On the rational stability of autonomous dynamical systems. Applications to control chained systems

Applied Mathematics and Computation 219 (2013) 10158–10171

Contents lists available at SciVerse ScienceDirect

Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

On the rational stability of autonomous dynamical systems. Applications to control chained systems Chaker Jammazi ⇑, Maâli Zaghdoudi Faculté des Sciences de Bizerte, Département de Mathématiques and Laboratoire d’Ingénierie Mathématiques, Ecole Polytechnique de Tunisie, Université de Carthage, Tunisia

a r t i c l e

i n f o

Keywords: Asymptotic estimation Hölderian feedbacks Chained system Rational partial stabilizability Backstepping

a b s t r a c t In this paper, several sufficient conditions for rational stability are provided. We applied our conditions to show that all chained systems are rationally stabilized-by Hölderian feedback laws – in partial sense. In addition, we show that the backstepping techniques can be extended to rational stabilizability theory. Ó 2013 Elsevier Inc. All rights reserved.

1. Introduction Stability of dynamical system in Lyapunov sense plays a major role in control theory, and becomes a challenging problems both in theory and applications, for more details, the reader is referred to [9,42,28,26,43] and references therein. In this paper, more attention is given to the so called ‘‘rational stability’’. Indeed; since the book developed by Hahn [19] in the sixties of last century, where the author showed that the rational stability can be characterized by means of Lyapunov functions, this notion did not known any intense progress like other tools of the stability theory. As its name connotes, the rational stability means that the solutions of dynamical system decrease like tca where a > 0. Recently Bacciotti and Rosier [1] have paid more attention in this field. Some analytic sufficient conditions are revised. In [34], the author speaks about another point of view of ‘‘rational stability’’. It is another approach of stabilization of nonlinear control systems by using induced norm of a ‘‘scaling operator’’ that maps one solution of the system to another. A natural question can be asked: why the rational stability? What is the advantage of this ‘‘stability’’? It is known that in nonlinear autonomous dynamical system of the form x_ ¼ XðxÞ with Xð0Þ ¼ 0, the linearization techniques is one of useful methods to study the stability or the asymptotic stability of the trivial solution. This techniques becomes insufficient if the matrix A :¼ DXð0Þ (approximation of x_ ¼ XðxÞ near zero) is not Hurwitz. In this case the solution does not decrease exponentially. Instead, it may sometimes be proved that the solutions decrease like tca with a > 0 called the rate decay of the solution. The real a measure the velocity of convergence of solutions which is important in many practical engineering (as well as satellite systems, unicycle systems, underwater, transport equation, string networks etc.). Moreover, it is shown in [3] that stabilization of nonlinear controllable systems by regular state feedback laws is impossible-in more general situations – which make the stabilization of such systems by non-ordinary feedback law is challenging problems [24,33,10,14,40,38,39,27]. In order to cope with this difficulty, a great effort by the control community was provided. Various sets of novel ideas and strategies have been developed by a number of authors: 1. time-varying feedback laws ([10,7,29,31,32,36,39,11,8,44,17]), and the references therein. 2. Discontinuous feedback laws ([13,45,5]), and the references therein. ⇑ Corresponding author. E-mail address: [email protected] (C. Jammazi). 0096-3003/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2013.03.096

C. Jammazi, M. Zaghdoudi / Applied Mathematics and Computation 219 (2013) 10158–10171

10159

It seems an attractive idea to construct, for control systems of n-dimension, stabilizing feedback laws such that in closed loop the system is rationally stable with respect to ðn  1Þ partial states ðx1 ; x2 ; . . . ; xn1 Þ0 and the last one converges. This concept is defined in the paper as ‘‘rational partial stabilizability’’. In this paper, we give a rigorous construction of the rational stability of continuous autonomous systems, followed by several examples of control systems. The basic idea of the rational stabilizability of control system of the form x_ ¼ Xðx; uÞ, is the construction of stabilizing feedback laws uðxÞ, vanishing in the equilibrium point such that all solutions xðtÞ can be estimated by jxðtÞj  tca ; a > 0 (the notation  means asymptotically equivalent). Several sufficient Lyapunov-like conditions that make all dynamical systems rationally stable are given. As an application, we show that we can stabilize rationally ðn  1Þ components of chained or multi-chained systems while the remaining state converges. Note that our partial rational stabilizability have a physical meaning, for convenience, we cite the example of unicycle system or the knife edge (e.g. Example 4.1). The partial rational stabilizability of the unicycle means that we can place the vehicle in the equilibrium position without taking into consideration his orientation (this later is constant for a large time). In addition, we provide an extension of the backstepping techniques to rational stabilizability. Two problems of partial orientation of underactuated systems are treated. The first one is the hovercraft and the second one is the satellite with two controllers. For both systems, we exploit our result obtained for double integrator to show how we can stabilize rationally five states while only one converges. The two examples are followed by computer simulations. 2. Notations and definition In this paper, we adopt the notations: j  j denotes the Euclidean norm on Rn ; h; i denotes the scalar product on Rn ; L1 ½0; þ1Þ is the Lebesgue space, L2 ð½0; þ1ÞÞ :¼ ff : ½0; þ1Þ ! R measurable, such that jf j2 2 L1 ½0; þ1Þg,’ is the symbol of transposition, LðRn ; RÞ is the space of linear maps and ‘‘sgn’’ is the sign function with sgnð0Þ ¼ 0. We start by recalling the notion of the rational partial stability. Let the dynamical systems in finite dimension be in the following form

x_ 1 ¼ f1 ðx1 ; x2 Þ;

x_ 2 ¼ f2 ðx1 ; x2 Þ;

ð1Þ p

np

where f ¼ ðf1 ; f2 Þ is a smooth vector field which defined on R  R 0 < p 6 n. We assume that

p

; x1 2 R , x2 2 R

f1 ð0; x2 Þ ¼ 0 and f 2 ð0; ;x2 Þ ¼ 0 8x2 2 Rnp :

np

and p is an integer such that

ð2Þ

Let us remark, that due to the existence of geometric integral for the system (1) or the asymptotic stability of the whole system is not possible (due, for example, to Brockett necessary condition for stabilizability [3]), in this case stabilization should be treated in the sense of partial asymptotic stability; see, for instance, [21–24]. Note that, our partial asymptotic stability definition is different from those given and used in [18,47,48], where the authors focused on a part of the system and supposed that the rest is bounded. For these reasons our next definition concerns the partial rational stability of dynamical systems. Definition 2.1 (p-rational partial stability [24]). The system (1) is said to be p-rational partially stable if the following properties are satisfied  the origin of the system (1) is Lyapunov stable.  there exist positive numbers M; k; g and r with g 6 1 such that if

9r > 0 : ðjx1 ð0Þj þ jx2 ð0Þj 6 rÞ )

8 > > < jx1 ðtÞj 6

Mjxð0Þjg

ð1 þ jxð0Þjk tÞk > > : lim x2 ðtÞ ¼ aðxð0ÞÞ;

;

8t P 0;

t!þ1

where aðxð0ÞÞ is a constant vector depending on initial conditions. The control system



x_ ¼ Xðx; uÞ; Xð0; x2 ; 0Þ ¼ 0;

is p-rationally partially stabilizable if there exists a continuous feedback x # uðxÞ such that, for every x2 2 Rnp ; uð0; x2 Þ ¼ 0, and such that ð0; 0Þ 2 Rp  Rnp is p-rationally partially stable for the closed loop system x_ ¼ Xðx; uðxÞÞ. Remark 1.  The case n ¼ p correspond to ‘‘complete’’ rational stability of system (1).  Obviously, rational stability implies asymptotic stability, but the converse is not true, for example x_ ¼ x is asymptotically stable but is not rationally stable.

10160

C. Jammazi, M. Zaghdoudi / Applied Mathematics and Computation 219 (2013) 10158–10171

3. Asymptotic estimation of solutions In some cases, the decay of the energy or the Lyapunov function is not exponential, but can be polynomial. Our aim here is to give a sufficient conditions that yield the explicit decay rate. Now we state our main results. Proposition 3.1. Let the dynamical system be

x_ ¼ gðxÞ;

and g 2 C 1 ðRn ; Rn Þ:

gð0Þ ¼ 0; 1

ð3Þ

We assume that there exist C -function V : R ! R, some positive constants c1 ; c2 such that 1. there exists

n

e > 0, such that for every jxj < e; V satisfies

r1

c1 jxj 6 VðxÞ 6 c2 jxjr2 2. there exist c > 0 and a > 0 such that

V_ þ cV 1þa 6 0:

ð4Þ

Then x_ ¼ gðxÞ is rationally stable. Proof. We assume that the assumptions of Proposition 3.1 hold, conditions 1 and 2 of the proposition imply that (3) is asymptotically stable. By integration of the differential inequality (4), we have for V – 0,

d a V P ac; dt

ð5Þ

then by integration the inequality (5) along ½0; t, we easily get

V6

1 ðat þ bÞ

1

where a ¼ ac

;

and b ¼

a

1 ; Vðxð0ÞÞa

r1

since c1 jxj 6 VðxÞ, then we get the estimation

jxðtÞj 6

 1=r1 1 1 ; 1 c1 ðat þ bÞar1

which implies the rational stability of system (3). h Corollary 3.1. Let the dynamical system (3), we assume that there exists a C 1 -function V : Rn ! R satisfying: (i) there exist a real numbers c1 ; c2 ; c3 ; r1 ; r 2 ; r3 2 ð0; þ1Þ and r 3 > r 2 , such that for every x 2 Rn such that jxj < e,

c1 jxjr1 6 VðxÞ 6 c2 jxjr2 ;

ð6Þ

n

(ii) for every x 2 R such that jxj < e,

_ VðxÞ 6 c3 jxjr3 ;

ð7Þ

then, system (3) is rationally stable.

Proof. We assume that the assumptions of Corollary 3.1 hold, conditions (6) and (7) imply that 0 2 Rn is Lyapunov stable for the system x_ ¼ gðxÞ. By combining the assertions (i and ii), we obtain constants c3 > 0 and r 3 > r2 > 0 such that r

3 V_ 6 cV r2 ;

Since r3 > r 2 , then

c¼ r3 r2

c3 r =r2

c23

:

2 ¼ 1 þ a, where a ¼ r3rr > 0, then, (8) is equivalent to 2

V_ 6 cV 1þa : Hence, from Proposition 3.1, system (3) is rationally stable. h

ð8Þ

C. Jammazi, M. Zaghdoudi / Applied Mathematics and Computation 219 (2013) 10158–10171

10161

Remark 2. 1. The proposition still holds if the map g and the Lyapunov candidate function are only continuous. In this case, the time derivative of V is defined in Dini sense by

_ 0 Þ :¼ Dþ VðxðtÞÞ ¼ lim sup þ VðxðtÞÞ  Vðx0 Þ ; Vðx t!0 t the inequality (4) becomes Dþ V 6 cV 1þa ; a > 0. 2. The result of Proposition 3.1 is necessary for the rational stability, for convenience see [1] Theorem 5.3. The next proposition gives rational stability of system (3) by comparison functions. Proposition 3.2. We consider the system (3) and we assume there exist C 1 -function V : Rn ! R, some positive constants c1 ; c2 such that 1. there exists

e > 0, such that for every jxj < e; V satisfies

r1

c1 jxj 6 VðxÞ 6 c2 jxjr2 ; 2. there exists function r : ½0; þ1Þ ! ½0; þ1Þ positive definite such that V_ þ rðVÞ 6 0, [ rðtÞ 3. there exists a > 0 such that limt!0 aþ1 ¼ l 2 ð0; þ1Þ fþ1g. t Then, the equilibrium point 0 of (3) is rationally stable. Proof.  First case: 0 < l < þ1. We assume that the assumptions of Proposition 3.2 hold, conditions 1 and 2 imply that (3) is asymptotically stable and

lim VðxðtÞÞ ¼ 0:

t!þ1

rðtÞ Moreover, if we pick up a 2 ð0; þ1Þ such that limt!0 aþ1 ¼ l > 0. Therefore, by using limit definition, there exists t0 > 0 such t that for every 0 6 t 6 t0 , we have

rðtÞ P

l aþ1 t : 2

Now, for this t 0 > 0, there exists t > 0 such that for every t P t one gets VðxðtÞÞ 6 t0 and therefore for t P t  ,

rðVÞ P

l aþ1 V ; 2

the last inequality implies for t is large,

l V_ 6  V aþ1 ; 2 which in turn implies, by using, Proposition 3.1 the rational stability of (3).  The proof of the case l ¼ þ1 is similar. h

Corollary 3.2. We consider the Assumptions 1 and 2 of the above Proposition and we assume that the function r is differentiable and satisfy the inequality

r_ P cr 1k ;

c > 0;

k 2 ð0; 1Þ:

ð9Þ

Then (3) is rationally stable. _ ¼ cyðtÞ1k , we get yðtÞ ¼ yð0Þ þ ðkcÞ1=k t1=k . Then the solution of (9) satisfies Proof. We integrate the differential equation yðtÞ 1=k 1=k rðtÞ P ðkcÞ t ; t P 0. In this case, the inequality V_ 6 rðVÞ becomes V_ 6 CV 1=k with 1k > 1 and C ¼ ðkcÞ1=k which, together with Proposition 3.1, concludes the proof of Corollary 3.2. h Proposition 3.3. We consider the Assumptions 1 and 2 of the Proposition 3.2 and we assume that r : R ! R is a continuous function such that the scalar system y_ ¼ rðyÞ is rationally stable. Then (3) is rationally stable.

10162

C. Jammazi, M. Zaghdoudi / Applied Mathematics and Computation 219 (2013) 10158–10171

Proof. The fact that y_ ¼ rðyÞ is rationally stable, then by converse Lyapunov function, see Remark 2, there exist a candidate _ 6 cW 1þa . Therefore the solution y can be estimated by Lyapunov function W, two reals c > 0 and a > 0 such that W a _ jyðtÞj 6 ð1þbtÞ k , where k > 0; a and b are two positive constants depending on initial conditions. Since V 6 rðVÞ, then V sata h isfies V 6 ð1þbtÞ k , which with Assumption 1 concludes the rational stability of system (3).

The next result deals with the robustness condition of the rational stability, more precisely, we study the systems of the form

x_ ¼ gðxÞ þ hðxÞ; x 2 Rn

and g; h 2 C 1 ;

ð10Þ

where x_ ¼ gðxÞ is the nominal system and h denotes the perturbation term results, in general cases, from disturbances, uncertainties, parameter variations, or modeling errors. In the following we investigate the sensitivity to perturbations of systems with a rational stability equilibrium by studying the behavior of solutions of the perturbed system (10) in a neighborhood of the equilibrium of the nominal system x_ ¼ gðxÞ. Proposition 3.4. We assume that origin of the nominal system x_ ¼ gðxÞ is rationally stable and limjxj!0 jhðxÞj jgðxÞj ¼ 0. Then the origin of the perturbed system (10) is locally rationally stable. Proof. Without loss of generality, we assume that n ¼ 1. The nominal system x_ ¼ gðxÞ is rationally stable, then there exist V : R ! R satisfying (6), c > 0 and a > 0 such that V_ þ cV 1þa 6 0. By taking the time derivative of V along the system x_ ¼ gðxÞ þ hðxÞ, we get V_ ¼ @V gðxÞ þ @V hðxÞ. Since limjxj!0 jhðxÞj ¼ 0, then for any e > 0, there exists r > 0 such that if jxj 6 r then jgðxÞj @x @x @V @V _ jhðxÞj 6 ejgðxÞj. Therefore, we get for jxj 6 r; V 6 gðxÞ þ ej jjgðxÞj 6 @V gðxÞð1 þ esgnð@V gðxÞÞÞ, where sgnð:Þ is the usual sign @x

function. By choosing (10). h

e ¼ 1=2, we get V_ 6 2c @V gðxÞ 6  2c V @x

@x

1þa

@x

@x

, which implies the local rational stability of the perturbed system

4. Applications to control systems 4.1. Chains of integrators For finite-dimensional control systems, chains of integrators or cascaded systems are key systems that can appear in many mechanical systems. For this reason, the stabilization of such systems is an interesting area of many works [12,2,35]. In this subsection, we give an explicit construction of Hölderian feedback laws that make chains of integrators rationally stable. We shall start by the double integrator. Proposition 4.1. The double integrator

x_ ¼ y;

y_ ¼ u;

ð11Þ

is rationally stable under the of Hölderian feedback law

 1þ2pk u ¼ ð2p þ 1Þx2p y  x2k1  y þ x2pþ1 ; k; p 2 N are odd integers:

ð12Þ

Moreover, if p is odd rational number such that 0 < p < 1=2 and k > maxð1; 2pÞ, then the solutions xðtÞ and yðtÞ are Lebesgueintegrable. Proof. Step 1: We start by studying the reduced system x_ ¼ u which is rationally stabilized under the feedback u ¼ x2pþ1 , where p is a positive integer or nonnegative odd rational (i.e. p ¼ pp1 where p1 ; p2 are odd integers). 2

Step 2: We apply the backstepping techniques, let k be an odd integer, and consider the candidate Lyapunov function V defined by

V¼

2 1 2k 1  x þ y þ x2pþ1 : 2k 2

ð13Þ

The time derivative of V along the system (11) is given by

V_ ¼ x2k1 y þ ðy þ x2pþ1 Þðu þ ð2p þ 1Þx2p yÞ: 2pþ1

Let z ¼ y þ x . Then y ¼ z  x A simple calculation yields

2pþ1

ð14Þ

.

V_ ¼ x2ðkþpÞ þ zðu þ x2k1 þ ð2p þ 1Þx2p yÞ:

ð15Þ

C. Jammazi, M. Zaghdoudi / Applied Mathematics and Computation 219 (2013) 10158–10171

10163

Then, under the feedback u given in (12), the time derivative of V along the double integrator becomes

V_ ¼ x2ðpþkÞ  ðy þ x2pþ1 Þ2þ2p=k :

ð16Þ

From (16) we have k _ kþp ðVÞ P x2k

k _ kþp and ðVÞ P ðy þ x2pþ1 Þ2 ;

therefore we get k _ kþp 2ðVÞ P x2k þ ðy þ x2pþ1 Þ2 P V;

ð17Þ

and p

c :¼ ð1=2Þ1þk > 0;

V_ 6 cV 1þp=k ;

and by integration of the last differential inequality we get

V6

1 ðat þ bÞk=p

ð18Þ

;

where a and b are two constants depending on initial conditions. Now, from (13) we have

jxj 6 ð2kVÞ1=2k and jy þ x2pþ1 j 6 ð2VÞ1=2 ;

ð19Þ

we incorporate (19) in (18), we get the following estimations

pffiffiffiffiffiffi 2k 2k

jxðtÞj 6

ðat þ bÞ1=2p

ð20Þ

;

2p

jyðtÞj 6 jy þ x j þ jx

2pþ1

j6

pffiffiffi 2 ðat þ bÞk=2p

þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2k ð2kÞð2pþ1Þ ðat þ bÞð2pþ1Þ=2p

ð21Þ

;

R þ1 we know that 1 dt=ta converges if and only if a > 1. Then by integral comparison Theorem, we get from (20) and (21), that if the odd rational p is in the open interval ð0; 1=2Þ and k > maxð1; 2pÞ, then the state variables xðtÞ and yðtÞ are Lebesgueintegrable. h Proposition 4.2. Let be k and p nonnegative odd integers supposed to be large and p > k. Then the triple integrator x_ ¼ y; y_ ¼ z; z_ ¼ v , is rationally stable under the Hölderian feedback law

v¼



y þ x2pþ1

1þ2pk

_ yÞ  ðz  uðx; yÞÞ1þ2p=k ; þ uðx;

k;

p 2 N ;

ð22Þ

_ yÞ is the time derivative of u. where uðx; yÞ is the feedback given in Proposition 4.1 and uðx; Proof. The proof is similar to the last one by taking the candidate Lyapunov function defined as

W :¼

2 1 1 2k 1  x þ y þ x2pþ1 þ ðz  uðx; yÞÞ2 ; 2k 2 2

where uðx; yÞ is the feedback given in the Proposition 4.1. h This result can be generalized by induction to all systems with n-integrators. Proposition 4.3. The chain of integrator

x_ 1 ¼ x2 ;

x_ 2 ¼ x3 ;

x_ 3 ¼ x4 ; . . . ;

x_ n2 ¼ xn1 ;

x_ n1 ¼ xn ;

x_ n ¼ u;

ð23Þ

is rationally stabilizable by Hölderian feedback laws. The Proposition 4.3 leads to the following result which states that all controllable linear system can be rationally stabilizable. Proposition 4.4. In Rn , all linear controllable system of the form x_ ¼ Ax þ Bu is rationally stabilizable by Hölderian feedback laws. Proof. We consider the controllable linear system x_ ¼ Ax þ Bu. This system is a feedback equivalent to Brunovsky canonical form which is a family of independently chain of integrator. Then Proposition 4.3 allows to conclude the result. h

10164

C. Jammazi, M. Zaghdoudi / Applied Mathematics and Computation 219 (2013) 10158–10171

4.2. Chained systems It is known that all chained system cannot be stabilized by regular feedback laws in usual sense, this is due to Brockett’s necessary condition [3]. A natural question can be asked: can chained systems be partially rationally stabilized by state of feedback laws? In the sequel, we give a positive answer to this alternative. Proposition 4.5. The nonlinear chained systems

x_ 1 ¼ x2

x_ 2 ¼ x3

u1 ;

u1 ; . . . ; x_ n2 ¼ xn1 u1 ;

x_ n1 ¼ u2 ;

x_ n ¼ u1 ;

ð24Þ

is ðn  1Þ-partially rationally stabilizable by means of Hölderian feedback laws. Proof. Get partitioned input u2 with the following form u2 ¼ uu1 , where u is a suitable feedback law which stabilizes in finite-time the chain of integrators

x_ 1 ¼ x2 ;

x_ 2 ¼ x3 ;

x_ 3 ¼ x4 ; . . . ; x_ n3 ¼ xn2 ;

x_ n2 ¼ xn1 ;

x_ n1 ¼ u:

ð25Þ

Clearly, all the assumptions of [20, Theorem 1] hold, then (25) is finite-time stabilizable. Indeed, for the control system (25), there exists a continuous, feedback law x # uðxÞ, a Lyapunov function V constructed with an explicit recursive design-for the construction see [20] – and two real numbers a 2 ð0; 1Þ and c > 0 such that a _ Vj ð25Þ þ cV 6 0;

a¼

2ðn  1Þ 2 ð0; 1Þ; 2ðn  1Þ þ 1

_ where Vj ð25Þ denotes the time derivative of V along the system (25). The time derivative of the function V for system (24) is given by

_ V_ ¼ u1 Vj ð25Þ :

ð26Þ

If u1 P 0, then (26) becomes a _ V_ ¼ u1 Vj ð25Þ 6 cu1 V :

ð27Þ

Now, we choose the feedback u1 with the following form

u1 ¼ V 1þk ; where k > 0:

ð28Þ

In this case, we get

V_ 6 cV 1þaþk : By Proposition 3.1, we conclude the rational stability of the closed loop system (24) with respect to ðx1 ; x2 ; . . . ; xn1 Þ0 . According to proof of Proposition 3.1, there exist two constants a and b, depending on initial conditions such that the function V can be estimated by

1

V6

ð29Þ

:

1

ðat þ bÞaþk From (28) and (29), we get the estimation

0 6 u1 6

1 1þk

ð30Þ

;

ðat þ bÞaþk since 0 < a < 1, then 0 < a þ k < 1 þ k and

1þk

> 1. Hence the map t #

1

is Lebesgue integrable on ½0; þ1Þ. 1þk ðat þ bÞaþk Now, since xn ðtÞ ¼ xn ð0Þ þ 0 u1 ðxðsÞÞds, and t # u1 ðxðtÞÞ 2 L ½0; þ1Þ then limt!þ1 xn ðtÞ exists, i.e xn converges. h Rt

aþk

1

The following theorem extends Proposition 4.5 to all multi-chained systems [4]. Theorem 4.1. Let m be a positive integer, let n1 ; . . . nm be m nonnegative integers such that n ¼ 1 þ m þ chain single-generator chained form:

Pm

j¼1 nj .

The following m-

8 z_ 1;0 ¼ v 1 ; z_ 2;0 ¼ v 2 ; z_ m;0 ¼ v m ; > > > > _ _ _ < z1;1 ¼ v 0 z1;0 ; z2;1 ¼ v 0 z2;0 ; zm;1 ¼ v 0 zm;0 ; . .. .. > . > . . ... . > > : z_ m;nm ¼ v 0 zm;nm 1 ; z_ n ¼ v 0 ; z_ 1;n1 ¼ v 0 z1;n1 1 ; z_ 2;n2 ¼ v 0 z2;n2 1 ; is ðn  1Þ-partially rationally stabilizable by Hölderian state feedback laws (the last component of the state being zn ).

C. Jammazi, M. Zaghdoudi / Applied Mathematics and Computation 219 (2013) 10158–10171

10165

Proof. The considered system is a collection of m-chain of integrators, then the construction of Hölderian stabilizing feedback law of the multi-chained system of Murray and Sastry follows from Proposition 4.5. h Example 4.1. We consider the Brockett’s integrator which is presented in the following form

x_ 1 ¼ v 1 x2 ; x_ 2 ¼ v 2 ; x_ 3 ¼ v 1 :

ð31Þ

The state is given by x ¼: ðx1 ; x2 ; x3 Þ0 2 R3 and the control is u :¼ ðv 1 ; v 2 Þ 2 R2 . The system (31) models several physical situations. As shown in [3], the stabilization of (31) by regular state feedback laws is impossible; which make the stabilization of (31) by non-ordinary feedback law is challenging. The aim of this example is to build tow feedback laws such that in closed loop, (31) is rationally stable with respect to ðx1 ; x2 Þ and x3 converges. To this end, let a 2 ð0; 1Þ, and let the feedback [2]

v ¼ sgnðx2 Þjx2 ja  sgnð/ðx1 ; x2 ÞÞj/ðx1 ; x2 Þj

a

2a

ð32Þ

;

where

/ðx1 ; x2 Þ ¼ x1 þ

1 sgnðx2 Þjx2 j2a : 2a

Let r; s > 0 be two real-numbers and let the candidate Lyapunov function

Vðx1 ; x2 Þ ¼

3a 2a r j/ðx1 ; x2 Þj2a þ sx2 /ðx1 ; x2 Þ þ jx2 j3a : 3a 3a

Then ð0; 0Þ 2 R2  R is 2- rationally stable for the closed loop system (31) with

v 1 ðxÞ :¼ ðVðx1 ; x2 ÞÞ1þk ; v 2 ðxÞ :¼ v 1 ðxÞv ðxÞ:

v 1 and v 2 defined by

k > 0;

Indeed, the time derivative of V along (31) satisfies V_ ¼ v 1 V_ jð33Þ , where V_ jð33Þ is the time derivative of V along (33),

x_ 1 ¼ x2 ;

x_ 2 ¼ v ;

ð33Þ

by using [2] Proposition 1, the function V_ jð33Þ satisfies for all ðx1 ; x2 Þ 2 R2 the inequality 2 V_ jð33Þ 6 cV 3a ;

c > 0;

then, under the choice of

with 0 <

2 < 1; 3a

v 1 :¼ V 1þk P 0, we get

2 V_ 6 cV 1þkþ3a ;

which together with Proposition 3.1 concludes the rational stability of (31) with respect to ðx1 ; x2 Þ. Now, we turn to study the convergence of the state x3 . As in the proof of Proposition 4.5, the function V can be estimated by

1

V6

1

; 2

ðat þ bÞkþ3a where a and b depending on initial conditions, then we get

1

0 < v1 6 ðat þ

1þk kþ 2 bÞ 3a

;

by the same argument as above one concludes that

v 1 2 L1 ½0; þ1Þ. Since x_ 3 ¼ v 1

then x3 converges. h

4.3. Backstepping techniques The backstepping techniques is a popular methods for the conception of stabilizing feedback laws of cascaded systems and becomes one of useful methods for solving stabilization problems [12,9,30,6]. Next, we extend the previous Proposition 4.1, for the double integrator, to all nonlinear control systems. Given the control system x_ ¼ f ðx; uÞ; ðx; uÞ 2 Rn  Rm , we assume that f 2 C 1 ðRn  Rm ; Rn Þ and f ð0; 0Þ ¼ 0. Without loss of generality, we assume that m ¼ 1. Theorem 4.2. Given the control system

x_ ¼ f ðx; uÞ;

ð34Þ 1

we assume that system (34) is rationally stabilizable by a C -feedback law uðxÞ. Then the extended dynamical system

10166

C. Jammazi, M. Zaghdoudi / Applied Mathematics and Computation 219 (2013) 10158–10171

x_ ¼ f ðx; yÞ y_ ¼ u;

ð35Þ

is rationally stabilizable by C 0 state feedback law. Proof. Since (34) is rationally stabilizable by a C 1 feedback law uðxÞ, then, by converse Lyapunov theorem see Remark 2, there exists a candidate Lyapunov function V and nonnegative real number a such that in closed loop, the function V satisfies

V_ 6 cV 1þa :

ð36Þ

We consider the candidate Lyapunov function W defined by

1 W :¼ V þ ðy  uðxÞÞ2 : 2 We compute the time derivative of W along the system (35), we obtain

_ ¼ @V f ðx; yÞ þ ðy  uðxÞÞðu  u _ ðxÞÞ: W @x

ð37Þ

Since f 2 C 1 ðRn  R; Rn Þ, then by Taylor’s expansion, there exists a matrix G 2 C 0 ðRn  R  R; LðR; Rn ÞÞ such that f ðx; yÞ ¼ f ðx; uðxÞÞ þ Gðx; y; uðxÞÞðy  uðxÞÞ, 8ðx; yÞ 2 Rn  R. _ becomes According to (36), W

 _ 6 cV 1þa þ ðy  uðxÞÞðu  u _ ðxÞÞ þ Gðx; y; /ðxÞÞ; @V W ðy  uðxÞÞ @x    _ ðxÞ þ Gðx; y; uðxÞÞ; @V 6 cV 1þa þ ðy  uðxÞÞ u  u : @x

ð38Þ

In this case, as in the proof of Proposition 4.1, we choose the following feedback

_ ðxÞ  Gðx; y; uðxÞÞ; uðx; yÞ ¼ u

@V @x



 ðy  uðxÞÞ1þ2p=k ;

ð39Þ

where k; p 2 N are odd integers or odds rationals. Hence, under (39), the function W satisfies

_ 6 cV 1þa  ðy  uðxÞÞ2þ2p=k 6 0: W

ð40Þ

_ is negative definite, then, in closed loop (35) is globally asymptotically stable. Therefore Vðx; yÞ ! 0 as t ! þ1. Clearly W Without loss of generality, we assume that V < 1, in this case we can select odd rational numbers p and k such that a < pk. Then we get V a P V p=k . (40) becomes

_ 6 cV 1þp=k  ðy  uðxÞÞ2þ2p=k ; W

ð41Þ

a straightforward computation leads to

_ 6 C W 1þp=k ; W

where C a positive constant;

ð42Þ

we integrate (42) we easily obtain

W6

1 ðat þ bÞk=p

;

where a and b are two constants depending on initial conditions, which gives the desired result. h Remark 3. We consider the assumption of Theorem 4.2. If the control input u in Rm , then we change the Lyapunov function W as follows

1 W :¼ V þ jy  uðxÞj2 : 2 In this case, the feedback u takes the form

 @V _ ðxÞ  Gðx; y; uðxÞÞ;  jy  uðxÞj2a ðy  uðxÞÞ; uðx; yÞ ¼ u @x and the function W satisfies the inequality

_ 6 cW 1þa : W Thus allows to conclude the rational stability of the augmented system (35).

C. Jammazi, M. Zaghdoudi / Applied Mathematics and Computation 219 (2013) 10158–10171

10167

Example 4.2 [23]. Considering the following system

x_ 1 ¼ u1 ; x_ 2 ¼ u2 ; x_ 3 ¼ x1 x2 :

ð43Þ

where k; p 2 N or k; p are two non-negative odd rational numbers, such that 2p > 1 and k > p. The sate is given by x ¼ ðx1 ; x2 ; x3 Þ0 2 R3 and u ¼ ðu1 ; u2 Þ 2 R2 is the control. This system represents the angular velocity control of a rigid spacecraft with two controllers. We have shown in [23], that, if k > 2p > 1, then, the following Hölder feedback laws



1þ2p k 2p u1 ¼ 2px32p1 x1 x2  ðx2 þ x3 Þx32k1 þ x2k ; 3  a 1 x1  x3 2ðkþpÞ1

u2 ¼ x1 x2  x3

2p

 a2 ðx2 þ x3 Þ1þ k ;

ai > 1; i ¼ 1; 2:

render the system (43) rationally partially stable. Indeed, the rational partial asymptotic stabilizability of the system (43) is obtained with the help of the candidate Lyapunov function

V¼

2 1 1 2k 1 x3 þ x1  x2p þ ðx2 þ x3 Þ2 ; 3 2k 2 2

and, it was shown that under the feedback laws u1 and u2 , the Lyapunov function V satisfies the differential inequality p

V_ 6 cV 1þk ;

c > 0:

Example 4.3 [24]. We consider the Brockett’s integrator

x_ 1 ¼ v 1 x2 ;

x_ 2 ¼ v 2 ;

x_ 3 ¼ v 1 :

ð44Þ

The main idea, is to use the backstepping techniques for building Hölderian stabilizing control laws that make (44) rationally partially stabilizable. Let

v 1 ðxÞ ¼ x1 , and let k 2 N . We consider the candidate Lyapunov function V defined by V¼

2 1 2k 1 x1 þ x2 þ x2p : 1 2k 2

ð45Þ

The time derivative of V along the system (44) is given by



2ðkþpÞ 2k V_ ¼ x1 : þ y v 2 þ 2px2p 1 x2 þ x1

ð46Þ

It is then sufficient to choose 2p k

2k 1þ v 2 ¼ 2px2p 1 x2  x1  a y

;

a > 1:

ð47Þ

Thus we get 2p 2ðkþpÞ V_ ¼ x1  ay2þ k ;

and the global asymptotic stabilization of (44) with respect to ðx1 ; x2 Þ follows from first Lyapunov Theorem. Clearly the stabilization of ðx1 ; x2 Þ is not sufficient to conclude the convergence of the ‘‘uncontrolled’’ state x3 , but if we know how convergence of ðx1 ; x2 Þ i.e. like tca one concludes the convergence of x3 . In this case we will show that the controller v 1 :¼ x1 is Lebesgue-integrable. One easily see that there exists a constant c > 0 such that p

V_ 6 cV 1þk : This leads to the existence of constants a and b depending on initial conditions such that

V6

1 k

ð48Þ

:

ðat þ bÞp From (45), we have x2k 1 6 2kV, then by using (48), we get

jx1 ðtÞj 6

ck ðat þ bÞ1=2p

; ck ¼ ð2kÞ1=2k :

ð49Þ

R þ1 It is clear that if 0 < 2p < 1, then the state x1 is Lebesgue integrable on Rþ . Indeed, we have 1 dt=t a converges if and only if R þ1 1 a > 1. Since, 2p > 1, then from integral comparison Theorem, we get 1 jx1 ðtÞjdt < þ1, therefore

10168

C. Jammazi, M. Zaghdoudi / Applied Mathematics and Computation 219 (2013) 10158–10171

Z

þ1

jx1 ðtÞjdt ¼

0

Z

1

jx1 ðtÞjdt þ

Z

0

þ1

jx1 ðtÞjdt < þ1:

1

Since, the state x3 ðtÞ satisfies the equation x_ 3 ¼ v 1 ¼ x1 and x1 2 L1 ½0; þ1Þ, then x3 converges. 4.4. Application to partial attitude control of underactuated hovercraft The problem of stabilization of underactuated hovercraft is the subject of numerous papers. Let us recall that this system cannot be stabilized by means of pure state feedback laws; this is due to Brockett condition [3]. Various feedbacks (continuous time-varying feedback laws or discontinuous feedback laws) are derived to overcome this obstruction, we cite for example the works [36,37,41,16]. In [21] the partial exponential stabilizability of the system is also solved via backstepping techniques and under smooth feedback laws. In this section, we show how we can obtain the rational partial stabilizability of the hovercraft. The hovercraft derived from [41] is modeled by

8 > h_ ¼ u cosw  v sinw > > > > > /_ ¼ u sinw þ v cosw > > > < w_ ¼ r > u_ ¼ v r þ su > > > > > > v_ ¼ ur  bv > > :_ r ¼ sr ;

ð50Þ

where h; /; w denote the velocities in surge, sway and yaw respectively and u; v ; r denote the position and orientation of the ship in the earth frame and su and sr are the the control force in surge and sway respectively and b is a nonnegative constant. € ¼ sr , in this case we can choose for example feedbacks given in Proposition 4.1 for the rational Clearly, w satisfies w stabilization;



sr :¼ ð2p þ 1Þw2p r  w2k1  r þ w2pþ1

1þ2pk

;

then under sr , the states w and r are rationally stable. Since wðtÞ ! 0 as t ! þ1, then cosðwðtÞÞ ’ 1, therefore, a simple approximation of the system (50) around ðh; v ; uÞ0 we get

h_ ¼ u; For (51),

su

v_ ¼ bv ;

u_ ¼ su :

ð51Þ

v is exponentially stable since v_ ¼ bv ;

b > 0. Similar to

sr we can choose su as follow:



1þ2pk :¼ ð2p þ 1Þh2p u  h2k1  u þ h2pþ1 ;

then, we get the rational stability of h and u. _ 6 juj þ jv j and by choosing in the feedback Now, we turn to studying the convergence of the state /. We have j/j su ; k > maxð1; 2pÞ and 0 < p < 1=2 then u 2 L1 ½0; þ1Þ. Also due to exponential stability of v we get v 2 L1 ½0; þ1Þ. Hence /_ 2 L1 ½0; þ1Þ and therefore / converges. Numerical simulations. In the following charts, we present numerical simulations to validate our feedback laws constructed bellow. The initial condition is x0 ¼ ð0:5; 0:9; 0:4; 0:5; 0:7; 0:5Þ0 . These simulations show how the system (50) is rationally partially stabilizable. Clearly the state / converges to value ’ 1:4. 1.5

0.8

1

θ φ ψ

0.5

0.6 0.4

0

0.2

−0.5

0

−1

0

5

10

15 time (s)

20

25

30

u v r

−0.2

0

5

10

15 time (s)

20

25

30

10169

C. Jammazi, M. Zaghdoudi / Applied Mathematics and Computation 219 (2013) 10158–10171

4.5. Partial orientation of satellite with two controllers The problem of attitude control of satellite is well known [15,11,31]. Let us recall that satellite with two controllers is not stabilizable in usual sense by regular state feedback laws. A great effort has been deployed to overcome this problem [11,31,21]. In this subsection, we shall apply Proposition 4.1 to study the partial orientation of the satellite with two controllers. We generate state feedback laws that make the satellite five-partially rationally stable. For many reasons explain in [15,46], we choose the Euler-Poisson parameterizations ([46–48]) to describe the motion of rigid spacecraft. The dynamic of satellite with two controllers [48,47] is presented as follows

8 x_ 1 ¼ u1 > > > > > x_ 2 ¼ u2 > > > < x_ 3 ¼ x1 x2 > > m_ 1 ¼ x3 m2  x2 m3 > > > > > m_ 2 ¼ x1 m3  x3 m1 > : m_ 3 ¼ x2 m1  x1 m2 ;

ð52Þ

where xi (i ¼ 1; 2; 3) are the coordinates of the angular velocity with respect to the principal axes of inertia; mi the coordinates of the fixed unit vector m with respect to the principal axes of inertia; and u1 ; u2 the jet control torque. We keep to the following equilibrium: xi ¼ 0, m1 ¼ m2 ¼ 0 and m3 ¼ 1. The system (52) admits the following integrals:

m21 þ m22 þ m23 ¼ 1;

ð53Þ

by using the implicit function Theorem [25], the identity (53) becomes, if we choose the hemisphere

m3

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi ¼ 1  m21 þ m22 :

m3 > 0, ð54Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi Now, we consider the unit open ball Bð0; 1Þ and the function rðm1 ; m2 Þ ¼ 1  m21 þ m22 .

r is smooth on the open ball Bð0; 1Þ, we develop r in the first order by Taylor’s formula in the neighborhood of ð0; 0Þ we get the expansion

rðm1 ; m2 Þ ¼ 1 þ gðm1 ; m2 Þ; where the function g is smooth and satisfies gð0; 0Þ ¼ g 0 ðm1 ; m2 Þð0; 0Þ ¼ 0. By replacing the state m3 by 1 þ gðm1 ; m2 Þ we get

8 x_ 1 ¼ u1 > > > > > x_ 2 ¼ u2 > > > < x_ 3 ¼ x1 x2 > m_ 1 ¼ x3 m2  x2 ð1 þ gðm1 ; m2 ÞÞ > > > > > > m_ 2 ¼ x1 ð1 þ gðm1 ; m2 ÞÞ  x3 m1 > : m_ 3 ¼ x2 m1  x1 m2 :

ð55Þ

By taking the time derivative of m_ 1 and m_ 2 respectively, we get

(

m€1 ¼ u2 þ dtd ðx2 m2  x2 gðm1 ; m2 ÞÞ m€2 ¼ u1 þ dtd ðx1 gðm1 ; m2 Þ  x3 m1 Þ;

the terms dtd ðx2 m2  x2 gðm1 ; m2 ÞÞ and lowing feedback linearization

d ð dt

ð56Þ

x1 gðm1 ; m2 Þ  x3 m1 Þ represent higher order nonlinear terms. Then by taking the fol-

 1 :¼ u2 þ dtd ðx2 m2  x2 gðm1 ; m2 ÞÞ u  2 :¼ u1 þ dtd ðx1 gðm1 ; m2 Þ  x3 m1 Þ; u in this case we can choose feedbacks given in Proposition 4.1 for the rational stabilization;



1þ2p k 2k1  1 :¼ ð2p þ 1Þm2p þ x3 m2  x2 m3 þ m12pþ1 ; u 1 ðx3 m2  x2 m3 Þ þ m1

1þ2p k 2k1  2 :¼ ð2p þ 1Þm2p  x1 m3  x3 m1 þ m2pþ1 ; u 2 ðx1 m3  x2 m1 Þ  m2 1

10170

C. Jammazi, M. Zaghdoudi / Applied Mathematics and Computation 219 (2013) 10158–10171

with the considerations p is odd rational number such that 0 < p < 1=2 and k > maxð1; 2pÞ, then the solutions mi ðtÞ and m_ i ðtÞ are rationally stable and Lebesgue-integrable for i ¼ 1; 2. In this case we get m3 ðtÞ ! 1 as t ! þ1. For t is large enough, we have m3 ðtÞ ’ 1 and mi ðtÞ ’ 0; i ¼ 1; 2. Then we have

m_ 1 ¼ x1 ; m_ 2 ¼ x2 ;

ð57Þ

thus we get xi (i ¼ 1; 2) are asymptotically stable, moreover xi are Lebesgue-integrable and satisfies the inequality (20). 1 1 More precisely jxi ðtÞj 6 þ . A simple calculation shows that, for i 2 f1; 2g, we have ð2pþ1Þ=2p ðat þ bÞk=2p ðat þ bÞ 2 1 xi 2 L ½0; þ1Þ. Since x_ 3 ¼ x1 x2 , then x_ 3 2 L ½0; þ1Þ and therefore the state x3 converges. Numerical simulations. In the following figures, we present numerical simulations to validate our feedback laws constructed bellow. The initial condition is x0 ¼ ð1:5; 1:5; 0:5; 1; 0; 0Þ0 . These simulations show how the system (52) is rationally partially stabilizable. Clearly the state x3 converges to value ’ 0:25. 3

1.2

ω 1 ω2 ω

2

1

3

0.8

1

0.6

0

0.4

ν 1 ν 2 ν3

0.2 −1 −2

0 0

2

4

time (s)

6

8

10

−0.2

0

2

4

time (s)

6

8

10

5. Conclusion In this paper, rational stability and and stabilization are investigated for systems described by ordinary differential equations. Several sufficient conditions are given for scalar and n-dimensional systems. These conditions are illustrated by some examples of control systems. Clearly, the partial rational stability is an alternative to overcome the Brockett’s condition for stabilizability by regular feedback laws. This partial rational stabilizability is practical in many situations, and seems sufficient when some physical variables are not important, especially when the objective is to control some coordinates e.g. orientation control. Acknowledgements The first author would like to thank the Associate Editor and the Referee that read the work with great care, and made interesting remarks and suggestions to improve the quality of the paper. References [1] A. Bacciotti, L. Rosier, Liapunov functions and stability in control theory, Commun. Contr. Eng. (2005) Springer-Verlag. [2] S.P. Bhat, D.S. Bernstein, Continuous finite-time stabilization of the translational and rotational double integrators, IEEE Trans. Automat. Contr. 43 (5) (1998) 678–682. [3] R.W. Brockett, Asymptotic stability and feedback stabilization differential geometric control theory, Prog. Math. 27 (1983) 181–191. [4] L.G. Bushnell, D.M. Tilbury, S.S. Sastry, Steering three-input nonholonomic systems: the fire truck example, Int. J. Robot. Res. 9 (August) (1994). [5] F.M. Ceragioli, Discontinuous ordinary differential equations and stabilization, in: Tesi di dottorato di ricerca in matematica Consorzio delle universit‘a di Cagliari, Firenze Modena Perugia e Siena, 1999. [6] F. Chen, L. Chen, W. Zhang, Stabilization of parameters perturbation chaotic system via backstepping, Appl. Math. Comput. 200 (1) (2008) 101–109. [7] J.-M. Coron, Global asymptotic stabilization for controllable systems without drift, Math. Contr. Sign. Syst. 5 (1992) 295–312. [8] J.-M. Coron, Stabilization in finite time of locally controllable systems by means of continuous time-varying feedback laws, SIAM J. Contr. Optimiz. 33 (3) (1995) 804–833. [9] J.-M. Coron, Control and Nonlinearity, Mathematical Surveys and Monographs, vol. 136, 2007. [10] J.-M. Coron, B. d’Andréa Novel, Smooth stabilizing time-varying control laws for a class of nonlinear systems, Applications to mobile robots, in Michel Fliess, (Ed.), IFAC Nonlinear Control Systems Design, Bordeaux, France, pp. 413–418. 1992. [11] J.-M. Coron, E.Y. Keraï, Explicit feedbacks stabilizing the attitude of a rigid spacecraft with two torques, Automatica 32 (1996) 669–677. [12] J.-M. Coron, L. Praly, Adding an integrator for the stabilization problem, Syst. Contr. Lett. 17 (1991) 89–104. [13] J.-M. Coron, L. Rosier, A relation between continuous time-varying and discontinuous feedback stabilization, J. Math. Syst. Estimat. Contr. 4 (1994) 67– 84.

C. Jammazi, M. Zaghdoudi / Applied Mathematics and Computation 219 (2013) 10158–10171

10171

[14] C. Canudas de Wit, O.J. Sørdalen, Exponential stabilization of mobile robots with nonholonomic constraints, IEEE Trans. Automat. Contr 37 (11) (1992) 1791–1797. [15] A. El-Gohary, On the orientation of rigid body using point masses, Appl. Math. Comput. 151 (2004) 163–173. [16] L. Fantoni, R. Lozano, F. Mazenc, K.Y. Pettersen, Stabilization of a nonlinear underactuated hovercraft, in: Proceedings of the 38th Conference on Decision & Control, 1999, pp. 2533–2538. [17] X.-S. Ge, L.-Q. Chen, Optimal motion planning for nonholonomic systems using genetic algorithm with wavelet approximation, Appl. Math. Comput. 180 (2006) 76–85. [18] W. Haddad, V. Chellaboina, S. Nersesov, A unification between partial stability of state-dependent impulsive systems and stability theory for timedependent impulsive systems, Proc. Am. Contr. Conf. (2003) 4004–4009. [19] W. Hahn, Theory and Application of Liapunov’s Direct Method, Englewood Cliffs NJ, 1963. [20] X. Huang, W. Lin, B. Yang, Global finite-time stabilization of a class of uncertain nonlinear systems, Automatica 41 (2005) 881–888. [21] C. Jammazi, Backstepping and partial asymptotic stabilization. Applications to partial attitude control, Int. J. Contr. Automat. Syst. 6 (6) (2008) 859– 872. [22] C. Jammazi, Finite-time partial stabilizability of chained systems, CR Acad. Sci. Paris Ser. I 346 (17–18) (2008) 975–980. [23] C. Jammazi, On a sufficient condition for finite-time partial stability and stabilization: applications, IMA J. Math. Contr. 27 (1) (2010) 29–56. [24] C. Jammazi, A discussion on the Hölder and robust finite-time partial stabilizability of Brockett’s Integrator, ESAIM: Contr. Optimisat. Calc. Var. 18 (2) (2012) 360–382. [25] C. Jammazi, A. Abichou, A stability analysis and control of constrained mechanical systems, Adv. Model. Anal. – A General Math. Model. 38 (2001) 1–21. [26] H.K. Khalil, Nonlinear Systms, third ed., Prentice Hall, 2002. [27] Y-W. Liang, D.-C. Liaw, On asymptotic stabilization of driftless systems, Appl. Math. Comput. 114 (2000) 303–314. [28] Der-Cherng Liaw, Application of center manifold reduction to nonlinear system stabilization, Appl. Math. Comput. 91 (1998) 243–258. [29] R.T. M’Closkey, R.M. Murray, Exponential stabilization of driftless nonlinear control systems using homogeneous feedback, IEEE Trans. Automat. Contr. 42 (1997) 614–628. [30] P. Morin, Stabilisation de systèmes non linéaires critiques et application la commande de véhicules, Habilitation à diriger des recherches L’université de Nice-Sofia Antipolis, 2004. [31] P. Morin, C. Samson, Time-varying exponential stabilization of a rigid spacecraft with two control torques, IEEE Trans. Automat. Cont. 42 (1997) 528– 534. [32] P. Morin, C. Samson, J-B. Pomet, Z-Ping Jiang, Time-varying feedback stabilization of the attitude of a rigid spacecraft with two controls, Syst. Contr. Lett. 25 (1995) 375–385. [33] R. Murray, Z. Li, S.S. Sasrty, A Mathematical Introduction to Robotic Manipulation, Library of Congress Cataloging-in-Publication Data CRC edition, 1993. [34] R. Olfati-Saber, Rational stability and characterization of stabilizability of nonlinear systems, Proc. Am. Contr. Conf. (2000) 3096–3100. [35] Y. Orlov, Discontinuous systems – Lyapunov analysis and robust synthesis under uncertainty conditions, Commun. Contr. Eng. (2009) Springer-Verlag. [36] K.Y. Pettersen, O. Egeland, Exponential stabilization of an underactuated surface vessel, in: Proc. 35th IEEE Conf. on Decision and Control, Kobe, Japan, December 1996. [37] K.Y. Pettersen, H. Nijmeijer, Global practical stabilization and tracking for an underactuated ship. A combined averaging and backstepping approch, Model. Ident. Contr. 20 (4) (1999) 189–199. [38] J.B. Pomet, Explicit design of time-varying stabilizing control laws for a class of controllable systems without drift, Syst. Contr. Lett. 18 (1992) 147–158. [39] C. Samson, Velocity and torque feedback control of a nonholonomic cart, in: Proceeding of International Workshop on Nonlinear and Adaptive Control, Advanced Robot Control, vol. 162, Springer-Verlag, November 1991, pp. 125–151. [40] C. Samson, Control of chained systems: application to path following and time-varying point-stabilization of mobile robots, IEEE Trans. Automat. Contr. 40 (1) (1995) 64–77. [41] H. Sira-Ramirez, Dynamic second-order sliding mode control of the hovercraft vessel, IEEE Trans. Contr. Syst. Technol. 10 (6) (2002) 860–866. [42] X. Song, S. Li, An Li, Practical stability of nonlinear differential equation with initial time difference, Appl. Math. Comput. 203 (2008) 157–162. [43] E.D. Sontag, Mathematical control theory: determinstic finite dimensional systems, Text Appl. Math. 6 (1998). Springer-Verlag. [44] E.D. Sontag, H.J. Sussmann, Remarks on continuous feedback, IEEE CDC Albuquerque 2 (1980) 916–921. [45] H.J. Sussmann, Subanalytic sets and feedback control, J. Diff. Eq. 31 (1) (1979) 31–52. [46] P. Tsiotras, On the choice of coordinates for control problems on so(3), in: 30th Annual Conference on Information Sciences and Systems, Princeton University, 1996, pp. 20–22. [47] V.I. Vorotnikov, Partial Stability and Control, Birkhäuser, 1998. [48] A.L. Zuyev, On partial stabilization of nonlinear autonomous systems: sufficient conditions and examples, in: Proc. of the European Control Conference ECC’01 Porto (Portugal), 2001, pp. 1918–1922.