Converse Lyapunov theorem and Input-to-State Stability

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1 Converse Lyapunov theorem and Input-to-State Stability April 6, Converse Lyapunov theorem In the previous lecture, we have discussed few examples of nonlinear control systems and stability concepts using Lyapunov theory. In this lecture, we will cover the Converse Lyapunov theorem (i.e., the existence of (smooth Lyapunov function and the concept of input-to-state stability. The course material is mainly based on the expositions presented in [3, 4, 5]. Terminologies (from the previous lecture. Consider a nonlinear system described by ẋ = f(t, x, x(t 0 = x 0 R n (1 where f : [0, R n R n is piecewise-continuous in t and locally Lipschitz in x. The point x R n is an equilibrium point for (1 if f(t, x = 0 for all t t 0. Definition 1.1. The origin ( x = 0 is 1. stable if for every ɛ > 0, there exists δ(ɛ, t 0 > 0 such that x 0 < δ(ɛ, t 0 x(t, x 0 < ɛ t t 0 ; 2. asymptotically stable if it is stable and there exists c(t 0 > 0 such that x 0 < c(t 0 lim t x(t, x 0 = 0; 3. exponentially stable if it is stable and there exist k, λ, c(t 0 > 0 such that x 0 < c(t 0 x(t, x 0 k x 0 e λ(t t 0 ; 4. uniformly stable if for every ɛ > 0, there exists δ(ɛ > 0 such that x 0 < δ(ɛ x(t, x 0 < ɛ t t 0 ; 1

2 5. uniformly asymptotically stable if it is uniformly stable and there exists c > 0 such that x 0 < c lim t x(t, x 0 = 0 uniformly in t uniformly exponentially stable if it is uniformly stable and there exist k, λ, c > 0 such that x 0 < c x(t, x 0 k x 0 e λ(t t 0 uniformly in t 0. Consider a continuously-differentiable (C 1 function V : [0, B d R which satisfies α 1 ( x V (t, x α 2 ( x (t, x [0, B d, where α 1, α 2 K. Using such V, it has been shown in the previous lecture that the following implications hold: (P1 t + f(t, x 0 (t, x [0, B d the origin of (1 is uniformly stable; (P2 t + f(t, x α( x (t, x [0, B d, α K the origin of (1 is uniformly asymptotically stable. When we restrict ourselves to the autonomous system ẋ = f(x with the associated function V : B d R satisfying α 1 ( x V (x α 2 ( x x B d, where α 1, α 2 K, then the following implications hold: (P3 f(x 0 x B d the origin is stable; (P4 f(x α( x x B d, α K the origin is asymptotically stable. Note that if the function V is radially unbounded, then the above implications on the asymptotic stability hold globally. In the present note, we will discuss the reverse implications. More precisely, we are interested to establish that if the origin of (1 is uniformly asymptotically stable then there exists a smooth function V : [0, B d [0, such that the antecedent in (P2 holds. Similar reverse implication can be made with regards to the asymptotic stability of an autonomous system (c.f., (P4 2

3 Such converse results are useful when one wants to deduce or obtain other stability properties that are derived from the asymptotic stability property. For example, they have been used to obtain robustness property of nonlinear systems via the input-to-state stability (ISS concept which we will discuss further later. However, we still need to obtain or to calculate such a Lyapunov function when we are going to analyze the stability of an interconnected systems using Lyapunov analysis. In the past decade, a numerical algorithm based on the sum-of-squares programming has been developed and widely used to find a polynomial Lyapunov function, see, for example, [1]. The following theorem provides the converse Lyapunov theorem for systems that are exponentially stable. Theorem 1.2. Consider the nonlinear system in (1 with x = 0 and with bounded Jacobian matrix f/ on B d. If the system is uniformly locally exponentially stable, i.e., there exist positive constants k, λ, d 0 such that x(t k x(t 0 e λ(t t 0 x(t 0 B d0, d 0 < d, t t 0, then there exists a C 1 function V : [0, B d0 [0, satisfying c 1 x 2 V (t, x c 2 x 2 + t f(t, x c 3 x 2 c4 x, with c 1, c 2, c 3, c 4 > 0. Moreover, if d = and the origin is uniformly globally exponentially stable, then the function V is defined and satisfies the above inequalities on R n. Proof. We will show that the Lyapunov function V (t, x satisfying the above inequalities is constructed based on the solutions of the differential equation. Let V (t, x = t+δ t z(τ, t, x 2 dτ, where δ is a positive constant (to be chosen later and z(τ, t, x is the solution of (1 with x 0 = x B d0 and t 0 = t. By the exponential stability of the solution, it follows that t+δ V (t, x x 2 k 2 e 2λ(τ t dτ = k2 2λ t ( 1 e 2λδ x 2. By the boundedness of the Jacobian matrix of f and by the assumption that f(t, 0 = 0, we have that f(t, x L x for some L > 0 and for all x B d0. This implies that z(τ, t, x 2 x 2 e 2L(τ t. 3

4 In other words, V (t, x x 2 t+δ t e 2L(τ t dτ = 1 ( 1 e 2Lδ x 2 2L holds for all x B d0. Hence the first inequalities hold with c 1 = 1 e 2Lδ 2L c 2 = k2 (1 e 2λδ. 2λ Now, it remains to calculate the time derivative of V (t, x. By direct computation, we have t + f(t, x = z(t + δ, t, x 2 z(t, t, x 2 + t+δ t+δ t 2z T (τ, t, x + 2z T z(τ, t, x (τ, t, x dτf(t, x t = z(t + δ, t, x 2 x 2 (1 k 2 e 2λδ x 2. and z(τ, t, x dτ t By taking δ = ln(2k 2 /(2λ, the second inequality in the theorem holds with c 3 = 1/2. The last claim on the boundedness of the gradient of V can be obtained in a similar way (see also the proof of Theorem 4.15 in [3]. As an example, consider the following system ẋ = x 3 x, x(0 = x 0. The analytical solution of the system s trajectory is given by 1 x(t = x 0 (e 2t (x x 2 0. It is immediate to see that the trajectory x(t converges exponentially to zero for all initial conditions x 0, i.e., the origin is exponentially stable. Using a simple quadratic function V (x = 1 2 x2 as a Lyapunov function, we can also conclude the exponential convergence of the system to zero. Let us construct the Lyapunov function as used in the proof of the converse Lyapunov theorem above. For this purpose, the corresponding constants k, λ are given by k = 1, λ = 1. Now if we use the same Lyapunov construction as in the converse Lyapunov theorem, then the corresponding Lyapunov function is given by V (x = δ 0 x 2 (e 2τ (x x 2 dτ 1 The solution can be found by using standard numerical software such as Mathematica, Maple, Matlab, or the free web-based symbolic tools in 4

5 where δ = ln(2/2. A routine computation to the time-derivative of this Lyapunov function with x B d, d > 0 gives us δ (x ( x3 x = ( x 3 x ( δ = ( x 3 x 0 x 2 (e 2τ (x x 2 dτ 2x 0 (e 2τ (x x 2 x2 (2xe 2τ 2x (e 2τ (x x 2 dτ 2 ( δ = ( 2x 4 2x (e 2τ (x x 2 x 2 (e 2τ 1 (e 2τ (x x 2 dτ 2 ( δ = ( 2x 4 2x 2 e 2τ 0 (e 2τ (x x 2 dτ 2 ( δ ( 2x 4 2x 2 1 (e 2δ (x x 2 dτ 2 = ( 2x 4 2x 2 ln(2/2 (x c 3 x 2, 0 where c 3 = ln(2. (d Theorem 1.2 shows the converse Lyapunov theorem for exponentially stable systems. In the following theorems, we will present two standard results on the converse Lyapunov theorems. The first one is related to the existence of Lyapunov function for uniform asymptotic stable (non-autonomous systems and the second one is related to the existence of radially unbounded Lyapunov function for asymptotic stable (autonomous systems. Theorem 1.3. [3, Theorem 4.16] Consider the nonlinear system in (1 with x = 0 and with bounded Jacobian matrix f/ on B d (uniformly in t. Let β be a KL function and d 0 > 0 is such that β(d 0, 0 < d. If the system is uniformly asymptotically stable, i.e., x(t β( x(t 0, t t 0 x(t 0 B d0, t t 0, then there exists a C 1 function V : [0, B d0 [0, satisfying α 1 ( x V (t, x α 2 ( x + f(t, x α t 3( x α 4 ( x, with α 1, α 2, α 3, α 4 are K functions defined on [0, d 0 ]. Theorem 1.4. [3, Theorem 4.17] Consider an autonomous system ẋ = f(x with a locally Lipschitz f. If the origin is asymptotically stable with the domain of attraction of R A then there exists a smooth positive definite function V (x and a continuous positive definite function W (x, both defined for all x R A, such that V (x as x R A f(x W (x x R A 5

6 and for any c > 0, the set {x V (x c} is a compact subset of R A. The function V (x is radially unbounded if R A = R n. Other related result on the converse Lyapunov theorem can be found in [4, 8]. In [4], the authors consider stability with respect to a compact set, rather than an equilibrium point. In [8], the converse theorem is discussed for nonlinear systems whose state convergence is described by a modified (but more general KL estimate. 2 Input-to-State Stability So far, we have discussed the asymptotic stability characterization of an equilibrium of nonlinear systems which can be time-varying. Such characterization is fairly limited (for the control design since we do not have any estimation on the properties of the state trajectories when external (such as, reference, disturbance or control signals are affecting the systems. Of course, for a given signal, we can write the systems as a time-varying system and analyze the stability of the equilibrium as before. However, this does not allow us to conclude the stability of the equilibrium if a different external signal is applied. On the other hand, one can naively think that if an exponentially converging external input is applied affinely to an autonomous system whose origin is globally exponentially stable (and hence, there exists a radially unbounded Lyapunov function then the origin remains globally attractive. This claim is not true in general and is shown in the following simple counter-example. Example 2.1. [7] Consider the following affine nonlinear system { ẋ1 = g(x Σ : 1 x 2 x 1 ẋ 2 = 2x 2 + u, where x 1, x 2, u R. Let the function g be continuous, g(s 1 for all s, g(s = 1 for all s < 0.5 and for all s > 1.5, and g(1 = 1. Under such an assumption, it has been shown in [7] that if d = 0, the system Σ is GES. In particular, it satisfies x(t 9e t x(0. Now, if u(t = x 2 (0e t, x 1 (0 0 and x 2 (0 = x 1 (0 1, we have that the solution to the differential equation is given by x 1 (t = e t x 1 (0 x 2 (t = e t x 2 (0. This shows that the state x 1 becomes unbounded as t. 6

7 This example shows that an exponential exogeneous signal, which is typically not expected to destroy the stability of a system, can create a problem to a GES system, which is intuitively a robust system. It is shown above that even in the neighborhood of the origin, we can introduce a small exponentially decaying input signals (where the initial condition u(0 is equal to x 2 (0 such that the system becomes unstable. Let us now consider a nonlinear system with external input described by where x(t R n, u(t R m and f is locally Lipschitz. ẋ = f(x, u x(t 0 = x 0, (2 As shown before that the global exponential stability property of the autonomous system (i.e., when u = 0 does not guarantee the stability of the origin when an exponentially converging input u is applied to the system. We will now study the input-to-state property by investigating whether one of the following properties is satisfied. u L x L u L 2 x L. For linear systems described by ẋ = Ax + Bu with x(t 0 = x 0, the input-to-state properties above are described in the following ways: u L x L u L 2 x L : x(t, x 0 k x 0 e λ(t t0 + c sup s [t0,t] u(s : x(t, x 0 k x 0 e λ(t t0 + t t 0 c u(s 2 ds. The first term of the inequality describes the influence of initial condition to the state estimate which decays exponentially to zero. When the influence of the initial condition has become small, the last term of the inequality, which represents the influence of the input to the state estimate, will become dominant. A natural generalization of the first notion to the nonlinear systems is to consider a KL function that substitutes the exponential decaying term, and a K function that replaces the constant gain c in the second term. In other words, we have the following notion of input-to-state stability, or ISS, in short. Definition 2.2. The system (2 is input-to-state stable if there exist a KL function β and a K function γ such that the trajectory x(t of (2 satisfies ( for all t t 0. x(t β( x 0, t t 0 + γ sup u(s s [t 0,t] Similarly, for the second notion, we have the generalization of L 2 L from the linear systems to the nonlinear ones by replacing the exponential decaying term by a KL function and the constant c by a K function. 7

8 Definition 2.3. The system (2 is integral input-to-state stable (or iiss if there exist a KL function β, a K function α and a K function γ such that the trajectory x(t of (2 satisfies ( t x(t β( x 0, t t 0 + α γ( u(s ds t 0 for all t 0. The following theorems give the Lyapunov characterization of ISS and iiss systems. Theorem 2.4. [5, 6] The nonlinear system (2 is ISS if and only if there exists a smooth ISS Lyapunov function V (x satisfying α 1 ( x V (x α 2 ( x f(x, u α 3( x + γ( u for all x R n, u R m where α 1, α 2, α 3 and γ are K functions. (3 Proof. Sufficiency part. By the bounds on V (x, it follows directly that α 3 ( x α 3 (α 1 2 (V (x. Hence, where α := α 3 α 1 2. V (x α(v (x + γ( u, For all x M := {x R n 1 α(v (x γ( u }, we have 2 V (x 1 ( 2 α(v (x + γ( u 1 2 α(v (x 1 α(v (x. 2 By the comparison lemma, it follows immediately that for all x 0 M, there exists a KL function β such that V (x(t β(v (x 0, t. On the other hand, for all x(t such that 1 α(v (x(t < γ( u(t, we have 2 V (x(t α 1 (2γ( u(t α (2γ( 1 sup u(s s [t 0,t] Combining the above two inequalities, we arrive at ( {β(v (x 0, t, α 1 V (x(t max 8 2γ( sup u(s s [t 0,t]. }.

9 Using the lower- and upper estimate of V (x, { x(t α 1 1 (V (x(t max β( x 0, t + γ ( sup u(s s [t 0,t] β( x 0, t, γ ( sup u(s s [t 0,t] where β(s, t := α 1 1 (β(α 2 (s, t and γ(s := α 1 1 ( α 1 (2γ(s. Note that in the above theorem, the ISS Lyapunov function under consideration is assumed to be smooth. However, as can be seen in the proof of sufficiency, it is sufficient to assume that V (x is continuously differentiable. The theorem provides in fact a strong statement that for ISS systems, there exists a smooth (and not only continuously differentiable ISS Lyapunov function. Theorem 2.5. [2, 5] The nonlinear system (2 is iiss if and only if there exists a smooth iiss Lyapunov function V (x satisfying α 1 ( x V (x α 2 ( x f(x, u α 3( x + γ( u for all x R n, u R m where α 1, α 2, γ are K functions and the function α 3 : [0, [0, is positive-definite (thus, it does not have to be of K. Consider the following simple example. Let the system be given by } (4 ẋ = x 3 + u. (5 By using V (x = x 2, it can be checked that the origin of the autonomous part of the system, i.e., ẋ = x 3, is globally asymptotically stable. Now, if we use the same function V (x = x 2 as a candidate for an ISS Lyapunov function, the time-derivative of V along the trajectory of the system is (x ( x3 = 2 x 4 + 2xu ( x 2 x x u 4/3, u 4/3 4 where the second inequality is due to the application of Young s inequality. It shows that the function V (x is an ISS Lyapunov function where α 1 (s = α 2 (s = s 2, α 3 (s = 3 2 s4 and γ(s = 3 2 s4/3. In other words, the system (5 is ISS. Another simple example is the following simple mechanical systems with a nonlinear spring and is described by mÿ + dẏ + ky + ka 2 y 3 = u, (6 9

10 where m, d, k are the mass, damping and spring constants, respectively and a > 0 is a positive constant. The system can be written into state-space equations as follows ẋ 1 = x 2 ẋ 2 = 1 ( dx m 2 kx 1 ka 2 x u, (7 where x 1 = y and x 2 = ẏ. If we consider u = 0, the origin is globally asymptotically stable with the Lyapunov function given by V (x { = 1 2 kx ka2 x c 1 x 1 x mk, 2 mx2 2 2mkd where c 1 > 0 is chosen such that c 1 < min }. Using the same 2mk+d 2 function V (x, it can be checked that it can be used as an ISS Lyapunov function. In the above results, we have not established the relation between the global asymptotic stability of the autonomous system to the ISS property. The following result from [6] has established an important result towards this endeavour. Proposition 2.6. [6, Theorem 1] The nonlinear system (2 is ISS if and only if the origin of the autonomous system is globally asymptotically stable and there exist a continuous, non-decreasing function γ : [0, [0, with γ(0 = 0 such that for all x 0 R n and for all bounded input signal u, lim x(t γ( sup u(s. t s [0, The last property is usually called in the literature as the asymptotic gain property. References [1] A.A. Ahmadi, P.A. Parrilo, Stability of Polynomial Differential Equations: Complexity and Converse Lyapunov Questions, [2] D. Angeli, E. Sontag, Y. Wang, A characterization of integral input-to-state stability, IEEE Trans. Aut. Contr., vol. 45, no. 6, pp , [3] H.K. Khalil, Nonlinear Systems, Prentice-Hall, Upper Saddle River, NJ, [4] Y. Lin, E.D. Sontag, Y. Wang, A smooth converse Lyapunov theorem for robust stability, SIAM J. Contr. Optim., vol. 34, pp , [5] E.D. Sontag, Input-to-state stability: basic concepts and results, Nonlinear and Optimal Control Theory, P. Nistri & G. Stefani (eds., Prentice-Hall, pp , Springer-Verlag, Berlin, [6] E.D. Sontag, Y. Wang, New characterizations of input-to-state stability, IEEE Trans. Aut. Contr., vol. 41, no. 9, pp ,

11 [7] A.R. Teel, J. Hespanha, Examples of GES systems that can be driven to infinity by arbitrarily small additive decaying exponentials, IEEE Trans. Aut. Contr., vol. 49, no. 8, pp , [8] A.R. Teel, L. Praly, A smooth Lyapunov function from a class-kl estimage involving two positive semidefinite functions, ESAIM: Contr. Opt. Calc. Var., vol. 5, pp ,