]
uNDI (119905) = [g (x (119905))]minus1
[xdes (119905) minus f (x (119905))] (15)
where xdes(119905) represents the desired dynamicsIt is worth noting that the requirement for existing NDI
control law in (15) is that [g(x)]minus1 must exist Howeverthe matrix g(x) is not generally a full rank for nonflatsystems These systems have a larger number of states thanthe number of control inputs Hence the number of statesthat can be inverted is less than or equal to the numberof inputs One way of achieving [g(x)]minus1 is to formulatethe existence problem as a two-time scale problem A two-time scale problem makes the system states separate into fastand slow dynamics The fast dynamics comprise a group ofstates affected directly by the control inputs On the otherhand the slow dynamics consists of states influenced by thefast dynamics Separating the system states into two groupsreduces the system order enough to provide a chance toexist more number of inputs than the number of the fastdynamic states Consequently [g(x)]minus1 can be obtained [4ndash6] In contrast for an overactuated system that adopts a largernumber of inputs than the number of states the inverse canbe obtained by reducing the number of inputs to be equal tothe number of states using control allocation [8 9]
By substituting (15) into (8) the inner-loop dynamics inFigure 5 controlled by conventional NDI yields
xdes = x (16)
Hence NDI controller replaces the original dynamics intothe user-selected desired dynamics by inverting the originaldynamics Consequently the control system can guaranteethe stability without gain-scheduling However it is impos-sible to achieve perfect cancellation of the original dynamicsdue to uncertainties such asmodelmismatches disturbancesand measurement noises which leads to poor robustnessof NDI controller Hence robustness issues against uncer-tainties must be considered when designing NDI control-ler
To achieve the robustness of NDI the equivalent controlmethod used to design sliding surface is considered Fordesired trajectory xdes isin R119899 let xlowast = xdes minus x isin R119899
6 Mathematical Problems in Engineering
Select a set of nonlinear smooth functions 120590(xlowast) = [1205901(xlowast)
1205902(xlowast) 120590
119898(xlowast)]119879 = 0 as a sliding surface If the system
states intercept the sliding surface at 119905 = 1199050and the sliding
mode exists for 119905 ge 1199050 then (119889119889119905)120590(x(119905)) = 0 and 120590(x(119905)) = 0
for all 119905 ge 1199050 Then the equivalent input ueq that the state
trajectory stays on the sliding surface can be analyzed asfollows [13ndash15]
(xlowast) = 120597120590
120597xlowastxlowast = 120597120590
120597xlowast(xdes minus x) = 0
997904rArr120597120590
120597xlowastxdes minus
120597120590
120597xlowast(f (x) + g (x) ueq) = 0
997904rArr ueq = (120597120590
120597xlowastg (x))
minus1
(120597120590
120597xlowastxdes minus
120597120590
120597xlowastf (x))
(17)
If 119898 = 119899 and 120597120590120597xlowast = I119899 then (17) can be represented as a
form of NDI input shown in (15)
ueq = g(x)minus1 [xdes minus f (x)] = uNDI (18)
Hence if the sliding surface is designed by an identity matrixthen the equivalent input of SMC can be represented as NDIinput
Definition 1 Let 120590NDI(x) be a set of smooth functions120590NDI119894(x) (119894 = 1 2 119899) that is 120590NDI(x)=[120590NDI1(x)120590NDI2(x) 120590NDI119899(x)]
119879 Then 120590NDI(x) = x is defined as theNDI surface that is the NDI surface is the sliding surfacedesigned by the identity matrix
It is worth noting that the ideal sliding dynamics shownin (13) yields the desired dynamics as follows
x = f (x) + g (x)ueq
= [f (x) + g (x) ( 120597120590120597xlowast
g (x))minus1
(120597120590
120597xlowastxdes minus
120597120590
120597xlowastf (x))]
= xdes(19)
In (19) the states on the identity sliding surface are governedby the desired dynamics Moreover the performances ofthe control system in sliding mode are determined only bythe desired dynamics This suggests that the difficulties indesigning a sliding mode controller to satisfy the specificsystem performances are converted to the desired dynamicsdesigning problem
33 Design of the Robust Nonlinear Dynamic Inversion Simi-lar to the designmethodof SMC each entry119906
119894(119905) (119894 = 1 2
119898) of the RNDI control law can be obtained by introducingthe following values
119906119894(119905) =
119906+
119894 if 120590
119894(119909des119894 minus 119909119894) = 119909des119894 minus 119909119894 gt 0
119906minus
119894 if 120590
119894(119909des119894 minus 119909119894) = 119909des119894 minus 119909119894 lt 0
(20)
where 119906+119894= 119906minus
119894satisfies the following for 119891
119894(x) the 119894th entry
of f(x) and for 119892119894119895(x) the (119894 119895)-entry of g(x)
lim119909119894rarr119909
+
des119894
119891119894(119909) +
119899
sum
119895=1
119895 = 119894
119892119894119895(119909) 119906119895+ 119892119894119894(119909) 119906minus
119894lt des119894
lim119909119894rarr119909
minus
des119894
119891119894(119909) +
119899
sum
119895=1
119895 = 119894
119892119894119895(119909) 119906119895+ 119892119894119894(119909) 119906+
119894gt des119894
(21)
where the superscripts + and minus denote the right- and left-hand limit respectivelyThe following theorem and corollarypropose the method for designing the RNDI controller andprovide the stability analytically
Theorem 2 Consider the following input
u = u119873119863119868
+ gminus1 (x) u119904119908 (22)
where u119873119863119868
is the conventional NDI input and u119904119908
isthe switching input that satisfies the following for u
119904119908=
[1199061199041199081 1199061199041199082 119906
119904119908119899]119879
119906119904119908119894
gt 0 119894119891 120590119894(x119889119890119904minus x) = 119909
119889119890119904119894minus 119909119894gt 0
119906119904119908119894
lt 0 119894119891 120590119894(x119889119890119904minus x) = 119909
119889119890119904119894minus 119909119894lt 0
(23)
Then the closed-loop system controlled by (22) is globally stable
Proof Select a Lyapunov candidate as
119881 =1
2120590(xdes minus x)119879120590 (xdes minus x) (24)
From (22) and (23)
= 120590(xdes minus x)119879 (xdes minus x)
= 120590(xdes minus x)119879 (xdes minus x)
= 120590(xdes minus x)119879 [xdes minus f (x) + g (x) u]
= 120590(xdes minus x)119879
times [xdes minus f (x) + g (x) (uNDI + gminus1 (x) usw)]
= 120590(xdes minus x)119879
times [xdes minus f (x) + g (x) uNDI minus usw]
= minus120590(xdes minus x)119879usw
= minus
119899
sum
119894=1
120590119894(xdes minus x) 119906sw119894
(25)
From (25) if 120590119894(xdes minus x) gt 0 then 119906sw119894 gt 0 and lt 0
Similarly if 120590119894(xdes minus x) lt 0 then 119906sw119894 lt 0 and lt 0 Hence
the closed-loop system controlled by (22) is globally stable
Mathematical Problems in Engineering 7
NDI surface
fi(x) +n
sumj=1jnei
gij(x)uj + gii(x)u+i
fi(x) +n
sumj=1jnei
gij(x)uj + gii(x)uminusi
120590i( minus ) gt 0x xdes
120590i( minus x) lt 0xdes
120590i( minus ) = 0xdes x
Figure 6 State trajectory on the NDI surface
Since the system controlled by (22) is globally stable fromTheorem 2 the states are forced to the NDI surface as shownin Figure 6
Corollary 3 If the switching input denoted as u119904119908= [1199061199041199081
1199061199041199082 119906
119904119908119899]119879 is designed with 119906
119904119908119894= 119896119894sgn(120590
119894(x119889119890119904minus x))
for a positive 119896119894 then the control system is globally stable
Proof If 120590119894(xdes minus x) gt 0 then 119906sw119894 = 119896119894 gt 0 and conversely
if 120590119894(xdes minus x) lt 0 then 119906sw119894 = minus119896119894 lt 0 Hence according to
Theorem 2 the control system is globally stable
Definition 4 The control law proposed inTheorem 2 is calledthe robust nonlinear dynamic inversion (RNDI) law that is fora diagonal matrix K = diag(119896
119894) (119894 = 1 2 119899) with 119896
119894gt 0
the RNDI control law yields
uRNDI = uNDI + usw
= g(x)minus1 [xdes minus f (x) + K sgn (xdes minus x)] (26)
It is worth noting that the form of the switching inputrepresented in Corollary 3 is a relay with constant gains formin SMC as mentioned in Section 31 Actually the switchingfunction satisfying (23) can be obtained directly from SMCdesign method as follows
(i) relay with constant gains 119906sw119894 = 119896119894 sgn(120590119894(xdes minus x))with 119896
119894gt 0
(ii) relay with state dependent gains 119906sw119894 = 119896119894(xdes minus
x) sgn(120590119894(xdes minus x)) with 119896
119894(sdot) gt 0
(iii) univector nonlinearity with scale factor usw =
119896119894120590(xdes minus x)120590(xdes minus x) with 119896
119894gt 0
It is also remarkable that using the sgn function whiledesigning the switching input leads to unfavorable resultsin control systems such as the chattering problem Thechattering problem makes the system have unmodeled highfrequencies or actuator saturation In sliding mode controltheory several methods have been proposed to overcomethis problem by designing a continuous control input insteadof a discontinuous switching input This method forces the
controlled system to follow the approximated sliding motionin some boundaries instead of the ideal sliding motionFollowing the similar manner the chattering problem in theRNDI control can be solved using the continuous controlinput For example replace sgn(xdes minus x) into sat(xdes minus x) =[sat1(119909des1 minus 1199091) sat2(119909des2 minus 1199092) sat119899(119909des119899 minus 119909119899)]
119879 inorder to design a continuous switching input as follows for apositive 120575
119894
sat119894(119909des119894 minus 119909119894
120575119894
) =
1 if (119909des119894 minus 119909119894) gt 120575119909des119894 minus 119909119894
120575119894
if 1003816100381610038161003816119909des119894 minus 1199091198941003816100381610038161003816 lt 120575
minus1 if (119909des119894 minus 119909119894) lt minus120575(27)
Then 119906sw119894 = 119896119894sat119894((119909des119894minus119909119894)120575119894) In (27) plusmn120575 is the boundarylayer in which the states are governed by the followingdynamics for sat((xdes minus x)120575) = [sat
1((119909des1 minus 1199091)1205751)
sat2((119909des2 minus 1199092)1205752) sat119899((119909des119899 minus 119909119899)120575119899)]
119879 and K =
diag(119896119894) (119894 = 1 2 119899) with 119896
119894gt 0
x = f (x) + g (x)u
= f (x) + g (x)
times [(g (x))minus1 (xdes minus f (x))
+(g (x))minus1Ksat((xdes minus x)
120575)]
= xdes + Ksat((xdes minus x)
120575)
= xdes
+ [1198961(119909des1 minus 1199091)
1205751
1198962(119909des2 minus 1199092)
1205752
sdot sdot sdot
119896119899(119909des119899 minus 119909119899)
120575119899
]
119879
(28)
8 Mathematical Problems in Engineering
uf
RefTime
Angular
Position of Referencetrajectorygenerator
the leader spacecraft
Reffcn
PositionControl
Dynamics of the Robust nonlinear dynamic inversion follower spacecraft
PositionScope
DisturbancesFd
++
t
ww
plplinputs
velocity
Figure 7 Block diagram using MATLAB Simulink
34 Stability of the Proposed RNDI Controller In this paperthe stability of the closed-loop system controlled by theproposed RNDI is analyzed by a Lyapunov stability criterionConsider the following nonlinear system including boundedmodel uncertainties Δf(x(119905)) isin R119899 and Δg(x(119905)) isin R119899 andbounded disturbance d(119905) isin R119899
x (119905) = [f (x (119905)) + Δf (x (119905))]
+ [g (x (119905)) + Δg (x (119905))] u (119905) + d (119905) (29)
It is assumed that 120585(119905 x(119905) u(119905)) = Δf(x(119905)) + Δg(x(119905)) + d(119905)satisfying 120585(119905 x(119905) u(119905))
2lt 120582min(K) where 120582min(K) is the
minimum eigenvalue of K For the desired state xdes isin 119877119899denote xlowast(119905) = xdesminusx To analyze the stability of the proposedRNDI controller select a Lyapunov candidate as follows
119881 (xlowast (119905)) = 12xlowast(119905)119879xlowast (119905) (30)
Then the derivative of 119881(xlowast(119905)) yields
(xlowast (119905))
= xlowast(119905)119879xlowast (119905)
= xlowast(119905)119879 [xdes (119905) minus [f (x (119905)) + Δf (x (119905))]
+ [g (x (119905)) + Δg (x (119905))] u (119905) + d (119905) ]
= xlowast(119905)119879 [xdes (119905) minus (f (x (119905)) + g (x (119905)) u (119905))
minus (Δf (x (119905)) + Δg (x (119905)) u (119905) + d (119905)) ]
= xlowast(119905)119879 [minus120585 (119905 x (119905) u (119905)) minus K sgn (xlowast (119905))]
le minusxlowast(119905)119879120585 (119905 x (119905) u (119905)) minus 120582min (K)1003817100381710038171003817xlowast
(119905)10038171003817100381710038171
le 120585 (119905 x (119905) u (119905))1
1003817100381710038171003817119909lowast
(119905)10038171003817100381710038171 minus 120582min (K)
1003817100381710038171003817xlowast
(119905)10038171003817100381710038171
le [120585 (119905 x (119905) u (119905))2minus 120582min (K)]
1003817100381710038171003817xlowast
(119905)10038171003817100381710038171
(31)
By the hypothesis 120585(119905 x(119905) u(119905))2lt 120582min(K) the derivative
of 119881(xlowast(119905)) is always negative Hence the system controlledby the proposed RNDI controller is globally stable againstdisturbances noises and model mismatches
Table 1 Parameters for spacecraft formation flying [1 3]
Symbol Value Unit119866 6673 times 10minus11 m3kgsdots2
119872 5974 times 1024 kg119898119891
410 kg119898119897
1550 kgr119897
[4224 times 1024 0 0]119879 m1205960
7272 times 10minus5 rads
4 Simulation Results
In this section numerical simulations are conducted to eva-luate the performance of the proposed robust nonlineardynamic inversion controller
41 Simulation Description The aim of SFF is to design thefeasible control input u
119905(119905) such that p
119897rarr p119889as 119905 rarr infin
for a given reference relative position trajectory p119889isin R3
of the follower spacecraft with respect to the leader space-craft To evaluate the performance of the proposed RNDIcontroller MATLAB Simulink is selected as a simulationtool Figure 7 shows the block diagram of the overall SFFstructure using MATLAB Simulink In this paper it isassumed that the reference trajectory is set to p
119889=
[100 sin(41205960119905) 100 cos(4120596
0119905) 0]119879 that is the follower space-
craft tracks a circular orbit centered at the leader spacecraftwith a radius of 100 meters on a plane generated by e
119903and e120579
with an angular velocity 41205960 The initial relative position and
velocity are assumed to be respectively as follows
p119897(0) = [10 90 minus20]
119879
k119897(0) = [0 0 0]
119879
(32)
Table 1 lists some parameters and their values used in thissimulation Some uncertainties such as disturbances and sen-sor noises of velocity and position are induced in this simu-lation Disturbances acting on the follower spacecraft areassumed such that F
119889= [290532 31775 minus112298]
119879 (N)And it is also assumed that maximally 20 of random velo-city sensor noises in three axes create difficulties in allowingthe follower spacecraft to obtain accurate velocity informa-tion Moreover maximally 20 and 40 of random positionsensor noises in e
119903and e
ℎaxes are also considered in this
simulation
Mathematical Problems in Engineering 9
0 10 20 30 40 50 60 70 80 90 100
0
5
10
Time (s)
No disturbanceDisturbance
minus5
minus10
minus15
e r-d
irect
ion
(m)
(a) Position tracking error in e119903-direction
0 10 20 30 40 50 60 70 80 90 100
0
2
Time (s)
No disturbanceDisturbance
minus2
minus4
minus6
minus8
minus10
e 120579-d
irect
ion
(m)
(b) Position tracking error in e120579-direction
0 10 20 30 40 50 60 70 80 90 100
0
5
10
Time (s)
No disturbanceDisturbance
minus5
minus10
minus15
minus20
e h-d
irect
ion
(m)
(c) Position tracking error in eℎ-direction
Figure 8 Relative position errors of no disturbance case and disturbance injected case The black thin line represents the trajectories of thespacecraft without considering any disturbances The blue thick line shows the results of the trajectories of the disturbed spacecraft
42 Simulation Results Figure 8 describes the position track-ing error results of the no disturbance case and disturbancesinjected case The objective of this simulation is to verifythe performance degradation of the disturbances injectedspacecraft In both cases NDI with the proportional type ofthe desired dynamics is designed as a primary controller forSFF As shown in Figure 8 if no disturbance is acting on thespacecraft then the position errors between the referencedposition and the follower position are zero in 30 secHoweverthe lack of robustness in NDI controller leads a significantperformance degradation of the disturbed spacecraft so ittracks the references with a tracking error of almost 8m 1mand 4m in e
119903 e120579 and e
ℎaxes respectively
Figures 9 and 10 show the trajectories of the followerspacecraft controlled by NDI and RNDI controllers Asmentioned in Figure 8 the position errors between thereferenced position and follower position controlled by NDIcannot be zero due to the induced uncertainties However bycombining the switching input into conventional NDI con-troller the RNDI control law attempts to reduce the effect ofthe uncertainties Consequently the state can track the refer-ence signal within 1m error as shown in Figure 9The controlinputs generated by NDI and RNDI controllers are described
in Figure 10 To compensate the induced uncertainties thespacecraft controlled by RNDI generates a large input force ineℎaxis around 27 sec (Figure 10(b)) Figure 11 shows the tra-
jectory of the spacecraft controlled by the RNDI controllerThe spacecraft maneuvers around a circle with a 100-meterradius Although the spacecraft is vibrated due to the induceduncertainties the error distance between the reference trajec-tory and the position of the follower is less than 1m as shownin Figure 9 Hence the follower spacecraft controlled by theproposed RNDI can track the reference trajectory
5 Conclusion
In this paper the robust nonlinear dynamic inversion controlstrategy that improves the robustness of NDI has been pro-posed NDI is a nonlinear control technique that eliminatesthe need of linearizing and designing gain-scheduled con-trollers for each equilibrium point by canceling the originaldynamics Because NDI removes the system nonlinearitiesdirectly it provides an opportunity to overcome the difficul-ties of ensuring stabilities and performances between variousoperational points while gain-scheduling However the maindrawback of designing NDI controller is the weak robustness
10 Mathematical Problems in Engineering
0 10 20 30 40 50 60 70 80 90 100
0
5
10
Time (m)
NDIRNDI
e r-d
irect
ion
(m)
minus5
minus15
minus10
(a) Position tracking error in e119903-direction
0
2
0 10 20 30 40 50 60 70 80 90 100Time (s)
NDIRNDI
e 120579-d
irect
ion
(m)
minus6
minus2
minus4
minus8
minus10
(b) Position tracking error in e120579-direction
0 10 20 30 40 50 60 70 80 90 100
0
5
10
Time (s)
NDIRNDI
e h-d
irect
ion
(m)
minus20
minus15
minus10
minus5
(c) Position tracking error in eℎ-direction
Figure 9 Simulation results of the position tracking error of the spacecrafts controlled by NDI and RNDIThe black thin line represents thetrajectories of the spacecraft controlled by NDI while the blue thick line represents the trajectories of the spacecraft controlled by RNDI
20 30 40 50 60 70 80 90 100
0
100
200
300
400
Time (s)
Inpu
t for
ces (
N)
minus200
minus100
ux along with er-axis
uz along with eh-axis
uy along with e120579-axis
(a) Control inputs generated by NDI
20 30 40 50 60 70 80 90 100
0
100
200
300
400
Time (s)
Inpu
t for
ces (
N)
minus200
minus100
ux along with er-axis
uz along with eh-axis
uy along with e120579-axis
(b) Control inputs generated by RNDI
Figure 10 Control inputs generated by NDI and RNDI
due to difficulties in obtaining accurate information of thenonlinear system dynamics In contrast to NDI SMC isconsidered a robust nonlinear control technique that usesswitching values as inputs to force the system states intoa sliding surface and to maintain the states on it Theproposed RNDI controller is designed by combining the
conventional NDI controller and switching inputs analyzedin SMC Hence RNDI takes the advantages of both NDI andSMC such that it
(i) is easy to design and implement(ii) eliminates the need of gain-scheduling
Mathematical Problems in Engineering 11
0 50 100 150050100150
0
10
e120579 axis er axis
e hax
is
minus50minus50minus100minus100 minus150minus150
minus10
minus20
Figure 11 Trajectory of the follower spacecraft controlled by theproposed RNDI controller
(iii) is easy to achieve the system required performancedue to the similar concept to classical control meth-ods and
(iv) is inherently robust
However the drawback of SMC such as the chatteringproblem still remains as a problem of RNDI control It isexpected that various chattering-free methods can be usedto reduce the effect of chattering phenomenon for examplereplacing the sign function into the saturation function
Moreover the equivalence of NDI input and equivalentinput designed by the identity sliding surface is also analyzedin this paper The reduced order dynamics on the slidingsurface can be converted to the user-selected desired dynam-ics As a result the difficulties in designing a sliding modecontroller for satisfying the specific system performances areconverted to a designing problem of the desired dynamicsHence RNDI provides an opportunity to solve the difficultiesin selecting the sliding surface of SMC The effectiveness ofthe proposed RNDI controller is confirmed by a set of simu-lations with application to the spacecraft formation flying
Nomenclature
G Universal gravity constantM Mass of the Earth119898119891 Mass of the follower spacecraft
119898119897 Mass of the leader spacecraft
r119891 Distance from the center of the Earth to
the center of the follower spacecraftr119897 Distance from the center of the Earth to
the center of the leader spacecraftxdes Desired dynamics
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research was supported by the MSIP (Ministry ofScience ICT amp Future Planning) Korea under the C-ITRC(Convergence Information Technology Research Center)
support program (NIPA-2014-H0401-14-1004) supervised bythe NIPA (National IT Industry Promotion Agency)
References
[1] Y Lv Q Hu G Ma and J Zhou ldquo6 DOF synchronized con-trol for spacecraft formation flying with input constraint andparameter uncertaintiesrdquo ISA Transactions vol 50 no 4 pp573ndash580 2011
[2] D Forta F Bordi and C Scolese ldquoLow-cost minimum sizesatellites for demonstration of formation flying modes at smallkilometer-size distancerdquo in Proceedings of the 13th AIAAUSUConference on Small Satellite SSC99-VI-3 pp 1ndash15 1999
[3] M S De Queiroz V Kapila and Q Yan ldquoAdaptive nonlinearcontrol ofmultiple spacecraft formation flyingrdquo Journal of Guid-ance Control and Dynamics vol 23 no 3 pp 385ndash390 2000
[4] J Georgie and J Valasek ldquoEvaluation of longitudinal desireddynamics for dynamic-inversion controlled generic reentryvehiclesrdquo Journal of Guidance Control and Dynamics vol 26no 5 pp 811ndash819 2003
[5] D Ito J Georgie J Valasek and D T Ward Reentry VehicleFlight Controls Design Guidelines Dynamic Inversion NASATPmdash2002ndash210771 2002
[6] A J Ostroff and B J Bacon ldquoForce and moment approachfor achievable dynamics using nonlinear dynamic inversionrdquo inProceedings of theAIAAGuidance Navigation andControl Con-ference and Exhibit pp 424ndash434 Reston Va USA 1999
[7] D Enns D Bugajski R Hendrick and G Stein ldquoDynamicinversion an evolving methodology for flight control designrdquoInternational Journal of Control vol 59 no 1 pp 71ndash91 1994
[8] J A Richard J M Buffington A G Sparks and S S BandaRobust Multivariable Flight Control Springer New York NYUSA 1994
[9] R J Adams and S S Banda ldquoRobust flight control design usingdynamic inversion and structured singular value synthesisrdquoIEEE Transactions on Control Systems Technology vol 1 no 2pp 80ndash92 1993
[10] M B McFarland and S M Hoque ldquoRobustness of a nonlinearmissile autopilot designed using dynamic inversionrdquo in Pro-ceedings of the AIAA Guidance Navigation and Control Confer-ence and Exhibit AIAA-2000-3970 Denver Colo USA August2000
[11] G Looye and H-D Joos ldquoDesign of robust dynamic inversioncontrol laws usingmulti-objective optimizationrdquo in Proceedingsof the AIAA Guidance Navigation and Control Conference andExhibit AIAA-2001-4285 2001
[12] P B Acquatella W Falkena E van Kampen and Q P ChuldquoRobust nonlinear spacecraft attitude control using incrementalnonlinear dynamic inversionrdquo in Proceedings of the AIAAGuid-ance Navigation and Control Conference and Exhibit Min-neapolis Minn USA August 2012
[13] P K Menon V R Iragavarapu and E J Ohlmeyer ldquoNonlinearMissile Autopilot Design using Time-Scale Separationrdquo in Pro-ceedings of the AIAA Guidance Navigation and Control Confer-ence and Exhibit AIAA-1997-3765 1997
[14] A Steinicke and G Michalka ldquoImproving transient perfor-mance of dynamic inversion missile autopilot by use of back-steppingrdquo in Proceedings of the AIAAGuidance Navigation andControl Conference and Exhibit AIAA-2002-4658 MontereyCalif USA August 2002
12 Mathematical Problems in Engineering
[15] I Yang D Kim and D Lee ldquoA flight control strategy usingrobust dynamic inversion based on sliding mode controlrdquo inproceeding of the AIAA Guidance Navigation and ControlConference August 2012
[16] H Sira-Ramirez ldquoDifferential geometric methods in variable-structure controlrdquo International Journal of Control vol 48 no4 pp 1359ndash1390 1988
[17] V I Utkin ldquoVariable structure systems with sliding modesrdquoIEEE Transactions on Automatic Control vol 22 no 2 pp 212ndash222 1977
[18] R A DeCarlo S H Zak and G P Matthews ldquoVariable struc-ture control of nonlinear multivariable systems a tutorialrdquo Pro-ceedings of the IEEE vol 76 no 3 pp 212ndash232 1988
[19] J Y HungWGao and J CHung ldquoVariable structure control asurveyrdquo IEEE Transactions on Industrial Electronics vol 40 no1 pp 2ndash22 1993
[20] G Papageorgiou andM Polansky ldquoTuning a dynamic inversionpitch axis autopilot using McFarlane-Glover loop shapingrdquoOptimal Control Applications ampMethods vol 30 no 3 pp 287ndash308 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 5
considered to analyze the similar characteristics betweenSMC and NDI If the system states intercept the slidingsurface at 119905 = 119905
0and a sliding mode exists for 119905 ge 119905
0 then
(119889119889119905)120590(x(119905)) = 0 and 120590(x(119905)) = 0 for all 119905 ge 1199050 Then the
equivalent input ueq that forces the state trajectory to stay onthe sliding surface can be analyzed as follows [19]
(x) = 120597120590120597x(f (x) + g (x)ueq) = 0
997904rArr ueq = minus(120597120590
120597xg(x))
minus1
(120597120590
120597xf (x))
(12)
Substituting (12) into (1) the system dynamics on the slidingsurface for 119905 ge 119905
0is governed by
x = f (x) + g (x) ueq = [I minus g (x) (120597120590120597x
g(x))minus1
(120597120590
120597x)] f (x)
(13)
The dynamics represented in (13) is defined as the idealsliding dynamics [16] Consequently the closed-loop systemdynamics in sliding mode are specified by the sliding surface
One of the widely used designing methods of the slidingmode controller combines switching inputs and the equiv-alent inputs as represented in (12) that is if the slidingsurface is determined then the sliding mode controller canbe obtained by adding switching input (119906sw119894) of which entry119906119894(119905) yields [18]
119906119894= 119906eq119894 + 119906sw119894 (119894 = 1 2 119898) (14)
where 119906eq119894 is the 119894th entry of the equivalent input analyzedin (12) And 119906sw119894 is the switching input that forces the systemstates tomove towards the sliding surface In [18] the possiblecandidates for designing the switching input are introducedfor example
(i) relay with constant gains 119906sw119894(x) = 120572119894 sgn(120590119894(x))with120572119894lt 0
(ii) relay with state dependent gains 119906sw119894(x) =
120572119894(x) sgn(120590
119894(x)) with 120572
119894(sdot) lt 0
(iii) univector nonlinearity with scale factorusw(x)=120588120590(x)120590(x) with 120588 lt 0
32 Similar Characteristics between Sliding Mode Control andNonlinear Dynamic Inversion Before analyzing the similarcharacteristics between SMC andNDI the conventional NDIcontroller is firstly introduced NDI controller consists oftwo blocks (Figure 5) desired dynamic block and dynamicinversion (DI) block [4ndash6] In the desired dynamic blockthe control variables are defined and the rate commandsof the selected control variables are generated And in theDI block proper control inputs are generated by invertingthe plant dynamics in order to make the inner closed-loop transfer function as an integrator [20] Hence thesystem states controlled by the dynamic inversion block willfollow the user-selected control variables For this reason thedesired dynamics are usually designed to satisfy the specific
dynamicsDynamicinversion
x x
xPlant
xcmd asymp1
s
Figure 5 Structure of dynamic inversion [4]
performancesrequirements so that the performance of thecontrolled system can achieve the system requirements In[4] some forms of desired dynamics are introduced propor-tional proportional integral flying quality ride quality andso forth
For the nonlinear system shown in (8) NDI control inputuNDI(119905) can be given by [4]
uNDI (119905) = [g (x (119905))]minus1
[xdes (119905) minus f (x (119905))] (15)
where xdes(119905) represents the desired dynamicsIt is worth noting that the requirement for existing NDI
control law in (15) is that [g(x)]minus1 must exist Howeverthe matrix g(x) is not generally a full rank for nonflatsystems These systems have a larger number of states thanthe number of control inputs Hence the number of statesthat can be inverted is less than or equal to the numberof inputs One way of achieving [g(x)]minus1 is to formulatethe existence problem as a two-time scale problem A two-time scale problem makes the system states separate into fastand slow dynamics The fast dynamics comprise a group ofstates affected directly by the control inputs On the otherhand the slow dynamics consists of states influenced by thefast dynamics Separating the system states into two groupsreduces the system order enough to provide a chance toexist more number of inputs than the number of the fastdynamic states Consequently [g(x)]minus1 can be obtained [4ndash6] In contrast for an overactuated system that adopts a largernumber of inputs than the number of states the inverse canbe obtained by reducing the number of inputs to be equal tothe number of states using control allocation [8 9]
By substituting (15) into (8) the inner-loop dynamics inFigure 5 controlled by conventional NDI yields
xdes = x (16)
Hence NDI controller replaces the original dynamics intothe user-selected desired dynamics by inverting the originaldynamics Consequently the control system can guaranteethe stability without gain-scheduling However it is impos-sible to achieve perfect cancellation of the original dynamicsdue to uncertainties such asmodelmismatches disturbancesand measurement noises which leads to poor robustnessof NDI controller Hence robustness issues against uncer-tainties must be considered when designing NDI control-ler
To achieve the robustness of NDI the equivalent controlmethod used to design sliding surface is considered Fordesired trajectory xdes isin R119899 let xlowast = xdes minus x isin R119899
6 Mathematical Problems in Engineering
Select a set of nonlinear smooth functions 120590(xlowast) = [1205901(xlowast)
1205902(xlowast) 120590
119898(xlowast)]119879 = 0 as a sliding surface If the system
states intercept the sliding surface at 119905 = 1199050and the sliding
mode exists for 119905 ge 1199050 then (119889119889119905)120590(x(119905)) = 0 and 120590(x(119905)) = 0
for all 119905 ge 1199050 Then the equivalent input ueq that the state
trajectory stays on the sliding surface can be analyzed asfollows [13ndash15]
(xlowast) = 120597120590
120597xlowastxlowast = 120597120590
120597xlowast(xdes minus x) = 0
997904rArr120597120590
120597xlowastxdes minus
120597120590
120597xlowast(f (x) + g (x) ueq) = 0
997904rArr ueq = (120597120590
120597xlowastg (x))
minus1
(120597120590
120597xlowastxdes minus
120597120590
120597xlowastf (x))
(17)
If 119898 = 119899 and 120597120590120597xlowast = I119899 then (17) can be represented as a
form of NDI input shown in (15)
ueq = g(x)minus1 [xdes minus f (x)] = uNDI (18)
Hence if the sliding surface is designed by an identity matrixthen the equivalent input of SMC can be represented as NDIinput
Definition 1 Let 120590NDI(x) be a set of smooth functions120590NDI119894(x) (119894 = 1 2 119899) that is 120590NDI(x)=[120590NDI1(x)120590NDI2(x) 120590NDI119899(x)]
119879 Then 120590NDI(x) = x is defined as theNDI surface that is the NDI surface is the sliding surfacedesigned by the identity matrix
It is worth noting that the ideal sliding dynamics shownin (13) yields the desired dynamics as follows
x = f (x) + g (x)ueq
= [f (x) + g (x) ( 120597120590120597xlowast
g (x))minus1
(120597120590
120597xlowastxdes minus
120597120590
120597xlowastf (x))]
= xdes(19)
In (19) the states on the identity sliding surface are governedby the desired dynamics Moreover the performances ofthe control system in sliding mode are determined only bythe desired dynamics This suggests that the difficulties indesigning a sliding mode controller to satisfy the specificsystem performances are converted to the desired dynamicsdesigning problem
33 Design of the Robust Nonlinear Dynamic Inversion Simi-lar to the designmethodof SMC each entry119906
119894(119905) (119894 = 1 2
119898) of the RNDI control law can be obtained by introducingthe following values
119906119894(119905) =
119906+
119894 if 120590
119894(119909des119894 minus 119909119894) = 119909des119894 minus 119909119894 gt 0
119906minus
119894 if 120590
119894(119909des119894 minus 119909119894) = 119909des119894 minus 119909119894 lt 0
(20)
where 119906+119894= 119906minus
119894satisfies the following for 119891
119894(x) the 119894th entry
of f(x) and for 119892119894119895(x) the (119894 119895)-entry of g(x)
lim119909119894rarr119909
+
des119894
119891119894(119909) +
119899
sum
119895=1
119895 = 119894
119892119894119895(119909) 119906119895+ 119892119894119894(119909) 119906minus
119894lt des119894
lim119909119894rarr119909
minus
des119894
119891119894(119909) +
119899
sum
119895=1
119895 = 119894
119892119894119895(119909) 119906119895+ 119892119894119894(119909) 119906+
119894gt des119894
(21)
where the superscripts + and minus denote the right- and left-hand limit respectivelyThe following theorem and corollarypropose the method for designing the RNDI controller andprovide the stability analytically
Theorem 2 Consider the following input
u = u119873119863119868
+ gminus1 (x) u119904119908 (22)
where u119873119863119868
is the conventional NDI input and u119904119908
isthe switching input that satisfies the following for u
119904119908=
[1199061199041199081 1199061199041199082 119906
119904119908119899]119879
119906119904119908119894
gt 0 119894119891 120590119894(x119889119890119904minus x) = 119909
119889119890119904119894minus 119909119894gt 0
119906119904119908119894
lt 0 119894119891 120590119894(x119889119890119904minus x) = 119909
119889119890119904119894minus 119909119894lt 0
(23)
Then the closed-loop system controlled by (22) is globally stable
Proof Select a Lyapunov candidate as
119881 =1
2120590(xdes minus x)119879120590 (xdes minus x) (24)
From (22) and (23)
= 120590(xdes minus x)119879 (xdes minus x)
= 120590(xdes minus x)119879 (xdes minus x)
= 120590(xdes minus x)119879 [xdes minus f (x) + g (x) u]
= 120590(xdes minus x)119879
times [xdes minus f (x) + g (x) (uNDI + gminus1 (x) usw)]
= 120590(xdes minus x)119879
times [xdes minus f (x) + g (x) uNDI minus usw]
= minus120590(xdes minus x)119879usw
= minus
119899
sum
119894=1
120590119894(xdes minus x) 119906sw119894
(25)
From (25) if 120590119894(xdes minus x) gt 0 then 119906sw119894 gt 0 and lt 0
Similarly if 120590119894(xdes minus x) lt 0 then 119906sw119894 lt 0 and lt 0 Hence
the closed-loop system controlled by (22) is globally stable
Mathematical Problems in Engineering 7
NDI surface
fi(x) +n
sumj=1jnei
gij(x)uj + gii(x)u+i
fi(x) +n
sumj=1jnei
gij(x)uj + gii(x)uminusi
120590i( minus ) gt 0x xdes
120590i( minus x) lt 0xdes
120590i( minus ) = 0xdes x
Figure 6 State trajectory on the NDI surface
Since the system controlled by (22) is globally stable fromTheorem 2 the states are forced to the NDI surface as shownin Figure 6
Corollary 3 If the switching input denoted as u119904119908= [1199061199041199081
1199061199041199082 119906
119904119908119899]119879 is designed with 119906
119904119908119894= 119896119894sgn(120590
119894(x119889119890119904minus x))
for a positive 119896119894 then the control system is globally stable
Proof If 120590119894(xdes minus x) gt 0 then 119906sw119894 = 119896119894 gt 0 and conversely
if 120590119894(xdes minus x) lt 0 then 119906sw119894 = minus119896119894 lt 0 Hence according to
Theorem 2 the control system is globally stable
Definition 4 The control law proposed inTheorem 2 is calledthe robust nonlinear dynamic inversion (RNDI) law that is fora diagonal matrix K = diag(119896
119894) (119894 = 1 2 119899) with 119896
119894gt 0
the RNDI control law yields
uRNDI = uNDI + usw
= g(x)minus1 [xdes minus f (x) + K sgn (xdes minus x)] (26)
It is worth noting that the form of the switching inputrepresented in Corollary 3 is a relay with constant gains formin SMC as mentioned in Section 31 Actually the switchingfunction satisfying (23) can be obtained directly from SMCdesign method as follows
(i) relay with constant gains 119906sw119894 = 119896119894 sgn(120590119894(xdes minus x))with 119896
119894gt 0
(ii) relay with state dependent gains 119906sw119894 = 119896119894(xdes minus
x) sgn(120590119894(xdes minus x)) with 119896
119894(sdot) gt 0
(iii) univector nonlinearity with scale factor usw =
119896119894120590(xdes minus x)120590(xdes minus x) with 119896
119894gt 0
It is also remarkable that using the sgn function whiledesigning the switching input leads to unfavorable resultsin control systems such as the chattering problem Thechattering problem makes the system have unmodeled highfrequencies or actuator saturation In sliding mode controltheory several methods have been proposed to overcomethis problem by designing a continuous control input insteadof a discontinuous switching input This method forces the
controlled system to follow the approximated sliding motionin some boundaries instead of the ideal sliding motionFollowing the similar manner the chattering problem in theRNDI control can be solved using the continuous controlinput For example replace sgn(xdes minus x) into sat(xdes minus x) =[sat1(119909des1 minus 1199091) sat2(119909des2 minus 1199092) sat119899(119909des119899 minus 119909119899)]
119879 inorder to design a continuous switching input as follows for apositive 120575
119894
sat119894(119909des119894 minus 119909119894
120575119894
) =
1 if (119909des119894 minus 119909119894) gt 120575119909des119894 minus 119909119894
120575119894
if 1003816100381610038161003816119909des119894 minus 1199091198941003816100381610038161003816 lt 120575
minus1 if (119909des119894 minus 119909119894) lt minus120575(27)
Then 119906sw119894 = 119896119894sat119894((119909des119894minus119909119894)120575119894) In (27) plusmn120575 is the boundarylayer in which the states are governed by the followingdynamics for sat((xdes minus x)120575) = [sat
1((119909des1 minus 1199091)1205751)
sat2((119909des2 minus 1199092)1205752) sat119899((119909des119899 minus 119909119899)120575119899)]
119879 and K =
diag(119896119894) (119894 = 1 2 119899) with 119896
119894gt 0
x = f (x) + g (x)u
= f (x) + g (x)
times [(g (x))minus1 (xdes minus f (x))
+(g (x))minus1Ksat((xdes minus x)
120575)]
= xdes + Ksat((xdes minus x)
120575)
= xdes
+ [1198961(119909des1 minus 1199091)
1205751
1198962(119909des2 minus 1199092)
1205752
sdot sdot sdot
119896119899(119909des119899 minus 119909119899)
120575119899
]
119879
(28)
8 Mathematical Problems in Engineering
uf
RefTime
Angular
Position of Referencetrajectorygenerator
the leader spacecraft
Reffcn
PositionControl
Dynamics of the Robust nonlinear dynamic inversion follower spacecraft
PositionScope
DisturbancesFd
++
t
ww
plplinputs
velocity
Figure 7 Block diagram using MATLAB Simulink
34 Stability of the Proposed RNDI Controller In this paperthe stability of the closed-loop system controlled by theproposed RNDI is analyzed by a Lyapunov stability criterionConsider the following nonlinear system including boundedmodel uncertainties Δf(x(119905)) isin R119899 and Δg(x(119905)) isin R119899 andbounded disturbance d(119905) isin R119899
x (119905) = [f (x (119905)) + Δf (x (119905))]
+ [g (x (119905)) + Δg (x (119905))] u (119905) + d (119905) (29)
It is assumed that 120585(119905 x(119905) u(119905)) = Δf(x(119905)) + Δg(x(119905)) + d(119905)satisfying 120585(119905 x(119905) u(119905))
2lt 120582min(K) where 120582min(K) is the
minimum eigenvalue of K For the desired state xdes isin 119877119899denote xlowast(119905) = xdesminusx To analyze the stability of the proposedRNDI controller select a Lyapunov candidate as follows
119881 (xlowast (119905)) = 12xlowast(119905)119879xlowast (119905) (30)
Then the derivative of 119881(xlowast(119905)) yields
(xlowast (119905))
= xlowast(119905)119879xlowast (119905)
= xlowast(119905)119879 [xdes (119905) minus [f (x (119905)) + Δf (x (119905))]
+ [g (x (119905)) + Δg (x (119905))] u (119905) + d (119905) ]
= xlowast(119905)119879 [xdes (119905) minus (f (x (119905)) + g (x (119905)) u (119905))
minus (Δf (x (119905)) + Δg (x (119905)) u (119905) + d (119905)) ]
= xlowast(119905)119879 [minus120585 (119905 x (119905) u (119905)) minus K sgn (xlowast (119905))]
le minusxlowast(119905)119879120585 (119905 x (119905) u (119905)) minus 120582min (K)1003817100381710038171003817xlowast
(119905)10038171003817100381710038171
le 120585 (119905 x (119905) u (119905))1
1003817100381710038171003817119909lowast
(119905)10038171003817100381710038171 minus 120582min (K)
1003817100381710038171003817xlowast
(119905)10038171003817100381710038171
le [120585 (119905 x (119905) u (119905))2minus 120582min (K)]
1003817100381710038171003817xlowast
(119905)10038171003817100381710038171
(31)
By the hypothesis 120585(119905 x(119905) u(119905))2lt 120582min(K) the derivative
of 119881(xlowast(119905)) is always negative Hence the system controlledby the proposed RNDI controller is globally stable againstdisturbances noises and model mismatches
Table 1 Parameters for spacecraft formation flying [1 3]
Symbol Value Unit119866 6673 times 10minus11 m3kgsdots2
119872 5974 times 1024 kg119898119891
410 kg119898119897
1550 kgr119897
[4224 times 1024 0 0]119879 m1205960
7272 times 10minus5 rads
4 Simulation Results
In this section numerical simulations are conducted to eva-luate the performance of the proposed robust nonlineardynamic inversion controller
41 Simulation Description The aim of SFF is to design thefeasible control input u
119905(119905) such that p
119897rarr p119889as 119905 rarr infin
for a given reference relative position trajectory p119889isin R3
of the follower spacecraft with respect to the leader space-craft To evaluate the performance of the proposed RNDIcontroller MATLAB Simulink is selected as a simulationtool Figure 7 shows the block diagram of the overall SFFstructure using MATLAB Simulink In this paper it isassumed that the reference trajectory is set to p
119889=
[100 sin(41205960119905) 100 cos(4120596
0119905) 0]119879 that is the follower space-
craft tracks a circular orbit centered at the leader spacecraftwith a radius of 100 meters on a plane generated by e
119903and e120579
with an angular velocity 41205960 The initial relative position and
velocity are assumed to be respectively as follows
p119897(0) = [10 90 minus20]
119879
k119897(0) = [0 0 0]
119879
(32)
Table 1 lists some parameters and their values used in thissimulation Some uncertainties such as disturbances and sen-sor noises of velocity and position are induced in this simu-lation Disturbances acting on the follower spacecraft areassumed such that F
119889= [290532 31775 minus112298]
119879 (N)And it is also assumed that maximally 20 of random velo-city sensor noises in three axes create difficulties in allowingthe follower spacecraft to obtain accurate velocity informa-tion Moreover maximally 20 and 40 of random positionsensor noises in e
119903and e
ℎaxes are also considered in this
simulation
Mathematical Problems in Engineering 9
0 10 20 30 40 50 60 70 80 90 100
0
5
10
Time (s)
No disturbanceDisturbance
minus5
minus10
minus15
e r-d
irect
ion
(m)
(a) Position tracking error in e119903-direction
0 10 20 30 40 50 60 70 80 90 100
0
2
Time (s)
No disturbanceDisturbance
minus2
minus4
minus6
minus8
minus10
e 120579-d
irect
ion
(m)
(b) Position tracking error in e120579-direction
0 10 20 30 40 50 60 70 80 90 100
0
5
10
Time (s)
No disturbanceDisturbance
minus5
minus10
minus15
minus20
e h-d
irect
ion
(m)
(c) Position tracking error in eℎ-direction
Figure 8 Relative position errors of no disturbance case and disturbance injected case The black thin line represents the trajectories of thespacecraft without considering any disturbances The blue thick line shows the results of the trajectories of the disturbed spacecraft
42 Simulation Results Figure 8 describes the position track-ing error results of the no disturbance case and disturbancesinjected case The objective of this simulation is to verifythe performance degradation of the disturbances injectedspacecraft In both cases NDI with the proportional type ofthe desired dynamics is designed as a primary controller forSFF As shown in Figure 8 if no disturbance is acting on thespacecraft then the position errors between the referencedposition and the follower position are zero in 30 secHoweverthe lack of robustness in NDI controller leads a significantperformance degradation of the disturbed spacecraft so ittracks the references with a tracking error of almost 8m 1mand 4m in e
119903 e120579 and e
ℎaxes respectively
Figures 9 and 10 show the trajectories of the followerspacecraft controlled by NDI and RNDI controllers Asmentioned in Figure 8 the position errors between thereferenced position and follower position controlled by NDIcannot be zero due to the induced uncertainties However bycombining the switching input into conventional NDI con-troller the RNDI control law attempts to reduce the effect ofthe uncertainties Consequently the state can track the refer-ence signal within 1m error as shown in Figure 9The controlinputs generated by NDI and RNDI controllers are described
in Figure 10 To compensate the induced uncertainties thespacecraft controlled by RNDI generates a large input force ineℎaxis around 27 sec (Figure 10(b)) Figure 11 shows the tra-
jectory of the spacecraft controlled by the RNDI controllerThe spacecraft maneuvers around a circle with a 100-meterradius Although the spacecraft is vibrated due to the induceduncertainties the error distance between the reference trajec-tory and the position of the follower is less than 1m as shownin Figure 9 Hence the follower spacecraft controlled by theproposed RNDI can track the reference trajectory
5 Conclusion
In this paper the robust nonlinear dynamic inversion controlstrategy that improves the robustness of NDI has been pro-posed NDI is a nonlinear control technique that eliminatesthe need of linearizing and designing gain-scheduled con-trollers for each equilibrium point by canceling the originaldynamics Because NDI removes the system nonlinearitiesdirectly it provides an opportunity to overcome the difficul-ties of ensuring stabilities and performances between variousoperational points while gain-scheduling However the maindrawback of designing NDI controller is the weak robustness
10 Mathematical Problems in Engineering
0 10 20 30 40 50 60 70 80 90 100
0
5
10
Time (m)
NDIRNDI
e r-d
irect
ion
(m)
minus5
minus15
minus10
(a) Position tracking error in e119903-direction
0
2
0 10 20 30 40 50 60 70 80 90 100Time (s)
NDIRNDI
e 120579-d
irect
ion
(m)
minus6
minus2
minus4
minus8
minus10
(b) Position tracking error in e120579-direction
0 10 20 30 40 50 60 70 80 90 100
0
5
10
Time (s)
NDIRNDI
e h-d
irect
ion
(m)
minus20
minus15
minus10
minus5
(c) Position tracking error in eℎ-direction
Figure 9 Simulation results of the position tracking error of the spacecrafts controlled by NDI and RNDIThe black thin line represents thetrajectories of the spacecraft controlled by NDI while the blue thick line represents the trajectories of the spacecraft controlled by RNDI
20 30 40 50 60 70 80 90 100
0
100
200
300
400
Time (s)
Inpu
t for
ces (
N)
minus200
minus100
ux along with er-axis
uz along with eh-axis
uy along with e120579-axis
(a) Control inputs generated by NDI
20 30 40 50 60 70 80 90 100
0
100
200
300
400
Time (s)
Inpu
t for
ces (
N)
minus200
minus100
ux along with er-axis
uz along with eh-axis
uy along with e120579-axis
(b) Control inputs generated by RNDI
Figure 10 Control inputs generated by NDI and RNDI
due to difficulties in obtaining accurate information of thenonlinear system dynamics In contrast to NDI SMC isconsidered a robust nonlinear control technique that usesswitching values as inputs to force the system states intoa sliding surface and to maintain the states on it Theproposed RNDI controller is designed by combining the
conventional NDI controller and switching inputs analyzedin SMC Hence RNDI takes the advantages of both NDI andSMC such that it
(i) is easy to design and implement(ii) eliminates the need of gain-scheduling
Mathematical Problems in Engineering 11
0 50 100 150050100150
0
10
e120579 axis er axis
e hax
is
minus50minus50minus100minus100 minus150minus150
minus10
minus20
Figure 11 Trajectory of the follower spacecraft controlled by theproposed RNDI controller
(iii) is easy to achieve the system required performancedue to the similar concept to classical control meth-ods and
(iv) is inherently robust
However the drawback of SMC such as the chatteringproblem still remains as a problem of RNDI control It isexpected that various chattering-free methods can be usedto reduce the effect of chattering phenomenon for examplereplacing the sign function into the saturation function
Moreover the equivalence of NDI input and equivalentinput designed by the identity sliding surface is also analyzedin this paper The reduced order dynamics on the slidingsurface can be converted to the user-selected desired dynam-ics As a result the difficulties in designing a sliding modecontroller for satisfying the specific system performances areconverted to a designing problem of the desired dynamicsHence RNDI provides an opportunity to solve the difficultiesin selecting the sliding surface of SMC The effectiveness ofthe proposed RNDI controller is confirmed by a set of simu-lations with application to the spacecraft formation flying
Nomenclature
G Universal gravity constantM Mass of the Earth119898119891 Mass of the follower spacecraft
119898119897 Mass of the leader spacecraft
r119891 Distance from the center of the Earth to
the center of the follower spacecraftr119897 Distance from the center of the Earth to
the center of the leader spacecraftxdes Desired dynamics
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research was supported by the MSIP (Ministry ofScience ICT amp Future Planning) Korea under the C-ITRC(Convergence Information Technology Research Center)
support program (NIPA-2014-H0401-14-1004) supervised bythe NIPA (National IT Industry Promotion Agency)
References
[1] Y Lv Q Hu G Ma and J Zhou ldquo6 DOF synchronized con-trol for spacecraft formation flying with input constraint andparameter uncertaintiesrdquo ISA Transactions vol 50 no 4 pp573ndash580 2011
[2] D Forta F Bordi and C Scolese ldquoLow-cost minimum sizesatellites for demonstration of formation flying modes at smallkilometer-size distancerdquo in Proceedings of the 13th AIAAUSUConference on Small Satellite SSC99-VI-3 pp 1ndash15 1999
[3] M S De Queiroz V Kapila and Q Yan ldquoAdaptive nonlinearcontrol ofmultiple spacecraft formation flyingrdquo Journal of Guid-ance Control and Dynamics vol 23 no 3 pp 385ndash390 2000
[4] J Georgie and J Valasek ldquoEvaluation of longitudinal desireddynamics for dynamic-inversion controlled generic reentryvehiclesrdquo Journal of Guidance Control and Dynamics vol 26no 5 pp 811ndash819 2003
[5] D Ito J Georgie J Valasek and D T Ward Reentry VehicleFlight Controls Design Guidelines Dynamic Inversion NASATPmdash2002ndash210771 2002
[6] A J Ostroff and B J Bacon ldquoForce and moment approachfor achievable dynamics using nonlinear dynamic inversionrdquo inProceedings of theAIAAGuidance Navigation andControl Con-ference and Exhibit pp 424ndash434 Reston Va USA 1999
[7] D Enns D Bugajski R Hendrick and G Stein ldquoDynamicinversion an evolving methodology for flight control designrdquoInternational Journal of Control vol 59 no 1 pp 71ndash91 1994
[8] J A Richard J M Buffington A G Sparks and S S BandaRobust Multivariable Flight Control Springer New York NYUSA 1994
[9] R J Adams and S S Banda ldquoRobust flight control design usingdynamic inversion and structured singular value synthesisrdquoIEEE Transactions on Control Systems Technology vol 1 no 2pp 80ndash92 1993
[10] M B McFarland and S M Hoque ldquoRobustness of a nonlinearmissile autopilot designed using dynamic inversionrdquo in Pro-ceedings of the AIAA Guidance Navigation and Control Confer-ence and Exhibit AIAA-2000-3970 Denver Colo USA August2000
[11] G Looye and H-D Joos ldquoDesign of robust dynamic inversioncontrol laws usingmulti-objective optimizationrdquo in Proceedingsof the AIAA Guidance Navigation and Control Conference andExhibit AIAA-2001-4285 2001
[12] P B Acquatella W Falkena E van Kampen and Q P ChuldquoRobust nonlinear spacecraft attitude control using incrementalnonlinear dynamic inversionrdquo in Proceedings of the AIAAGuid-ance Navigation and Control Conference and Exhibit Min-neapolis Minn USA August 2012
[13] P K Menon V R Iragavarapu and E J Ohlmeyer ldquoNonlinearMissile Autopilot Design using Time-Scale Separationrdquo in Pro-ceedings of the AIAA Guidance Navigation and Control Confer-ence and Exhibit AIAA-1997-3765 1997
[14] A Steinicke and G Michalka ldquoImproving transient perfor-mance of dynamic inversion missile autopilot by use of back-steppingrdquo in Proceedings of the AIAAGuidance Navigation andControl Conference and Exhibit AIAA-2002-4658 MontereyCalif USA August 2002
12 Mathematical Problems in Engineering
[15] I Yang D Kim and D Lee ldquoA flight control strategy usingrobust dynamic inversion based on sliding mode controlrdquo inproceeding of the AIAA Guidance Navigation and ControlConference August 2012
[16] H Sira-Ramirez ldquoDifferential geometric methods in variable-structure controlrdquo International Journal of Control vol 48 no4 pp 1359ndash1390 1988
[17] V I Utkin ldquoVariable structure systems with sliding modesrdquoIEEE Transactions on Automatic Control vol 22 no 2 pp 212ndash222 1977
[18] R A DeCarlo S H Zak and G P Matthews ldquoVariable struc-ture control of nonlinear multivariable systems a tutorialrdquo Pro-ceedings of the IEEE vol 76 no 3 pp 212ndash232 1988
[19] J Y HungWGao and J CHung ldquoVariable structure control asurveyrdquo IEEE Transactions on Industrial Electronics vol 40 no1 pp 2ndash22 1993
[20] G Papageorgiou andM Polansky ldquoTuning a dynamic inversionpitch axis autopilot using McFarlane-Glover loop shapingrdquoOptimal Control Applications ampMethods vol 30 no 3 pp 287ndash308 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Mathematical Problems in Engineering
Select a set of nonlinear smooth functions 120590(xlowast) = [1205901(xlowast)
1205902(xlowast) 120590
119898(xlowast)]119879 = 0 as a sliding surface If the system
states intercept the sliding surface at 119905 = 1199050and the sliding
mode exists for 119905 ge 1199050 then (119889119889119905)120590(x(119905)) = 0 and 120590(x(119905)) = 0
for all 119905 ge 1199050 Then the equivalent input ueq that the state
trajectory stays on the sliding surface can be analyzed asfollows [13ndash15]
(xlowast) = 120597120590
120597xlowastxlowast = 120597120590
120597xlowast(xdes minus x) = 0
997904rArr120597120590
120597xlowastxdes minus
120597120590
120597xlowast(f (x) + g (x) ueq) = 0
997904rArr ueq = (120597120590
120597xlowastg (x))
minus1
(120597120590
120597xlowastxdes minus
120597120590
120597xlowastf (x))
(17)
If 119898 = 119899 and 120597120590120597xlowast = I119899 then (17) can be represented as a
form of NDI input shown in (15)
ueq = g(x)minus1 [xdes minus f (x)] = uNDI (18)
Hence if the sliding surface is designed by an identity matrixthen the equivalent input of SMC can be represented as NDIinput
Definition 1 Let 120590NDI(x) be a set of smooth functions120590NDI119894(x) (119894 = 1 2 119899) that is 120590NDI(x)=[120590NDI1(x)120590NDI2(x) 120590NDI119899(x)]
119879 Then 120590NDI(x) = x is defined as theNDI surface that is the NDI surface is the sliding surfacedesigned by the identity matrix
It is worth noting that the ideal sliding dynamics shownin (13) yields the desired dynamics as follows
x = f (x) + g (x)ueq
= [f (x) + g (x) ( 120597120590120597xlowast
g (x))minus1
(120597120590
120597xlowastxdes minus
120597120590
120597xlowastf (x))]
= xdes(19)
In (19) the states on the identity sliding surface are governedby the desired dynamics Moreover the performances ofthe control system in sliding mode are determined only bythe desired dynamics This suggests that the difficulties indesigning a sliding mode controller to satisfy the specificsystem performances are converted to the desired dynamicsdesigning problem
33 Design of the Robust Nonlinear Dynamic Inversion Simi-lar to the designmethodof SMC each entry119906
119894(119905) (119894 = 1 2
119898) of the RNDI control law can be obtained by introducingthe following values
119906119894(119905) =
119906+
119894 if 120590
119894(119909des119894 minus 119909119894) = 119909des119894 minus 119909119894 gt 0
119906minus
119894 if 120590
119894(119909des119894 minus 119909119894) = 119909des119894 minus 119909119894 lt 0
(20)
where 119906+119894= 119906minus
119894satisfies the following for 119891
119894(x) the 119894th entry
of f(x) and for 119892119894119895(x) the (119894 119895)-entry of g(x)
lim119909119894rarr119909
+
des119894
119891119894(119909) +
119899
sum
119895=1
119895 = 119894
119892119894119895(119909) 119906119895+ 119892119894119894(119909) 119906minus
119894lt des119894
lim119909119894rarr119909
minus
des119894
119891119894(119909) +
119899
sum
119895=1
119895 = 119894
119892119894119895(119909) 119906119895+ 119892119894119894(119909) 119906+
119894gt des119894
(21)
where the superscripts + and minus denote the right- and left-hand limit respectivelyThe following theorem and corollarypropose the method for designing the RNDI controller andprovide the stability analytically
Theorem 2 Consider the following input
u = u119873119863119868
+ gminus1 (x) u119904119908 (22)
where u119873119863119868
is the conventional NDI input and u119904119908
isthe switching input that satisfies the following for u
119904119908=
[1199061199041199081 1199061199041199082 119906
119904119908119899]119879
119906119904119908119894
gt 0 119894119891 120590119894(x119889119890119904minus x) = 119909
119889119890119904119894minus 119909119894gt 0
119906119904119908119894
lt 0 119894119891 120590119894(x119889119890119904minus x) = 119909
119889119890119904119894minus 119909119894lt 0
(23)
Then the closed-loop system controlled by (22) is globally stable
Proof Select a Lyapunov candidate as
119881 =1
2120590(xdes minus x)119879120590 (xdes minus x) (24)
From (22) and (23)
= 120590(xdes minus x)119879 (xdes minus x)
= 120590(xdes minus x)119879 (xdes minus x)
= 120590(xdes minus x)119879 [xdes minus f (x) + g (x) u]
= 120590(xdes minus x)119879
times [xdes minus f (x) + g (x) (uNDI + gminus1 (x) usw)]
= 120590(xdes minus x)119879
times [xdes minus f (x) + g (x) uNDI minus usw]
= minus120590(xdes minus x)119879usw
= minus
119899
sum
119894=1
120590119894(xdes minus x) 119906sw119894
(25)
From (25) if 120590119894(xdes minus x) gt 0 then 119906sw119894 gt 0 and lt 0
Similarly if 120590119894(xdes minus x) lt 0 then 119906sw119894 lt 0 and lt 0 Hence
the closed-loop system controlled by (22) is globally stable
Mathematical Problems in Engineering 7
NDI surface
fi(x) +n
sumj=1jnei
gij(x)uj + gii(x)u+i
fi(x) +n
sumj=1jnei
gij(x)uj + gii(x)uminusi
120590i( minus ) gt 0x xdes
120590i( minus x) lt 0xdes
120590i( minus ) = 0xdes x
Figure 6 State trajectory on the NDI surface
Since the system controlled by (22) is globally stable fromTheorem 2 the states are forced to the NDI surface as shownin Figure 6
Corollary 3 If the switching input denoted as u119904119908= [1199061199041199081
1199061199041199082 119906
119904119908119899]119879 is designed with 119906
119904119908119894= 119896119894sgn(120590
119894(x119889119890119904minus x))
for a positive 119896119894 then the control system is globally stable
Proof If 120590119894(xdes minus x) gt 0 then 119906sw119894 = 119896119894 gt 0 and conversely
if 120590119894(xdes minus x) lt 0 then 119906sw119894 = minus119896119894 lt 0 Hence according to
Theorem 2 the control system is globally stable
Definition 4 The control law proposed inTheorem 2 is calledthe robust nonlinear dynamic inversion (RNDI) law that is fora diagonal matrix K = diag(119896
119894) (119894 = 1 2 119899) with 119896
119894gt 0
the RNDI control law yields
uRNDI = uNDI + usw
= g(x)minus1 [xdes minus f (x) + K sgn (xdes minus x)] (26)
It is worth noting that the form of the switching inputrepresented in Corollary 3 is a relay with constant gains formin SMC as mentioned in Section 31 Actually the switchingfunction satisfying (23) can be obtained directly from SMCdesign method as follows
(i) relay with constant gains 119906sw119894 = 119896119894 sgn(120590119894(xdes minus x))with 119896
119894gt 0
(ii) relay with state dependent gains 119906sw119894 = 119896119894(xdes minus
x) sgn(120590119894(xdes minus x)) with 119896
119894(sdot) gt 0
(iii) univector nonlinearity with scale factor usw =
119896119894120590(xdes minus x)120590(xdes minus x) with 119896
119894gt 0
It is also remarkable that using the sgn function whiledesigning the switching input leads to unfavorable resultsin control systems such as the chattering problem Thechattering problem makes the system have unmodeled highfrequencies or actuator saturation In sliding mode controltheory several methods have been proposed to overcomethis problem by designing a continuous control input insteadof a discontinuous switching input This method forces the
controlled system to follow the approximated sliding motionin some boundaries instead of the ideal sliding motionFollowing the similar manner the chattering problem in theRNDI control can be solved using the continuous controlinput For example replace sgn(xdes minus x) into sat(xdes minus x) =[sat1(119909des1 minus 1199091) sat2(119909des2 minus 1199092) sat119899(119909des119899 minus 119909119899)]
119879 inorder to design a continuous switching input as follows for apositive 120575
119894
sat119894(119909des119894 minus 119909119894
120575119894
) =
1 if (119909des119894 minus 119909119894) gt 120575119909des119894 minus 119909119894
120575119894
if 1003816100381610038161003816119909des119894 minus 1199091198941003816100381610038161003816 lt 120575
minus1 if (119909des119894 minus 119909119894) lt minus120575(27)
Then 119906sw119894 = 119896119894sat119894((119909des119894minus119909119894)120575119894) In (27) plusmn120575 is the boundarylayer in which the states are governed by the followingdynamics for sat((xdes minus x)120575) = [sat
1((119909des1 minus 1199091)1205751)
sat2((119909des2 minus 1199092)1205752) sat119899((119909des119899 minus 119909119899)120575119899)]
119879 and K =
diag(119896119894) (119894 = 1 2 119899) with 119896
119894gt 0
x = f (x) + g (x)u
= f (x) + g (x)
times [(g (x))minus1 (xdes minus f (x))
+(g (x))minus1Ksat((xdes minus x)
120575)]
= xdes + Ksat((xdes minus x)
120575)
= xdes
+ [1198961(119909des1 minus 1199091)
1205751
1198962(119909des2 minus 1199092)
1205752
sdot sdot sdot
119896119899(119909des119899 minus 119909119899)
120575119899
]
119879
(28)
8 Mathematical Problems in Engineering
uf
RefTime
Angular
Position of Referencetrajectorygenerator
the leader spacecraft
Reffcn
PositionControl
Dynamics of the Robust nonlinear dynamic inversion follower spacecraft
PositionScope
DisturbancesFd
++
t
ww
plplinputs
velocity
Figure 7 Block diagram using MATLAB Simulink
34 Stability of the Proposed RNDI Controller In this paperthe stability of the closed-loop system controlled by theproposed RNDI is analyzed by a Lyapunov stability criterionConsider the following nonlinear system including boundedmodel uncertainties Δf(x(119905)) isin R119899 and Δg(x(119905)) isin R119899 andbounded disturbance d(119905) isin R119899
x (119905) = [f (x (119905)) + Δf (x (119905))]
+ [g (x (119905)) + Δg (x (119905))] u (119905) + d (119905) (29)
It is assumed that 120585(119905 x(119905) u(119905)) = Δf(x(119905)) + Δg(x(119905)) + d(119905)satisfying 120585(119905 x(119905) u(119905))
2lt 120582min(K) where 120582min(K) is the
minimum eigenvalue of K For the desired state xdes isin 119877119899denote xlowast(119905) = xdesminusx To analyze the stability of the proposedRNDI controller select a Lyapunov candidate as follows
119881 (xlowast (119905)) = 12xlowast(119905)119879xlowast (119905) (30)
Then the derivative of 119881(xlowast(119905)) yields
(xlowast (119905))
= xlowast(119905)119879xlowast (119905)
= xlowast(119905)119879 [xdes (119905) minus [f (x (119905)) + Δf (x (119905))]
+ [g (x (119905)) + Δg (x (119905))] u (119905) + d (119905) ]
= xlowast(119905)119879 [xdes (119905) minus (f (x (119905)) + g (x (119905)) u (119905))
minus (Δf (x (119905)) + Δg (x (119905)) u (119905) + d (119905)) ]
= xlowast(119905)119879 [minus120585 (119905 x (119905) u (119905)) minus K sgn (xlowast (119905))]
le minusxlowast(119905)119879120585 (119905 x (119905) u (119905)) minus 120582min (K)1003817100381710038171003817xlowast
(119905)10038171003817100381710038171
le 120585 (119905 x (119905) u (119905))1
1003817100381710038171003817119909lowast
(119905)10038171003817100381710038171 minus 120582min (K)
1003817100381710038171003817xlowast
(119905)10038171003817100381710038171
le [120585 (119905 x (119905) u (119905))2minus 120582min (K)]
1003817100381710038171003817xlowast
(119905)10038171003817100381710038171
(31)
By the hypothesis 120585(119905 x(119905) u(119905))2lt 120582min(K) the derivative
of 119881(xlowast(119905)) is always negative Hence the system controlledby the proposed RNDI controller is globally stable againstdisturbances noises and model mismatches
Table 1 Parameters for spacecraft formation flying [1 3]
Symbol Value Unit119866 6673 times 10minus11 m3kgsdots2
119872 5974 times 1024 kg119898119891
410 kg119898119897
1550 kgr119897
[4224 times 1024 0 0]119879 m1205960
7272 times 10minus5 rads
4 Simulation Results
In this section numerical simulations are conducted to eva-luate the performance of the proposed robust nonlineardynamic inversion controller
41 Simulation Description The aim of SFF is to design thefeasible control input u
119905(119905) such that p
119897rarr p119889as 119905 rarr infin
for a given reference relative position trajectory p119889isin R3
of the follower spacecraft with respect to the leader space-craft To evaluate the performance of the proposed RNDIcontroller MATLAB Simulink is selected as a simulationtool Figure 7 shows the block diagram of the overall SFFstructure using MATLAB Simulink In this paper it isassumed that the reference trajectory is set to p
119889=
[100 sin(41205960119905) 100 cos(4120596
0119905) 0]119879 that is the follower space-
craft tracks a circular orbit centered at the leader spacecraftwith a radius of 100 meters on a plane generated by e
119903and e120579
with an angular velocity 41205960 The initial relative position and
velocity are assumed to be respectively as follows
p119897(0) = [10 90 minus20]
119879
k119897(0) = [0 0 0]
119879
(32)
Table 1 lists some parameters and their values used in thissimulation Some uncertainties such as disturbances and sen-sor noises of velocity and position are induced in this simu-lation Disturbances acting on the follower spacecraft areassumed such that F
119889= [290532 31775 minus112298]
119879 (N)And it is also assumed that maximally 20 of random velo-city sensor noises in three axes create difficulties in allowingthe follower spacecraft to obtain accurate velocity informa-tion Moreover maximally 20 and 40 of random positionsensor noises in e
119903and e
ℎaxes are also considered in this
simulation
Mathematical Problems in Engineering 9
0 10 20 30 40 50 60 70 80 90 100
0
5
10
Time (s)
No disturbanceDisturbance
minus5
minus10
minus15
e r-d
irect
ion
(m)
(a) Position tracking error in e119903-direction
0 10 20 30 40 50 60 70 80 90 100
0
2
Time (s)
No disturbanceDisturbance
minus2
minus4
minus6
minus8
minus10
e 120579-d
irect
ion
(m)
(b) Position tracking error in e120579-direction
0 10 20 30 40 50 60 70 80 90 100
0
5
10
Time (s)
No disturbanceDisturbance
minus5
minus10
minus15
minus20
e h-d
irect
ion
(m)
(c) Position tracking error in eℎ-direction
Figure 8 Relative position errors of no disturbance case and disturbance injected case The black thin line represents the trajectories of thespacecraft without considering any disturbances The blue thick line shows the results of the trajectories of the disturbed spacecraft
42 Simulation Results Figure 8 describes the position track-ing error results of the no disturbance case and disturbancesinjected case The objective of this simulation is to verifythe performance degradation of the disturbances injectedspacecraft In both cases NDI with the proportional type ofthe desired dynamics is designed as a primary controller forSFF As shown in Figure 8 if no disturbance is acting on thespacecraft then the position errors between the referencedposition and the follower position are zero in 30 secHoweverthe lack of robustness in NDI controller leads a significantperformance degradation of the disturbed spacecraft so ittracks the references with a tracking error of almost 8m 1mand 4m in e
119903 e120579 and e
ℎaxes respectively
Figures 9 and 10 show the trajectories of the followerspacecraft controlled by NDI and RNDI controllers Asmentioned in Figure 8 the position errors between thereferenced position and follower position controlled by NDIcannot be zero due to the induced uncertainties However bycombining the switching input into conventional NDI con-troller the RNDI control law attempts to reduce the effect ofthe uncertainties Consequently the state can track the refer-ence signal within 1m error as shown in Figure 9The controlinputs generated by NDI and RNDI controllers are described
in Figure 10 To compensate the induced uncertainties thespacecraft controlled by RNDI generates a large input force ineℎaxis around 27 sec (Figure 10(b)) Figure 11 shows the tra-
jectory of the spacecraft controlled by the RNDI controllerThe spacecraft maneuvers around a circle with a 100-meterradius Although the spacecraft is vibrated due to the induceduncertainties the error distance between the reference trajec-tory and the position of the follower is less than 1m as shownin Figure 9 Hence the follower spacecraft controlled by theproposed RNDI can track the reference trajectory
5 Conclusion
In this paper the robust nonlinear dynamic inversion controlstrategy that improves the robustness of NDI has been pro-posed NDI is a nonlinear control technique that eliminatesthe need of linearizing and designing gain-scheduled con-trollers for each equilibrium point by canceling the originaldynamics Because NDI removes the system nonlinearitiesdirectly it provides an opportunity to overcome the difficul-ties of ensuring stabilities and performances between variousoperational points while gain-scheduling However the maindrawback of designing NDI controller is the weak robustness
10 Mathematical Problems in Engineering
0 10 20 30 40 50 60 70 80 90 100
0
5
10
Time (m)
NDIRNDI
e r-d
irect
ion
(m)
minus5
minus15
minus10
(a) Position tracking error in e119903-direction
0
2
0 10 20 30 40 50 60 70 80 90 100Time (s)
NDIRNDI
e 120579-d
irect
ion
(m)
minus6
minus2
minus4
minus8
minus10
(b) Position tracking error in e120579-direction
0 10 20 30 40 50 60 70 80 90 100
0
5
10
Time (s)
NDIRNDI
e h-d
irect
ion
(m)
minus20
minus15
minus10
minus5
(c) Position tracking error in eℎ-direction
Figure 9 Simulation results of the position tracking error of the spacecrafts controlled by NDI and RNDIThe black thin line represents thetrajectories of the spacecraft controlled by NDI while the blue thick line represents the trajectories of the spacecraft controlled by RNDI
20 30 40 50 60 70 80 90 100
0
100
200
300
400
Time (s)
Inpu
t for
ces (
N)
minus200
minus100
ux along with er-axis
uz along with eh-axis
uy along with e120579-axis
(a) Control inputs generated by NDI
20 30 40 50 60 70 80 90 100
0
100
200
300
400
Time (s)
Inpu
t for
ces (
N)
minus200
minus100
ux along with er-axis
uz along with eh-axis
uy along with e120579-axis
(b) Control inputs generated by RNDI
Figure 10 Control inputs generated by NDI and RNDI
due to difficulties in obtaining accurate information of thenonlinear system dynamics In contrast to NDI SMC isconsidered a robust nonlinear control technique that usesswitching values as inputs to force the system states intoa sliding surface and to maintain the states on it Theproposed RNDI controller is designed by combining the
conventional NDI controller and switching inputs analyzedin SMC Hence RNDI takes the advantages of both NDI andSMC such that it
(i) is easy to design and implement(ii) eliminates the need of gain-scheduling
Mathematical Problems in Engineering 11
0 50 100 150050100150
0
10
e120579 axis er axis
e hax
is
minus50minus50minus100minus100 minus150minus150
minus10
minus20
Figure 11 Trajectory of the follower spacecraft controlled by theproposed RNDI controller
(iii) is easy to achieve the system required performancedue to the similar concept to classical control meth-ods and
(iv) is inherently robust
However the drawback of SMC such as the chatteringproblem still remains as a problem of RNDI control It isexpected that various chattering-free methods can be usedto reduce the effect of chattering phenomenon for examplereplacing the sign function into the saturation function
Moreover the equivalence of NDI input and equivalentinput designed by the identity sliding surface is also analyzedin this paper The reduced order dynamics on the slidingsurface can be converted to the user-selected desired dynam-ics As a result the difficulties in designing a sliding modecontroller for satisfying the specific system performances areconverted to a designing problem of the desired dynamicsHence RNDI provides an opportunity to solve the difficultiesin selecting the sliding surface of SMC The effectiveness ofthe proposed RNDI controller is confirmed by a set of simu-lations with application to the spacecraft formation flying
Nomenclature
G Universal gravity constantM Mass of the Earth119898119891 Mass of the follower spacecraft
119898119897 Mass of the leader spacecraft
r119891 Distance from the center of the Earth to
the center of the follower spacecraftr119897 Distance from the center of the Earth to
the center of the leader spacecraftxdes Desired dynamics
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research was supported by the MSIP (Ministry ofScience ICT amp Future Planning) Korea under the C-ITRC(Convergence Information Technology Research Center)
support program (NIPA-2014-H0401-14-1004) supervised bythe NIPA (National IT Industry Promotion Agency)
References
[1] Y Lv Q Hu G Ma and J Zhou ldquo6 DOF synchronized con-trol for spacecraft formation flying with input constraint andparameter uncertaintiesrdquo ISA Transactions vol 50 no 4 pp573ndash580 2011
[2] D Forta F Bordi and C Scolese ldquoLow-cost minimum sizesatellites for demonstration of formation flying modes at smallkilometer-size distancerdquo in Proceedings of the 13th AIAAUSUConference on Small Satellite SSC99-VI-3 pp 1ndash15 1999
[3] M S De Queiroz V Kapila and Q Yan ldquoAdaptive nonlinearcontrol ofmultiple spacecraft formation flyingrdquo Journal of Guid-ance Control and Dynamics vol 23 no 3 pp 385ndash390 2000
[4] J Georgie and J Valasek ldquoEvaluation of longitudinal desireddynamics for dynamic-inversion controlled generic reentryvehiclesrdquo Journal of Guidance Control and Dynamics vol 26no 5 pp 811ndash819 2003
[5] D Ito J Georgie J Valasek and D T Ward Reentry VehicleFlight Controls Design Guidelines Dynamic Inversion NASATPmdash2002ndash210771 2002
[6] A J Ostroff and B J Bacon ldquoForce and moment approachfor achievable dynamics using nonlinear dynamic inversionrdquo inProceedings of theAIAAGuidance Navigation andControl Con-ference and Exhibit pp 424ndash434 Reston Va USA 1999
[7] D Enns D Bugajski R Hendrick and G Stein ldquoDynamicinversion an evolving methodology for flight control designrdquoInternational Journal of Control vol 59 no 1 pp 71ndash91 1994
[8] J A Richard J M Buffington A G Sparks and S S BandaRobust Multivariable Flight Control Springer New York NYUSA 1994
[9] R J Adams and S S Banda ldquoRobust flight control design usingdynamic inversion and structured singular value synthesisrdquoIEEE Transactions on Control Systems Technology vol 1 no 2pp 80ndash92 1993
[10] M B McFarland and S M Hoque ldquoRobustness of a nonlinearmissile autopilot designed using dynamic inversionrdquo in Pro-ceedings of the AIAA Guidance Navigation and Control Confer-ence and Exhibit AIAA-2000-3970 Denver Colo USA August2000
[11] G Looye and H-D Joos ldquoDesign of robust dynamic inversioncontrol laws usingmulti-objective optimizationrdquo in Proceedingsof the AIAA Guidance Navigation and Control Conference andExhibit AIAA-2001-4285 2001
[12] P B Acquatella W Falkena E van Kampen and Q P ChuldquoRobust nonlinear spacecraft attitude control using incrementalnonlinear dynamic inversionrdquo in Proceedings of the AIAAGuid-ance Navigation and Control Conference and Exhibit Min-neapolis Minn USA August 2012
[13] P K Menon V R Iragavarapu and E J Ohlmeyer ldquoNonlinearMissile Autopilot Design using Time-Scale Separationrdquo in Pro-ceedings of the AIAA Guidance Navigation and Control Confer-ence and Exhibit AIAA-1997-3765 1997
[14] A Steinicke and G Michalka ldquoImproving transient perfor-mance of dynamic inversion missile autopilot by use of back-steppingrdquo in Proceedings of the AIAAGuidance Navigation andControl Conference and Exhibit AIAA-2002-4658 MontereyCalif USA August 2002
12 Mathematical Problems in Engineering
[15] I Yang D Kim and D Lee ldquoA flight control strategy usingrobust dynamic inversion based on sliding mode controlrdquo inproceeding of the AIAA Guidance Navigation and ControlConference August 2012
[16] H Sira-Ramirez ldquoDifferential geometric methods in variable-structure controlrdquo International Journal of Control vol 48 no4 pp 1359ndash1390 1988
[17] V I Utkin ldquoVariable structure systems with sliding modesrdquoIEEE Transactions on Automatic Control vol 22 no 2 pp 212ndash222 1977
[18] R A DeCarlo S H Zak and G P Matthews ldquoVariable struc-ture control of nonlinear multivariable systems a tutorialrdquo Pro-ceedings of the IEEE vol 76 no 3 pp 212ndash232 1988
[19] J Y HungWGao and J CHung ldquoVariable structure control asurveyrdquo IEEE Transactions on Industrial Electronics vol 40 no1 pp 2ndash22 1993
[20] G Papageorgiou andM Polansky ldquoTuning a dynamic inversionpitch axis autopilot using McFarlane-Glover loop shapingrdquoOptimal Control Applications ampMethods vol 30 no 3 pp 287ndash308 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 7
NDI surface
fi(x) +n
sumj=1jnei
gij(x)uj + gii(x)u+i
fi(x) +n
sumj=1jnei
gij(x)uj + gii(x)uminusi
120590i( minus ) gt 0x xdes
120590i( minus x) lt 0xdes
120590i( minus ) = 0xdes x
Figure 6 State trajectory on the NDI surface
Since the system controlled by (22) is globally stable fromTheorem 2 the states are forced to the NDI surface as shownin Figure 6
Corollary 3 If the switching input denoted as u119904119908= [1199061199041199081
1199061199041199082 119906
119904119908119899]119879 is designed with 119906
119904119908119894= 119896119894sgn(120590
119894(x119889119890119904minus x))
for a positive 119896119894 then the control system is globally stable
Proof If 120590119894(xdes minus x) gt 0 then 119906sw119894 = 119896119894 gt 0 and conversely
if 120590119894(xdes minus x) lt 0 then 119906sw119894 = minus119896119894 lt 0 Hence according to
Theorem 2 the control system is globally stable
Definition 4 The control law proposed inTheorem 2 is calledthe robust nonlinear dynamic inversion (RNDI) law that is fora diagonal matrix K = diag(119896
119894) (119894 = 1 2 119899) with 119896
119894gt 0
the RNDI control law yields
uRNDI = uNDI + usw
= g(x)minus1 [xdes minus f (x) + K sgn (xdes minus x)] (26)
It is worth noting that the form of the switching inputrepresented in Corollary 3 is a relay with constant gains formin SMC as mentioned in Section 31 Actually the switchingfunction satisfying (23) can be obtained directly from SMCdesign method as follows
(i) relay with constant gains 119906sw119894 = 119896119894 sgn(120590119894(xdes minus x))with 119896
119894gt 0
(ii) relay with state dependent gains 119906sw119894 = 119896119894(xdes minus
x) sgn(120590119894(xdes minus x)) with 119896
119894(sdot) gt 0
(iii) univector nonlinearity with scale factor usw =
119896119894120590(xdes minus x)120590(xdes minus x) with 119896
119894gt 0
It is also remarkable that using the sgn function whiledesigning the switching input leads to unfavorable resultsin control systems such as the chattering problem Thechattering problem makes the system have unmodeled highfrequencies or actuator saturation In sliding mode controltheory several methods have been proposed to overcomethis problem by designing a continuous control input insteadof a discontinuous switching input This method forces the
controlled system to follow the approximated sliding motionin some boundaries instead of the ideal sliding motionFollowing the similar manner the chattering problem in theRNDI control can be solved using the continuous controlinput For example replace sgn(xdes minus x) into sat(xdes minus x) =[sat1(119909des1 minus 1199091) sat2(119909des2 minus 1199092) sat119899(119909des119899 minus 119909119899)]
119879 inorder to design a continuous switching input as follows for apositive 120575
119894
sat119894(119909des119894 minus 119909119894
120575119894
) =
1 if (119909des119894 minus 119909119894) gt 120575119909des119894 minus 119909119894
120575119894
if 1003816100381610038161003816119909des119894 minus 1199091198941003816100381610038161003816 lt 120575
minus1 if (119909des119894 minus 119909119894) lt minus120575(27)
Then 119906sw119894 = 119896119894sat119894((119909des119894minus119909119894)120575119894) In (27) plusmn120575 is the boundarylayer in which the states are governed by the followingdynamics for sat((xdes minus x)120575) = [sat
1((119909des1 minus 1199091)1205751)
sat2((119909des2 minus 1199092)1205752) sat119899((119909des119899 minus 119909119899)120575119899)]
119879 and K =
diag(119896119894) (119894 = 1 2 119899) with 119896
119894gt 0
x = f (x) + g (x)u
= f (x) + g (x)
times [(g (x))minus1 (xdes minus f (x))
+(g (x))minus1Ksat((xdes minus x)
120575)]
= xdes + Ksat((xdes minus x)
120575)
= xdes
+ [1198961(119909des1 minus 1199091)
1205751
1198962(119909des2 minus 1199092)
1205752
sdot sdot sdot
119896119899(119909des119899 minus 119909119899)
120575119899
]
119879
(28)
8 Mathematical Problems in Engineering
uf
RefTime
Angular
Position of Referencetrajectorygenerator
the leader spacecraft
Reffcn
PositionControl
Dynamics of the Robust nonlinear dynamic inversion follower spacecraft
PositionScope
DisturbancesFd
++
t
ww
plplinputs
velocity
Figure 7 Block diagram using MATLAB Simulink
34 Stability of the Proposed RNDI Controller In this paperthe stability of the closed-loop system controlled by theproposed RNDI is analyzed by a Lyapunov stability criterionConsider the following nonlinear system including boundedmodel uncertainties Δf(x(119905)) isin R119899 and Δg(x(119905)) isin R119899 andbounded disturbance d(119905) isin R119899
x (119905) = [f (x (119905)) + Δf (x (119905))]
+ [g (x (119905)) + Δg (x (119905))] u (119905) + d (119905) (29)
It is assumed that 120585(119905 x(119905) u(119905)) = Δf(x(119905)) + Δg(x(119905)) + d(119905)satisfying 120585(119905 x(119905) u(119905))
2lt 120582min(K) where 120582min(K) is the
minimum eigenvalue of K For the desired state xdes isin 119877119899denote xlowast(119905) = xdesminusx To analyze the stability of the proposedRNDI controller select a Lyapunov candidate as follows
119881 (xlowast (119905)) = 12xlowast(119905)119879xlowast (119905) (30)
Then the derivative of 119881(xlowast(119905)) yields
(xlowast (119905))
= xlowast(119905)119879xlowast (119905)
= xlowast(119905)119879 [xdes (119905) minus [f (x (119905)) + Δf (x (119905))]
+ [g (x (119905)) + Δg (x (119905))] u (119905) + d (119905) ]
= xlowast(119905)119879 [xdes (119905) minus (f (x (119905)) + g (x (119905)) u (119905))
minus (Δf (x (119905)) + Δg (x (119905)) u (119905) + d (119905)) ]
= xlowast(119905)119879 [minus120585 (119905 x (119905) u (119905)) minus K sgn (xlowast (119905))]
le minusxlowast(119905)119879120585 (119905 x (119905) u (119905)) minus 120582min (K)1003817100381710038171003817xlowast
(119905)10038171003817100381710038171
le 120585 (119905 x (119905) u (119905))1
1003817100381710038171003817119909lowast
(119905)10038171003817100381710038171 minus 120582min (K)
1003817100381710038171003817xlowast
(119905)10038171003817100381710038171
le [120585 (119905 x (119905) u (119905))2minus 120582min (K)]
1003817100381710038171003817xlowast
(119905)10038171003817100381710038171
(31)
By the hypothesis 120585(119905 x(119905) u(119905))2lt 120582min(K) the derivative
of 119881(xlowast(119905)) is always negative Hence the system controlledby the proposed RNDI controller is globally stable againstdisturbances noises and model mismatches
Table 1 Parameters for spacecraft formation flying [1 3]
Symbol Value Unit119866 6673 times 10minus11 m3kgsdots2
119872 5974 times 1024 kg119898119891
410 kg119898119897
1550 kgr119897
[4224 times 1024 0 0]119879 m1205960
7272 times 10minus5 rads
4 Simulation Results
In this section numerical simulations are conducted to eva-luate the performance of the proposed robust nonlineardynamic inversion controller
41 Simulation Description The aim of SFF is to design thefeasible control input u
119905(119905) such that p
119897rarr p119889as 119905 rarr infin
for a given reference relative position trajectory p119889isin R3
of the follower spacecraft with respect to the leader space-craft To evaluate the performance of the proposed RNDIcontroller MATLAB Simulink is selected as a simulationtool Figure 7 shows the block diagram of the overall SFFstructure using MATLAB Simulink In this paper it isassumed that the reference trajectory is set to p
119889=
[100 sin(41205960119905) 100 cos(4120596
0119905) 0]119879 that is the follower space-
craft tracks a circular orbit centered at the leader spacecraftwith a radius of 100 meters on a plane generated by e
119903and e120579
with an angular velocity 41205960 The initial relative position and
velocity are assumed to be respectively as follows
p119897(0) = [10 90 minus20]
119879
k119897(0) = [0 0 0]
119879
(32)
Table 1 lists some parameters and their values used in thissimulation Some uncertainties such as disturbances and sen-sor noises of velocity and position are induced in this simu-lation Disturbances acting on the follower spacecraft areassumed such that F
119889= [290532 31775 minus112298]
119879 (N)And it is also assumed that maximally 20 of random velo-city sensor noises in three axes create difficulties in allowingthe follower spacecraft to obtain accurate velocity informa-tion Moreover maximally 20 and 40 of random positionsensor noises in e
119903and e
ℎaxes are also considered in this
simulation
Mathematical Problems in Engineering 9
0 10 20 30 40 50 60 70 80 90 100
0
5
10
Time (s)
No disturbanceDisturbance
minus5
minus10
minus15
e r-d
irect
ion
(m)
(a) Position tracking error in e119903-direction
0 10 20 30 40 50 60 70 80 90 100
0
2
Time (s)
No disturbanceDisturbance
minus2
minus4
minus6
minus8
minus10
e 120579-d
irect
ion
(m)
(b) Position tracking error in e120579-direction
0 10 20 30 40 50 60 70 80 90 100
0
5
10
Time (s)
No disturbanceDisturbance
minus5
minus10
minus15
minus20
e h-d
irect
ion
(m)
(c) Position tracking error in eℎ-direction
Figure 8 Relative position errors of no disturbance case and disturbance injected case The black thin line represents the trajectories of thespacecraft without considering any disturbances The blue thick line shows the results of the trajectories of the disturbed spacecraft
42 Simulation Results Figure 8 describes the position track-ing error results of the no disturbance case and disturbancesinjected case The objective of this simulation is to verifythe performance degradation of the disturbances injectedspacecraft In both cases NDI with the proportional type ofthe desired dynamics is designed as a primary controller forSFF As shown in Figure 8 if no disturbance is acting on thespacecraft then the position errors between the referencedposition and the follower position are zero in 30 secHoweverthe lack of robustness in NDI controller leads a significantperformance degradation of the disturbed spacecraft so ittracks the references with a tracking error of almost 8m 1mand 4m in e
119903 e120579 and e
ℎaxes respectively
Figures 9 and 10 show the trajectories of the followerspacecraft controlled by NDI and RNDI controllers Asmentioned in Figure 8 the position errors between thereferenced position and follower position controlled by NDIcannot be zero due to the induced uncertainties However bycombining the switching input into conventional NDI con-troller the RNDI control law attempts to reduce the effect ofthe uncertainties Consequently the state can track the refer-ence signal within 1m error as shown in Figure 9The controlinputs generated by NDI and RNDI controllers are described
in Figure 10 To compensate the induced uncertainties thespacecraft controlled by RNDI generates a large input force ineℎaxis around 27 sec (Figure 10(b)) Figure 11 shows the tra-
jectory of the spacecraft controlled by the RNDI controllerThe spacecraft maneuvers around a circle with a 100-meterradius Although the spacecraft is vibrated due to the induceduncertainties the error distance between the reference trajec-tory and the position of the follower is less than 1m as shownin Figure 9 Hence the follower spacecraft controlled by theproposed RNDI can track the reference trajectory
5 Conclusion
In this paper the robust nonlinear dynamic inversion controlstrategy that improves the robustness of NDI has been pro-posed NDI is a nonlinear control technique that eliminatesthe need of linearizing and designing gain-scheduled con-trollers for each equilibrium point by canceling the originaldynamics Because NDI removes the system nonlinearitiesdirectly it provides an opportunity to overcome the difficul-ties of ensuring stabilities and performances between variousoperational points while gain-scheduling However the maindrawback of designing NDI controller is the weak robustness
10 Mathematical Problems in Engineering
0 10 20 30 40 50 60 70 80 90 100
0
5
10
Time (m)
NDIRNDI
e r-d
irect
ion
(m)
minus5
minus15
minus10
(a) Position tracking error in e119903-direction
0
2
0 10 20 30 40 50 60 70 80 90 100Time (s)
NDIRNDI
e 120579-d
irect
ion
(m)
minus6
minus2
minus4
minus8
minus10
(b) Position tracking error in e120579-direction
0 10 20 30 40 50 60 70 80 90 100
0
5
10
Time (s)
NDIRNDI
e h-d
irect
ion
(m)
minus20
minus15
minus10
minus5
(c) Position tracking error in eℎ-direction
Figure 9 Simulation results of the position tracking error of the spacecrafts controlled by NDI and RNDIThe black thin line represents thetrajectories of the spacecraft controlled by NDI while the blue thick line represents the trajectories of the spacecraft controlled by RNDI
20 30 40 50 60 70 80 90 100
0
100
200
300
400
Time (s)
Inpu
t for
ces (
N)
minus200
minus100
ux along with er-axis
uz along with eh-axis
uy along with e120579-axis
(a) Control inputs generated by NDI
20 30 40 50 60 70 80 90 100
0
100
200
300
400
Time (s)
Inpu
t for
ces (
N)
minus200
minus100
ux along with er-axis
uz along with eh-axis
uy along with e120579-axis
(b) Control inputs generated by RNDI
Figure 10 Control inputs generated by NDI and RNDI
due to difficulties in obtaining accurate information of thenonlinear system dynamics In contrast to NDI SMC isconsidered a robust nonlinear control technique that usesswitching values as inputs to force the system states intoa sliding surface and to maintain the states on it Theproposed RNDI controller is designed by combining the
conventional NDI controller and switching inputs analyzedin SMC Hence RNDI takes the advantages of both NDI andSMC such that it
(i) is easy to design and implement(ii) eliminates the need of gain-scheduling
Mathematical Problems in Engineering 11
0 50 100 150050100150
0
10
e120579 axis er axis
e hax
is
minus50minus50minus100minus100 minus150minus150
minus10
minus20
Figure 11 Trajectory of the follower spacecraft controlled by theproposed RNDI controller
(iii) is easy to achieve the system required performancedue to the similar concept to classical control meth-ods and
(iv) is inherently robust
However the drawback of SMC such as the chatteringproblem still remains as a problem of RNDI control It isexpected that various chattering-free methods can be usedto reduce the effect of chattering phenomenon for examplereplacing the sign function into the saturation function
Moreover the equivalence of NDI input and equivalentinput designed by the identity sliding surface is also analyzedin this paper The reduced order dynamics on the slidingsurface can be converted to the user-selected desired dynam-ics As a result the difficulties in designing a sliding modecontroller for satisfying the specific system performances areconverted to a designing problem of the desired dynamicsHence RNDI provides an opportunity to solve the difficultiesin selecting the sliding surface of SMC The effectiveness ofthe proposed RNDI controller is confirmed by a set of simu-lations with application to the spacecraft formation flying
Nomenclature
G Universal gravity constantM Mass of the Earth119898119891 Mass of the follower spacecraft
119898119897 Mass of the leader spacecraft
r119891 Distance from the center of the Earth to
the center of the follower spacecraftr119897 Distance from the center of the Earth to
the center of the leader spacecraftxdes Desired dynamics
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research was supported by the MSIP (Ministry ofScience ICT amp Future Planning) Korea under the C-ITRC(Convergence Information Technology Research Center)
support program (NIPA-2014-H0401-14-1004) supervised bythe NIPA (National IT Industry Promotion Agency)
References
[1] Y Lv Q Hu G Ma and J Zhou ldquo6 DOF synchronized con-trol for spacecraft formation flying with input constraint andparameter uncertaintiesrdquo ISA Transactions vol 50 no 4 pp573ndash580 2011
[2] D Forta F Bordi and C Scolese ldquoLow-cost minimum sizesatellites for demonstration of formation flying modes at smallkilometer-size distancerdquo in Proceedings of the 13th AIAAUSUConference on Small Satellite SSC99-VI-3 pp 1ndash15 1999
[3] M S De Queiroz V Kapila and Q Yan ldquoAdaptive nonlinearcontrol ofmultiple spacecraft formation flyingrdquo Journal of Guid-ance Control and Dynamics vol 23 no 3 pp 385ndash390 2000
[4] J Georgie and J Valasek ldquoEvaluation of longitudinal desireddynamics for dynamic-inversion controlled generic reentryvehiclesrdquo Journal of Guidance Control and Dynamics vol 26no 5 pp 811ndash819 2003
[5] D Ito J Georgie J Valasek and D T Ward Reentry VehicleFlight Controls Design Guidelines Dynamic Inversion NASATPmdash2002ndash210771 2002
[6] A J Ostroff and B J Bacon ldquoForce and moment approachfor achievable dynamics using nonlinear dynamic inversionrdquo inProceedings of theAIAAGuidance Navigation andControl Con-ference and Exhibit pp 424ndash434 Reston Va USA 1999
[7] D Enns D Bugajski R Hendrick and G Stein ldquoDynamicinversion an evolving methodology for flight control designrdquoInternational Journal of Control vol 59 no 1 pp 71ndash91 1994
[8] J A Richard J M Buffington A G Sparks and S S BandaRobust Multivariable Flight Control Springer New York NYUSA 1994
[9] R J Adams and S S Banda ldquoRobust flight control design usingdynamic inversion and structured singular value synthesisrdquoIEEE Transactions on Control Systems Technology vol 1 no 2pp 80ndash92 1993
[10] M B McFarland and S M Hoque ldquoRobustness of a nonlinearmissile autopilot designed using dynamic inversionrdquo in Pro-ceedings of the AIAA Guidance Navigation and Control Confer-ence and Exhibit AIAA-2000-3970 Denver Colo USA August2000
[11] G Looye and H-D Joos ldquoDesign of robust dynamic inversioncontrol laws usingmulti-objective optimizationrdquo in Proceedingsof the AIAA Guidance Navigation and Control Conference andExhibit AIAA-2001-4285 2001
[12] P B Acquatella W Falkena E van Kampen and Q P ChuldquoRobust nonlinear spacecraft attitude control using incrementalnonlinear dynamic inversionrdquo in Proceedings of the AIAAGuid-ance Navigation and Control Conference and Exhibit Min-neapolis Minn USA August 2012
[13] P K Menon V R Iragavarapu and E J Ohlmeyer ldquoNonlinearMissile Autopilot Design using Time-Scale Separationrdquo in Pro-ceedings of the AIAA Guidance Navigation and Control Confer-ence and Exhibit AIAA-1997-3765 1997
[14] A Steinicke and G Michalka ldquoImproving transient perfor-mance of dynamic inversion missile autopilot by use of back-steppingrdquo in Proceedings of the AIAAGuidance Navigation andControl Conference and Exhibit AIAA-2002-4658 MontereyCalif USA August 2002
12 Mathematical Problems in Engineering
[15] I Yang D Kim and D Lee ldquoA flight control strategy usingrobust dynamic inversion based on sliding mode controlrdquo inproceeding of the AIAA Guidance Navigation and ControlConference August 2012
[16] H Sira-Ramirez ldquoDifferential geometric methods in variable-structure controlrdquo International Journal of Control vol 48 no4 pp 1359ndash1390 1988
[17] V I Utkin ldquoVariable structure systems with sliding modesrdquoIEEE Transactions on Automatic Control vol 22 no 2 pp 212ndash222 1977
[18] R A DeCarlo S H Zak and G P Matthews ldquoVariable struc-ture control of nonlinear multivariable systems a tutorialrdquo Pro-ceedings of the IEEE vol 76 no 3 pp 212ndash232 1988
[19] J Y HungWGao and J CHung ldquoVariable structure control asurveyrdquo IEEE Transactions on Industrial Electronics vol 40 no1 pp 2ndash22 1993
[20] G Papageorgiou andM Polansky ldquoTuning a dynamic inversionpitch axis autopilot using McFarlane-Glover loop shapingrdquoOptimal Control Applications ampMethods vol 30 no 3 pp 287ndash308 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Mathematical Problems in Engineering
uf
RefTime
Angular
Position of Referencetrajectorygenerator
the leader spacecraft
Reffcn
PositionControl
Dynamics of the Robust nonlinear dynamic inversion follower spacecraft
PositionScope
DisturbancesFd
++
t
ww
plplinputs
velocity
Figure 7 Block diagram using MATLAB Simulink
34 Stability of the Proposed RNDI Controller In this paperthe stability of the closed-loop system controlled by theproposed RNDI is analyzed by a Lyapunov stability criterionConsider the following nonlinear system including boundedmodel uncertainties Δf(x(119905)) isin R119899 and Δg(x(119905)) isin R119899 andbounded disturbance d(119905) isin R119899
x (119905) = [f (x (119905)) + Δf (x (119905))]
+ [g (x (119905)) + Δg (x (119905))] u (119905) + d (119905) (29)
It is assumed that 120585(119905 x(119905) u(119905)) = Δf(x(119905)) + Δg(x(119905)) + d(119905)satisfying 120585(119905 x(119905) u(119905))
2lt 120582min(K) where 120582min(K) is the
minimum eigenvalue of K For the desired state xdes isin 119877119899denote xlowast(119905) = xdesminusx To analyze the stability of the proposedRNDI controller select a Lyapunov candidate as follows
119881 (xlowast (119905)) = 12xlowast(119905)119879xlowast (119905) (30)
Then the derivative of 119881(xlowast(119905)) yields
(xlowast (119905))
= xlowast(119905)119879xlowast (119905)
= xlowast(119905)119879 [xdes (119905) minus [f (x (119905)) + Δf (x (119905))]
+ [g (x (119905)) + Δg (x (119905))] u (119905) + d (119905) ]
= xlowast(119905)119879 [xdes (119905) minus (f (x (119905)) + g (x (119905)) u (119905))
minus (Δf (x (119905)) + Δg (x (119905)) u (119905) + d (119905)) ]
= xlowast(119905)119879 [minus120585 (119905 x (119905) u (119905)) minus K sgn (xlowast (119905))]
le minusxlowast(119905)119879120585 (119905 x (119905) u (119905)) minus 120582min (K)1003817100381710038171003817xlowast
(119905)10038171003817100381710038171
le 120585 (119905 x (119905) u (119905))1
1003817100381710038171003817119909lowast
(119905)10038171003817100381710038171 minus 120582min (K)
1003817100381710038171003817xlowast
(119905)10038171003817100381710038171
le [120585 (119905 x (119905) u (119905))2minus 120582min (K)]
1003817100381710038171003817xlowast
(119905)10038171003817100381710038171
(31)
By the hypothesis 120585(119905 x(119905) u(119905))2lt 120582min(K) the derivative
of 119881(xlowast(119905)) is always negative Hence the system controlledby the proposed RNDI controller is globally stable againstdisturbances noises and model mismatches
Table 1 Parameters for spacecraft formation flying [1 3]
Symbol Value Unit119866 6673 times 10minus11 m3kgsdots2
119872 5974 times 1024 kg119898119891
410 kg119898119897
1550 kgr119897
[4224 times 1024 0 0]119879 m1205960
7272 times 10minus5 rads
4 Simulation Results
In this section numerical simulations are conducted to eva-luate the performance of the proposed robust nonlineardynamic inversion controller
41 Simulation Description The aim of SFF is to design thefeasible control input u
119905(119905) such that p
119897rarr p119889as 119905 rarr infin
for a given reference relative position trajectory p119889isin R3
of the follower spacecraft with respect to the leader space-craft To evaluate the performance of the proposed RNDIcontroller MATLAB Simulink is selected as a simulationtool Figure 7 shows the block diagram of the overall SFFstructure using MATLAB Simulink In this paper it isassumed that the reference trajectory is set to p
119889=
[100 sin(41205960119905) 100 cos(4120596
0119905) 0]119879 that is the follower space-
craft tracks a circular orbit centered at the leader spacecraftwith a radius of 100 meters on a plane generated by e
119903and e120579
with an angular velocity 41205960 The initial relative position and
velocity are assumed to be respectively as follows
p119897(0) = [10 90 minus20]
119879
k119897(0) = [0 0 0]
119879
(32)
Table 1 lists some parameters and their values used in thissimulation Some uncertainties such as disturbances and sen-sor noises of velocity and position are induced in this simu-lation Disturbances acting on the follower spacecraft areassumed such that F
119889= [290532 31775 minus112298]
119879 (N)And it is also assumed that maximally 20 of random velo-city sensor noises in three axes create difficulties in allowingthe follower spacecraft to obtain accurate velocity informa-tion Moreover maximally 20 and 40 of random positionsensor noises in e
119903and e
ℎaxes are also considered in this
simulation
Mathematical Problems in Engineering 9
0 10 20 30 40 50 60 70 80 90 100
0
5
10
Time (s)
No disturbanceDisturbance
minus5
minus10
minus15
e r-d
irect
ion
(m)
(a) Position tracking error in e119903-direction
0 10 20 30 40 50 60 70 80 90 100
0
2
Time (s)
No disturbanceDisturbance
minus2
minus4
minus6
minus8
minus10
e 120579-d
irect
ion
(m)
(b) Position tracking error in e120579-direction
0 10 20 30 40 50 60 70 80 90 100
0
5
10
Time (s)
No disturbanceDisturbance
minus5
minus10
minus15
minus20
e h-d
irect
ion
(m)
(c) Position tracking error in eℎ-direction
Figure 8 Relative position errors of no disturbance case and disturbance injected case The black thin line represents the trajectories of thespacecraft without considering any disturbances The blue thick line shows the results of the trajectories of the disturbed spacecraft
42 Simulation Results Figure 8 describes the position track-ing error results of the no disturbance case and disturbancesinjected case The objective of this simulation is to verifythe performance degradation of the disturbances injectedspacecraft In both cases NDI with the proportional type ofthe desired dynamics is designed as a primary controller forSFF As shown in Figure 8 if no disturbance is acting on thespacecraft then the position errors between the referencedposition and the follower position are zero in 30 secHoweverthe lack of robustness in NDI controller leads a significantperformance degradation of the disturbed spacecraft so ittracks the references with a tracking error of almost 8m 1mand 4m in e
119903 e120579 and e
ℎaxes respectively
Figures 9 and 10 show the trajectories of the followerspacecraft controlled by NDI and RNDI controllers Asmentioned in Figure 8 the position errors between thereferenced position and follower position controlled by NDIcannot be zero due to the induced uncertainties However bycombining the switching input into conventional NDI con-troller the RNDI control law attempts to reduce the effect ofthe uncertainties Consequently the state can track the refer-ence signal within 1m error as shown in Figure 9The controlinputs generated by NDI and RNDI controllers are described
in Figure 10 To compensate the induced uncertainties thespacecraft controlled by RNDI generates a large input force ineℎaxis around 27 sec (Figure 10(b)) Figure 11 shows the tra-
jectory of the spacecraft controlled by the RNDI controllerThe spacecraft maneuvers around a circle with a 100-meterradius Although the spacecraft is vibrated due to the induceduncertainties the error distance between the reference trajec-tory and the position of the follower is less than 1m as shownin Figure 9 Hence the follower spacecraft controlled by theproposed RNDI can track the reference trajectory
5 Conclusion
In this paper the robust nonlinear dynamic inversion controlstrategy that improves the robustness of NDI has been pro-posed NDI is a nonlinear control technique that eliminatesthe need of linearizing and designing gain-scheduled con-trollers for each equilibrium point by canceling the originaldynamics Because NDI removes the system nonlinearitiesdirectly it provides an opportunity to overcome the difficul-ties of ensuring stabilities and performances between variousoperational points while gain-scheduling However the maindrawback of designing NDI controller is the weak robustness
10 Mathematical Problems in Engineering
0 10 20 30 40 50 60 70 80 90 100
0
5
10
Time (m)
NDIRNDI
e r-d
irect
ion
(m)
minus5
minus15
minus10
(a) Position tracking error in e119903-direction
0
2
0 10 20 30 40 50 60 70 80 90 100Time (s)
NDIRNDI
e 120579-d
irect
ion
(m)
minus6
minus2
minus4
minus8
minus10
(b) Position tracking error in e120579-direction
0 10 20 30 40 50 60 70 80 90 100
0
5
10
Time (s)
NDIRNDI
e h-d
irect
ion
(m)
minus20
minus15
minus10
minus5
(c) Position tracking error in eℎ-direction
Figure 9 Simulation results of the position tracking error of the spacecrafts controlled by NDI and RNDIThe black thin line represents thetrajectories of the spacecraft controlled by NDI while the blue thick line represents the trajectories of the spacecraft controlled by RNDI
20 30 40 50 60 70 80 90 100
0
100
200
300
400
Time (s)
Inpu
t for
ces (
N)
minus200
minus100
ux along with er-axis
uz along with eh-axis
uy along with e120579-axis
(a) Control inputs generated by NDI
20 30 40 50 60 70 80 90 100
0
100
200
300
400
Time (s)
Inpu
t for
ces (
N)
minus200
minus100
ux along with er-axis
uz along with eh-axis
uy along with e120579-axis
(b) Control inputs generated by RNDI
Figure 10 Control inputs generated by NDI and RNDI
due to difficulties in obtaining accurate information of thenonlinear system dynamics In contrast to NDI SMC isconsidered a robust nonlinear control technique that usesswitching values as inputs to force the system states intoa sliding surface and to maintain the states on it Theproposed RNDI controller is designed by combining the
conventional NDI controller and switching inputs analyzedin SMC Hence RNDI takes the advantages of both NDI andSMC such that it
(i) is easy to design and implement(ii) eliminates the need of gain-scheduling
Mathematical Problems in Engineering 11
0 50 100 150050100150
0
10
e120579 axis er axis
e hax
is
minus50minus50minus100minus100 minus150minus150
minus10
minus20
Figure 11 Trajectory of the follower spacecraft controlled by theproposed RNDI controller
(iii) is easy to achieve the system required performancedue to the similar concept to classical control meth-ods and
(iv) is inherently robust
However the drawback of SMC such as the chatteringproblem still remains as a problem of RNDI control It isexpected that various chattering-free methods can be usedto reduce the effect of chattering phenomenon for examplereplacing the sign function into the saturation function
Moreover the equivalence of NDI input and equivalentinput designed by the identity sliding surface is also analyzedin this paper The reduced order dynamics on the slidingsurface can be converted to the user-selected desired dynam-ics As a result the difficulties in designing a sliding modecontroller for satisfying the specific system performances areconverted to a designing problem of the desired dynamicsHence RNDI provides an opportunity to solve the difficultiesin selecting the sliding surface of SMC The effectiveness ofthe proposed RNDI controller is confirmed by a set of simu-lations with application to the spacecraft formation flying
Nomenclature
G Universal gravity constantM Mass of the Earth119898119891 Mass of the follower spacecraft
119898119897 Mass of the leader spacecraft
r119891 Distance from the center of the Earth to
the center of the follower spacecraftr119897 Distance from the center of the Earth to
the center of the leader spacecraftxdes Desired dynamics
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research was supported by the MSIP (Ministry ofScience ICT amp Future Planning) Korea under the C-ITRC(Convergence Information Technology Research Center)
support program (NIPA-2014-H0401-14-1004) supervised bythe NIPA (National IT Industry Promotion Agency)
References
[1] Y Lv Q Hu G Ma and J Zhou ldquo6 DOF synchronized con-trol for spacecraft formation flying with input constraint andparameter uncertaintiesrdquo ISA Transactions vol 50 no 4 pp573ndash580 2011
[2] D Forta F Bordi and C Scolese ldquoLow-cost minimum sizesatellites for demonstration of formation flying modes at smallkilometer-size distancerdquo in Proceedings of the 13th AIAAUSUConference on Small Satellite SSC99-VI-3 pp 1ndash15 1999
[3] M S De Queiroz V Kapila and Q Yan ldquoAdaptive nonlinearcontrol ofmultiple spacecraft formation flyingrdquo Journal of Guid-ance Control and Dynamics vol 23 no 3 pp 385ndash390 2000
[4] J Georgie and J Valasek ldquoEvaluation of longitudinal desireddynamics for dynamic-inversion controlled generic reentryvehiclesrdquo Journal of Guidance Control and Dynamics vol 26no 5 pp 811ndash819 2003
[5] D Ito J Georgie J Valasek and D T Ward Reentry VehicleFlight Controls Design Guidelines Dynamic Inversion NASATPmdash2002ndash210771 2002
[6] A J Ostroff and B J Bacon ldquoForce and moment approachfor achievable dynamics using nonlinear dynamic inversionrdquo inProceedings of theAIAAGuidance Navigation andControl Con-ference and Exhibit pp 424ndash434 Reston Va USA 1999
[7] D Enns D Bugajski R Hendrick and G Stein ldquoDynamicinversion an evolving methodology for flight control designrdquoInternational Journal of Control vol 59 no 1 pp 71ndash91 1994
[8] J A Richard J M Buffington A G Sparks and S S BandaRobust Multivariable Flight Control Springer New York NYUSA 1994
[9] R J Adams and S S Banda ldquoRobust flight control design usingdynamic inversion and structured singular value synthesisrdquoIEEE Transactions on Control Systems Technology vol 1 no 2pp 80ndash92 1993
[10] M B McFarland and S M Hoque ldquoRobustness of a nonlinearmissile autopilot designed using dynamic inversionrdquo in Pro-ceedings of the AIAA Guidance Navigation and Control Confer-ence and Exhibit AIAA-2000-3970 Denver Colo USA August2000
[11] G Looye and H-D Joos ldquoDesign of robust dynamic inversioncontrol laws usingmulti-objective optimizationrdquo in Proceedingsof the AIAA Guidance Navigation and Control Conference andExhibit AIAA-2001-4285 2001
[12] P B Acquatella W Falkena E van Kampen and Q P ChuldquoRobust nonlinear spacecraft attitude control using incrementalnonlinear dynamic inversionrdquo in Proceedings of the AIAAGuid-ance Navigation and Control Conference and Exhibit Min-neapolis Minn USA August 2012
[13] P K Menon V R Iragavarapu and E J Ohlmeyer ldquoNonlinearMissile Autopilot Design using Time-Scale Separationrdquo in Pro-ceedings of the AIAA Guidance Navigation and Control Confer-ence and Exhibit AIAA-1997-3765 1997
[14] A Steinicke and G Michalka ldquoImproving transient perfor-mance of dynamic inversion missile autopilot by use of back-steppingrdquo in Proceedings of the AIAAGuidance Navigation andControl Conference and Exhibit AIAA-2002-4658 MontereyCalif USA August 2002
12 Mathematical Problems in Engineering
[15] I Yang D Kim and D Lee ldquoA flight control strategy usingrobust dynamic inversion based on sliding mode controlrdquo inproceeding of the AIAA Guidance Navigation and ControlConference August 2012
[16] H Sira-Ramirez ldquoDifferential geometric methods in variable-structure controlrdquo International Journal of Control vol 48 no4 pp 1359ndash1390 1988
[17] V I Utkin ldquoVariable structure systems with sliding modesrdquoIEEE Transactions on Automatic Control vol 22 no 2 pp 212ndash222 1977
[18] R A DeCarlo S H Zak and G P Matthews ldquoVariable struc-ture control of nonlinear multivariable systems a tutorialrdquo Pro-ceedings of the IEEE vol 76 no 3 pp 212ndash232 1988
[19] J Y HungWGao and J CHung ldquoVariable structure control asurveyrdquo IEEE Transactions on Industrial Electronics vol 40 no1 pp 2ndash22 1993
[20] G Papageorgiou andM Polansky ldquoTuning a dynamic inversionpitch axis autopilot using McFarlane-Glover loop shapingrdquoOptimal Control Applications ampMethods vol 30 no 3 pp 287ndash308 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 9
0 10 20 30 40 50 60 70 80 90 100
0
5
10
Time (s)
No disturbanceDisturbance
minus5
minus10
minus15
e r-d
irect
ion
(m)
(a) Position tracking error in e119903-direction
0 10 20 30 40 50 60 70 80 90 100
0
2
Time (s)
No disturbanceDisturbance
minus2
minus4
minus6
minus8
minus10
e 120579-d
irect
ion
(m)
(b) Position tracking error in e120579-direction
0 10 20 30 40 50 60 70 80 90 100
0
5
10
Time (s)
No disturbanceDisturbance
minus5
minus10
minus15
minus20
e h-d
irect
ion
(m)
(c) Position tracking error in eℎ-direction
Figure 8 Relative position errors of no disturbance case and disturbance injected case The black thin line represents the trajectories of thespacecraft without considering any disturbances The blue thick line shows the results of the trajectories of the disturbed spacecraft
42 Simulation Results Figure 8 describes the position track-ing error results of the no disturbance case and disturbancesinjected case The objective of this simulation is to verifythe performance degradation of the disturbances injectedspacecraft In both cases NDI with the proportional type ofthe desired dynamics is designed as a primary controller forSFF As shown in Figure 8 if no disturbance is acting on thespacecraft then the position errors between the referencedposition and the follower position are zero in 30 secHoweverthe lack of robustness in NDI controller leads a significantperformance degradation of the disturbed spacecraft so ittracks the references with a tracking error of almost 8m 1mand 4m in e
119903 e120579 and e
ℎaxes respectively
Figures 9 and 10 show the trajectories of the followerspacecraft controlled by NDI and RNDI controllers Asmentioned in Figure 8 the position errors between thereferenced position and follower position controlled by NDIcannot be zero due to the induced uncertainties However bycombining the switching input into conventional NDI con-troller the RNDI control law attempts to reduce the effect ofthe uncertainties Consequently the state can track the refer-ence signal within 1m error as shown in Figure 9The controlinputs generated by NDI and RNDI controllers are described
in Figure 10 To compensate the induced uncertainties thespacecraft controlled by RNDI generates a large input force ineℎaxis around 27 sec (Figure 10(b)) Figure 11 shows the tra-
jectory of the spacecraft controlled by the RNDI controllerThe spacecraft maneuvers around a circle with a 100-meterradius Although the spacecraft is vibrated due to the induceduncertainties the error distance between the reference trajec-tory and the position of the follower is less than 1m as shownin Figure 9 Hence the follower spacecraft controlled by theproposed RNDI can track the reference trajectory
5 Conclusion
In this paper the robust nonlinear dynamic inversion controlstrategy that improves the robustness of NDI has been pro-posed NDI is a nonlinear control technique that eliminatesthe need of linearizing and designing gain-scheduled con-trollers for each equilibrium point by canceling the originaldynamics Because NDI removes the system nonlinearitiesdirectly it provides an opportunity to overcome the difficul-ties of ensuring stabilities and performances between variousoperational points while gain-scheduling However the maindrawback of designing NDI controller is the weak robustness
10 Mathematical Problems in Engineering
0 10 20 30 40 50 60 70 80 90 100
0
5
10
Time (m)
NDIRNDI
e r-d
irect
ion
(m)
minus5
minus15
minus10
(a) Position tracking error in e119903-direction
0
2
0 10 20 30 40 50 60 70 80 90 100Time (s)
NDIRNDI
e 120579-d
irect
ion
(m)
minus6
minus2
minus4
minus8
minus10
(b) Position tracking error in e120579-direction
0 10 20 30 40 50 60 70 80 90 100
0
5
10
Time (s)
NDIRNDI
e h-d
irect
ion
(m)
minus20
minus15
minus10
minus5
(c) Position tracking error in eℎ-direction
Figure 9 Simulation results of the position tracking error of the spacecrafts controlled by NDI and RNDIThe black thin line represents thetrajectories of the spacecraft controlled by NDI while the blue thick line represents the trajectories of the spacecraft controlled by RNDI
20 30 40 50 60 70 80 90 100
0
100
200
300
400
Time (s)
Inpu
t for
ces (
N)
minus200
minus100
ux along with er-axis
uz along with eh-axis
uy along with e120579-axis
(a) Control inputs generated by NDI
20 30 40 50 60 70 80 90 100
0
100
200
300
400
Time (s)
Inpu
t for
ces (
N)
minus200
minus100
ux along with er-axis
uz along with eh-axis
uy along with e120579-axis
(b) Control inputs generated by RNDI
Figure 10 Control inputs generated by NDI and RNDI
due to difficulties in obtaining accurate information of thenonlinear system dynamics In contrast to NDI SMC isconsidered a robust nonlinear control technique that usesswitching values as inputs to force the system states intoa sliding surface and to maintain the states on it Theproposed RNDI controller is designed by combining the
conventional NDI controller and switching inputs analyzedin SMC Hence RNDI takes the advantages of both NDI andSMC such that it
(i) is easy to design and implement(ii) eliminates the need of gain-scheduling
Mathematical Problems in Engineering 11
0 50 100 150050100150
0
10
e120579 axis er axis
e hax
is
minus50minus50minus100minus100 minus150minus150
minus10
minus20
Figure 11 Trajectory of the follower spacecraft controlled by theproposed RNDI controller
(iii) is easy to achieve the system required performancedue to the similar concept to classical control meth-ods and
(iv) is inherently robust
However the drawback of SMC such as the chatteringproblem still remains as a problem of RNDI control It isexpected that various chattering-free methods can be usedto reduce the effect of chattering phenomenon for examplereplacing the sign function into the saturation function
Moreover the equivalence of NDI input and equivalentinput designed by the identity sliding surface is also analyzedin this paper The reduced order dynamics on the slidingsurface can be converted to the user-selected desired dynam-ics As a result the difficulties in designing a sliding modecontroller for satisfying the specific system performances areconverted to a designing problem of the desired dynamicsHence RNDI provides an opportunity to solve the difficultiesin selecting the sliding surface of SMC The effectiveness ofthe proposed RNDI controller is confirmed by a set of simu-lations with application to the spacecraft formation flying
Nomenclature
G Universal gravity constantM Mass of the Earth119898119891 Mass of the follower spacecraft
119898119897 Mass of the leader spacecraft
r119891 Distance from the center of the Earth to
the center of the follower spacecraftr119897 Distance from the center of the Earth to
the center of the leader spacecraftxdes Desired dynamics
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research was supported by the MSIP (Ministry ofScience ICT amp Future Planning) Korea under the C-ITRC(Convergence Information Technology Research Center)
support program (NIPA-2014-H0401-14-1004) supervised bythe NIPA (National IT Industry Promotion Agency)
References
[1] Y Lv Q Hu G Ma and J Zhou ldquo6 DOF synchronized con-trol for spacecraft formation flying with input constraint andparameter uncertaintiesrdquo ISA Transactions vol 50 no 4 pp573ndash580 2011
[2] D Forta F Bordi and C Scolese ldquoLow-cost minimum sizesatellites for demonstration of formation flying modes at smallkilometer-size distancerdquo in Proceedings of the 13th AIAAUSUConference on Small Satellite SSC99-VI-3 pp 1ndash15 1999
[3] M S De Queiroz V Kapila and Q Yan ldquoAdaptive nonlinearcontrol ofmultiple spacecraft formation flyingrdquo Journal of Guid-ance Control and Dynamics vol 23 no 3 pp 385ndash390 2000
[4] J Georgie and J Valasek ldquoEvaluation of longitudinal desireddynamics for dynamic-inversion controlled generic reentryvehiclesrdquo Journal of Guidance Control and Dynamics vol 26no 5 pp 811ndash819 2003
[5] D Ito J Georgie J Valasek and D T Ward Reentry VehicleFlight Controls Design Guidelines Dynamic Inversion NASATPmdash2002ndash210771 2002
[6] A J Ostroff and B J Bacon ldquoForce and moment approachfor achievable dynamics using nonlinear dynamic inversionrdquo inProceedings of theAIAAGuidance Navigation andControl Con-ference and Exhibit pp 424ndash434 Reston Va USA 1999
[7] D Enns D Bugajski R Hendrick and G Stein ldquoDynamicinversion an evolving methodology for flight control designrdquoInternational Journal of Control vol 59 no 1 pp 71ndash91 1994
[8] J A Richard J M Buffington A G Sparks and S S BandaRobust Multivariable Flight Control Springer New York NYUSA 1994
[9] R J Adams and S S Banda ldquoRobust flight control design usingdynamic inversion and structured singular value synthesisrdquoIEEE Transactions on Control Systems Technology vol 1 no 2pp 80ndash92 1993
[10] M B McFarland and S M Hoque ldquoRobustness of a nonlinearmissile autopilot designed using dynamic inversionrdquo in Pro-ceedings of the AIAA Guidance Navigation and Control Confer-ence and Exhibit AIAA-2000-3970 Denver Colo USA August2000
[11] G Looye and H-D Joos ldquoDesign of robust dynamic inversioncontrol laws usingmulti-objective optimizationrdquo in Proceedingsof the AIAA Guidance Navigation and Control Conference andExhibit AIAA-2001-4285 2001
[12] P B Acquatella W Falkena E van Kampen and Q P ChuldquoRobust nonlinear spacecraft attitude control using incrementalnonlinear dynamic inversionrdquo in Proceedings of the AIAAGuid-ance Navigation and Control Conference and Exhibit Min-neapolis Minn USA August 2012
[13] P K Menon V R Iragavarapu and E J Ohlmeyer ldquoNonlinearMissile Autopilot Design using Time-Scale Separationrdquo in Pro-ceedings of the AIAA Guidance Navigation and Control Confer-ence and Exhibit AIAA-1997-3765 1997
[14] A Steinicke and G Michalka ldquoImproving transient perfor-mance of dynamic inversion missile autopilot by use of back-steppingrdquo in Proceedings of the AIAAGuidance Navigation andControl Conference and Exhibit AIAA-2002-4658 MontereyCalif USA August 2002
12 Mathematical Problems in Engineering
[15] I Yang D Kim and D Lee ldquoA flight control strategy usingrobust dynamic inversion based on sliding mode controlrdquo inproceeding of the AIAA Guidance Navigation and ControlConference August 2012
[16] H Sira-Ramirez ldquoDifferential geometric methods in variable-structure controlrdquo International Journal of Control vol 48 no4 pp 1359ndash1390 1988
[17] V I Utkin ldquoVariable structure systems with sliding modesrdquoIEEE Transactions on Automatic Control vol 22 no 2 pp 212ndash222 1977
[18] R A DeCarlo S H Zak and G P Matthews ldquoVariable struc-ture control of nonlinear multivariable systems a tutorialrdquo Pro-ceedings of the IEEE vol 76 no 3 pp 212ndash232 1988
[19] J Y HungWGao and J CHung ldquoVariable structure control asurveyrdquo IEEE Transactions on Industrial Electronics vol 40 no1 pp 2ndash22 1993
[20] G Papageorgiou andM Polansky ldquoTuning a dynamic inversionpitch axis autopilot using McFarlane-Glover loop shapingrdquoOptimal Control Applications ampMethods vol 30 no 3 pp 287ndash308 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Mathematical Problems in Engineering
0 10 20 30 40 50 60 70 80 90 100
0
5
10
Time (m)
NDIRNDI
e r-d
irect
ion
(m)
minus5
minus15
minus10
(a) Position tracking error in e119903-direction
0
2
0 10 20 30 40 50 60 70 80 90 100Time (s)
NDIRNDI
e 120579-d
irect
ion
(m)
minus6
minus2
minus4
minus8
minus10
(b) Position tracking error in e120579-direction
0 10 20 30 40 50 60 70 80 90 100
0
5
10
Time (s)
NDIRNDI
e h-d
irect
ion
(m)
minus20
minus15
minus10
minus5
(c) Position tracking error in eℎ-direction
Figure 9 Simulation results of the position tracking error of the spacecrafts controlled by NDI and RNDIThe black thin line represents thetrajectories of the spacecraft controlled by NDI while the blue thick line represents the trajectories of the spacecraft controlled by RNDI
20 30 40 50 60 70 80 90 100
0
100
200
300
400
Time (s)
Inpu
t for
ces (
N)
minus200
minus100
ux along with er-axis
uz along with eh-axis
uy along with e120579-axis
(a) Control inputs generated by NDI
20 30 40 50 60 70 80 90 100
0
100
200
300
400
Time (s)
Inpu
t for
ces (
N)
minus200
minus100
ux along with er-axis
uz along with eh-axis
uy along with e120579-axis
(b) Control inputs generated by RNDI
Figure 10 Control inputs generated by NDI and RNDI
due to difficulties in obtaining accurate information of thenonlinear system dynamics In contrast to NDI SMC isconsidered a robust nonlinear control technique that usesswitching values as inputs to force the system states intoa sliding surface and to maintain the states on it Theproposed RNDI controller is designed by combining the
conventional NDI controller and switching inputs analyzedin SMC Hence RNDI takes the advantages of both NDI andSMC such that it
(i) is easy to design and implement(ii) eliminates the need of gain-scheduling
Mathematical Problems in Engineering 11
0 50 100 150050100150
0
10
e120579 axis er axis
e hax
is
minus50minus50minus100minus100 minus150minus150
minus10
minus20
Figure 11 Trajectory of the follower spacecraft controlled by theproposed RNDI controller
(iii) is easy to achieve the system required performancedue to the similar concept to classical control meth-ods and
(iv) is inherently robust
However the drawback of SMC such as the chatteringproblem still remains as a problem of RNDI control It isexpected that various chattering-free methods can be usedto reduce the effect of chattering phenomenon for examplereplacing the sign function into the saturation function
Moreover the equivalence of NDI input and equivalentinput designed by the identity sliding surface is also analyzedin this paper The reduced order dynamics on the slidingsurface can be converted to the user-selected desired dynam-ics As a result the difficulties in designing a sliding modecontroller for satisfying the specific system performances areconverted to a designing problem of the desired dynamicsHence RNDI provides an opportunity to solve the difficultiesin selecting the sliding surface of SMC The effectiveness ofthe proposed RNDI controller is confirmed by a set of simu-lations with application to the spacecraft formation flying
Nomenclature
G Universal gravity constantM Mass of the Earth119898119891 Mass of the follower spacecraft
119898119897 Mass of the leader spacecraft
r119891 Distance from the center of the Earth to
the center of the follower spacecraftr119897 Distance from the center of the Earth to
the center of the leader spacecraftxdes Desired dynamics
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research was supported by the MSIP (Ministry ofScience ICT amp Future Planning) Korea under the C-ITRC(Convergence Information Technology Research Center)
support program (NIPA-2014-H0401-14-1004) supervised bythe NIPA (National IT Industry Promotion Agency)
References
[1] Y Lv Q Hu G Ma and J Zhou ldquo6 DOF synchronized con-trol for spacecraft formation flying with input constraint andparameter uncertaintiesrdquo ISA Transactions vol 50 no 4 pp573ndash580 2011
[2] D Forta F Bordi and C Scolese ldquoLow-cost minimum sizesatellites for demonstration of formation flying modes at smallkilometer-size distancerdquo in Proceedings of the 13th AIAAUSUConference on Small Satellite SSC99-VI-3 pp 1ndash15 1999
[3] M S De Queiroz V Kapila and Q Yan ldquoAdaptive nonlinearcontrol ofmultiple spacecraft formation flyingrdquo Journal of Guid-ance Control and Dynamics vol 23 no 3 pp 385ndash390 2000
[4] J Georgie and J Valasek ldquoEvaluation of longitudinal desireddynamics for dynamic-inversion controlled generic reentryvehiclesrdquo Journal of Guidance Control and Dynamics vol 26no 5 pp 811ndash819 2003
[5] D Ito J Georgie J Valasek and D T Ward Reentry VehicleFlight Controls Design Guidelines Dynamic Inversion NASATPmdash2002ndash210771 2002
[6] A J Ostroff and B J Bacon ldquoForce and moment approachfor achievable dynamics using nonlinear dynamic inversionrdquo inProceedings of theAIAAGuidance Navigation andControl Con-ference and Exhibit pp 424ndash434 Reston Va USA 1999
[7] D Enns D Bugajski R Hendrick and G Stein ldquoDynamicinversion an evolving methodology for flight control designrdquoInternational Journal of Control vol 59 no 1 pp 71ndash91 1994
[8] J A Richard J M Buffington A G Sparks and S S BandaRobust Multivariable Flight Control Springer New York NYUSA 1994
[9] R J Adams and S S Banda ldquoRobust flight control design usingdynamic inversion and structured singular value synthesisrdquoIEEE Transactions on Control Systems Technology vol 1 no 2pp 80ndash92 1993
[10] M B McFarland and S M Hoque ldquoRobustness of a nonlinearmissile autopilot designed using dynamic inversionrdquo in Pro-ceedings of the AIAA Guidance Navigation and Control Confer-ence and Exhibit AIAA-2000-3970 Denver Colo USA August2000
[11] G Looye and H-D Joos ldquoDesign of robust dynamic inversioncontrol laws usingmulti-objective optimizationrdquo in Proceedingsof the AIAA Guidance Navigation and Control Conference andExhibit AIAA-2001-4285 2001
[12] P B Acquatella W Falkena E van Kampen and Q P ChuldquoRobust nonlinear spacecraft attitude control using incrementalnonlinear dynamic inversionrdquo in Proceedings of the AIAAGuid-ance Navigation and Control Conference and Exhibit Min-neapolis Minn USA August 2012
[13] P K Menon V R Iragavarapu and E J Ohlmeyer ldquoNonlinearMissile Autopilot Design using Time-Scale Separationrdquo in Pro-ceedings of the AIAA Guidance Navigation and Control Confer-ence and Exhibit AIAA-1997-3765 1997
[14] A Steinicke and G Michalka ldquoImproving transient perfor-mance of dynamic inversion missile autopilot by use of back-steppingrdquo in Proceedings of the AIAAGuidance Navigation andControl Conference and Exhibit AIAA-2002-4658 MontereyCalif USA August 2002
12 Mathematical Problems in Engineering
[15] I Yang D Kim and D Lee ldquoA flight control strategy usingrobust dynamic inversion based on sliding mode controlrdquo inproceeding of the AIAA Guidance Navigation and ControlConference August 2012
[16] H Sira-Ramirez ldquoDifferential geometric methods in variable-structure controlrdquo International Journal of Control vol 48 no4 pp 1359ndash1390 1988
[17] V I Utkin ldquoVariable structure systems with sliding modesrdquoIEEE Transactions on Automatic Control vol 22 no 2 pp 212ndash222 1977
[18] R A DeCarlo S H Zak and G P Matthews ldquoVariable struc-ture control of nonlinear multivariable systems a tutorialrdquo Pro-ceedings of the IEEE vol 76 no 3 pp 212ndash232 1988
[19] J Y HungWGao and J CHung ldquoVariable structure control asurveyrdquo IEEE Transactions on Industrial Electronics vol 40 no1 pp 2ndash22 1993
[20] G Papageorgiou andM Polansky ldquoTuning a dynamic inversionpitch axis autopilot using McFarlane-Glover loop shapingrdquoOptimal Control Applications ampMethods vol 30 no 3 pp 287ndash308 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 11
0 50 100 150050100150
0
10
e120579 axis er axis
e hax
is
minus50minus50minus100minus100 minus150minus150
minus10
minus20
Figure 11 Trajectory of the follower spacecraft controlled by theproposed RNDI controller
(iii) is easy to achieve the system required performancedue to the similar concept to classical control meth-ods and
(iv) is inherently robust
However the drawback of SMC such as the chatteringproblem still remains as a problem of RNDI control It isexpected that various chattering-free methods can be usedto reduce the effect of chattering phenomenon for examplereplacing the sign function into the saturation function
Moreover the equivalence of NDI input and equivalentinput designed by the identity sliding surface is also analyzedin this paper The reduced order dynamics on the slidingsurface can be converted to the user-selected desired dynam-ics As a result the difficulties in designing a sliding modecontroller for satisfying the specific system performances areconverted to a designing problem of the desired dynamicsHence RNDI provides an opportunity to solve the difficultiesin selecting the sliding surface of SMC The effectiveness ofthe proposed RNDI controller is confirmed by a set of simu-lations with application to the spacecraft formation flying
Nomenclature
G Universal gravity constantM Mass of the Earth119898119891 Mass of the follower spacecraft
119898119897 Mass of the leader spacecraft
r119891 Distance from the center of the Earth to
the center of the follower spacecraftr119897 Distance from the center of the Earth to
the center of the leader spacecraftxdes Desired dynamics
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research was supported by the MSIP (Ministry ofScience ICT amp Future Planning) Korea under the C-ITRC(Convergence Information Technology Research Center)
support program (NIPA-2014-H0401-14-1004) supervised bythe NIPA (National IT Industry Promotion Agency)
References
[1] Y Lv Q Hu G Ma and J Zhou ldquo6 DOF synchronized con-trol for spacecraft formation flying with input constraint andparameter uncertaintiesrdquo ISA Transactions vol 50 no 4 pp573ndash580 2011
[2] D Forta F Bordi and C Scolese ldquoLow-cost minimum sizesatellites for demonstration of formation flying modes at smallkilometer-size distancerdquo in Proceedings of the 13th AIAAUSUConference on Small Satellite SSC99-VI-3 pp 1ndash15 1999
[3] M S De Queiroz V Kapila and Q Yan ldquoAdaptive nonlinearcontrol ofmultiple spacecraft formation flyingrdquo Journal of Guid-ance Control and Dynamics vol 23 no 3 pp 385ndash390 2000
[4] J Georgie and J Valasek ldquoEvaluation of longitudinal desireddynamics for dynamic-inversion controlled generic reentryvehiclesrdquo Journal of Guidance Control and Dynamics vol 26no 5 pp 811ndash819 2003
[5] D Ito J Georgie J Valasek and D T Ward Reentry VehicleFlight Controls Design Guidelines Dynamic Inversion NASATPmdash2002ndash210771 2002
[6] A J Ostroff and B J Bacon ldquoForce and moment approachfor achievable dynamics using nonlinear dynamic inversionrdquo inProceedings of theAIAAGuidance Navigation andControl Con-ference and Exhibit pp 424ndash434 Reston Va USA 1999
[7] D Enns D Bugajski R Hendrick and G Stein ldquoDynamicinversion an evolving methodology for flight control designrdquoInternational Journal of Control vol 59 no 1 pp 71ndash91 1994
[8] J A Richard J M Buffington A G Sparks and S S BandaRobust Multivariable Flight Control Springer New York NYUSA 1994
[9] R J Adams and S S Banda ldquoRobust flight control design usingdynamic inversion and structured singular value synthesisrdquoIEEE Transactions on Control Systems Technology vol 1 no 2pp 80ndash92 1993
[10] M B McFarland and S M Hoque ldquoRobustness of a nonlinearmissile autopilot designed using dynamic inversionrdquo in Pro-ceedings of the AIAA Guidance Navigation and Control Confer-ence and Exhibit AIAA-2000-3970 Denver Colo USA August2000
[11] G Looye and H-D Joos ldquoDesign of robust dynamic inversioncontrol laws usingmulti-objective optimizationrdquo in Proceedingsof the AIAA Guidance Navigation and Control Conference andExhibit AIAA-2001-4285 2001
[12] P B Acquatella W Falkena E van Kampen and Q P ChuldquoRobust nonlinear spacecraft attitude control using incrementalnonlinear dynamic inversionrdquo in Proceedings of the AIAAGuid-ance Navigation and Control Conference and Exhibit Min-neapolis Minn USA August 2012
[13] P K Menon V R Iragavarapu and E J Ohlmeyer ldquoNonlinearMissile Autopilot Design using Time-Scale Separationrdquo in Pro-ceedings of the AIAA Guidance Navigation and Control Confer-ence and Exhibit AIAA-1997-3765 1997
[14] A Steinicke and G Michalka ldquoImproving transient perfor-mance of dynamic inversion missile autopilot by use of back-steppingrdquo in Proceedings of the AIAAGuidance Navigation andControl Conference and Exhibit AIAA-2002-4658 MontereyCalif USA August 2002
12 Mathematical Problems in Engineering
[15] I Yang D Kim and D Lee ldquoA flight control strategy usingrobust dynamic inversion based on sliding mode controlrdquo inproceeding of the AIAA Guidance Navigation and ControlConference August 2012
[16] H Sira-Ramirez ldquoDifferential geometric methods in variable-structure controlrdquo International Journal of Control vol 48 no4 pp 1359ndash1390 1988
[17] V I Utkin ldquoVariable structure systems with sliding modesrdquoIEEE Transactions on Automatic Control vol 22 no 2 pp 212ndash222 1977
[18] R A DeCarlo S H Zak and G P Matthews ldquoVariable struc-ture control of nonlinear multivariable systems a tutorialrdquo Pro-ceedings of the IEEE vol 76 no 3 pp 212ndash232 1988
[19] J Y HungWGao and J CHung ldquoVariable structure control asurveyrdquo IEEE Transactions on Industrial Electronics vol 40 no1 pp 2ndash22 1993
[20] G Papageorgiou andM Polansky ldquoTuning a dynamic inversionpitch axis autopilot using McFarlane-Glover loop shapingrdquoOptimal Control Applications ampMethods vol 30 no 3 pp 287ndash308 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
12 Mathematical Problems in Engineering
[15] I Yang D Kim and D Lee ldquoA flight control strategy usingrobust dynamic inversion based on sliding mode controlrdquo inproceeding of the AIAA Guidance Navigation and ControlConference August 2012
[16] H Sira-Ramirez ldquoDifferential geometric methods in variable-structure controlrdquo International Journal of Control vol 48 no4 pp 1359ndash1390 1988
[17] V I Utkin ldquoVariable structure systems with sliding modesrdquoIEEE Transactions on Automatic Control vol 22 no 2 pp 212ndash222 1977
[18] R A DeCarlo S H Zak and G P Matthews ldquoVariable struc-ture control of nonlinear multivariable systems a tutorialrdquo Pro-ceedings of the IEEE vol 76 no 3 pp 212ndash232 1988
[19] J Y HungWGao and J CHung ldquoVariable structure control asurveyrdquo IEEE Transactions on Industrial Electronics vol 40 no1 pp 2ndash22 1993
[20] G Papageorgiou andM Polansky ldquoTuning a dynamic inversionpitch axis autopilot using McFarlane-Glover loop shapingrdquoOptimal Control Applications ampMethods vol 30 no 3 pp 287ndash308 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of