Automatica 44 (2008) 2981–2984

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Automatica

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Technical communique

Stabilization via homogeneous feedback controlsI

Emmanuel MoulayIRCCyN (UMR-CNRS 6597) – Ecole Centrale de Nantes, 1 rue de la Noë – B.P. 92 101, 44321 Nantes cedex 3, France

a r t i c l e i n f o

Article history:Received 22 August 2007Received in revised form23 January 2008Accepted 26 May 2008Available online 24 September 2008

Keywords:Control Lyapunov functionFeedback stabilizationHomogeneous system

a b s t r a c t

In this paper, we provide an explicit homogeneous feedback control with the requirement that a controlLyapunov function exists for an affine control system and satisfies an homogeneous condition. We use amodified version of the Sontag formula to achieve ourmain goal.Moreover,we prove that the existence ofan homogeneous control Lyapunov function for an homogeneous affine system leads to an homogeneousclosed-loop system by using the previous feedback control.

© 2008 Elsevier Ltd. All rights reserved.

1. Introduction

The weighted homogeneity, which has been introduced byRothschild and Stein (1976), plays an important role in nonlinearcontrol theory. This notion was extensively used:

(1) for the construction of a nilpotent approximating systemwhich is small time local controllable and homogeneous(see Hermes (1991)),

(2) for homogeneous feedback stabilization in Kawski (1990,1991) when searching an homogeneous closed-loop system,

(3) for finite time stabilization (see Bhat and Bernstein (2005)).

The stabilization of control systems by homogeneous feedbackcontrol has also been developed in Coron and Praly (1991) byusing backstepping technics and in Grüne (2000) by using ahomogeneous control Lyapunov function for controllable systems.The paper deals with finding homogeneous continuous feed-

back laws by using control Lyapunov functions.We develop amod-ified version of the Sontag formula in order to obtain an explicithomogeneous feedback control. In a first way, it is interesting tofind an homogeneous feedback control even if homogeneous feed-back stabilization is not possible. Indeed, homogeneous functionspossess very interesting properties which allow to restrict theirstudy to compact manifolds (see Rosier (1992, Lemma 1)). In asecond way, finding an homogeneous feedback control is often

I This work was supported in part by AtlanSTIC (CNRS FR 2819). This paper wasnot presented at any IFACmeeting. This paper was recommended for publication inrevised form by Associate Editor Vladimir O. Nikiforov under the direction of EditorAndré L. Tits.E-mail address: Emmanuel.Moulay@irccyn.ec-nantes.fr.

0005-1098/$ – see front matter© 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.automatica.2008.05.003

the first step when dealing with the research of an homogeneousclosed-loop system.We can alsomention that certain homogeneous systems cannot

be stabilized by homogeneous feedback laws. This has been firstpointed out in Rosier (1993) and in Sepulchre and Aeyels (1996)for affine systems.The organization of this paper is as follows. Section 2 provides

some useful notations and definitions. In Section 3 we developour main result based on a modified version of the Sontagformula. We give an explicit homogeneous feedback controlfor affine systems which possess a control Lyapunov functionsatisfying an homogeneous condition. The closed-loop system isnot homogeneous in general. In Section 4, we give an explicithomogeneous feedback control for homogeneous affine systemswhich possess an homogeneous control Lyapunov function. In thiscase, the closed-loop system is homogeneous.

2. Notations and definitions

Throughout this paper, V will be a non empty neighborhood ofthe origin in Rn. Let us consider the system

x = f (x) , x ∈ Rn (1)

where f : Rn → Rn is a continuous function. If V : V → Rn is acontinuously differentiable function, the derivative of V along thesolutions of the system (1) is defined by

V (x) = 〈∇V (x), f (x)〉 .

As it is customary in control theory, a Lyapunov function V :V → Rn for the system (1) is a continuously differentiable positivedefinite function such that V is negative definite. The Lie derivativeof V : Rn → R along f : Rn → Rn is defined by:

2982 E. Moulay / Automatica 44 (2008) 2981–2984

Lf V : Rn → Rx 7→ 〈∇V (x), f (x)〉 .

A function V : Rn → R is homogeneous of degree dwith respect tothe weights (r1, . . . , rn) ∈ Rn>0 if

V (∆λ (x)) = λdV (x1, . . . , xn)

for all λ > 0 where∆λ (x) = (λr1x1, . . . , λrnxn). A vector field f ishomogeneous of degree dwith respect to the weights (r1, . . . , rn) ∈Rn>0 if for all 1 ≤ i ≤ n, the i-th component fi is a homogeneousfunction of degree ri + d, that is

fi (∆λ (x)) = λri+dfi (x1, . . . , xn)

for all λ > 0. The system (1) is said homogeneous of degree d withrespect to the weights (r1, . . . , rn) ∈ Rn>0 if f is homogeneous ofdegree dwith respect to the weights (r1, . . . , rn) ∈ Rn>0.

3. Stabilization by homogeneous feedback controls

Letm ∈ N∗, and consider the following affine system

x = f0(x)+m∑i=1

fi(x)ui, x ∈ Rn, u ∈ Rm (2)

where fi ∈ C (Rn) for all 0 ≤ i ≤ m and f0(0) = 0 and the closed-loop system

x = f0(x)+m∑i=1

fi(x)ui(x), x ∈ Rn. (3)

Let us recall the definition of stabilization. The control system (2)is stabilizable (respectively continuously stabilizable) if there existsa non empty neighborhood of the origin V in Rn and a feedbackcontrol law u ∈ C0 (V\ {0} ,Rm) (respectively u ∈ C0 (V,Rm))such that:

(1) u (0) = 0,(2) the origin of the system (3) is asymptotically stable.

We now recall some usual definitions. A continuously differen-tiable positive definite function V : V → R≥0 is acontrol Lyapunovfunction for the system (2) if for all x ∈ V\ {0},

infu∈Rm

(a (x)+ 〈B (x) , u〉) < 0

where

a (x) = Lf0V (x),bi(x) = LfiV (x), 1 ≤ i ≤ mB (x) = (b1(x), . . . , bm(x)) .

As usual, such a control Lyapunov function satisfies the smallcontrol property if for each ε > 0, there exist δ > 0 such that, ifx ∈ δBn, then there exist some u ∈ εBm such that

a (x)+ 〈B (x) , u〉 < 0.

Remark 1. Ifm = 1, the small control property is equivalent to

lim sup‖x‖n→0

a(x)|B (x)|

≤ 0.

The limit may very well be−∞.

We set b (x) = ‖B (x)‖2.If we want to use a control Lyapunov function to obtain an

homogeneous feedback control, we can use the followingmodifiedversion of the Sontag feedback control.

Lemma 2. If there exists a continuously differentiable controlLyapunov function V : V → R≥0 for the control system (2), then it is

stabilizable under the feedback control u (x) = (u1 (x) , . . . , um (x))defined by

ui (x) =

−bi (x) a(x)+(|a(x)|p + b(x)q

) 1p

b(x)if x ∈ V\ {0}

0 if x = 0(4)

where p > 1, q > 1 are positive real numbers. If furthermore Vsatisfies the small control property, then the feedback control (4) isalso continuous at the origin.

Proof. Suppose there exists a continuously differentiable controlLyapunov function V : V → R≥0. Let

E ={(x, y) ∈ R2 : x < 0 or y > 0

}and ϕ a function defined on E by

ϕ (x, y) =

x+

(|x|p + |y|q

) 1p

yif y 6= 0

0 if y = 0.

As ϕ is defined on E, we have to prove that

limy→0,x<0

x+(|x|p + |y|q

) 1p

y= 0.

limy→0,x<0

x+(|x|p + |y|q

) 1p

y= lim

y→0,x<0

x+ |x|(1+ |y|

q

|x|p

) 1p

y

= limy→0,x<0

−x |y|q

py |x|p

= limy→0,x<0

|y|q−1 sgn (y)p |x|p−1

= 0.

ϕ is continuous on E. The fact that V is a control Lyapunov functionimplies that (a(x), b (x)) ∈ E for all x ∈ V\ {0}. Thus

u (x) = (u1 (x) , . . . , um (x))

defined by

ui (x) = −bi (x) ϕ (a(x), b (x))

is continuous on V\ {0} and we obtain for all x ∈ V\ {0},⟨∇V (x) , f0 (x)+

m∑i=1

fi(x)ui(x)

⟩= −

(|a(x)|p + b (x)q

) 1p

< 0.

V is a Lyapunov function for the closed-loop system (3) and (4),and by using the Lyapunov theoremwe know that the origin of theclosed loop system (3) and (4) is asymptotically stable.The proof concerning the stabilization under the small control

property is similar to the one given in Sontag (1989, Theorem 1).�

A less general version of this lemma was proved in Moulay andPerruquetti (2006, Lemma 19). The Sontag control corresponds tothe case (p, q) = (2, 2) (see Sontag (1989)).We now give a result concerning single input systems.

Theorem 3. Consider the single input system (2)withm = 1. If thereexists:

(1) a continuously differentiable control Lyapunov function V : V →R≥0 for the control system (2) such that a (x) is homogeneous ofdegree d0 and B (x) is homogeneous of degree d1 with respect tothe weights (r1, . . . , rn) ∈ Rn>0,

E. Moulay / Automatica 44 (2008) 2981–2984 2983

(2) p > 1, q > 1 positive reals such that p = 2qd1d0,

then the feedback control

u (x) =

−a(x)+(|a(x)|p + b(x)q

) 1p

B(x)if x ∈ V\ {0}

0 if x = 0(5)

stabilizes the system (2) and is homogeneous of degree d0 − d1 withrespect to the weights (r1, . . . , rn). If furthermore V satisfies the smallcontrol property, then the feedback control (5) continuously stabilizesthe system (2).Proof. The fact that u (x) stabilizes the system (2) is given byLemma 2.As a (x) is homogeneous of degree d0 and B (x) is homogeneous

of degree d1 with respect to the weights (r1, . . . , rn) ∈ Rn>0, wehave:

u (∆λ (x)) = −λd0a (x)+

(λpd0 |a (x) |p + λ2d1qb (x)q

) 1p

λd1B (x)

= −λd0a (x)+

(λpd0

(|a (x) |p + b (x)q

)) 1p

λd1B (x)

= −

λd0

(a (x)+

(|a (x) |p + b (x)q

) 1p

)λd1B (x)

= λd0−d1u (x) . �

Let us give an example

Example 4. Consider the system{x1 = −x31 + x2x2 = −x1 + u

and the control Lyapunov function

V (x) =x21 + x

22

2.

a (x) = −x41 is homogeneous of degree 4 and B (x) = x2 ishomogeneous of degree 1 with respect to the weights (1, 1). As

a (x)|B (x)|

=−x41|x2|≤ 0

we deduce from Remark 1 that V satisfies the small controlproperty. Theorem 3 with 4p = 2q, for instance (p, q) =

( 32 , 3

),

implies that the system is continuously stabilizable by the feedbackcontrol

u (x) = −−x41 +

(x61 + x

62

) 23

x2which is homogeneous of degree 3.We now address a result for multi-input systems whose proof canbe easily deduced from the previous one.

Theorem 5. Consider the system (2). If there exists:(1) a continuously differentiable control Lyapunov function V : V →

R≥0 for the control system (2) such that a (x) is homogeneous ofdegree d0 and bi (x) are homogeneous of degree d1 with respect tothe weights (r1, . . . , rn) ∈ Rn>0 for all 1 ≤ i ≤ m,

(2) p > 1, q > 1 positive reals such that p = 2qd1d0,

then the feedback control (4) stabilizes the system (2) and itscomponents ui(x) are homogeneous of degree d0 − d1 with respect tothe weights (r1, . . . , rn) for all 1 ≤ i ≤ m. If furthermore V satisfiesthe small control property, then the feedback control (4) continuouslystabilizes the system (2).

4. Homogeneous feedback stabilization

By homogeneous stabilization, we mean that there exists anexplicit feedback control such that the closed-loop system ishomogeneous. This is the purpose of this section. The followinglemma can be found in Rosier (1992).

Lemma 6. If V is homogeneous of degree d1 with respect to theweights (r1, . . . , rn) ∈ Rn>0 and if f is homogeneous of degree d2 withrespect to the weights (r1, . . . , rn) ∈ Rn>0, thenLf V is homogeneousof degree d1 + d2 with respect to the weights (r1, . . . , rn) ∈ Rn>0.We can now give a theorem for homogeneous feedback stabiliza-tion using an homogeneous control Lyapunov function. Neverthe-less, it is more restrictive than Theorem 3.

Theorem 7. Consider the single input system (2) with m = 1 wheref0 (x) is homogeneous of degree d0 and f1 (x) homogeneous of degreed1 with respect to the weights (r1, . . . , rn) ∈ Rn>0. If there exists acontinuously differentiable control Lyapunov function V : V → R≥0for the control system (2) homogeneous of degree d with respect to theweights (r1, . . . , rn) ∈ Rn>0, then the feedback control (5) where

p =2q (d+ d1)d+ d0

, p > 1, q > 1,

is homogeneous of degree d0− d1 and the closed-loop system (3)–(5)is homogeneous of degree d0 with respect to the weights (r1, . . . , rn).If furthermore V satisfies the small control property, then the feedbackcontrol (5) continuously stabilizes the system (2).Proof. As V (x) is homogeneous of degree d, f0 (x) is homogeneousof degree d0 and f1 (x) is homogeneous of degree d1 with respectto the weights (r1, . . . , rn) ∈ Rn>0, Lemma 6 implies that a (x)is homogeneous of degree d + d0 and B (x) is homogeneous ofdegree d + d1. Theorem 3 implies that the feedback control (5) ishomogeneous of degree d0 − d1.The closed loop system satisfies

(f0 + f1u) (∆λ (x)) = λd0 f0 (x)+ λd1 f1 (x) λd0−d1u (x)= λd0 (f0 + f1u) (x) . �

Example 8. Consider the system{x1 = x1 − x2x2 = x1 + x2u

and the control Lyapunov function

V (x) =x21 + x

22

2homogeneous of degree 2 with respect to the weights (1, 1).f0 (x) =

(x1 − x2x1

)and f1 (x) =

(0x2

)are homogeneous of degree

0 with respect to the weights (1, 1). Theorem 7 with p = 2q, forinstance (p, q) = (4, 2), implies that the system is stabilizable bythe feedback control

u (x) = −x21 +

(x81 + x

82

) 14

x22which is homogeneous of degree 0 with respect to the weights(1, 1). Moreover, the closed-loop system is then homogeneous ofdegree 0 with respect to the weights (1, 1).As Theorem 5 generalized Theorem 3 to multi-input systems, it isalso possible to generalize Theorem 7 to multi-input systems.

5. Concluding remarks

The paper deals with the design of an homogeneous feedbackcontrol based on control Lyapunov functions satisfying anhomogeneous condition. The control law is derived from the well

2984 E. Moulay / Automatica 44 (2008) 2981–2984

known Sontag control. When the affine control system and thecontrol Lyapunov functions are homogeneous, the closed-loopsystem is also homogeneous.

Acknowledgements

The author is extremely grateful to Sergej Celikovsky for themany discussions he has had about the notion of homogeneity.

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