Input-to-State Stability of Switched Systems with Time Delays

Abstract

This chapter deals with the input-to-state stability of switched nonlinear systems with time delays. The proposed results demonstrate a connection between small-gain arguments in the context of input-to-state stability and the traditional Lyapunov-Razumikhin method for switched systems. By using the notion of average dwell time, it is shown that a switching among ISS systems with compatible Lyapunov-Razumikhin functions will not destroy the stability property if the switching is not too fast on average. Particularly, it is shown that the existence of a common Razumikhin function is sufficient to guarantee input-to-state stability for time-delayed systems under arbitrary switching.

Z.-P. Jiang—Work partially supported by the NSF grants DMS-0906659 and ECCS-1230040.

Y. Wang—Work partially supported by the NSF grant DMS-0906918.

Appendix

Although it is common in the study of the ISS property to consider inputs that take values in an Euclidean space ${\mathbb R}^m$ or a subspace in ${\mathbb R}^m$, the concept of input-to-state stability may also apply to the more general context when the inputs take values in a normed space which may be infinite dimensional. This is in particular the case in the contexts when Lyapunov-Razumikhin approach is used to deal with the delays on state variables (as in this work). Below we consider such a case when the input takes values in $C{[-\theta , 0]}$.

Consider a system as follows:

$$\begin{aligned} \dot{x}(t) = f(x(t), v(t), u(t)), \end{aligned}$$

(30)

where for each $t\ge 0$, we have $x(t)\in {\mathbb R}^n$, $v(t)\in C[-\theta , 0]$, and $u(t)\in {\mathbb R}^m$. Assume that $f:{\mathbb R}^n\times C[-\theta , 0]\times {\mathbb R}^m \rightarrow {\mathbb R}^n$ is completely continuous, and Lipschitz on each compact set. The input v is a continuous function from ${\mathbb R}_{\ge 0}$ to $C[-\theta , 0]$, and the input u is a measurable, locally essentially bounded function from ${\mathbb R}_{\ge 0}$ to ${\mathbb R}^m$.

For any given interval I, we define $||v||_{I} = \sup \{||v(t)||: \ t\in I\}$. Note that since $v: {\mathbb R}_{\ge 0}\rightarrow C[-\theta , 0]$ is continuous, we know that v(I) is compact if I is compact. Consequently, for any bounded interval I, we know that $||v||_{I}$ is finite. In the case when $I = [0, \infty )$, $||v||_{I}$ will be simply denoted by ||v||.

For each pair (v, u), the map $F(x, t)=f(x, v(t), u(t))$ satisfies the following:

• for each fixed x, $F(x, \cdot )$ is measurable and locally essentially bounded;

• F is locally Lipschitz on x in the sense that for any compact subset $K\subseteq {\mathbb R}^n$ and any interval $[a, b]\subseteq [0, \infty )$, there exists a locally essentially bounded function $\alpha (\cdot )$ such that

  $$ |F(x_1, t)-F(x_2, t)|\le \alpha (t)|x_1-x_2| $$

  for almost all $t\in [a, b]$ and all $x_1, x_2\in K$.

As a consequence, for each input pair (v, u) and each initial value condition $x(0)=x_0$, there exists a unique maximum solution of (30) defined on some interval [0, T); see e.g., [15]. Such a solution will be denoted by $x(t, x_0, v, u)$.

As in the case when the input functions take values in finite dimensional spaces, the ISS property for (30) can also be defined for a system as in (30):

Definition 1

([14]) A system as in (30) is input-to-state stable (ISS) provided there exist functions $\beta \in {\mathscr {K}\mathscr {L}}$, $\kappa \in {\mathscr {K}}$ and $\gamma \in {\mathscr {K}}$ such that

$$ |x(t, x_0, v, u)|\le \text {max}\left\{ \beta (|x_0|, t), \ \kappa (||v||), \ \gamma (||u||)\right\} \quad \forall \,t\ge 0 $$

holds for all inputs v and u.

Observe that by causality, the above inequality is equivalent to:

$$ |x(t, x_0, v, u)|\le \text {max}\left\{ \beta (|x_0|, t), \ \kappa (||v||_{[0, t)}), \ \gamma (||u||_{[0, t)})\right\} \quad \forall \,t\ge 0. $$

Also note that if $v(t) = x_t$, then $||v||_{[0, t]}= ||x||_{[-\theta , t]}$. (Recall that for each $t\ge 0$, the function $x_t: C[-\theta , 0]\rightarrow {\mathbb R}^n$ is defined by $x_t(s) = x(t+s)$.)

A $C^1$ function $V:{\mathbb R}^n\rightarrow {\mathbb R}_{\ge 0}$ is an ISS-Lyapunov function for the system (30) provided

• for some $\underline{\alpha }$ and $\overline{\alpha }$ in ${\mathscr {K}}_\infty$, we have

  $$ \underline{\alpha }(|x|)\le V(x)\le \overline{\alpha }(|x|)\qquad \forall \,x; $$

• for some $\chi \in {\mathscr {K}}$ and $\rho$ in ${\mathscr {K}}$ and some continuous positive definite function $\alpha$, the following holds for all $x\in {\mathbb R}^n, \ \mathrm{v}\in C[-\theta , 0]$, and $u\in {\mathbb R}^m$:

  $$\begin{aligned} V(x)\ge \text {max}\{\chi (||\mathrm{v}||_{[-\theta , 0]}), \ \rho (|u|)\}\ \Rightarrow \ DV(x)f(x, \mathrm{v}, u)\le -\alpha (|x|). \end{aligned}$$

  (31)

As in the case when the inputs only take values in finite dimensional spaces, the following result can be proven, by following exactly the same proof as given in [14]:

Proposition 3

A system as in (30) is ISS if it admits an ISS-Lyapunov function.

More precisely, if there is an ISS-Lyapunov function for (30) satisfying (31), then for some $\beta \in {\mathscr {K}\mathscr {L}}$, we have

$$ |V(x(t, x_0, v, u))|\le \text {max}\left\{ \beta (V(x_0), t), \ \chi (||v||_{[0, t)}), \ \rho (||u||_{[0, t)})\right\} \quad \forall \,t\ge 0. $$

We conjecture that the converse of Proposition 3 also holds. Many of the proofs of converse Lyapunov theorems in the literature (such as [11, 16]) may still work in the more general case when v takes values in a normed space (or Banach space), yet one cannot claim that the converse Lyapunov theorem holds for such systems before all details are validated.