Integral Characterizations of Asymptotic Stability

Integral characterizations of asymptotic stability

? Elena Panteley†, Antonio Lor´ıa• and Andrew R. Teel

†INRIA Rhoneˆ Alpes ZIRST - 655 Av. de l’Europe 38330 Montbonnot Saint-Martin, France [email protected] •C.N.R.S. UMR 5528 LAG-ENSIEG, B.P. 46, 38402, St. Martin d’Heres, France [email protected] ?Center for Control Engineering and Computation (CCEC) Department of Electrical and Computer Engineering University of California Santa Barbara, CA 93106-9560, USA [email protected]

Keywords: UGAS, stability of sets, Lyapunov stability ever this is too restrictive. For time-varying systems, Naren- analysis. dra and Anaswamy [4] proposed to use an observability type condition which is formulated as an integral condition on the derivative of the Lyapunov function. This approach was fur- ther developed in the papers of Aeyels and Peuteman [1], Abstract where a “difference” stability criterion was formulated as a condition on the decrease of the difference between the val- We present an integral characterization of uniform global ues of the “Lyapunov” function at two time instances; more asymptotic stability (UGAS) of sets and use this character- precisely it is required that that there exists T > 0 such that ization to provide a result on UGAS of sets for systems with for all t t the following inequality is satisfied: output injection. ≥ ◦ V (t + T, x(t + T )) V (t, x) γ( x(t) ), − ≤ | | 1 Introduction where γ . The advantage∈ K∞ of this formulation is firstly, that the func- Starting with the original paper by Lyapunov [3], the concept tion V (t, x) is not required to be continuously differentiable of stability of solutions of a differential equation x˙ = f(t, x) nor locally Lipschitz and secondly, this formulation allows a was formulated as a stability problem with respect to some certain increase of the derivative of the Lyapunov function at function F (x) of the state. The main stream of the results for some moments (in case it exists). stability of differential equations is devoted to the analysis of Another result to be mentioned is related to the ISS con- stability properties of the origin x = 0 assuming that F (x) = cept of Sontag. In the paper [5] for the case of time-invariant x . At the same time, stability of sets (both, compact or not) systems without inputs the author gives the following “inte- |was| extensively studied in 60’s and 70’s by several authors, gral” criterion of asymptotic stability (plus some other con- as well as stability with respect to part of the coordinates. ditions): The study of stability of sets finds applications in control problems for systems with cyclic coordinates, in output con- t trol, tracking control and stability analysis of the origin for α( x(τ, x ) )dτ γ( x ) t 0, | ◦ | ≤ | ◦| ∀ ≥ time-varying systems (cf. section 2). Z0 The classical Lyapunov approach to conclude (asymptotic) where the functions α, γ . Notice that there is no need stability of the origin consists in finding a positive definite of Lyapunov function in this∈ K formulation,∞ however this con- (Lyapunov) function of the state, which is strictly decreasing dition is to be verified at every time instant. along the system’s trajectories. The typical condition used In this paper we relax the last integral condition and for- to guarantee this property is that the time derivative of the mulate necessary and sufficient (“integral”) conditions for Lyapunov function is negative definite. In applications, how- uniform asymptotic stability of closed, not necessary compact sets for ordinary differential equations with non Lips- 1. The set is UGAS. chitz right hand side. In this sense, our result generalizes A those mentioned before, at the price of imposing an extra 2. (a) The set is UGS; assumption on uniform global stability (UGS). It must be A (b) there exists a continuous positive definite function mentioned that this assumption is reasonable in many appli- , and for each pair of strictly pos- cations of adaptive control, and when dealing with passive γ : R 0 R 0 itive real≥ → numbers≥ satisfying , there exists systems. This result is next used to formulate a criterion ν r such that for all ,≤ all and establishing invariance of asymptotic stability of nonlinear βrν > 0 t 0 x Br all solutions x(t, x ) we have≥ ◦ ∈ systems under output injection. ◦ t [γ( x(τ, x ) ) ν] dτ βrν . 2 Definitions | ◦ |A − ≤ Z0 In this paper we study differential equations x˙ = F (x) (1) where x Rn and F ( ) is continuous and throughout the Remark 1 In [5] it is mentioned that, for the case when paper we assume∈ that the· system (1) is forward complete, that F (x) is locally Lipschitz, if ν = 0 and γ the same ∈ K∞ is we assume that for each x Rn, every solution x(t, x ) result holds true. Furthermore, Sontag does not impose the of (1) is defined on [0, ). ◦ ∈ ◦ UGS assumption. The latter follows as a particular case of ∞ We will study stability of (1) with respect to closed, not Sontag’s property of ISS for a system with zero input. necessarily compact, sets Rn. Given a closed set A ⊂ A ⊂ Rn and x Rn, we define Next we consider the system: ∈ x := inf x z . (2) x˙ = F (x) + K(x) (6) | |A z | − | ∈A For the system (1), the set is said to be uniformly globally where K : Rn Rn is continuous. stable (UGS) if the systemA (1) is forward complete and there → exists ρ such that ∈ K∞ Lemma 2 If x(t, x ) ρ( x ) t 0 . (3) | ◦ |A ≤ | ◦|A ∀ ≥ 1. The set is UGS, For the system (1), the set is said to be uniformly globally A A asymptotically stable (UGAS) if the set is UGS and for 2. for the system x˙ = F (x) there exist a locally Lipschitz A n each r > 0, ε > 0 there exists T > 0 such that function V : R R 0 and class- functions αi, i = 1,..., 4, such→ that ≥ K∞ x r , t T = x(t, x ) ε . (4) | ◦|A ≤ ≥ ⇒ | ◦ |A ≤ α1( x ) V (x) α2( x ), Local versions of these definitions apply when there exists | |A ≤ ≤ | |A r > 0 such that the given bound on the trajectories, holds for n n and for almost all x R , all x r, where r = x R : x < r . ∈ ◦ ∈ B B { ∈ | |A } Note that UGAS of the origin for a time-varying differen- ∂V (x) ˙ n 1 F (x) α3( x ), tial equation X = f(X, t) with X R − is equivalent to ∂x ≤ − | |A UGAS of the set x := (XT , p)T ∈ n : X = 0 for the ∂V (x) R ∂x α4( x ), differential equation ∈ | | ≤ | |A X˙ f(X, p) n m x˙ = = =: F (x) . (5) 3. there exist a function h : R R , nondecreasing p˙ 1 functions β, k , k : →, a continuous, posi- 1 2 R 0 R 0 tive definite function γ ≥and→ a class-≥ function k such K∞ 3 Integral tools for Lyapunov stability analy- that sis K(x) k1( x )k( h(x) ), | | ≤ | |A | | In this section we present some results on UGAS for nonlin- h(x) k2( x ), | | ≤ | |A ear systems, stated using integrability conditions, opposed to t those expressed in terms of Lyapunov functions, more com- γ( h(x(t, x )) )dt β( x ) t 0 . | ◦ | ≤ | ◦|A ∀ ≥ monly used. Z0

Lemma 1 The following statements are equivalent: Then the set is UGAS for the system (6). A References

[1] D. Aeyels, and J. Peuteman, A new asymptotic stability criterion for non-linear time-variant differential equations, IEEE Trans. on Automat. Contr., 43(7):968–971, 1998.

[2] Y. Lin, E.D. Sontag, and Y. Wang, A smooth converse Lyapunov theorem for robust stability, SIAM J. on Contr. and Opt., 34:124–160, 1996.

[3] A. M. Lyapunov, Probleme` de la stabilite´ du mouvement, Annales de la faculte´ de sciences de Toulouse, 9:203- 474, 1907. (Translation from the original published in Comm. Soc. Math., Kharkov 1893, reprinted in Ann. Math. Studies 17, Princeton 1949). See also ”Stability of motion” Academic Press: NY, 1996. [4] K. Narendra and A. Annaswamy, Persistent excitation in adaptive systems, Int. J. of Contr., 45(1):127–160, 1987. [5] E. D. Sontag, Comments on integral variants of ISS, Sys- tems and Control Letters, 34: 93–100, 1998.