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INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROLInt. J. Robust Nonlinear Control (2009)Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/rnc.1494

Stabilization of n integrators in cascade with bounded input with experimentalapplication to a VTOL laboratory system

G. Sanahuja, P. Castillo∗,† and A. Sanchez

Heudiasyc Laboratory, UMR CNRS 6599, Universite de Technologie de Compiegne, Compiegne, France

SUMMARY

This paper presents a global stabilization of a chain of n integrators in cascade. The control strategy is obtained usingthe Lyapunov approach and separated saturation functions. Moreover, the stability analysis is obtained using the recurrencetheorem. This generalized control law is designed in order to quickly implement it on a system, as choosing a degree n givesall conditions to have a stable system. Moreover, in the proposed controller the saturation function bound only one state.This allows us to easily tune the control parameters. Simulations and real-time experiments are presented for the VTOLplatform represented as a chain of two and four integrators in cascade. Copyright q 2009 John Wiley & Sons, Ltd.

Received 14 November 2008; Revised 30 April 2009; Accepted 14 June 2009

KEY WORDS: nonlinear control; saturation functions; Lyapunov analysis; VTOL vehicle; real-time experiments

1. INTRODUCTION

The stabilization of chains of integrators has beenextensively studied in the last decades. In particular,the problem of stabilizing a chain of integrators usingbounded input with delays [1, 2] or without delays [3]was already treated in the literature.

Before 1992, the determination of explicit expres-sions of globally asymptotically stabilizing feed-backs for chains of integrators of dimension strictlylarger than two without delay was an open problem.

∗Correspondence to: P. Castillo, Heudiasyc Laboratory, UMRCNRS 6599, Universite de Technologie de Compiegne,Compiegne, France.

†E-mail: [email protected]

Teel in [3] has solved it by exhibiting a simple familyof stabilizing feedbacks for these systems. This pioneerresult has been very fruitful. First, it allowed to solvetheoretical issues (see [4, 5]) and more applied prob-lems (see [6]) and, second, it is at the origin of a newcontrol design methodology called forwarding (see, forexample, [7, 8]).

Forwarding is a systematic tool for the stabiliza-tion of general cascades. This methodology requires, ingeneral, the (approximate) solution of a partial differ-ential equation and tends to generate complex controllaws. Moreover, although forwarding tackles success-fully saturated inputs, it is not a low-amplitude design,so it does not impose restrictions on the control ampli-tude. The control Lyapunov function approach can alsoprovide control laws for stabilization in the presence ofinput constraints, with the use of universal formulae [9].

Copyright q 2009 John Wiley & Sons, Ltd.

G. SANAHUJA, P. CASTILLO AND A. SANCHEZ

Relevant results are general, however, large amountof the studies on low-amplitude designs is typicallybased on small-gain considerations. They also requirefull state feedback, and in some cases only semiglobalresults are provided [10–14].

From a structural point of view, for systemsdescribed by a generalized linear [3, 5, 11] or nonlinear[15, 16] chain of integrators, the control laws consistof a generalization of the nested saturations schemeof [3] or linear combinations of saturations [5]. Thesedesigns also make use of passivity, in the sense that,at each step of the procedure, the feedback consistsof a function of the state for which is a relativedegree one output. See also [17], where Teel’s nestedsaturation scheme is robustified against unmodeleddynamics.

Recently, in [18, 19] the authors proposed a controlalgorithm based on separated saturation functions. Inthis kind of controller the saturation arguments gener-ally depend on several state variables. The authors havecompared simulation in this algorithm with respect toa controller based on Teel’s results. They showed thatthe separated saturation controller in general is moreefficient than Teel’s one. On the other hand, the gainsand bounds of Teel’s controller are relatively complexto tune. Kaliora and Astolfi [20, 21] present the stabi-lization problem for a similar class of nonlinear feed-forward systems using bounded control. These resultshave inspired our research work.

The goals of this paper are to present a newmethod tostabilize a chain of integrators in cascade using boundedinput and to present the real-time experimental resultsobtained when applying the proposed control scheme.The difference with respect to the previous works is thatthe controller does not use the nested saturation scheme.Moreover, the controller synthesis procedure proposedin this paper is easier to understand than those onespresented in the literature. In addition, the proposedcontrol strategy and its robustness are shown in a real-time platform.

This paper is organized as follows: the systemand its stability analysis are described in Section 2.Section 3 gives some applications of the main result.In this section simulations and experimental results arealso presented. Finally, in Section 4, some conclusionsare provided.

2. STABILIZATION OF THE SYSTEM

A general system of n integrators in cascade can bewritten as follows:

x1 = �1x2

...

xn−1 = �n−1xn

xn = �nu

(1)

with �i �=0 constant, ∀i ∈[1,n]. Introducing thefollowing change of coordinates:

y1 = x1

y2 = �1x2

...

yn = �1 . . .�n−1xn

u′ = �1 . . .�nu

Then, system (1) becomes

y1 = y2

...

yn = u′(2)

Let us define the saturation function �bi (s) of thefollowing form (see Figure 1)

�bi (s)=

⎧⎪⎨⎪⎩

−bi for s<−bi

s for −bi�s�bi

bi for s>bi

Theorem 2.1The control law

u′ =−n∑

i=1�bi (ki yi) (3)

globally asymptotically stabilizes system (2) for allki>0 constant and bounds every state, yi , using thenonlinearity �bi (s).

Copyright q 2009 John Wiley & Sons, Ltd. Int. J. Robust Nonlinear Control (2009)DOI: 10.1002/rnc

INTEGRATORS IN CASCADE WITH BOUNDED INPUT

0

0

s

b i(s)

bi

bi

Figure 1. Saturation function.

ProofFirst, from (3) and using the Lyapunov analysis and therecurrence theorem, we will prove that ∀i ∈[1,n], ∃ atime Tn−i+1 large enough such that ∀t>Tn−i+1,

u′ =−n∑j=i

k j y j −�i and |yi | is bounded (4)

where �i is a bounded function, |�i |�b�i with b�i >0.Such that

�i =i−1∑j=1

�b j (k j y j ) (5)

Then, we will prove that using the Routh Hurtwitz crite-rion the system (2)–(4) remains stable and the states ofthe system converge to zero. �

2.1. Boundedness of the states

Equation (3) can also be written as

u′ =−n∑j=i

�b j (k j y j )−�i (6)

and �i can also be written as

�1 = 0

�i = �bi−1(ki−1yi−1)+�i−1 ∀i ∈[2,n] (7)

2.1.1. First step, i =n. We prove in this section that (4)is true for i =n. From (6) we have

u′ =−�bn (kn yn)−�n

Propose the following positive-definite function:

Vn = 12 z

2n

and define

zi = ki+1yi +zi+1 ∀i ∈[1,n−1]zn = yn

(8)

then,

Vn =u′yn =−yn(�bn (kn yn)+�n)

assuming that bn >b�n . Then, |kn yn|>b�n implies thatVn <0. Thus, there exists a time T1 such that ∀t>T1,

|yn|�b�n

kn

and the above implies that

u′ =−knyn−�n ∀t>T1 (9)

Note that (4) is true for i =n and ∀t>T1.

2.1.2. Second step. It is supposed that for a given i ,(4) is true ∀l ∈[i,n]. Now, we have to prove that (4) isalso true for i−1. From (4) to (9) we have that

u′ = −n∑j=l

k j y j −�l ∀t>Tn−l+1 ∀l ∈[i,n] and

|yl | is bounded

From (8), we have

n−1∑j=i−1

z j =n−1∑j=i−1

(k j+1y j +z j+1)

zi−1−zn =n−1∑j=i−1

k j+1y j

zi−1 =n−1∑j=i−1

k j+1y j + yn

(10)



then

zi−1=n−1∑j=i−1

k j+1y j+1+u′ =n∑j=i

k j y j +u′ =−�i

Propose the following positive-definite function:

Vi−1= 12 z

2i−1

then,

Vi−1 = −(

n−1∑j=i−1

k j+1y j + yn

)�i

= −(ki yi−1+

n−1∑j=i

k j+1y j + yn

)�i

= −(ki yi−1+zi )(�bi−1(ki−1yi−1)+�i−1)

Note that �1=0, then

V1=−(k2y1+z2)(�b1(k1y1))

Assuming that

bi−1>b�i−1

thus, if

|yi−1|>b�i−1

ki−1

then we obtain

sgn(�bi−1(ki−1yi−1)+�i−1)= sgn(yi−1)

as y j is bounded ∀ j ∈[i,n], then z j is also bounded(its bound will be called bz j ).

In addition, if

|yi−1|> bziki

then,

sgn(ki yi−1+zi )= sgn(yi−1)

and

|yi−1| >b�i−1

ki−1>

bziki

∀i ∈[2,n]

|y1| > b1k1

>bz2k2

implies that Vi−1<0 and ∃Tn−i+2>Tn−i+1 such that∀t>Tn−i+2

|yi−1| <b�i−1

ki−1∀i ∈[2,n]

|y1| <b1k1

(11)

and

u′ =−n∑

j=i−1k j y j −�i−1 ∀t>Tn

2.1.3. Third step: conclusion of the recurrence theorem.It was proved that (4) is true for i−1, then usingthe recurrence theorem (4) is also true ∀i ∈[1,n]. Thismeans that for i =1 there exists a time Tn such that

u′ =−n∑

i=1ki yi ∀t>Tn (12)

Note that the following conditions need to besatisfied:

b�i

ki>

bzi+1

ki+1∀i ∈[2,n−1]

b1k1

>bz2k2

bi > b�i ∀i ∈[2,n]

(13)

From (7),

b�i =i∑

j=2b j−1

From (10) and (11),

bzi =n−1∑j=i

k j+1b� j

k j+ b�n

kn

2.2. Convergence to zero

From (12) note that all the states are bounded and∀t>Tn the nonlinear control law, u′, yields to a linearcontrol law (without saturation functions) and thesystem remains stable. To make the convergence ofthe states to zero we use the Routh Hurtwitz criterionas follows.



The characteristic polynomial is given by the deter-minant of the following matrix:

A=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

−� 1 0 . . . 0

0 −� 1 . . . 0

......

0 . . . 1

−k1 −k2 . . . −kn−�

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

In order to compute the above we introduce thematrix Ai , ∀i ∈[1,n−1]:

Ai =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

−� 1 . . . 0

0 −� . . . 0

......

0 . . . 1

−ki −ki+1 . . . −kn−�

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

(14)

Note that Ai is a (n−i+1)×(n−i+1) matrix.Then, the determinant of Ai is obtained using thecofactor expansion on its first column

−�

∣∣∣∣∣∣∣∣∣∣∣∣∣∣

−� 1 . . . 0

0 −� . . . 0

......

0 . . . 1

−ki+1 −ki+2 . . . −kn−�

∣∣∣∣∣∣∣∣∣∣∣∣∣∣

+(−1)1+n−i+1(−ki)

∣∣∣∣∣∣∣∣∣∣∣

1 0 . . . 0

−� 1 0 . . . 0

......

0 . . . −� 1

∣∣∣∣∣∣∣∣∣∣∣The first determinant is obtained using the matrix

Ai+1, the second is obtained by a triangular matrix andit is equal to the product of each term on the diagonal.Then,

det(Ai )=−�det(Ai+1)+(−1)n−i+1ki (15)

Now, we use the recurrence theorem to prove that

det(Ai ) = (−�)n−i+1+(−1)n−i+1

×n−i∑j=0

� j ki+ j ∀i∈[1,n−1] (16)

The first step is to prove that (16) is true for i =n−1.From the above we have

det(An−1)=�2+1∑j=0

� j kn−1+ j =�2+kn−1+�kn

Moreover, from (14)

det(An−1) =∣∣∣∣∣

−� 1

−kn−1 −kn−�

∣∣∣∣∣= −�(−kn−�)+kn−1=�2+kn−1+�kn

Thus (16) is true for i =n−1.The next step is to suppose that (16) is true for a

given i and prove that it implies that (16) is also truefor i−1. From (15) we obtain

det(Ai−1)=−�det(Ai )+(−1)n−i ki−1

Note that (16) is true for i , then

det(Ai−1) = (−�)n−i+2+(−1)n−in−i∑j=0

� j+1ki+ j

+(−1)n−i ki−1

= (−�)n−i+2+(−1)n−in−i+1∑j=1

� j ki+ j−1

+(−1)n−i ki−1

= (−�)n−i+2+(−1)n−in−i+1∑j=0

� j ki+ j−1

and this implies that (16) is also true for i−1. Usingthe recurrence theorem, (16) is true ∀i ∈[1,n−1], thenfor i =1 we obtain

det(A1)=det(A)= (−�)n+(−1)nn−1∑j=0

� j k j+1 (17)



Then, using the Routh Hurwitz criterion on thisdeterminant, the ki parameters can be chosen in order tosatisfy the stability of the system (2) and guarantee theconvergence of the states to zero. Moreover, boundedfunctions �i and ki coefficients must be chosen in orderto satisfy (13).

3. APPLICATION

This section presents a real-time application to vali-date the proposed control scheme. The platform usedis derived from a simple aircraft named planar verticaltake off and landing (PVTOL). To ensure displacementsin a 2D space (like the PVTOL), the vehicle is notflying but rolling (using balls) on an inclined plan (seeFigures 2 and 3).

The vehicle moves using two motors equipped withblades, this characteristics give the PVTOL to havethree degrees of freedom (dof). The PVTOL is anunderactuated system because it has 3 dof and onlytwo control inputs. From Figure 3, note that the vehiclemakes a constant angle � �=0 with the horizontal. In thefollowing, the platform will still be called PVTOL.

Figure 2. The PVTOL aircraft (top view).

Figure 3. The PVTOL aircraft (side view).

3.1. Dynamic model of the PVTOL

The PVTOL aircraft is represented in Figures 2 and 3.In these figures, x and y denote the position of theaircraft center of mass G and � is the roll angle of theaircraft. M is the mass of the vehicle and J representsthe moment of inertia. The control inputs u1 and u2 arethe thrust (directed out of the bottom of the aircraft)and the angular acceleration (rolling moment). For ourcase, we suppose that there is no friction between thevehicle and the ground and only the gravity and motorsforces are considered.

Using Newton’s equation gives

Mx = − sin(�)u1

My = cos(�)u1−Mg sin(�)

J � = u2

(18)

3.2. Control strategy

To stabilize the aircraft the following control laws areproposed:

u1 = u′1+Mg sin(�)

cos�M (19)

u2 = Ju′2 (20)

where u′1 and u

′2 are the new control inputs to guarantee

the convergence to zero. Introducing the above into (18)and assuming cos� �=0, that is, �∈]−�/2,�/2[, weobtain

x = −(u′1+Mg sin(�)) tan�

y = u′1

� = u′2

(21)



Vertical position in the above equation can be written as

y1 = y2

y2 = u′1

(22)

where [y1 y2]T=[y y]T. Now the new control input, u′1,

is appled to ‘force’ the vertical displacement to have asuitable behavior. Then, using Theorem 2.1 we have

u′1=−�a(ka(y− yd))−�b(kb y) (23)

where ka , kb>0, yd is the desired altitude, |�a(·)|�baand |�b(·)|�bb with ba , bb>0. Moreover, from (13)

bb > ba

k2b > ka(24)

Introducing (23) into (22) and using Theorem 2.1, itfollows that ∃ a finite time t>T2>T1>0 large enough,such that

y=−ka(y− yd)−kb y

From the above it follows that, if yd is constant, theny− yd converges to the origin as t goes to infinity (thepolynomial s2+kbs+ka is stable). Therefore y andy→0, which, by using the above equation, implies thaty→ yd . Thus, after a time large enough, |u′

1|�� forsome constant �>0 arbitrarily small.

Then, (21) reduces to

x = −(�+Mg sin(�)) tan�

� = u′2

(25)

A nonlinear control based on saturation functions willalso be implemented to this system. This type of controlallows in the limit a guarantee of arbitrary bounds for�, �, x and x . To further simplify the analysis we willmake Kg = �+Mg sin(�).

Note from Theorem 2.1 that the amplitudes of thesaturation functions can be chosen in such a way that,after a finite time T ′, the roll angle lies in the interval−1rad��1rad. Moreover, there is not singularity in(19) provided that |�(0)|<�/2. Therefore, a very smallupper bound on |�| is imposed in such a way that

the difference tan(�)−� is arbitrarily small. Thus, thesubsystem (25) reduces to

x1 = x2

x2 = −Kg�1

�1 = �2

�2 = u′2

(26)

where [x x]T=[x1 x2]T and [� �]T=[�1 �2]T. Notethat Kg is constant and (26) represents four integratorsin cascade. Then, from Section 2 for n=4, we propose

u′2 = 1

Kg(�1(k1x1)+�2(k2x2)

−�3(k3Kg�1)−�4(k4Kg�2)) (27)

where k1, k2, k3, k4>0 and |�i (·)|�bi ∀i=1, . . .,4.Introducing (27) into (26) we have that ∀t>T6>T5>

T4>T3>T2 large enough, x , x , �, � are bounded (moredetails in Section 2).

From (17) we have

det(A)=�4+�3k4+�2k3+�k2+k1

then, the Routh Hurwitz criterion gives

k3k4 > k2

k2k3k4 > k1k24+k22

(28)

Table I. Parameters values used for simulations.

Parameter Value

k1 0.25k2 2.20k3 25.00k4 26.00b1 0.80b2 1.10b3 11.00b4 22.00ka 1.0kb 2.0ba 1.0bb 2.0� 30◦M 0.4



0 5 10 15 20 250

2

4

6x1 (t),x1(0)=5 x2 (t),x2(0)=1

t (s)

x 1 (

m)

x 2 (

m/s

)0 5 10 15 20 25

0

1

t (s)

0 5 10 15 20 25

0

10

20θ1 (t), θ1(0)=20

t (s)

θ 1 (

deg)

θ 2 (

deg)

0 5 10 15 20 25

0

5θ2 (t), θ2(0)=0

t (s)

0 5 10 15 20 250

5

10y(t),y(0)=5

t (s)

y (m

)

0 5 10 15 20 25

0

1

t (s)

dy (

m/s

)

0 5 10 15 20 25

0

2

4

u1

t (s)

u 1 u 2

0 5 10 15 20 25

0

u2

t (s)

Figure 4. Simulation results of the PVTOL.

To satisfy (13) we obtain

b4 > b1+b2+b3

b3 > b1+b2

b2 > b1

b1k1

>k3k22

b1+ k4k2k3

(b1+b2)+ b1+b2+b3k2k4

b1k2

>k4k23

(b1+b2)+ b1+b2+b3k3k4

(29)

b1+b2k3

>b1+b2+b3

k24



From (18) to (27) with the conditions (24), (28)and (29) we have that for a finite time large enough x ,x , �, �→0.

3.3. Simulation

To verify the efficacy of the proposed scheme, simu-lations were performed on the complete PVTOLdynamics (18). The values of the parameters are shownin Table I. The initial conditions are [y(0), x(0),�(0)]=[5m,5m,20◦] and [y(0), x(0), �(0)]=[−2m/s,1m/s,0◦/s]. Note that these conditions are so far of thedesired positions.

Notice from Figure 4 that the control strategy inclosed-loop system performs well even if initial condi-tions are ‘aggressive’.

3.4. Experimental results

In this section we present the real-time experimentalresults obtained when applying the controller proposedin the previous section to a two rotors vehicle (seeFigure 5). This platform is composed of a vehicle withtwo rotors moving on an inclined plane, a vision sensorto compute its position and orientation and a groundstation.

The control law is implemented on a PC runningMatlab’s Xpc Target and equipped with an output cardcontrolling the two motors. Position and orientation ofthe PVTOL are measured with a fixed camera system

Figure 5. Experimental platform.

placed above the PVTOL. The image plane of thecamera is parallel to the inclined plane. The PVTOLhas two powerful LEDs to help its localization with thecamera. Images are processed on a computer linked tothe Xpc Target via a serial cable. The experimentationscheme is represented in Figure 6.

The control strategies (23) and (27) were imple-mented on Matlab using in a first time parameters ofTable I. It is obvious that the exact model of the PVTOLplatform is not well known then the parameters weretuned manually to obtain a good behavior. The valuesof the parameters used in the real-time experiences areshown in Table II.

Figures 7 and 9 show the experimental results whenapplying the control strategy to the vehicle. Notice from

Figure 6. Scheme of the real-time implementation.

Table II. Parameters values used for the real-timeapplication.

Parameter Value

k1 0.001k2 0.002k3 0.100k4 0.001ka 0.800kb 0b1 1.0b2 5.0b3 1.0b4 5.0ba 2.0bb 2.1



0 500 1000 15000

10

20

30

40

50

60

70

80

Time (s)

y (p

ix)

Perturbations

Figure 7. y-position of the PVTOL aircraft.

0 500 1000 1500180

185

190

195

200

205

Time (s)

x (p

ix)

Figure 8. x-position of the PVTOL aircraft.

Figures 7 and 8 that the x and y-displacements aregiven in pixels (i.e. our experience was done in theimage-plane).

The control objective is to stabilize the PVTOLaircraft in (x,�)= (200pix,0◦) and to follow a smalltrajectory in the y-position. We add manually aggres-sive perturbations to prove the robustness of thecontrol scheme. From these figures we can observe that

0 500 1000 1500

0

1

2

Figure 9. -position of the PVTOL aircraft.

the controller performs well in practice even in thepresence of manual perturbations.

4. CONCLUSION

Global stabilization of a chain of n integrators incascade was presented in this paper. The controlstrategy was obtained using the Lyapunov approachand separated saturation functions. One characteristicof this controller is that it is easier to apply and totune in real-time applications with respect to the othercontrollers using saturation functions, like the oneusing nested saturation. The proposed control schemewas validated in simulation and in real-time applicationto case of 2 and 4 integrators in cascade (case: PVTOLaircraft). Simulations and experimental results showthat the control strategy performs well in practice evenin the presence of manual perturbations.

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