International audienceWe present a new characterization of exponential stability for nonlinear systems in the form of Lyapunov functions which may be upper and lower bounded by monotonic functions satisfying a growth order relationship rather than being polynomials of the state's norm. In particular, one may allow for Lyapunov functions with arbitrary weakly homogeneous bounds

Loria, Antonio

Panteley, Elena

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A new characterisation of exponential stabilityElena Panteley, Antonio LoriaTo cite this version:Elena Panteley, Antonio Loria. A new characterisation of exponential stability. MTNS 2010,Jul 2010, Budapest, Hungary. pp.1487-1489, 2010. <hal-00831375>HAL Id: hal-00831375https://hal-supelec.archives-ouvertes.fr/hal-00831375Submitted on 6 Jun 2013HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.A new characterisation of exponential stabilityElena Panteley Antonio Lorı´aAbstract—We present a new characterization of exponentialstability for nonlinear systems in the form of Lyapunov func-tions which may be upper and lower bounded by monotonicfunctions satisfying a growth order relationship rather thanbeing polynomials of the state’s norm. In particular, one mayallow for Lyapunov functions with arbitrary weakly homoge-neous bounds.I. INTRODUCTIONConsider the ordinary differential equationx˙ = f(t, x) (1)where f is continuous in t and locally Lipschitz in xuniformly in t. Let x = 0 be an equilibrium point of thelatter and denote the solutions with initial conditions t◦ ∈ R+and x◦ ∈ Rn, by x(t, x◦, t◦). We recall that the trivialsolution x = 0 is uniformly exponentially stable if thereexist constants k, λ an r such that|x◦| < r, t ∈ R+ ⇒ |x(t, x◦, t◦)| ≤ k|x◦|e−λ(t−t◦) .(2)We say that the origin is uniformly globally exponentiallystable if r = ∞.Exponential stability of nonlinear systems describedby ordinary differential equations dates back at least toKrasovskii’s work in the late 1950s –cf. [4, Theorem 11.1].The classical characterisation of uniform global exponentialstability involves a Lyapunov function which satisfies upperand lower quadratic bounds of |x|. This has been extendedto bounds that are polynomial of any order –cf. [2], [10], [3]:Theorem 1 Let x = 0 be an equilibrium point for x˙ =f(t, x), where f is a locally Lipschitz function and D ⊂ Rnbe a domain containing x = 0. Then x = 0 is uniformlyexponentially stable if and only if there exist a continuouslydifferentiable function V : [0,∞)×D → R+ such thatk1|x|p ≤ V (t, x) ≤ k2|x|p, (3a)∂V∂t+∂V∂xf(x) ≤ −k3|x|p (3b)for all t ≥ t◦ ≥ 0 and all x ∈ D, where p and ki, i = 1, 2, 3are positive constants. If the assumptions hold globally, thenx = 0 is uniformly globally exponentially stable.Equivalent characterizations, in terms of a Lyapunov func-tions decreasing at sampling times have been reported in thecontext of adaptive control. See for instance [6], [3, Theorem4.5] and [1]. For time-varying differential equations withThe authors are with the C.N.R.S. at LSS-SUPELEC,3, Rue Joliot Curie, 91192 Gif s/Yvette, France E-mail:panteley[loria]@lss.supelec.frlocally Lipschitz right-hand side, the following is essentiallycontained in [5]:[Integral characterization of UGES] For the dynami-cal system x˙ = f(x, t) with f locally Lipschitz andsupx6=0 |f(x, t)|/|x| < ∞ the origin is uniformly globallyexponentially stable if and only if there exists γ > 0 andp ≥ 1 such that∫ ∞0|x(t, x◦, t◦)|pdt ≤ γ|x◦|p . (4)In [9] exponential stability is characterized by integralconditions for systems described by differential inclusions(convex, upper semi-continuous). Such characterization isuseful to establish UGES from inputoutput interconnectionproperties involving, e.g. L2 bounds. In this short notewe give new differential characterisation of exponentialstability equivalent to (3) but which does not rely on explicitpolynomial bounds but functions satisfying a growth-orderrelation. Further results are established for weakly homoge-neous systems.II. MAIN RESULTSConsider the systemx˙ = f(t, x), f(t, 0) = 0 (5)Theorem 2 Let Br := {x ∈ Rn : |x| < r} and suppose thatf(t, ·) is Lipschitz on Br uniformly in t. Let V : R+×Br →R+ be a continuously differentiable function such that for allx ∈ Br, all t ≥ t◦ and all t◦ ∈ R+α1(|x|) ≤ V (t, x) ≤ α2(|x|) (6)V˙ (t, x) ≤ −µV (t, x), (7)where µ > 0 is a constant and α1, α2 ∈ K∞ (functionsR+ → R+, strictly increasing, zero at zero and proper).Then, the origin of (5) is exponentially stable on Br if andonly if there exist constants c > 0, c1 > 0 and c2 ∈ (0, 1)such that the following inequalities hold for all s ≥ 0:α−11 ◦ α2(s) ≤ cs (8)α−11 ◦ (c1α2(s)) ≤ c2s . (9)If all the conditions of the theorem are satisfied globally (i.e.ifr = +∞ then the origin is uniformly globally exponentiable.Proof of Theorem 2I. Sufficiency.Following the arguments of the proof of [3, Theorem 3.8]we can show that, for any given D and any constants ρ > 0and r > 0 which satisfy Br ⊂ D and ρ < α1(r), all theProceedings of the 19th International Symposium on Mathematical Theory of Networks and Systems – MTNS 2010 •  5–9 July, 2010 • Budapest, HungaryISBN 978-963-311-370-7 1487solutions of (5) starting from the set of initial conditions x0 ∈{x ∈ Bα−12(ρ)}are well defined and moreover x(x0, t) ∈ Brfor all t ≥ 0.Now, let T = 1−c1µc1 if c1 < 1 and T = 1 otherwise.From (7) it follows that V (x(t)) ≤ V (x(τ))e−µ(t−τ) for allt ≥ τ ≥ 0 hence, V (x(t)) ≤ V (x(τ)) for all τ ∈ [0, t).Integrating (6) from t to t + T (with T defined above) weobtainV (x(t + T ))− V (x(t)) ≤ −µ∫ t+TtV (x(τ))dτ≤ −µ∫ t+TtV (x(t + T ))dτ≤ −µTV (x(t + T )) ∀ t ≥ 0,(1 )where in the second step above we used the fact thatV (x(τ)) ≥ V (x(t + T )) for all τ ∈ [t, t + T ]. Hence,(1+µT )V (x(t+T )) ≤ V (x(t)) for all t ≥ 0. From this weobtainV (x(t + T )) ≤11 + µTV (x(t))≤ c1V (x(t)) if c1 ≤ 1V (x(t + T )) ≤11 + µTV (x(t))≤ V (x(t)) ≤ c1V (x(t)) if c1 ≥ 1 .(11)Therefore, V (x(t + T )) ≤ c1V (x(t)) for all t ≥ 0. Usingthe bounds (11) and (6) we obtainα1(|x(t + T )|) ≤ c1α2(|x(t)|) ∀ t ≥ 0 .Then, from (9) it follows that for all t ≥ 0|x(t + T )| ≤ c2|x(t)|. (12)The rest follows the proof guidelines of [3, Theorem 4.5].For any t ≥ 0 let N = ⌊ tT ⌋, where ⌊ · ⌋ stands for the lowerinteger part. Divide the interval [t − NT, t] into N equalsubintervals then,|x(t)| ≤ c2|x(t− T )|≤ c22|x(t− 2T )|...≤ cN2 |x(t−NT )|. (13)Since for all t ≥ 0 we have V (x(t)) ≤ V (x0) therefore,|x(t)| ≤ α−11 α2(|x0|) ∀ t ∈ [0, T ] (14)consequently, from (8) it follows that |x(t)| ≤ c|x0| for allt ∈ [0, T ]. Combining the last bound with (13) we obtain|x(x0, t)| ≤ cN2 c|x0| ≤ cct/T2 |x0| = c|x0|e−bT ,where b = 1T ln1c2. In case conditions (11)-(9) are satisfiedglobally we obtain that the system is UGES.II. NecessityAssume that the system (5) is (globally) exponentiallystable for all x ∈ D (x ∈ Rn) i.e., there exist k, γ > 0such that the trajectories of (5) satisfy|x(t, x0)| ≤ k|x0|e−γt ∀ t ≥ 0, x0 ∈ D (x0 ∈ Rn).Then, from the converse theorem on exponential stability(see for example [3]) it follows that there exists a Lyapunovfunction V : D → R+ ( V : Rn → R+) that satisfies theinequalitiesa1|x|2 ≤ V (x) ≤ a2|x|2dVdxf(x) ≤ −a3|x|2for some positive constants ai, i = 1, 2, 3.Hence, the functions αi (i = 1, 2, 3) in (11), (7) are givenby αi(s) = ais2. Simple calculations show that inequalities(8), (9) are satisfied. Indeed, α−11 (s) =(sa1)1/2therefore,for arbitrary c > 0 we haveα−11 (c2α2(s)) =(cα2(s)a1)1/2=(ca2s2a1)1/2=√ca2a1s .From this it follows trivially that (8) is satisfied with c =√a2/a1 and (9) is satisfied for arbitrary c2 ∈ (0, 1) withc1 =a1c22a2. Remark 1• If all conditions of the theorem 2 are satisfied except(8), then proceeding as before we obtain from (13) and(14) that|x(t, x0)| ≤ α(|x0|)e−bt• The first part of the statement i.e., without requiring (8),(9) is equivalent to UGAS –cf. [8], [7].III. EXPONENTIAL STABILITY FROM WEAKLYHOMOGENEOUS BOUNDSDefinition 1 A real function f : Rn → R is said to behomogeneous of order k if for any constant α ≥ 0 thefollowing inequality holds:f(αx) = αkf(x).Classical conditions imposed on the bounds of V (x) andits derivative to insure exponential stability are formulatedin terms of powers of x which are evidently homogeneousfunctions. In this section we show that this classical resultcan be extended to the class of systems with weakly homo-geneous bounds on V (x) and its derivative.Our second theorem is stated in terms of Lyapunov func-tions satisfying weakly homogeneous bounds. We recall thatreal function f : R+ → R+ is weakly homogeneous if forsome function L ∈ K∞ one hasf(λx) ≤ L(λ)f(x) (15)for all λ > 0.E. Panteley and A. Loría • A New Characterisation of Exponential Stability 1488Theorem 3 The origin of x˙ = f(x) is a uniformly ex-ponentially stable equilibrium if and only if there existscontinuously differentiable function V : Br → R+, a weaklyhomogeneous function α ∈ K∞ such that for all x ∈ D andall t ≥ 0a1α(|x|) ≤ V (x) ≤ a2α(|x|) (16)V˙ (x) ≤ −a3V, (17)for some positive constants a1, a2, a3. If all the conditionsof the theorem are satisfied globally (with r = +∞) then thesystem (5) is uniformly globaly exponentially stable.We wrap up this note with an example for which uniformglobal exponential stability is difficult to conclude fromTheorem 1 yet, it may be concluded from our main results.Example. Consider the systemx˙1 = −x1 +x2g(x)1 + x21x˙2 = −x2 +x1g(x)1 + x22where g is any locally Lipschitz function. It is easy to showthat the origin is uniformly globally exponentially stable byinvoking Theorem 3 and using the Lyapunov functionV (x) =12(x41 + x42) + x21 + x22 .Indeed the total time derivative of V yields V˙ ≤ −2V . Notethat this function does not have lower nor upper polynomialbounds however, the function α1(s) :=12V (s) is of the samegrowth order as α2(s) := 2V (s). ⋄Proof. I. Sufficiency.From theorem 2 it follows that we need only to verifythat conditions (8) and (9) are satisfied. To simplify thecalculations let us take a1 = 1 (what can be always donejust by choosing e.g. Vnew(x) = V (x)/a1). Since α(s) isa weakly homogeneous function, then there exists l ∈ K∞such that α(λs) ≥ l(λ)α(s) for all λ > 0. Then, for anyc1 > 0, defining k = l−1(c1a2) we haveα−1(c1a2α(s)) ≤ α−1 (l(k)α(s)) ≤ α−1 (α(ks)) = ks(18)Inequality (18) is valid for any c1 > 0, therefore it is validfor c1 = 1, hence (8) is satisfied. Moreover, we can alwayschoose c1 so that k = l−1(c1a2) < 1, so that (9) is satisfiedas well. Therefore all the conditions of the theorem 2 aresatisfied and therefore the system (5) is exponentially stable.II. NecessityThe “only if” part of the proof follows directly the stepsof the proof of Theorem 2. IV. CONCLUSIONSWe have presented a new Lyapunov-like characterizationof exponential stability for ordinary differential equations.Such characterization covers naturally sufficient and neces-sary conditions for particular cases such as K-exponentialstability and for homogeneous systems.REFERENCES[1] D. Aeyels and J. Peuteman. A note on the liapunov approach to thestudy of asymptotic stability of differential equations. In Proc. 4th.European Contr. Conf., 1997. paper no. 226.[2] W. Hahn. Stability of motion. Springer-Verlag, New York, 1967.[3] H. Khalil. Nonlinear systems. Macmillan Publishing Co., 2nd ed.,New York, 1996.[4] N. N. Krasovskii. Problems of the theory of stability of motion.Stanford Univ. Press, 1963. Translation of Russian edition, Moscow1959.[5] A. Megretski and A. Rantzer. System analysis via integral quadraticconstraints. IEEE Trans. on Automat. Contr., 42(6):819–830, June1997.[6] K. Narendra and A. Annaswamy. Persistent excitation in adaptivesystems. Int. J. of Contr., 45(1):127–160, 1987.[7] E. Panteley and A. Lorı´a. Growth rate conditions for stability ofcascaded time-varying systems. Automatica, 37(3):453–460, 2001.[8] L. Praly and Y. Wang. Stabilization in spite of matched unmodelleddynamics and an equivalent definition of input-to-state stability. Math.of Cont. Sign. and Syst., 9:1–33, 1996.[9] A. Teel, E. Panteley, and A. Lorı´a. Integral characterizations ofuniform asymptotic and exponential stability with applications. Math.of Cont. Sign. and Syst., 15:177–201, 2002.[10] M. Vidyasagar. Nonlinear systems analysis. Prentice Hall, New Jersey,1993.Proceedings of the 19th International Symposium on Mathematical Theory of Networks and Systems – MTNS 2010 •  5–9 July, 2010 • Budapest, Hungary1489

A new characterisation of exponential stability

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A new characterisation of exponential stability

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