Proceedings of the 41st IEEE Conference on Decision and Control Las Vegas, Nevada USA, December 2002
WeM11-6
Nonsmooth functions and uniform limits of smooth control Lyapunov functions Ludovic Faubourg LAAO, Universit´e de Bourgogne
[email protected] Abstract When is a (non-smooth) function the limit of a sequence of smooth (continuously differentiable) control Lyapunov functions ? It is known that a “non-smooth control Lyapunov function in the sense of Clarke (convex) gradient” is indeed the limit of a sequence of smooth control Lyapunov functions; we show that the converse is not true by exhibiting an counter example. We also give a condition under which a function cannot be the limit of a sequence of smooth control Lyapunov functions. Of course, a (non smooth) function that satisfies this condition cannot either be a control Lyapunov function in the sense of Clarke gradient. 1 Introduction Consider the following affine control system : x˙ = X0 (x) +
m X
ui Xi (x)
(1)
k=1
where X0 , . . . , Xm are smooth vector fields on Rn . This note deals with locally asymptotic stabiliztion of the origin x = 0 by a continuous feedback. Definition 1 (CLF: control Lyapunov function) A function V : Rn → R is said to be a CLF for (1) if it is continuously differentiable, positive definite, radially unbounded and satisfies for all x ∈ Rn \{0}, ¶ µ ∂V ∂V Xi (x) = 0, (i = 1 . . . m) ⇒ X0 (x) < 0 (2) ∂x ∂x The interest of CLFs is that existence of a continuous stabilizing feedback implies existence of a CLF, and in turn, when a CLF is explicitely known for the system (1), there are universal ways to design asymptotically stabilizing feedback controls, see [6] for example. In [4], we designed control Lyapunov functions for Judjevic-Quinn type systems. If the value function of an optimal control problem is smooth (plus other minor hypothesis) then it is a CLF.
0-7803-7516-5/02/$17.00 ©2002 IEEE
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Jean-Baptiste Pomet INRIA, Sophia Antipolis, France
[email protected] Unfortunately a value function is often nonsmooth. One natural question is: can this nonsmooth function be “reshaped” into a smooth CLF ? Or in other words : is it the uniform limit of a sequence of CLFs ? This is not true in general. For instance, the following system : x˙ 1 = x33 , x˙ 2 = x3 , x˙ 3 = u ,
(3)
is not (continuously) asymptotically stabilizable at x = 0 because it does not satisfy the Brockett condition (see [1]), hence there exists no CLF for this system. Yet, because the system is globally controllable we can build a value function for a reasonnable problem of optimal control. So we can claim that this value function can not be approached by a sequence of CLFs. The question raised by this analysis is to characterize the set of non-smooth functions that are uniform limits of CLFs for a fixed system (1). We shall give partial answers to that question, requiring two definitions. If V : Rn → R is not continuously differentiable, there are many substitutes to classical differentiation [2]. One of them is : Definition 2 (Clarke gradient) Let V : Rn → R be locally Lipchitz. Its Clarke gradient at x ∈ Rn is (DV stands for the subset of Rn where V is differentiable and “co” for the convex hull) : ½ ¾ ∂V (xk ) : xk → x, xk ∈ DV ∂V (x) = co lim (4) ∂x Definition 3 (CG-CLF) (CLF in the sense of Clarke gradient) A locally Lipchitz, positive definite and radially unbounded function V is a CG-CLF for the system (1) if it satisfies for all x ∈ Rn \{0}, ζ ∈ ∂V (x), ζXi (x) = 0, (i = 1 . . . m) ⇒ ζX0 (x) < 0 .
(5)
2 Uniform limit of CLFs versus CG-CLF In [5], it is stated that existence of a CG-CLF implies existence of a CLF. This result is proved there by providing a sequence of CLFs that tends uniformly to the initial CG-CLF. Hence we have :
Theorem 1 (Rifford) Any CG-CLF is the uniform limit of a sequnce of CLFs. If the converse to this result were true, then the concept of CG-CLF would be the right concept to answer the question raised in the introduction. This is not the case, as can be seen from the following example of a function that is the uniform limit of a sequence of CLFs for some control system, but fails to satisfy the condition to be a CG-CLF.
1. V1 (x0 ) = V2 (x0 ),
Consider the following affine control system in R2 : x˙ y˙
= =
−u(5 + y) −y + ux
(6)
Let us now define another function V : R2 → R, which is obtained by modifying a CLF for (6), V = x2 + y 2 in a neighborhood of the point (0, −5): V (x, y)
=
α(r)(x2 + y 2 ) ¶ µ 4x2 − y + 10 (1 − α(r)) τ0 (r)
is defined in a neighborhood of a singular point by the minimum of two differentiable functions. In this situation we give a necessary condition for a function to be a uniform limit of a sequence of CLFs. We consider the single input (m = 1) affine system x˙ = X0 (x)+uX1 (x). We have the following proposition : Proposition 2 Let V1 , V2 : Rn → R be two functions continuously differentiable in a neighborhood of a point x0 . Let V : Rn → R be defined by V = min(V1 , V2 ). Moreover suppose that :
(7)
p with r = x2 + (y + 5)2 , τ0 : R+ → R+ and α : R+ → [0, 1] defined by ( ³ ρ´ if ρ ∈ [0, 2] ρ 1− (8) τ0 (ρ) = 4 1 if ρ > 2 if ρ ≤ 2 0 1 2 α(ρ) = (ρ − 2) (5 − ρ) if 2 < ρ ≤ 4 (9) 4 1 if ρ > 4 The function V is continuous on R2 , and continuously differentiable on R2 \{(0, −5)}. Proposition 1 V satisfies the following properties: 1. V is a uniform limit of control Lyapunov functions for the system (6). 2. V is positive definite and radially unbounded with a unique minimum at (0, 0). 3. V is not a CG-CLF. Proof: See [3], or a detailed publication by the authors (under preparation), for a comprehensive proof.
3 A necessary condition to be a uniform limit of CLFs In the section we study the first type of nondifferentiability, that means the case where a nonsmooth function
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∂V2 ∂V1 X1 (x0 ) > 0, X1 (x0 ) < 0 ∂x ∂x µ ¶ ∂V1 ∂V2 ∂V2 ∂V1 X1 X0 − X1 X0 (x0 ) > 0 3. ∂x ∂x ∂x ∂x 2.
Then for all neighborhood Ω of x0 , there exists ² > 0, such that : for any continuously differentiable function S : Ω → R satisfying sup |S(x) − V (x)| < ²,
x∈Ω
there is a point x ¯ ∈ Ω such that property (2) fails for x=x ¯. Proof: See [3], or a detailed publication by the authors (under preparation), for a comprehensive proof. The conclusion of this proposition obviously prevents the function V from being the uniform limit of a sequence of CLFs. References [1] R. W. Brockett. Linear feedback systems and the groups of Galois and Lie. Linear Algebra Appl., 50:45– 60, 1983. [2] F. H. Clarke, Yu. S. Ledyaev, R. J. Stern, and P. R. Wolenski. Nonsmooth analysis and control theory. Springer-Verlag, New York, 1998. [3] Ludovic Faubourg. Construction de Fonctions de Lyapunov contrˆ ol´ees et stabilisation non lin´eaire. PhD thesis, Universit´e de Nice Sophia Antipolis, 2001. [4] Ludovic Faubourg and Jean-Baptiste Pomet. Control Lyapunov functions for homogeneous “Jurdjevic-Quinn” systems. ESAIM Control Optim. Calc. Var., 5:293–311 (electronic), 2000. [5] Ludovic Rifford. Probl`eme de stabilisation en th´eorie du contrˆ ole. PhD thesis, Universit´e Claude Bernard-Lyon 1, 2000. [6] Eduardo D. Sontag. A “universal” construction of Artstein’s theorem on nonlinear stabilization. Systems Control Lett., 13(2):117–123, 1989.
