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Sliding-mode control design for a class of systems withnon-matching nonlinearities and disturbancesBill M. Diong aa Department of Electrical and Computer Engineering , The University of Texas at EI Paso ,EI Paso, Texas 79968, USA E-mail:Published online: 23 Feb 2007.

To cite this article: Bill M. Diong (2004) Sliding-mode control design for a class of systems with non-matching nonlinearitiesand disturbances, International Journal of Systems Science, 35:8, 445-455, DOI: 10.1080/00207720410001723671

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International Journal of Systems Sciencevolume 35, number 8, 10 July 2004, pages 445–455

Sliding-mode control design for a class of systems

with non-matching nonlinearities and disturbances

BILL M. DIONG{

A novel sliding-mode control design is proposed for the class of single-input nonlinearsystems with non-matching nonlinearities and/or disturbances. This class of systemsincludes dual-spin spacecraft and rotational proof mass actuators that are used forsuppressing translational oscillation. The sliding-mode control law is designed via anL2-norm bounding technique that ensures, under certain conditions, asymptotic stabilityof the system when the disturbances satisfy the matching condition or are zero; other-wise, it ensures system stability with an a priori bound on the transmission gain of thedisturbances through the system for non-zero non-matching disturbances. Simulationresults for a rotational proof mass actuator or dual-spin spacecraft example illustratethe excellent responses obtained by applying the control laws derived from this proposeddesign procedure.

1. Introduction

Most real-world systems exhibit nonlinear dynamicbehaviour, have only a limited amount of controleffort available to them and may be subject to externaldisturbances. In particular, this is true for dual-spinspacecraft (Kinsey et al. 1992) and for rotational proofmass actuators for suppressing translational oscilla-tion (Bupp et al. 1994, Wan et al. 1996, Bupp andBernstein 1998), both of which can be modelled as anoscillating eccentric rotor system. Such a model exem-plifies the class of single-input nonlinear systems withnon-matching nonlinearities and/or disturbances.In this paper, we propose a new sliding-mode-based

approach to achieving asymptotic stability (when dis-turbances are absent) and a guaranteed level of distur-bance rejection (when present) for this particular classof dynamic systems. Sliding-mode control systems weresome of the first systems designed with the specific objec-tive of reducing system sensitivity to external distur-bances and internal parameter uncertainties. In suchsystems, the state trajectories are forced on to aninvariant manifold in finite time by a control law that

essentially overpowers the inherent system dynamicseither locally in a neighbourhood around the mani-fold or globally, with the ensuring result that the systemdynamics become disengaged from those disturbancesand parameter uncertainties that satisfy the matchingcondition (Drazenovic 1969). The theory of sliding-mode control systems (see DeCarlo et al. 1988, fora tutorial) has been applied successfully to treat suchproblems as adaptive regulation, trajectory trackingand model reference following. Modern industrialapplications that rely on sliding-mode control includepower electronic converters and electric motor drives(Utkin 1993).

However, sliding-mode control systems do have short-comings. Greater stress and wear and tear of equipmentoccurs as a result of the prescribed switching controlaction. In addition, because practical systems do notswitch infinitely quickly, a phenomenon known aschattering occurs during the sliding mode of operation,which may excite the unmodelled high-frequencydynamics of the controlled system. Since sliding-modecontrol systems are otherwise attractive because of theadvantages mentioned above, substantial research hasbeen performed to yield several methods to alleviatethis shortcoming, mainly by using approximation andfiltering techniques. Another drawback of sliding-modecontrol systems is that, in the presence of non-matchingdisturbances and parameter uncertainties, they donot become disengaged from those disturbances and

Received 12 November 2002. Revised 29 March 2004. Accepted 14

May 2004.

{Department of Electrical and Computer Engineering, The

University of Texas at EI Paso, EI Paso, Texas 79968, USA. e-mail:

bdiong@ece.utep.edu

International Journal of Systems Science ISSN 0020–7721 print/ISSN 1464–5319 online � 2004 Taylor & Francis Ltdhttp://www.tandf.co.uk/journals

DOI: 10.1080/00207720410001723671

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parameter uncertainties while operating on the invariant(sliding) manifold. Hence sliding-mode control designfor such systems needs to address specifically the effectof these non-matching items on system stability andperformance, which is an area of ongoing research. Asan example, for the case of linear systems affected bynon-matching disturbances and parameter uncertainties,Doing and Medanic (1997) have described results onstability and disturbance rejection for sliding mode con-trol of such systems based on the application of H1-norm bounding concepts. The robustness of sliding-mode control of nonlinear systems has also been atopic of recent growing research interest. Examples ofsuch investigations include those by Zinober and Liu(1996) and Lu and Spurgeon (1999) addressing non-linearities that satisfy various assumptions, such asthe matching condition. In addition, Edwards andSpurgeon (1998) presented a result on robustness analy-sis, which can be performed after a controller has beendesigned, for a system with non-matching nonlinearitiesalthough it did not provide any results on the directsynthesis of sliding-mode controllers for such systems.This present paper’s contribution is a novel sliding-

mode control design for the class of single-input non-linear systems with non-matching nonlinearities and/ordisturbances, as exemplified by the oscillating eccent-ric rotor model. The available knowledge about theunmatched nonlinearities allows those nonlinearitiesto be bounded only in certain directions (projections)of the state rather than uniformly in every direction,for ‘easier’ synthesis of a stabilizing controller. The pro-posed design procedure yields a sliding manifold thatensures asymptotic stability of the system when the dis-turbances satisfy the matching condition or are zero;otherwise, it ensures system stability with an a prioribound on the transmission gain of the disturbancesthrough the system for non-zero non-matching distur-bances, as the effects of those disturbances can onlybe limited but not eliminated. The results presented inSection 2 thus extend to the nonlinear systems case,those derived by Diong and Medanic (1997). Finally,simulation results for the oscillating eccentric rotormodel with a sliding mode controller designed usingthe proposed procedure are described in Section 3 todemonstrate its efficacy for system stabilization anddisturbance rejection.

2. Sliding-mode control design

The design of a sliding-mode control system is typi-cally performed in two stages. The first stage consistsof designing a manifold described by �(x)¼ 0 so thatthe dynamics of the system, when it has been restrictedto operating (in a sliding mode) on that manifold, pos-

sess some desired stability and performance characteris-tics; this is the sliding-phase design. For the secondstage, the form and parameters of the control law thatwill force the state trajectories of the closed-loopsystem to reach and slide along the surface �(x)¼ 0are chosen; this is the reaching-phase design. Both slid-ing-phase design and reaching-phase design can beaccomplished in several different ways, depending onthe given system’s characteristics and its control require-ments, including those described by DeCarlo et al.(1988), Diong and Medanic (1992, 1997) and Baidaand Izosimov (1985).

One particular approach to sliding-phase designfor the control of linear systems with non-matchingdisturbances and parameter uncertainties, as recentlydescribed by Diong and Medanic (1997), is based onthe application of H1-norm bounding concepts. Thedesign yields a sliding manifold that ensures asymptoticstability of the (undisturbed) system and an a prioribound on the transmission gain of any non-zero non-matching disturbances through the system.

This present paper proposes a related approach,based on L2-norm bounding, for the control of single-input nonlinear systems described by

_xx ¼ AþXli¼1

GiLiðxÞHi

!xþ Buþ Gwþ B½DAðxÞ

þ DBðxÞuþ DGðxÞw�, ð1 aÞ

y ¼ Hx, ð1 bÞ

where x 2 Rn and u 2 R are the state vector and control

respectively of the system, w 2 R is a disturbancebelonging to both L2e and L1, and y 2 R

q is theoutput vector to be regulated. We assume that (A,B)is a controllable pair and that G, H, Gi, Hi, i¼ 1, . . . , l,are known constant matrices, while the Li(x),i¼ 1, . . . , l, are continuously differentiable nonlinearfunctions of the state. In addition, DAðxÞ : R

n! R,

DBðxÞ : Rn! R and DGðxÞ : R

n! R are also nonlinear

functions of the state such that DBðxÞ and DGðxÞare bounded for all x, with jDBðxÞj < �B < 1 andjDGðxÞj < �G.

This particular system model captures the structuralinformation about the ‘location’ of each nonlinearityLi(x), which is employed to achieve a less conservativecontrol design. The BDAðxÞ, BDBðxÞu and BDGðxÞwterms in the model represent those system nonlinearitiesand disturbances that are in the range space of the con-trol matrix B; that is, they satisfy the matching conditionand so can either be cancelled or ‘overpowered’ directlyby an appropriate u. On the other hand, matrices G andGi, i¼ 1, . . . , l, do not satisfy the matching condition.Boundedness of DBðxÞ and DGðxÞ is assumed because

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a bounded control law will be proposed for this system.In particular, the bound |DBðxÞ|<1 is needed to avoidsign changes of the control matrix B[1þDBðxÞ] as xchanges.One system that has a model fitting the above descrip-

tion is the dual-spin spacecraft. Other examples includemechanical systems that exhibit some form of rotationalmotion, such as the rotational proof mass actuatorsystem for suppressing translational oscillation (Buppet al. 1994, Wan et al. 1996, Bupp and Bernstein 1998)as well as the flexible-joint manipulators described byAstorga (1992).

2.1. Sliding-phase design

Suppose that the desired sliding manifold is to bea linear function of the state and so described by�(x)¼Sx¼ 0. Selecting full rank N 2 R

ðn�1Þ�n such that

NB ¼ 0, ð2Þ

and assuming that S 2 R1�n satisfies, without loss

of design generality (Diong and Medanic 1992), thecondition

SB ¼ 1 ð3Þ

allows all permissible S to be parametrized byK 2 R

1�ðn�1Þ as

S ¼ B# þ KN, ð4Þ

where B# is the Moore–Penrose inverse of B. Then, if weperform a change in state variables on (1) to ðxTr �TÞ

T

with

xr�

� �¼

NS

� �x, x ¼ ðN# � BK B Þ

xr�

� �, ð5Þ

and since � ¼ _�� ¼ 0 when the system has been restrictedto the sliding manifold, this yields the sliding modedynamic equations as

_xxr ¼ As � BsK þXli¼1

NGiLiðxr,0ÞHiðN# � BKÞ

!xr

þNGw,

y ¼ HðN# � BKÞxr, ð6Þ

with

As ¼ NAN#, Bs ¼ NAB: ð7Þ

It is clear from (6) that, as long as the systemmaintains the sliding mode of operation, those lastthree terms in (1a) satisfying the matching conditionhave no effect on the system’s dynamics. On the otherhand, since G and Gi, i¼ 1, . . . , l, do not satisfythe matching condition, the nonlinearities and distur-bances associated with them continue to have an(usually negative) effect on the system dynamics duringthe sliding mode of operation. So the problem nowbecomes one of how to choose the matrix K in orderto guarantee asymptotic stability and disturbance rejec-tion during the sliding mode for the nonlinear systemdescribed by (6).

Let us consider the following result.

Theorem 1: For a system described by (6), define

Gþ ¼ ðG G1 . . .GlÞ, HTþ ¼ ðHT HT

1 . . .HTl Þ, ð8Þ

E ¼ I þ BTHTþHþB, ð9Þ

~AAs ¼ As � BsE�1BTHT

þHþN#: ð10Þ

Given some value of the real number �>0, if there existsa symmetric positive semidefinite solution P 2 R

ðn�1Þxðn�1Þ

to the equation

~AATs Pþ P ~AAs � PBsE

�1BTs Pþ

1

�2PNGþG

TþN

TP

þN#THTþðI �HþBE

�1BTHTþÞHþN

# ¼ 0 ð11Þ

and there is a domain D � Rn�1 on the sliding manifold

containing the origin xr¼ 0 such that

LTi ðxr,0ÞLiðxr,0Þ4

1

�2I for i ¼ 1, . . . , l,

and for all xr in D, ð12Þ

then for K¼BsTP,

(i) the origin of the system described by (6), withw¼ 0, is asymptotically stable

(ii) the system described by (6) is small-signalL1 input–output stable and

(iii) also small-signal finite-gain L2 stable withÐ �00y

TðtÞ�yðtÞ dt4 �2

Ð �00w

TðtÞwðtÞ dtþ xTr ð00ÞPxrð0

0Þ for theinitial sliding-mode condition xr(0

0) in some neigh-bourhood of the origin contained within D.

Proof: Suppose, for a particular value of �>0,that (11) has a symmetric positive semidefinitesolution P. Substituting (10) into (11), then expanding

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and collecting terms yield

AcðxrÞTPþPAcðxrÞþ

1

�2PNGGTNTP

þðN#�BKÞTHTHðN#�BKÞ

¼�PBsBTs P

�Xli¼1

ðN#�BKÞTHT

i ½I � �2LTi ðxrÞLiðxrÞ�HiðN

#�BKÞ

�Xli¼1

1

�PNGi � �ðN#�BKÞ

THTi L

Ti ðxrÞ

� �

�1

�GT

i NTP� �LiðxrÞHiðN

#�BKÞ

� �, ð13Þ

where AcðxrÞ¼As�BsKþPl

i¼1NGiLiðxr,0ÞHi ðN#�BKÞ

and K¼BTs P.

Now, denoting the left-hand side of (13) as R(P)for simplicity, note that R(P)4 0 for those xr in thedomain D defined by condition (12). Furthermore,(1/�2)PNGGTNTPþ (N#

�BK)THTH(N#�BK)5 0,

which implies that

AxðxrÞTPþ PAcðxrÞ40 for all xr in D: ð14Þ

Clearly, the origin xr¼ 0 is an equilibrium point for thesliding-mode dynamics (6) when w¼ 0. Then LaSalle’stheorem allows us to conclude that _xxr ¼ AcðxrÞxr, withK ¼ BT

s P, is asymptotically stable with respect to xr¼ 0.In addition, it follows easily from a well-known result

(Khalil 1996) that asymptotic stability of the systemdescribed by (6) implies that it is also small-signal L1

input–output stable.To show that a uniform L2-norm bound on the trans-

mission gain between disturbance w and output y is alsoobtained, observe that R(P)4 0 implies that

xTr AcðxrÞTPxr � xTr PAcðxrÞxr þ

1

�2xTr PNGGTNTPxr

þ xTr ðN# � BKÞ

THTHðN# � BKÞxr40: ð15Þ

Replacing Ac(xr)xr by _xxr �NGw, we obtain

d

dtðxTr PxrÞ þ

1

�GT

i NTPxr � �w

� �T1

�GT

i NTPxr � �w

� �

� �2wTwþ yTy4 0: ð16Þ

Assuming initial (at t¼ 00) sliding-mode conditions of

xr(00) and integrating yields

xTr ð�ÞPxrð�Þ � xTr ð00ÞPxrð0

0Þ þ

ð�00eTðtÞeðtÞdt

þ

ð�00yTðtÞyðtÞ dt4 �2

ð�00wTðtÞwðtÞ dt: ð17Þ

where e ¼ ð1=�ÞGTi N

TPxr � �w. Since the right-handside of the inequality is well defined and bounded forall w 2 L2e and for � 2 ½00,1Þ, it follows that all termson the left-hand side are well defined and bounded for� 2 ½00,1Þ as long as xr(t), t 2 ½00,��, remains within Deven in the presence of w. Consequently, this yields thethird result stated in the theorem. œ

This theorem thus characterizes a family of matricesK(�) that guarantees asymptotic stability of xr¼ 0 forthe sliding-mode dynamics and also a transmissiongain bound � with respect to all disturbances belongingto L2e, provided that each and every nonlinearityLi(xr, 0) is bounded according to (12) over a state-space domain encompassing the origin. This characteri-zation, in turn, defines a set of permissible slidingmanifolds in the state space for the control system thatis being designed.

Remark 1: The recommended procedure for obtaininga suitable � is to start with a (large) value of � that yieldsa solution P5 0 to (11), which is an algebraic Riccatiequation, then iteratively to decrease � and to solve(11) as long as the solution obtained remains positivesemidefinite. This defines a range of values for � suchthat Theorem 1 applies; however, the specific value of� ultimately chosen, which is usually not the minimumadmissible value, needs to take into account otherdesign objectives such as transient performance andcontrol effort. œ

2.2. Reaching-phase design

Consider next the design of the control u so that thesystem is forced into a sliding mode on the manifold�(x)¼Sx¼ (B#

þKN)x¼ 0 with K selected, for exam-ple, as proposed above. A sufficient condition for thisto occur is if the control results in

�ðxÞ _��ðxÞ < 0 ð18Þ

either locally in some neighbourhood around the desiredsliding surface or globally (DeCarlo et al. 1988). Asimple control law for achieving (18) in the single-input case is given by

u ¼ �� sgnð�Þ ¼ �� sgnðSxÞ, � > 0, ð19Þ

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where sgn(.) denotes the signum function, with theconstant � having to satisfy the condition

� >

���� 1

1þ DBðxÞ

�S

�Aþ

Xli¼1

GiLiðxÞHi

�x

þ DAðxÞ þ SGwþ DGðxÞw

����� ð20Þ

since (1 a) and (19) imply that

�ðxÞ _��ðxÞ ¼ � S AþXli¼1

GiLiðxÞHi

!xþ DAðxÞ þ SGw

"

þDGðxÞw� �½1þ DBðxÞ�sgnð�Þ

#

and noting that � sgn(�)¼ |�|. This particular choiceof u also has the advantage of placing an explicit con-straint on the control effort, although this generallyresults in only local and not global attractiveness ofthe sliding surface. Equation (20) clearly shows thatthe required size of the discontinuous control (19)depends on the L1 bound of the disturbance andon the desired size of the domain of attraction aboutthe sliding surface, as well as the bounds on DB(x)and DG(x).In practice, a discontinuous control law such as (19)

usually leads to an undesirable chattering of system vari-ables because of the non-zero time required for controlswitching. However, the chattering can be eliminated,which is of great practical importance, by the use ofa continuous approximation of the control law (19)instead. One such approximation is (Khalil 1996,Exercise 13.30)

u ¼ �� tanhð��Þ ¼ ��e�� � e���

e�� þ e���, �,� > 0: ð21Þ

Note that the hyperbolic tangent function is continu-ously differentiable (with respect to �), and as �!1,and tanh( � )! sgn( � ), so (21) tends to (19) in thelimit. It is being used here to illustrate that a contin-uous approximation to the discontinuous sliding-modecontrol law can alleviate undesirable chattering withoutincurring a significant loss of the performance achievedby the original designed control law. Other approaches tochattering attenuation, such as described by Edwardsand Spurgeon (1998) and Levant (2001), can also beconsidered.If DA(x) and DB(x) are fully known, then another

control law for the system described by (6) to achieve

a sliding model is

u ¼1

1þ DBðxÞ�S Aþ

Xli¼1

GiLiðxÞHi

!x� DAðxÞ

"

� � sgnðSxÞ

�, ð22Þ

with the constant � having to satisfy the condition

� > j½SGþ DGðxÞ�wj ð23Þ

in order to guarantee (18), since (1a) and (22) imply that

�ðxÞ _��ðxÞ ¼ �½SGw þ DGðxÞw� �sgnð�Þ�

and noting that �sgn(�)¼ |�|. Although (22) is also dis-continuous and does not have an a priori constraint likethe two previous control laws, it causes a lower level ofchattering than (19) since the size of the discontinuouspart of the control now depends only on the L1

bound of the disturbance and the bound on DG(x).Finally, the hyperbolic tangent function can again beused as a continuous approximation to the signumfunction in (22) to eliminate chattering altogether.

3. Oscillating eccentric rotor control design

and comparison

To demonstrate the effectiveness of the proposed con-trol approach, we consider the oscillating eccentric rotorsystem that models the dynamics both of an idealizedunbalanced dual-spin spacecraft (Kinsey et al. 1992)and also of a rotational proof mass actuator for sup-pressing translational oscillation (Bupp et al. 1994,Wan et al. 1996, Bupp and Bernstein 1998). As shownin figure 1, the model consists of a cart and a rotor.The cart, of mass M, is connected to a fixed wall by alinear spring of stiffness k and is constrained to have

Μ

m

k

FΙ

e

Τ

θ

Figure 1. Diagram of the oscillating eccentric rotor model.

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one-dimensional travel. Attached to the cart is theeccentric rotor, which has mass m and moment of inertiaI about its centre of mass located a distance e from thepoint about which the rotor rotates. The control torqueapplied to the rotor is denoted by T, while F is the dis-turbance force on the cart. Gravity is ignored since themotion occurs in a horizontal plane.It is supposed that the controller to be designed for

this system has to ensure that

(a) the closed-loop system is stable,

(b) the closed-loop system exhibits good settlingresponse for a class of initial conditions,

(c) the closed-loop system exhibits good disturbancerejection compared to the uncontrolled system fora sinusoidal disturbance and

(d) the control effort should be reasonable (e.g. limitedexplicitly to some maximum value).

Let q and _qq denote the translational position andvelocity respectively of the cart, and � and _�� denotethe rotor’s angular position and velocity respectively,where �¼ 0 is perpendicular to the motion of the cart,and �¼ 90� is aligned with the positive q direction.Then the system’s dynamics are described by

ðM þmÞ €qqþ kq ¼ �með €�� cos � � _��2 sin �Þ þ F ,

ðI þme2Þ €�� ¼ �me €qq cos � þ T :ð24Þ

As was proposed by Kinsey et al. (1992), let us definethe normalized variables

� ¼M þm

I þme2

� �1=2

q, � ¼k

M þm

� �1=2

t,

u ¼M þm

kðI þme2ÞT , w ¼

1

k

M þm

I þme2

� �1=2

F ,

ð25Þ

and represent the coupling between the translational androtational motions by the parameter

" ¼me

½ðI þme2ÞðM þmÞ�1=2

: ð26Þ

In addition, define z ¼ ½z1 z2 z3 z4�T¼ ½� _�� � _���T and

then effect a partial feedback linearization by using thenew coordinates

x1 ¼ z1 þ " sin z3,

x2 ¼ z2 þ "z4 cos z3,

x3 ¼ z3,

x4 ¼ z4,

ð27Þ

to yield

_xx1 ¼ x2,

_xx2 ¼ �x1 þ " sin x3 þ w,

_xx3 ¼ x4,

_xx4 ¼" cos x3

1� "2 cos2 x3fx1 � "ð1þ x24Þ sin x3g

þ1

1� "2 cos2 x3u�

" cos x3

1� "2 cos2 x3w:

ð28Þ

Observe that there is only one non-matching non-linearity present in (28): the " sin x3 term in the equationfor _xx2. Furthermore, the disturbance w enters the systemdynamics through the equations for _xx2 and _xx4, so thatthere is both a non-matching and a matching distur-bance component respectively, and the class of sinu-soidal disturbances assumed to be affecting the systembelongs to both L2e and L1. Consequently, sliding-mode control laws that will either ‘overpower’ orcancel exactly the matching nonlinearities and thematching disturbance component while forcing thestate trajectory into a sliding mode of operation on aparticular manifold appear to be good design choicesfor this system. Since (28) conforms to the structure dis-played in (6), the sliding-mode control design describedin Section 2 can be effected.

Let the system parameters be M¼ 9 kg, m¼ 1 kg,e¼ 0.1m, I¼ 0.09 kgm2, k¼ 10Nm�1 and "¼ 0.1 (thesame as in the work of Bupp et al. (1994)), with theconstraint on the control being

jT j41Nm continuous ) juj410: ð29Þ

Then (28) can be rewritten in the form of (1) with

A¼

0 1 0 0

�1 0 0:6" 0

0 0 0 1

0 0 0 0

0BBBBBBB@

1CCCCCCCA

¼

0 1 0 0

�1 0 0:06 0

0 0 0 1

0 0 0 0

0BBBBBBB@

1CCCCCCCA, ð30aÞ

G1 ¼

0

1

0

0

0BBBBBBB@

1CCCCCCCA, L1ðxÞ ¼ "

sin x3

x3� 0:6

� �, ð30bÞ

H1 ¼ ð0 0 1 0 Þ,

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B¼

0

0

0

1

1� 0:5"2

0BBBBBBBB@

1CCCCCCCCA¼

0

0

0

1:005

0BBBBBBB@

1CCCCCCCA, G¼

0

1

0

0

0BBBBBBB@

1CCCCCCCA, ð30cÞ

DAðxÞ ¼ ð1� 0:5"2Þ"cosx3

1� "2 cos2 x3

� x1� " 1þx24� �

sinx3�

, ð30dÞ

DBðxÞ ¼1� 0:5"2

1� "2 cos2 x3� 1,

DGðxÞ ¼�ð1� 0:5"2Þ"cosx3

1� "2 cos2 x3, ð30eÞ

such that (A,B) is a controllable pair. Note also thatDB(x) and DG(x) are bounded by �B¼ 0.0051 and�G¼ 0.101 respectively.Furthermore, since the variable to be regulated is the

displacement �� x1� "x3, we set

H ¼ ð 1 0 �" 0Þ: ð31Þ

Then we selected N satisfying (2) to be

N ¼

1 0 0 00 1 0 00 0 1 0

0@

1A: ð32Þ

Now, since G¼G1, defining Gþ¼ (G G1) results in atransmission gain via those input channels through theoutput channels Hþ¼ (HT H1

T)T that is 21/2 times thetransmission gain via either G or G1 through Hþ.Hence, we set Gþ¼G instead of its original definition(8) to avoid unnecessary conservatism in the design.Then (11) was solved, via a well-known approachdescribed by Anderson and Moore (1990), for decreas-ing values of � to yield eventually

P ¼

71:4876 �1:8874 �2:1755�1:8874 76:4532 2:3657�2:1755 2:3657 1:1314

0@

1A > 0: ð33Þ

corresponding to �¼ 24.9 (with the minimum admissible�� 23.52). This value was chosen, instead of a smallervalue, owing to the desire to avoid a ‘high-gain control’situation whereupon the large magnitudes of the ele-ments of S would have a corresponding effect on therequired size of the discontinuous control throughcondition (20) or (23). Note that the selected value of

� implies that condition (12) is satisfied by the L1(x) in(30b) for � 2.6<x3<2.6, which is sufficient (but notnecessary) for asymptotic stability (absent the distur-bance) and also small-signal input–output stabilitywith a disturbance transmission gain limit of 24.9 ofthe sliding-mode dynamics. Subsequently, the controlparameter K(¼Bs

TP) was calculated to be

K ¼ ð�2:1864 2:3775 1:1371 Þ ð34Þ

and, finally, (4) yielded

S ¼ ð�2:1864 2:3775 1:1371 0:9950Þ ð35Þ

to complete the sliding-phase design.

Remark 2: Note that the A2,3 element has to be non-zero in order for (A,B) to be a controllable pair. Itsvalue was chosen as 0.6" in order to reduce the mini-mum admissible value of � and increase the range ofx3 for L1(x) to satisfy condition (12). œ

Remark 3: Note that the designed manifold is linearwith respect to the [x1, x2, x3, x4] coordinates butnonlinear with respect to the original system coordi-nates. Furthermore, for the given and designed valuesof this example, specifically the vector S, it is signifi-cant that the resulting (nonlinear) sliding-mode dynam-ics are left with only a single equilibrium point at theorigin. œ

Reaching-phase design with the control law given by(19) was simplified because a maximum control magni-tude was specified for this problem. Hence, while (20)implies the need for

� >1� "2 cos2 x3

1� 0:5"2ðs1x2 � s2x1 þ s2" sinx3 þ s3x4 þ s2wÞ

����þ" cosx3 ½x1 � "ð1þ x24Þ sinx3 � w�

����,where s1, s2, s3 and s4 represent the elements of S, � wasset to 10 so that

u ¼ �10 sgnðSxÞ ð36Þ

in order to obtain the largest possible domain of attrac-tion for the sliding manifold subject to the given controlconstraint.

With this design in hand, we first considered thestabilization of the system with the disturbance F setto zero. The system’s state equations were simulatedusing control (36), with S given by (35). Figure 2(light solid curve) shows the settling response of the nor-malized cart displacement z1 based on an initial condition

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for [z1 z2 z3 z4]T equal to (0.1 0 0 0)T. Observe that z1

shows a settling time (time for z1 to reach and remainless than 5% of its initial value) of 22.45 units of �.Next, to study the disturbance rejection afforded by

the control (36), we set the system’s initial conditions tozero and the disturbance force F equal to 0.111 sin(2�).This amplitude of F was chosen because it causes theopen-loop system to respond with a steady-state oscilla-tion of z1 with amplitude 0.1, which corresponds to thesize of the initial condition used for evaluating thesettling response. Figure 3 (light solid curve) showsthat control (36) forces z1 to settle down to a steady-state oscillation of amplitude 0.0443, which is a 55.7%reduction of the disturbance’s impact on the system.Figures 4 (a) and (b) show the control u when thesystem responds to the given initial condition and tothe given sinusoidal disturbance respectively. Clearly,the chosen (maximum possible) control magnitude ismore than sufficient to ensure a sliding mode of opera-tion in each case and results in persistent high-frequencyswitching control action.To alleviate the chattering problem caused by con-

trol (36), control law (21) was also considered for the

system. Both � and S were kept at their above values;only the value of � had to be selected. Our analysis ofthe effects of this approximation of (36) on the stabilityand performance of the closed-loop system indicatedthat a choice of �¼ 10 would be reasonable; so thesystem was simulated with control

u ¼ �10 tanhð10SxÞ: ð37Þ

Figure 2 (dark dashed curve) shows the response ofz1 for the system with control (37), again with the distur-bance F set to zero and with the initial condition for[z1 z2 z3 z4]

T equal to (0.1 0 0 0)T; it is virtually indis-tinguishable from the light solid curve representing thesystem response due to control (36), with the settlingtime increasing only to 22.52 units of �. For the casewhen F¼ 0.111 sin(2�) and zero initial conditions,Figure 3 (dark dashed curve) shows that z1 settlesdown to a steady-state amplitude of 0.0445, which repre-sents a less than 0.5% decrease in the amount of distur-bance rejection. Hence, z1 responds almost exactlylike when the discontinuous sliding-mode control(36) was used except that the applied control u is

0 5 10 15 20 25 30 35 40 45 50−0.1

−0.05

0

0.05

0.1

z1

τ

Figure 2. System responses to initial condition with control (36) and with control (37).

0 5 10 15 20 25 30 35 40 45 50−0.1

−0.05

0

0.05

0.1

z1

τ

Figure 3. System responses to sinusoidal disturbances with control (36) and with control (37).

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now smooth; Figures 5 (a) and (b) show the controlu when the system responds to an initial conditionand to a sinusoidal disturbance, respectively.Control law (22) was also considered for reducing the

discontinuous control component and chattering, which

led to

u ¼ �1� "2 cos2 x31� 0:5"2

½s1x2 � s2x1 þ s2" sin x3 þ s3x4

þ � sgnðSxÞ� � " cos x3½x1 � "ð1þ x24Þ sin x3�, ð38Þ

0 5 10 15 20 25 30 35 40 45 50−5

0

5

10(a)

(b)

u

τ

0 5 10 15 20 25 30 35 40 45 50−0.5

0

0.5

u

τ

Figure 5. Control (37) when the system responds (a) to the initial condition and (b) to a disturbance.

0 5 10 15 20 25 30 35 40 45 50

−10

−5

0

5

10(a)

(b)

u

0 5 10 15 20 25 30 35 40 45 50

−10

−5

0

5

10

u

τ

τ

Figure 4. Control (36) when the system responds (a) to the initial condition and (b) to a sinusoidal disturbance.

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with the condition that

� > s2 �ð1� 0:5"2Þ" cos x31� "2 cos2 x3

��������jwj: ð39Þ

For the system and control parameters of this example,(39) requires that �>0.2751; so we chose �¼ 0.276.With control law (38), the simulated responses of z1 tothe same initial condition and to the same disturbance asbefore were essentially the same as for control laws (36)and (37). This is understandable, even though the con-trol laws are different in structure, because these exam-ples had initial states such that � was either somewhatclose to or actually on the sliding surface (for the distur-bance case). Then the reaching-phase responses wereof very short duration (less than 25 milliunits of �) anddid not differ much; so the respective sliding-phaseresponses (with the state trajectory kept on the slidingsurface by each control law but otherwise independentof it and depending instead on the sliding surfacepoint initially reached and that surface’s orientation)were also similar to each other. However, Figure 6shows that the corresponding control has a reducedswitching amplitude (of 0.276 instead of 10) as expected,which should alleviate the chattering problem.Furthermore, the sustained control and chattering prob-lem can be eliminated altogether by replacing the dis-continuous component of (38) with its continuousapproximation.

4. Conclusion

A novel sliding-mode approach to achieving asymp-totic stability and disturbance rejection for a class ofnonlinear systems with non-matching nonlinearities and/or disturbances, as exemplified by dual-spin spacecraftand rotational actuators that are used for suppress-ing translational oscillation, has been presented. Theapproach combines a bang–bang control law (or itscontinuous approximation) with an L2-norm boundingdesign of the sliding-mode dynamics. The latter ensures,after the system has been forced into a sliding mode ofoperation by the control, that the resulting dynamicsare asymptotically stable with respect to the origin (foran undisturbed system) and that the transmission gainof the non-matching disturbances through the systemis less than some prescribed value.

Simulations of the oscillating eccentric rotor modelexample with the sliding-mode control laws designedas proposed resulted in excellent responses to a class ofinitial conditions and to sinusoidal disturbances whilerequiring reasonable levels of control effort, thus illus-trating the proposed controller’s efficacy.

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0

0.(a)

(b)

5

u

τ

0 5 10 15 20 25 30 35 40 45 50

−0.5

0

0.5

u

τ

Figure 6. Control (38) when the system responds (a) to the initial condition and (b) to a sinusoidal disturbance.

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