Practical uniform input-to-state stability of perturbed triangular systems

Practical uniform input-to-state stability of perturbed triangular systems Abstract In this article, we present a practical uniform input-to-state stability result for perturbed triangular systems depending on a parameter. We present sufficient conditions for which each of these notions is preserved under cascade interconnection. 1. Introduction The asymptotic stability analysis by Lyapunov’s second method requires the construction of a strict Lyapunov function. This direct approach may be particularly hard for large-scale nonlinear time-varying systems. A natural way of simplifying this problem consists in dividing the system into simpler interconnected subsystems and to analyse each subsystem separately (Mazenc et al., 1999; Benabdallah et al., 2009, 2011). The interconnection of two input-to-state stability (ISS) systems is ISS since a growth rate condition can be always satisfied (Seibert & Suárez, 1990). However, for broader classes of systems, stability of their cascade is not always guaranteed. Seibert & Suárez (1990) derived global asymptotic stability (GAS) of a cascade of two time-invariant systems from individual GAS properties of the driving system and the disconnected driven system assuming that all solutions are bounded. The ISS analysis problem was studied for the autonomous case. There are many contributions in the corresponding literature. Some related works in Edwards et al. (2000), Karafyllis & Tsinias (2004), Sontag (1989, 1998), Sontag & Wang (1995) and Sontag & Teel (1995) introduced the notion of ISS. In particular, Jiang et al. (1996) showed that ISS is closed under composition, i.e. the cascade interconnection of two ISS systems is again an ISS system. To study uncertain dynamical systems, Jiang et al. (1994), Ito & Jiang (2006a, 2009) and Teel (1996) generalized the concept of ISS to the concept of input to state practical stability (ISpS). They proved that the interconnection of two ISpS systems is again an ISpS system. All these results are applied only to autonomous nonlinear systems. In Vidyasagar (1993), under some conditions, it was shown that a cascade nonautonomous system is globally uniformly exponentially stable if and only if each isolated subsystem is globally uniformly exponentially stable. Roughly, the result in Panteley & Lorıa (1998, 2001) shows that integrability of the perturbing trajectory of the driving system is sufficient to ensure GAS of the cascade. This observation has been re-interpreted and re-written in Angeli et al. (2000), Arcak et al. (2002), Lin et al. (2005) and Malisoff & Mazenc (2005), in terms of integral input-to-state stable stability (iISS) for a time-invariant cascade in which an iISS system is driven by a GAS system. The required trade-off condition is that the iISS gain of driven system needs to be steep satisfactorily in the direction toward the equilibrium if the convergence of the driving system is slow. Note that the set of iISS systems is larger and contains the ISS systems as a subset. The idea of the growth order and decay-rate trade-off result has been improved further in Chaillet & Angelli (2008) and Ito (2006, 2010), which shows additional conditions and states the trade-off in terms of Lyapunov-like inequalities (dissipation inequalities) of the two individual subsystems In addition, we complete the main result in Dashkovskiy et al. (2009) and Ito & Jiang (2006b), by giving a sufficient condition for the cascade composed of an iISS driven by a GAS one to remain GAS in the case when explicit Lyapunov functions are known. Roughly, the definition of this property is recalled in the sequel. it is again required that the dissipation term of the GAS subsystem dominates the supply function of the iISS one around zero. This result may be useful in practice since the iISS and GAS properties are commonly established through Lyapunov arguments. Furthermore, this result naturally extends to multiple cascaded systems, i.e. series of cascaded iISS systems driven by a GAS one. This result extends in this article to perturbed triangular time varying systems depending on a non-negative parameter where we interested in the study of practical uniform ISS of the time varying systems. This article is organized as follows. In Section 2, we give some definitions and results about practical uniform ISS (PUISS). In Section 3, some sufficient conditions are given to guarantee the practically uniformly ISS of a nonlinear perturbed time-varying system: remains practically uniformly input-to-state stable when it is perturbed by the output of another practically uniformly input-to-state system. In Section 4, we show that if both perturbed subsystems are practically uniformly input-to-state stable, then they can be practically uniformly input-to-state stable under some restrictive conditions on the one hand of the nominal system and on the other hand of the perturbation term. 2. Mathematical preliminaries Let $$L^{m}_{\infty}:=\{u:\mathbb{R}_+ \longrightarrow \mathbb{R}^m \ \text{such that } \ u \ \ \text{is bounded}\}$$. We use $$\|.\|_{\infty}$$ to denote the $$L^{m}_{\infty}$$ norm of u as a function defined on $$L^{m}_{\infty} $$ $$\|u\|_{\infty}:=\sup{\{\|u(t)\|, \ t \in \mathbb{R}_+\}.}$$ A function $$\alpha(.) : \mathbb{R}_+ \longrightarrow \mathbb{R}_+$$ is of class $$\mathcal{K}$$ if it is continuous, positive definite and strictly increasing and is of class $$\mathcal{K}_{\infty}$$, if it is also unbounded. A function $$ \beta(.,.): \mathbb{R}_+ \times \mathbb{R}_ + \longrightarrow \mathbb{R}_+$$ is said to be of class $$\mathcal{KL}$$ if for each fixed $$t\geq 0$$, $$\beta(.,t)$$ is of class $$\mathcal{K}$$ and for each fixed $$s\geq 0,$$$$\beta(s, .)$$ decreases to $$0$$ as $$t \rightarrow +\infty.$$ Lemma 2.1 Let $$\alpha_1(.)$$ and $$\alpha_2(.)$$$$\in \mathcal{K}_{\infty}$$ defined on $$[0,+\infty[.$$ Then the inverse function $$\alpha_1^{-1}(.)$$ of $$\alpha_1(.)$$ is of class $$\mathcal{K}_\infty.$$ $$\alpha_1\circ\alpha_2$$ is of class $$\mathcal{K}_\infty.$$ $$\forall s_1, s_2 \in \mathbb{R}_+,$$ $$\max{\{\alpha_1( s_1), \alpha_1(s_2)\}}\leq \alpha_1(s_1+s_2)\leq \alpha_1(2 s_1)+ \alpha_1(2s_2).$$ There exist a function $$\alpha \in \mathcal{K}_\infty$$ such that $$\forall s_1, s_2 \in \mathbb{R}_+,$$ $$\alpha(s_1+s_2)\leq \alpha_1(s_1) +\alpha_2(s_2).$$ Before stating our main theorem in Section (3), we introduce in this section some stability notions and some basic results. Consider the following controlled dynamical system depends on a parameter $$\varepsilon>0:$$ \begin{equation} \label{def1}\dot x = F^{\varepsilon}(t,x,u), \quad x(t_0)=x_0, \end{equation} (2.1) where $$t\in \mathbb{R}_+$$, $$x\in \mathbb{R}^n $$ is the state, $$u \in L^{m}_{\infty}$$ inputs, denoted by $$u$$ are measurable, essentially bounded functions from $$\mathbb{R}_+$$ to $$\mathbb{R}^m$$ and the function $$F^{\varepsilon}(.,.,.) : \mathbb{R}_+ \times \mathbb{R}^n \times \mathbb{R}^m \longrightarrow \mathbb{R}^n$$ is continuous in t and locally Lipschitz in $$x$$ and in $$u$$. We use $$x(t, t_0 ,x_0, u )$$: to denote the trajectory of the system corresponding to the initial condition $$x(t_0) = x_0$$ and the input function $$u$$. The Lipschitzness imposed on $$F^{\varepsilon}$$ guarantees the existence of a unique maximal solution of (2.1) for locally essentially bounded $$u$$. We start by recalling the definition of a PUISS and global practical uniform asymptotic stability (GPUAS) and some properties that this concept naturally induces, as well as related notions. Definition 2.1 (PUISS) We say that (2.1) is practical uniform input-to-state stable with respect to $$u$$ if there exists $$\varepsilon^*>0,$$ such that for any $$0<\varepsilon< \varepsilon^*,$$ there exist functions $$\beta_{\varepsilon}(.,.) \in \mathcal{KL}$$, $$\gamma_{\varepsilon}(.)\in \mathcal{K}$$ and positive scalars $$\rho(\varepsilon),$$ such that for each initial condition $$x_0$$ at any initial time $$t_0$$ and each measurable essentially bounded control $$u(.)$$ defined on $$\mathbb{R}_+$$, the solution $$x(.)$$ of the system (2.1) exists on $$\mathbb{R}_+$$ and satisfies \begin{equation} \label{dp1} \|x(t)\| \leq \beta_{\varepsilon}(\|x_0\|,t-t_0)+ \gamma_{\varepsilon}(\|u\|_{\infty})+\rho(\varepsilon), \ \forall t \geq t_0, \end{equation} (2.2) with $$\rho(\varepsilon)\to 0$$ as $$\varepsilon \to 0.$$ When (2.2) is satisfied with $$\rho(\varepsilon)= 0,$$ the system (2.1) is said to be input-to-state stable (ISS), a notion originally introduced by Sontag (1989, 1990). Definition 2.2 The system \begin{equation*} \dot x = F^{\varepsilon}(t,x,0), \quad x(t_0)=x_0. \end{equation*}is said to be globally practically uniformly asymptotically stable (GPUAS), if there exists $$\varepsilon^*>0,$$ such that for all $$0<\varepsilon< \varepsilon^*,$$ there exist function $$\beta_{\varepsilon}(.,.) \in \mathcal{KL}$$ and positive scalars $$\rho(\varepsilon),$$ such that for each initial condition $$x_0$$ at any initial time $$t_0$$, the solution $$x(.)$$ of the system (2.1) exists on $$\mathbb{R}_+$$ and satisfies \begin{equation} \label{dp2} \|x(t)\| \leq \beta_{\varepsilon}(\|x_0\|,t-t_0)+ \rho(\varepsilon), \ \forall t \geq t_0, \end{equation} (2.3)with$$\rho(\varepsilon)\to 0$$as$$\varepsilon \to 0.$$ We also recall that the system (2.1) is said to be $$0-$$GPUAS if the origin of $$\dot{x} = f(t,x,0)$$ is GPUAS. Remark 1 Systems of the form (2.1) can result from the design of high gain observers (see Benabdallah et al., 2011) or closed loop control systems. $$\rho(\varepsilon)$$ can be regarded as a measure of the distance of the system behavior from that of exponential stability (see Benabdallah et al., 2011). Since $$\rho(\varepsilon)$$ can be made arbitrarily small, the stability is called practical. Definition 2.3 A smooth function $$ V_{\varepsilon}:\mathbb{R}_+ \times \mathbb{R}^n \longrightarrow \mathbb{R}_+$$ is said to be a PUISS-Lyapunov function for the system (2.1), if there exist $$\mathcal{K}_{\infty}-$$functions $$\alpha_{1\varepsilon}(.), \ \alpha_{2\varepsilon}(.)$$, $$\mathcal{K}-$$functions $$\alpha_{3\varepsilon}(.)$$, $$\gamma_{\varepsilon}(.)$$ and a positive scalars $$\kappa(\varepsilon)> 0,$$ where $$\kappa(\varepsilon)\to 0$$ as $$\varepsilon \to 0$$, such that $$\forall x\in \mathbb{R}^n,$$$$\forall u \in \mathbb{R}^m$$ and $$\forall t\geq t_0:$$ \begin{equation} \label{eth0} \alpha_{1\varepsilon}(\|x\|)\leq V_{\varepsilon}(t,x) \leq \alpha_{2\varepsilon}(\|x\|) \end{equation} (2.4) and \begin{equation} \label{eth1}V_{\varepsilon}(t,x) \geq \gamma_{\varepsilon}(\|u\|_{\infty}) \ \Longrightarrow \ {\dot{V}_{\varepsilon}(t,x) \leq -\alpha_{3\varepsilon}(V_{\varepsilon})+\kappa(\varepsilon)}. \end{equation} (2.5) When (2.5) holds with $$\kappa(\varepsilon)=0$$, $$V_{\varepsilon}$$ is called an ISS-Lyapunou function for the system (2.1). It was shown in Sontag & Wang (1995) that for a time-invariant system $$\dot{x}= f(x,u)$$, the ISS property is equivalent to the existence of an ISS-Lyapunov function $$V$$ which is independent of $$t$$ and property (2.5) is equivalent to the existence of a $$\mathcal{K}$$-function $$\alpha_{3\varepsilon}(.)$$ and $$\chi_{\varepsilon}(.)$$ such that \begin{equation} \label{eth2} \dot{V}_{\varepsilon} (t,x) \leq -\alpha_{3\varepsilon}(V_{\varepsilon} )+\chi_{\varepsilon}(\|u\|_{\infty})+ \kappa(\varepsilon). \end{equation} (2.6) Remark 2 Observe that this definition is slightly different from the original definition proposed by Sontag & Wang (1995) in that $$\alpha_{3\varepsilon}$$ depending on a parameters $$\varepsilon>0$$ rather than independent on parameters as in Sontag & Wang (1995). Remark 3 It is immediate that the system (2.1) admits an iSpS- (respectively, ISS-) Lyapunov function satisfying (2.4) and (2.5) if and only if it admits an iSpS- (respectively, ISS-) Lyapunov function satisfying (2.4) and (2.6) in Sontag & Wang (1995). Recently, the equivalence between the iSpS property and the existence of an iSpS-Lyapunov function was shown by Sontag & Wang (1995), i.e. the following was proved. Proposition 2.2 The system (2.1) is PUISS (respectively, ISS) if and only if it has an PUISS- (respectively, ISS-) Lyapunov function. 3. PUISS of perturbed systems Now we will study the PUISS of two classes of perturbed parameter-dependent systems. Stability of perturbed systems is investigated, for instance, in Khalil (1996) but the considered systems do not depend on parameters. The first one concerns systems with a nominal part that depends on a parameter $$\varepsilon>0.$$ Let $$\mathcal{H}=\{\omega\in \mathcal{K} \ \text{such that } \ (id-\omega)\in \mathcal{K} \}.$$ Theorem 3.1 Consider the perturbed system \begin{equation} \dot{x}=F^{\varepsilon}(t,x) +G(t,x,u), \end{equation} (3.1) where $$F^{\varepsilon}: {\mathbb{R}}_+ \times {\mathbb{R}}^n \rightarrow {\mathbb{R}}^n $$ and $$G: {\mathbb{R}}_+ \times {\mathbb{R}}^n \times {\mathbb{R}}^m \rightarrow {\mathbb{R}}^n$$ are continuous and locally Lipschitz in $$x$$ with $$F^{\varepsilon}$$ known and $$G$$ unknown ‘the perturbed term’. Suppose that the nominal system $$\dot{x} = F^{\varepsilon}(t,x)$$ is uniformly globally practically asymptotically stable, with a Lyapunov function $$V_{\varepsilon}:{\mathbb{R}}_+\times {\mathbb{R}}^n \to {\mathbb{R}}_+,$$ which satisfies \begin{gather*} \alpha_{1_{\varepsilon}}(\|x\|)\leq V_{\varepsilon}(t,x) \leq \alpha_{2_{\varepsilon}}(\|x\|) \\ \frac{\partial V_{\varepsilon}}{\partial t}(t,x)+\frac{\partial V_{\varepsilon}}{\partial x}(t,x)F^{\varepsilon}(t,x)\leq -\alpha_{3_{\varepsilon}}(V_{\varepsilon})+\delta(\varepsilon), \end{gather*} where $$\alpha_{1_{\varepsilon}}(.)$$ and $$\alpha_{{2}_{ \varepsilon}}(.)$$$$\in{ \mathcal{K}}_{\infty}$$, $$\alpha_{3_{\varepsilon}}(.)$$ is a $$\mathcal{K}-$$ function and $$\delta(\varepsilon)$$ is a positive constant with $$\delta(\varepsilon)\to 0$$ as $$\varepsilon \to 0).$$ We suppose also that there exist positive constants $$\delta_1(\varepsilon)$$, $$\delta_2(\varepsilon)$$$$>0$$ and $$0< \theta< 1,$$ such that: \[ \|\frac{\partial V_{\varepsilon}}{\partial x}G(t,x,u)\| \leq \delta_1(\varepsilon) \|x\|+ \delta_2(\varepsilon) \|u\|_{\infty}, \quad \forall (t,x,u)\in {\mathbb{R}}_+ \times {\mathbb{R}}^ n \times {\mathbb{R}}^m \] and \begin{equation} \label{gain} \displaystyle\frac{\delta_1(\varepsilon)} {\theta}(\alpha_{3_{\varepsilon}} \circ \alpha_{1_{\varepsilon}})^{-1} \in \mathcal{H}. \end{equation} (3.2) Then, the system (3.1) is practically uniformly input-to-state stable. Proof. The derivative of $$V_{\varepsilon}$$ along the trajectories of system (3.1) is given by $$\dot{V}_{\varepsilon}(t,x(t))\leq -\alpha_{3_{\varepsilon}}(V_{\varepsilon})+\delta(\varepsilon)+\delta_1(\varepsilon) \|x\|+ \delta_2(\varepsilon) \|u\|_{\infty}.$$ Then, $$\dot{V}_{\varepsilon}(t,x(t))\leq -\alpha_{3_{\varepsilon}}(V_{\varepsilon})+\delta(\varepsilon)+\delta_1(\varepsilon) \alpha^{-1}_{1_{\varepsilon}}(V_{\varepsilon})+ \delta_2(\varepsilon) \|u\|_{\infty}.$$ Thus, there exists $$0<\theta<1$$ such that $$\dot{V}_{\varepsilon}(t,x(t))\leq -(1-\theta) \alpha_{3_{\varepsilon}}(V_{\varepsilon})-\theta \alpha_{3_{\varepsilon}}(V_{\varepsilon})+\delta(\varepsilon)+\delta_1(\varepsilon) \alpha^{-1}_{1_{\varepsilon}}(V_{\varepsilon})+ \delta_2(\varepsilon) \|u\|_{\infty},$$ which implies that $$\dot{V}_{\varepsilon}(t,x(t))\leq -(1-\theta) \alpha_{3_{\varepsilon}}(V_{\varepsilon})+\delta(\varepsilon), \ \forall \ V_{\varepsilon}(t,x) \geq \gamma_{\varepsilon}(\|u\|_{\infty}), $$ with $$\gamma_{\varepsilon}(.)$$ defined by: $$\gamma_{\varepsilon}(s):= (\theta\alpha_{3_{\varepsilon}})^{-1}\circ \left( id-\displaystyle\frac{\delta_1(\varepsilon)} {\theta}(\alpha_{3_{\varepsilon}} \circ \alpha_{1_{\varepsilon}})^{-1}\right) ^{-1}(\delta_2(\varepsilon) s).$$ According to the hypothesis (3.2), $$\gamma_{\varepsilon}(.)$$ is a $$\mathcal{K}$$-function. Thus, \begin{equation} V_{\varepsilon}(t,x) \geq \gamma_{\varepsilon}(\|u\|_{\infty}) \ \Longrightarrow \ \dot{V}(t,x) \leq -(1-\theta) \alpha_{3\varepsilon}(V_{\varepsilon})+\delta(\varepsilon) . \end{equation} (3.3) We conclude that the perturbed system (3.1) is practically uniformly input-to-state stable. □ 4. PUISS of perturbed triangular systems The first result we recall here is concerned with cascade connection of two PUISS systems, in the case when an PUISS-Lyapunov function is explicitly known for each of them. For the sake of generality, it is allowed that the driven subsystem depends also on the external input. We therefore deal with systems of the form: \begin{eqnarray} {\dot x}_1&=& f_1^{\varepsilon}(t,x_1,x_2,u) \label{Pertcas3}\\ \end{eqnarray} (4.1) \begin{eqnarray} {\dot x}_2&=& f_2^{\varepsilon}(t,x_2,u), \label{Pertcas4} \end{eqnarray} (4.2) where $$t\geq 0,$$$$x_1\in \mathbb{R}^n,$$$$x_2\in \mathbb{R}^p,$$$$u: \mathbb{R}_+\longrightarrow \mathbb{R}^m,$$ is measurable locally essentially bounded, $$f_1^{\varepsilon},$$$$f_2^{\varepsilon}$$ are both assumed to be locally Lipschitz. We also assume that $$f_1^{\varepsilon}(t,0,0,0) = 0$$ and $$f_2^{\varepsilon}(t,0,0) = 0$$. We present here a cascade of a PUISS subsystem controlled by another subsystem PUISS to be PUISS. Theorem 4.1 (PUISS+PUISS) Let $$V_{1\varepsilon}$$ and $$V_{2\varepsilon}$$ be continuous positive definite radially unbounded functions, differentiable on $$\mathbb{R}^n$$ and $$\mathbb{R}^p,$$ respectively. Suppose that $$\forall x_1\in \mathbb{R}^n,$$$$\forall x_2\in \mathbb{R}^p,$$$$\forall u \in L^{m}_{\infty}$$ and $$\forall t\geq t_0,$$ one has \begin{gather*} \frac{\partial V_{1\varepsilon}}{\partial t}+\frac{\partial V_{1\varepsilon}}{\partial x} f_1^{\varepsilon}(t,x_1,x_2,u)\leq -\alpha_{1\varepsilon}(V_{1\varepsilon})+\alpha_{2\varepsilon}(\|x_2\|)+\gamma_{1\varepsilon}(\|u\|_{\infty})+ \rho_1(\varepsilon),\\ \delta_{1_{\varepsilon}}(\|x_2\|)\leq V_{2\varepsilon}(t,x_2) \leq \delta_{2_{\varepsilon}}(\|x_2\|) \end{gather*} and \[ \frac{\partial V_{2\varepsilon}}{\partial t}+\frac{\partial V_{2\varepsilon}}{\partial x} f_2^{\varepsilon}(t,x_2,u)\leq -\delta_{3\varepsilon}(V_{2\varepsilon})+\gamma_{2\varepsilon}(\|u\|_{\infty})+ \rho_2(\varepsilon).\] where $$\delta_{{1\varepsilon}}(.) $$ and $$\delta_{{2\varepsilon}}(.) $$ are $$\mathcal{K}_{\infty}-$$functions, $$\alpha_{1\varepsilon}(.)$$, $$\alpha_{{2\varepsilon}}(.), $$$$\gamma_{{1\varepsilon}}(.), $$$$\gamma_{{2\varepsilon}}(.)$$ and $$\delta_{{3\varepsilon}}(.)$$ are $$\mathcal{K}-$$functions and $$\rho_i(\varepsilon) \rightarrow 0$$ as $$\varepsilon \rightarrow 0$$, $$\forall i=1,2.$$ If \begin{equation} \label{casc5}\displaystyle\frac{1}{\theta}\alpha_{2\varepsilon}\circ (\delta_{3\varepsilon} \circ \delta_{{1\varepsilon}})^{-1} \in \mathcal{H}, \end{equation} (4.3) the cascade system (4.1)–(4.2) is PUISS. Proof. The derivative of $$V_{\varepsilon}:=V_{1\varepsilon}+V_{2\varepsilon}$$ along the trajectories of system (4.1)–(4.2) is given by $$\dot{V_{\varepsilon}}(t,x)\leq -\alpha_{1\varepsilon}(V_{1\varepsilon})-\delta_{3\varepsilon}(V_{2\varepsilon}) +\alpha_{2\varepsilon}(\|x_2\|)+\gamma_{1\varepsilon}(\|u\|_{\infty})+\gamma_{2\varepsilon}(\|u\|_{\infty})+ \rho_1(\varepsilon)+ \rho_2(\varepsilon) .$$ Thus, there exist $$0<\theta<1$$ such that: $$\dot{V_{\varepsilon}}(t,x)\,{\leq }\,{-}\,\alpha_{1\varepsilon}(V_{1\varepsilon})-(1-\theta) \delta_{3\varepsilon}(V_{2\varepsilon}) -\theta \delta_{3\varepsilon}(V_{2\varepsilon}) +\alpha_{2\varepsilon}\circ \delta_{1_{\varepsilon}}^{-1}(V_{2\varepsilon})+\gamma_{1\varepsilon}(\|u\|_{\infty})+\gamma_{2\varepsilon}(\|u\|_{\infty})+ \rho_1(\varepsilon)+ \rho_2(\varepsilon)\!.$$ Then, using Lemma 2.1, there exists $$\alpha_{\varepsilon}\in \mathcal{K}$$ such that: $$\dot{V_{\varepsilon}}(t,x)\leq -\alpha_{\varepsilon}(V_{\varepsilon})+[\alpha_{2\varepsilon}\circ \delta_{1_{\varepsilon}}^{-1}-\theta \delta_{3\varepsilon}](V_{2\varepsilon}) +\gamma_{\varepsilon}(\|u\|_{\infty})+ \rho(\varepsilon),$$ where $$\gamma_{\varepsilon}(s):= \gamma_{1\varepsilon}(s)+\gamma_{2\varepsilon}(s) \in \mathcal{K}$$ and $$\rho(\varepsilon):= \rho_1(\varepsilon)+ \rho_2(\varepsilon) \rightarrow 0$$ as $$\varepsilon \rightarrow 0,$$ which implies that $$\dot{V_{\varepsilon}}(t,x)\leq -\alpha_{\varepsilon}(V_{\varepsilon})+\rho(\varepsilon), \ \ \text{if} \ [\alpha_{2\varepsilon}\circ \delta_{1_{\varepsilon}}^{-1}-\theta \delta_{3\varepsilon}](V_{2\varepsilon}) +\gamma_{\varepsilon}(\|u\|_{\infty})\leq 0. $$ Then, using (4.3), there exists $$\chi_{\varepsilon}\in \mathcal{K}$$ defined by: $$\chi_{\varepsilon}(s):= (\theta \delta_{3\varepsilon})^{-1}\circ \left( id-\displaystyle\frac{1}{\theta}\alpha_{2\varepsilon}\circ (\delta_{3\varepsilon} \circ \delta_{{1\varepsilon}})^{-1}\right)^{-1}\circ \gamma_{\varepsilon}( s),$$ such that \begin{equation} V_{\varepsilon}(t,x) \geq \chi_{\varepsilon}(\|u\|_{\infty}) \ \Longrightarrow \ \dot{V_{\varepsilon}}(t,x)\leq -\alpha_{\varepsilon}(V_{\varepsilon})+\rho(\varepsilon). \end{equation} (4.4) We conclude that the cascade system (4.1)–(4.2) is practically uniformly input-to-state stable. □ Remark 4 While theorem 4.1 is stated for the cascade interconnection of two nonlinear subsystems, it easily extends to multiple cascades. 4.1. Perturbation terms depending on a parameter $$\varepsilon$$ Now we consider cascaded systems of the form with perturbations depending on a parameter $$\varepsilon>0$$. \begin{eqnarray} {\dot x}_1&=& f_1(t,x_1,x_2,u)+g_1^{\varepsilon}(t,x_1,x_2) \label{Pert3}\\ \end{eqnarray} (4.5) \begin{eqnarray} {\dot x}_2&=& f_2(t,x_2,u)+g_2^{\varepsilon}(t,x_2) \label{Pert4} \end{eqnarray} (4.6) where $$t\geq 0,$$$$x_1\in \mathbb{R}^n,$$$$x_2\in \mathbb{R}^p,$$$$u: \mathbb{R}_+\longrightarrow \mathbb{R}^m,$$ is measurable locally essentially bounded and both $$f_1,$$$$f_2$$ are assumed to be locally Lipschitz in the state and piecewise continuous in the time. We also assume that $$f_1^{\varepsilon}(t,0,0,0) = 0$$ and $$f_2^{\varepsilon}(t,0,0) = 0$$. $$g^{\varepsilon}_1$$ and $$g^{\varepsilon}_2$$ are locally Lipschitz in the state and piecewise continuous in the time for all parameters $${\varepsilon}$$ under consideration. ($$\mathcal{A}_3$$) The system \[{\dot x}_1= f_1(t,x_1,0,u)\] is PUISS, with a Lyapunov PUISS function $$V_{1}$$ be continuous positive definite radially unbounded function, differentiable on $$\mathbb{R}^n$$. Suppose that $$\forall x_1\in \mathbb{R}^n,$$$$\forall u \in L^{m}_{\infty}$$ and $$\forall t\geq t_0,$$ the following holds \begin{gather} \label{a1} a_1 \|x_1\|^2\leq V_1(t,x_1)\leq a_2 \|x_1\|^2\\ \frac{\partial V_1}{\partial t}+\frac{\partial V_1}{\partial x_1}f_1(t,x_1,0,u) \leq -\alpha_1(V_1)+\chi_1(\|u\|_{\infty})+\delta_1\notag\\ \|\frac{\partial V_1}{\partial x_1}\|\leq a_3 \|x_1\|\notag \end{gather} (4.7) and \[\|\frac{\partial V_1}{\partial x_1}g_1^{\varepsilon}(t,x_1,x_2)\| \leq r_1(\varepsilon),\] where $$a_1$$, $$a_2,$$$$a_3$$, $$\delta_1$$ and $$r_1(\varepsilon)$$ are positives constants with $$r_1(\varepsilon)\rightarrow 0$$ as $$\varepsilon \rightarrow 0$$, $$\alpha_1(.) \in \mathcal{K}$$ and $$\chi_1(.) \in \mathcal{K}$$. In addition, there exists a positive constant $$M<\infty$$ such that \[\|\frac {\partial f_1 }{\partial x_2}\|\leq M , \quad \forall (t,x_1,x_2,u)\in \mathbb{R}_+\times \mathbb{R}^n \times\mathbb{R}^p\times L^{m}_{\infty}\] ($$\mathcal{A}_4$$) The system \[\dot x _2 = f_2 (t,x_2,u) \] is PUISS, with a Lyapunov PUISS function $$V_{2}$$ be continuous positive definite radially unbounded function, differentiable on $$\mathbb{R}^p$$. Suppose that $$\forall x_2\in \mathbb{R}^p,$$$$\forall u \in L^{m}_{\infty}$$ and $$\forall t\geq t_0:$$ \begin{gather} \label{b1} b_1 \|x_2\|^2\leq V_2(t,x_2)\leq b_2\|x_2\|^2\\ \frac{\partial V_2}{\partial t}+\frac{\partial V_2}{\partial x_2}f_2(t,x_2,u) \leq -\alpha_2(V_2)+\chi_2(\|u\|_{\infty})+\delta_2,\notag\\ \|\frac{\partial V_2}{\partial x_2}g_2^{\varepsilon}(t,x_2)\| \leq r_2(\varepsilon),\notag \end{gather} (4.8) where $$b_1$$, $$b_2$$, $$\delta_2$$ and $$r_2(\varepsilon)$$$$\geq 0$$ with $$r_2(\varepsilon) \rightarrow 0$$ as $$\varepsilon \rightarrow 0$$, $$\alpha_2(.)\in \mathcal{K}, \ \chi_2(.)\in \mathcal{K}$$. We suppose also there exists a parameter $$0<\theta<1$$ such that: \begin{equation} \label{gain5} \displaystyle\frac{M a_3}{2\theta \min{(a_1,b_1)}} {{\alpha}}^{-1}\in \mathcal{H}, \end{equation} (4.9) where $$\alpha(s):=\min{\{\alpha_1(\frac{s}{2}), \ \alpha_2(\frac{s}{2})\}}.$$ The following result establishes PUISS of triangular systems under some assumption and provided the knowledge of a Lyapunov function for the driven subsystem. Theorem 4.2 Under the assumptions $$(\mathcal{A}_3)$$ and $$(\mathcal{A}_4)$$, the cascaded system (4.5)–(4.6) is PUISS. Proof. The derivative of $$V:=V_{1}+V_{2}$$ along the trajectories of system (4.5)–(4.6) is given by $$\dot{V}(t,x)\leq -\alpha_{1}(V_{1})-\alpha_{2}(V_{2}) +\chi_1(\|u\|_{\infty})+\delta_1+\chi_2(\|u\|_{\infty})+\delta_2+a_3 M\|x_1\|\|x_2\|+r_1(\varepsilon)+r_2(\varepsilon). $$ Then, using lemma 2.1, there exist $$\alpha\in \mathcal{K}$$ such that: $$\dot{V}(t,x)\leq -\alpha(V) + \chi(\|u\|_{\infty})+ r(\varepsilon)+\displaystyle\frac{a_3M}{2a_1}V_1+\displaystyle\frac{a_3M}{2b_1}V_2,$$ where $$\chi(s):= \chi_{1}(s)+\chi_{2}(s) \in \mathcal{K}$$, $$\delta= \delta_1+\delta_2$$ and $$r(\varepsilon):= r_1(\varepsilon)+ r_2(\varepsilon) \rightarrow 0$$ as $$\varepsilon \rightarrow 0.$$ Then, $$\dot{V}(t,x)\leq -\alpha(V) + \chi(\|u\|_{\infty})+ r(\varepsilon)+\delta+\displaystyle\frac{a_3M}{2 \min{(a_1,b_1)}}V.$$ Thus, there exists $$0<\theta<1$$ such that $$\dot{V}(t,x)\leq -(1-\theta)\alpha(V)- \theta \alpha(V)+\displaystyle\frac{a_3M}{2 \min{(a_1,b_1)}}V+ \chi(\|u\|_{\infty})+ r(\varepsilon)+\delta,$$ which implies that $$\dot{V}(t,x)\leq -(1-\theta)\alpha(V)+r(\varepsilon)+\delta, \ \ \text{if} \ [ - \theta \alpha+\displaystyle\frac{a_3M}{2 \min{(a_1,b_1)}}id](V)+ \chi(\|u\|_{\infty})\leq 0. $$ Then, using 4.9, there exist $$\gamma\in \mathcal{K}$$ defined by $$\gamma(s):= (\theta \alpha)^{-1}\circ \left( id- \displaystyle\frac{M a_3}{2\theta \min{(a_1,b_1)}} {{\alpha}}^{-1}\right)^{-1}\circ \chi( s),$$ such that \begin{equation} V(t,x) \geq \gamma(\|u\|_{\infty}) \ \Longrightarrow \ \dot{V}(t,x)\leq -(1-\theta)\alpha(V)+r(\varepsilon)+\delta. \end{equation} (4.10) We conclude that the cascade system (4.5)–(4.6) is practically uniformly input-to-state stable. □ 4.2. Nominal system depending on a parameter $$\varepsilon$$ Now, we consider triangular perturbed systems of the following form with a nominal systems depending on a parameter $$\varepsilon>0$$: \begin{eqnarray} \dot{x}_1&=&f_1^{\varepsilon}(t,x_1)+g(t,x,u) \label{R11} \\ \end{eqnarray} (4.11) \begin{eqnarray} \dot{x}_2&=&f_2^{\varepsilon}(t,x_2), \label{R12} \ \end{eqnarray} (4.12) where $$x_1\in\mathbb{R}^n, \ x_2\in\mathbb{R}^p$$, $$u: \mathbb{R}_+\longrightarrow \mathbb{R}^m,$$ is measurable locally essentially bounded, $$x:=(x_1,x_2)^T$$, $$f_1^{\varepsilon}(t,x_1)$$ and $$f_2^{\varepsilon}(t,x_2) $$ are differentiable functions for all parameters $${\varepsilon}$$ under consideration and $$g(t,x,u)$$ are locally Lipschitz in the state and piecewise continuous in the time. Introduce the following assumption: $$(H_1)$$ Suppose that \begin{equation} \label{cascade1} \dot{x}_1=f_1^{\varepsilon}(t,x_1) \end{equation} (4.13) is globally practically uniformly asymptotically stable, with a Lyapunov function $$V_{1 \varepsilon},$$$$\alpha_{1_{\varepsilon}}(.)$$ and $$\alpha_{{2}_{ \varepsilon}}(.)$$$$\in{ \mathcal{K}}_{\infty}$$, a $$\mathcal{K}-$$ function $$\alpha_{3_{\varepsilon}}(.)$$ and a positive constant $$\delta(\varepsilon)$$ with $$\delta(\varepsilon)\to 0$$ as $$\varepsilon \to 0).$$ We suppose also that there exist positive constants $$\rho_1(\varepsilon) \geq 0,$$ with $$ \rho_1(\varepsilon) \rightarrow 0$$ as $$\varepsilon \rightarrow 0$$, such that: \begin{gather}\label{R:35}\alpha_{1\varepsilon} (\|x_1\|)\leq V_{1\varepsilon}(t,x_1)\leq \alpha_{2\varepsilon}(\|x_1\|)\\ \end{gather} (4.14) \begin{gather} \label{R:36}\displaystyle\frac{\partial V_{1\varepsilon}}{\partial t} +\displaystyle\frac{\partial V_{1\varepsilon}}{\partial x_1} f_1^{\varepsilon}(t,x_1)\leq \ -\alpha_{3\varepsilon} (V_{1\varepsilon})+\rho_1(\varepsilon) \end{gather} (4.15) $$(H_2)$$ Suppose that \begin{equation} \label{cascade2} \dot{x}_2=f_2^{\varepsilon}(t,x_2) \end{equation} (4.16) is GPUAS, there exists a Lyapunov function $$V_{2\varepsilon}(t,x_1),$$$$\gamma_{1\varepsilon},$$$$\gamma_{2\varepsilon}$$$$\in{ \mathcal{K}}_{\infty}$$, a $$\mathcal{K}-$$ function $$\gamma_{3\varepsilon}$$ and $$\rho_2(\varepsilon)\geq 0,$$ with $$ \rho_2(\varepsilon) \rightarrow 0$$ as $$\varepsilon \rightarrow 0$$ such that: \begin{gather}\label{R:37}\gamma_{1\varepsilon} (\|x_2\|)\leq V_{2\varepsilon}(t,x_1)\leq \gamma_{2\varepsilon}(\|x_2\|)\\ \end{gather} (4.17) \begin{gather} \label{R:38}\displaystyle\frac{\partial V_{2\varepsilon}}{\partial t} +\displaystyle\frac{\partial V_{2\varepsilon}}{\partial x_2} f_2^{\varepsilon}(t,x_1)\leq \ -\gamma_{3\varepsilon} (V_{2\varepsilon})+\rho_2(\varepsilon) \end{gather} (4.18) $$(H_3)$$ We suppose the existence for scalars $$\delta_1(\varepsilon)$$, $$\delta_2(\varepsilon)$$$$\geq 0$$, a $$\mathcal{K}$$-function $$\alpha_{4\varepsilon}(.)$$ a constant $$\theta,$$$$0< \theta< 1$$ such that: \[\|\displaystyle\frac{\partial V_{1\varepsilon}}{\partial x_1} g(t,x,u)\| \leq \delta_1(\varepsilon) \|x_1\|+ \delta_2(\varepsilon) \|x_2\| +\alpha_{4\varepsilon}(\|u\|_{\infty}), \quad \forall (t,x_1,x_2,u)\in {\mathbb{R}}_+\times {\mathbb{R}}^ n\times {\mathbb{R}}^p \times {\mathbb{R}}^m .\] and \begin{equation} \label{gain4} \displaystyle\frac{1}{\theta} {\bar{\beta}}_{\varepsilon}\circ \sigma_{\varepsilon}^{-1} \in \mathcal{H}, \end{equation} (4.19) where \begin{align*} \bar{\beta}_{\varepsilon}(s)&:=\max{\{\delta_1(\varepsilon)\alpha_{1_{\varepsilon}}^{-1}(s), \ \delta_2(\varepsilon) \gamma_{1_{\varepsilon}}^{-1}(s)\}},\\ \sigma_{\varepsilon}(s)&:=\min{\{\alpha_{3_{\varepsilon}}(\frac{s}{2}), \ \gamma_{3_{\varepsilon}}(\frac{s}{2})\}}. \end{align*} Theorem 4.3 Under assumptions $$(H_1)$$, $$(H_2)$$ and $$(H_3),$$ the triangular perturbed system (4.11)–(4.12) is (PUISS). Proof. The derivative of $$V_{\varepsilon}:=V_{1\varepsilon}+V_{2\varepsilon}$$ along the trajectories of system (4.11)–(4.12) is given by $$\dot{V_{\varepsilon}}(t,x)\leq -\alpha_{3\varepsilon}(V_{1\varepsilon})-\gamma_{3\varepsilon}(V_{2\varepsilon}) + \delta_1(\varepsilon) \|x_1\|+ \delta_2(\varepsilon) \|x_2\| +\alpha_{4\varepsilon}(\|u\|_{\infty})+\rho_1(\varepsilon)+\rho_2(\varepsilon)$$ then, using lemma 2.1, there exist $$\sigma_{\varepsilon}(s):=\min{\{\alpha_{3_{\varepsilon}}(\frac{s}{2}), \ \gamma_{3_{\varepsilon}}(\frac{s}{2})\}}\in \mathcal{K}$$, such that: $$\dot{V_{\varepsilon}}(t,x)\leq -\sigma_{\varepsilon}(V_{\varepsilon})+ \delta_1(\varepsilon) \alpha_{1_{\varepsilon}}^{-1}(V_{1\varepsilon})+\delta_2(\varepsilon) \gamma_{1_{\varepsilon}}^{-1}(V_{2\varepsilon}) +\alpha_{4\varepsilon}(\|u\|_{\infty})+\rho_1(\varepsilon)+\rho_2(\varepsilon),$$ which implies that $$\dot{V_{\varepsilon}}(t,x)\leq -\sigma_{\varepsilon}(V_{\varepsilon})+ {\bar{\beta}}_{\varepsilon}(V_{\varepsilon}) +\alpha_{4\varepsilon}(\|u\|_{\infty})+\rho(\varepsilon),$$ where $${\bar{\beta}}_{\varepsilon}(s):=\max{\{\delta_1(\varepsilon)\alpha_{1_{\varepsilon}}^{-1}(s), \ \delta_2(\varepsilon) \gamma_{1_{\varepsilon}}^{-1}(s)\}}\in \mathcal{K}$$ and $$\rho(\varepsilon):= \rho_1(\varepsilon)+ \rho_2(\varepsilon) \rightarrow 0$$ as $$\varepsilon \rightarrow 0,$$ Thus, there exist $$0<\theta<1$$ such that: $$\dot{V_{\varepsilon}}(t,x)\leq -(1-\theta)\sigma_{\varepsilon}(V_{\varepsilon})- \theta \sigma_{\varepsilon}(V_{\varepsilon})+ {\bar{\beta}}_{\varepsilon}(V_{\varepsilon}) +\alpha_{4\varepsilon}(\|u\|_{\infty})+\rho(\varepsilon),$$ which implies that $$\dot{V_{\varepsilon}}(t,x)\leq -(1-\theta)\sigma_{\varepsilon}(V_{\varepsilon})+\rho(\varepsilon), \ \ \text{if} \ [ {\bar{\beta}}_{\varepsilon}-\theta \sigma_{\varepsilon}](V_{\varepsilon}) +\alpha_{4\varepsilon}(\|u\|_{\infty})\leq 0. $$ Then, using 4.19 there exist $$\chi_{\varepsilon}\in \mathcal{K}$$ defined by: $$\chi_{\varepsilon}(s):= (\theta \sigma_{\varepsilon})^{-1}\circ \left( id-\displaystyle\frac{1}{\theta} {\bar{\beta}}_{\varepsilon}\circ \sigma_{\varepsilon}^{-1}\right)^{-1}\circ \alpha_{4\varepsilon}( s),$$ such that \begin{equation} V_{\varepsilon}(t,x) \geq \chi_{\varepsilon}(\|u\|_{\infty}) \ \Longrightarrow \ \dot{V_{\varepsilon}}(t,x)\leq -(1-\theta)\sigma_{\varepsilon}(V_{\varepsilon})+\rho(\varepsilon). \end{equation} (4.20) We conclude that the cascade system (4.11)–(4.12) is practically uniformly input-to-state stable. □ Example 1 Consider the system \begin{eqnarray} {\dot x}_1&=& -\varepsilon x_1+ \displaystyle\frac{x_1x_2}{1+x_1^2}+ \displaystyle e^{-t} u \end{eqnarray} (4.21) \begin{eqnarray} {\dot x}_2&=& -x_2^3+\varepsilon \displaystyle\frac{x_2e^{-2t}}{1+x_2^2}. \label{exple3} \end{eqnarray} (4.22) Setting \begin{gather*} V_{1{\varepsilon}}:= \frac{\varepsilon}{2}x_1^2, \ V_{2{\varepsilon}}:= \frac{\varepsilon}{2}x_2^2\\ f_1^{\varepsilon}(t,x_1)=-\varepsilon x_1, \ \ f_2^{\varepsilon}(t,x_2)=-x_2^3+ \varepsilon \displaystyle\frac{x_2e^{-2t}}{1+x_2^2} \ \text{and} \ \ g(t,x,u)=\displaystyle\frac{x_1 x_2}{1+x_1^2}+ \displaystyle e^{-t} u. \end{gather*} We deduce that assumptions $$(H_1)$$, $$(H_2)$$ and $$(H_3)$$ are satisfied. Therefore, we can apply Theorem 4.3 to prove that system (4.21)–(4.22) is practically uniformly input-to-state stable. 5. Conclusion This article has investigated stability of cascade interconnection of subsystems that are not necessary UISS. References Angeli D. , Sontag E. D. & Wang Y. ( 2000 ) A characterization of integral input-to-state stability. IEEE Trans. Autom. Control , 45 , 1082 – 1097 . Google Scholar CrossRef Search ADS Arcak M. , Angeli D. & Sontag E. ( 2002 ) A unifying integral ISS framework for stability of nonlinear cascades. SIAM J. Control Optim. , 40 , 1888 – 1904 . Google Scholar CrossRef Search ADS Benabdallah A. , Ellouze I. & Hammami M. A. ( 2009 ) Practical stability of nonlinear time varying cascade systems. J. Dyn. Contr. Syst. , 15 , 45 – 62 . Google Scholar CrossRef Search ADS Benabdallah A. , Ellouze I. & Hammami M. A. ( 2011 ) Practical exponential stability of perturbed triangular systems and a separation principle. Asian J. Contr. , 13 , 445 – 448 . Google Scholar CrossRef Search ADS Chaillet A. & Angelli D. ( 2008 ) Integral input to state stable systems in cascade. Syst. Control Lett. , 57 , 519 – 527 . Google Scholar CrossRef Search ADS Dashkovskiy S. N. , Rüffer B. S. & Wirth F. R. ( 2009 ) Small gain theorems for large scale systems and construction of ISS Lyapunov functions [Online]. Available at http://arxiv.org/pdf/0901.1842. Edwards H. A. , Lin Y. & Wang Y. ( 2000 ) On input-to-state stability for time varying nonlinear systems. Proceedings of the 39th IEEE Conf. Decision Control , vol. 4 . pp. 3501 – 3506 . Ito H. ( 2006 ) State-dependent scaling problems and stability of interconnected iISS and ISS systems. IEEE Trans. Autom. Control , 51 , 1626 – 1643 . Google Scholar CrossRef Search ADS Ito H. ( 2010 ) A Lyapunov approach to cascade interconnection of integral input-to-state stable systems. IEEE Trans. Autom. Control , 55 , 702 – 708 . Google Scholar CrossRef Search ADS Ito H. & Jiang Z.-P. ( 2006a ) Nonlinear small-gain condition covering iISS systems: necessity and sufficiency from a Lyapunov perspective. Proc. of the 45th IEEE Conf. Decision Control . pp. 355 – 360 . Ito H. & Jiang Z.-P. ( 2006b ) On necessary conditions for stability of interconnected iISS systems. Proc. Amer. Control Conf. pp. 1499 – 1504 . Ito H. & Jiang Z.-P. ( 2009 ) Necessary and sufficient small gain conditions for integral input-to-state stable systems: a Lyapunov perspective. IEEE Trans. Autom. Control , 54 , 2389 – 2404 . Google Scholar CrossRef Search ADS Jiang Z.-P. , Mareels I. & Wang Y. ( 1996 ) A Lyapunov formulation of the nonlinear small-gain theorem for interconnected ISS systems. Automatica , 32 , 1211 – 1215 . Google Scholar CrossRef Search ADS Jiang Z.-P. , Teel A. & Praly L. ( 1994 ) Small-gain theorem for ISS systems and applications. Math. Contr. Signals Syst. , 7 , 95 – 120 . Google Scholar CrossRef Search ADS Karafyllis I. & Tsinias J. ( 2004 ) Non-uniform in time input-to-state stability and the small-gain theorem. IEEE Trans. Autom. Control , 49 , 196 – 216 . Google Scholar CrossRef Search ADS Khalil H. K. ( 1996 ) Nonlinear Systems , 2nd edn . New York : Mac-Millan . Lin Y. , Wang Y. & Cheng D. ( 2005 ) On non-uniform and semi-uniform input-to-state-stability for time varying systems. Proceedings of the 16th IFAC World Congress, Prague, Czech Republic [CD ROM] , pp. 312 – 317 . Malisoff M. & Mazenc F. ( 2005 ) Further remarks on strict input-to-state stable Lyapunov functions for time-varying systems. Automatica , 41 , 1973 – 1978 . Google Scholar CrossRef Search ADS Mazenc F. , Sepulchre R. & Jankovic M. ( 1999 ) Lyapunov functions for stable cascades and applications to global stabilization. IEEE Trans. Autom. Control , 44 , 1795 – 1800 . Google Scholar CrossRef Search ADS Panteley E. & Lorıa A. ( 1998 ) On global uniform asymptotic stability of nonlinear time-varying systems in cascade. Syst. Control Lett. , 33 , 131 – 138 . Google Scholar CrossRef Search ADS Panteley E. & Lorıa A. ( 2001 ) Growth rate conditions for uniform asymptotic stability of cascaded time-varying systems. Automatica , 37 , 453 – 460 . Google Scholar CrossRef Search ADS Seibert P. & Suárez R. ( 1990 ) Global stabilization of nonlinear cascaded systems. Syst. Control Lett. , 14 , 347 – 352 . Google Scholar CrossRef Search ADS Sontag E. D. ( 1989 ) Remarks on stabilization and input-to-state stability. Proc. IEEE Conf. Decision Control . pp. 1376 – 1378 . Sontag E. ( 1998 ) Comments on integral variants of ISS. Syst. Control Lett. , 34 , 93 – 100 . Google Scholar CrossRef Search ADS Sontag E. D. & Teel A. ( 1995 ) Changing supply functions in input/state stable systems. IEEE Trans. Autom. Control , 40 , 1476 – 1478 . Google Scholar CrossRef Search ADS Sontag E. & Wang Y. ( 1995 ) On characterizations of input-to-state stability property. Syst. Control Lett. , 24 , 351 – 359 . Google Scholar CrossRef Search ADS Sontag E. D. & Teel A. ( 1995 ) Changing supply functions in input/state stable systems. IEEE Trans. Autom. Control , 40 , 1476 – 1478 . Google Scholar CrossRef Search ADS Teel A. ( 1996 ) A nonlinear small gain theorem for the analysis of control systems with saturation. IEEE Trans. Autom. Control , 41 , 9 . Vidyasagar M. ( 1993 ) Nonlinear Systems Analysis . Englewood Cliffs : Prentice Hall . © The authors 2018. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png IMA Journal of Mathematical Control and Information Oxford University Press

Practical uniform input-to-state stability of perturbed triangular systems

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Oxford University Press
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© The authors 2018. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.
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0265-0754
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1471-6887
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Abstract

Abstract In this article, we present a practical uniform input-to-state stability result for perturbed triangular systems depending on a parameter. We present sufficient conditions for which each of these notions is preserved under cascade interconnection. 1. Introduction The asymptotic stability analysis by Lyapunov’s second method requires the construction of a strict Lyapunov function. This direct approach may be particularly hard for large-scale nonlinear time-varying systems. A natural way of simplifying this problem consists in dividing the system into simpler interconnected subsystems and to analyse each subsystem separately (Mazenc et al., 1999; Benabdallah et al., 2009, 2011). The interconnection of two input-to-state stability (ISS) systems is ISS since a growth rate condition can be always satisfied (Seibert & Suárez, 1990). However, for broader classes of systems, stability of their cascade is not always guaranteed. Seibert & Suárez (1990) derived global asymptotic stability (GAS) of a cascade of two time-invariant systems from individual GAS properties of the driving system and the disconnected driven system assuming that all solutions are bounded. The ISS analysis problem was studied for the autonomous case. There are many contributions in the corresponding literature. Some related works in Edwards et al. (2000), Karafyllis & Tsinias (2004), Sontag (1989, 1998), Sontag & Wang (1995) and Sontag & Teel (1995) introduced the notion of ISS. In particular, Jiang et al. (1996) showed that ISS is closed under composition, i.e. the cascade interconnection of two ISS systems is again an ISS system. To study uncertain dynamical systems, Jiang et al. (1994), Ito & Jiang (2006a, 2009) and Teel (1996) generalized the concept of ISS to the concept of input to state practical stability (ISpS). They proved that the interconnection of two ISpS systems is again an ISpS system. All these results are applied only to autonomous nonlinear systems. In Vidyasagar (1993), under some conditions, it was shown that a cascade nonautonomous system is globally uniformly exponentially stable if and only if each isolated subsystem is globally uniformly exponentially stable. Roughly, the result in Panteley & Lorıa (1998, 2001) shows that integrability of the perturbing trajectory of the driving system is sufficient to ensure GAS of the cascade. This observation has been re-interpreted and re-written in Angeli et al. (2000), Arcak et al. (2002), Lin et al. (2005) and Malisoff & Mazenc (2005), in terms of integral input-to-state stable stability (iISS) for a time-invariant cascade in which an iISS system is driven by a GAS system. The required trade-off condition is that the iISS gain of driven system needs to be steep satisfactorily in the direction toward the equilibrium if the convergence of the driving system is slow. Note that the set of iISS systems is larger and contains the ISS systems as a subset. The idea of the growth order and decay-rate trade-off result has been improved further in Chaillet & Angelli (2008) and Ito (2006, 2010), which shows additional conditions and states the trade-off in terms of Lyapunov-like inequalities (dissipation inequalities) of the two individual subsystems In addition, we complete the main result in Dashkovskiy et al. (2009) and Ito & Jiang (2006b), by giving a sufficient condition for the cascade composed of an iISS driven by a GAS one to remain GAS in the case when explicit Lyapunov functions are known. Roughly, the definition of this property is recalled in the sequel. it is again required that the dissipation term of the GAS subsystem dominates the supply function of the iISS one around zero. This result may be useful in practice since the iISS and GAS properties are commonly established through Lyapunov arguments. Furthermore, this result naturally extends to multiple cascaded systems, i.e. series of cascaded iISS systems driven by a GAS one. This result extends in this article to perturbed triangular time varying systems depending on a non-negative parameter where we interested in the study of practical uniform ISS of the time varying systems. This article is organized as follows. In Section 2, we give some definitions and results about practical uniform ISS (PUISS). In Section 3, some sufficient conditions are given to guarantee the practically uniformly ISS of a nonlinear perturbed time-varying system: remains practically uniformly input-to-state stable when it is perturbed by the output of another practically uniformly input-to-state system. In Section 4, we show that if both perturbed subsystems are practically uniformly input-to-state stable, then they can be practically uniformly input-to-state stable under some restrictive conditions on the one hand of the nominal system and on the other hand of the perturbation term. 2. Mathematical preliminaries Let $$L^{m}_{\infty}:=\{u:\mathbb{R}_+ \longrightarrow \mathbb{R}^m \ \text{such that } \ u \ \ \text{is bounded}\}$$. We use $$\|.\|_{\infty}$$ to denote the $$L^{m}_{\infty}$$ norm of u as a function defined on $$L^{m}_{\infty} $$ $$\|u\|_{\infty}:=\sup{\{\|u(t)\|, \ t \in \mathbb{R}_+\}.}$$ A function $$\alpha(.) : \mathbb{R}_+ \longrightarrow \mathbb{R}_+$$ is of class $$\mathcal{K}$$ if it is continuous, positive definite and strictly increasing and is of class $$\mathcal{K}_{\infty}$$, if it is also unbounded. A function $$ \beta(.,.): \mathbb{R}_+ \times \mathbb{R}_ + \longrightarrow \mathbb{R}_+$$ is said to be of class $$\mathcal{KL}$$ if for each fixed $$t\geq 0$$, $$\beta(.,t)$$ is of class $$\mathcal{K}$$ and for each fixed $$s\geq 0,$$$$\beta(s, .)$$ decreases to $$0$$ as $$t \rightarrow +\infty.$$ Lemma 2.1 Let $$\alpha_1(.)$$ and $$\alpha_2(.)$$$$\in \mathcal{K}_{\infty}$$ defined on $$[0,+\infty[.$$ Then the inverse function $$\alpha_1^{-1}(.)$$ of $$\alpha_1(.)$$ is of class $$\mathcal{K}_\infty.$$ $$\alpha_1\circ\alpha_2$$ is of class $$\mathcal{K}_\infty.$$ $$\forall s_1, s_2 \in \mathbb{R}_+,$$ $$\max{\{\alpha_1( s_1), \alpha_1(s_2)\}}\leq \alpha_1(s_1+s_2)\leq \alpha_1(2 s_1)+ \alpha_1(2s_2).$$ There exist a function $$\alpha \in \mathcal{K}_\infty$$ such that $$\forall s_1, s_2 \in \mathbb{R}_+,$$ $$\alpha(s_1+s_2)\leq \alpha_1(s_1) +\alpha_2(s_2).$$ Before stating our main theorem in Section (3), we introduce in this section some stability notions and some basic results. Consider the following controlled dynamical system depends on a parameter $$\varepsilon>0:$$ \begin{equation} \label{def1}\dot x = F^{\varepsilon}(t,x,u), \quad x(t_0)=x_0, \end{equation} (2.1) where $$t\in \mathbb{R}_+$$, $$x\in \mathbb{R}^n $$ is the state, $$u \in L^{m}_{\infty}$$ inputs, denoted by $$u$$ are measurable, essentially bounded functions from $$\mathbb{R}_+$$ to $$\mathbb{R}^m$$ and the function $$F^{\varepsilon}(.,.,.) : \mathbb{R}_+ \times \mathbb{R}^n \times \mathbb{R}^m \longrightarrow \mathbb{R}^n$$ is continuous in t and locally Lipschitz in $$x$$ and in $$u$$. We use $$x(t, t_0 ,x_0, u )$$: to denote the trajectory of the system corresponding to the initial condition $$x(t_0) = x_0$$ and the input function $$u$$. The Lipschitzness imposed on $$F^{\varepsilon}$$ guarantees the existence of a unique maximal solution of (2.1) for locally essentially bounded $$u$$. We start by recalling the definition of a PUISS and global practical uniform asymptotic stability (GPUAS) and some properties that this concept naturally induces, as well as related notions. Definition 2.1 (PUISS) We say that (2.1) is practical uniform input-to-state stable with respect to $$u$$ if there exists $$\varepsilon^*>0,$$ such that for any $$0<\varepsilon< \varepsilon^*,$$ there exist functions $$\beta_{\varepsilon}(.,.) \in \mathcal{KL}$$, $$\gamma_{\varepsilon}(.)\in \mathcal{K}$$ and positive scalars $$\rho(\varepsilon),$$ such that for each initial condition $$x_0$$ at any initial time $$t_0$$ and each measurable essentially bounded control $$u(.)$$ defined on $$\mathbb{R}_+$$, the solution $$x(.)$$ of the system (2.1) exists on $$\mathbb{R}_+$$ and satisfies \begin{equation} \label{dp1} \|x(t)\| \leq \beta_{\varepsilon}(\|x_0\|,t-t_0)+ \gamma_{\varepsilon}(\|u\|_{\infty})+\rho(\varepsilon), \ \forall t \geq t_0, \end{equation} (2.2) with $$\rho(\varepsilon)\to 0$$ as $$\varepsilon \to 0.$$ When (2.2) is satisfied with $$\rho(\varepsilon)= 0,$$ the system (2.1) is said to be input-to-state stable (ISS), a notion originally introduced by Sontag (1989, 1990). Definition 2.2 The system \begin{equation*} \dot x = F^{\varepsilon}(t,x,0), \quad x(t_0)=x_0. \end{equation*}is said to be globally practically uniformly asymptotically stable (GPUAS), if there exists $$\varepsilon^*>0,$$ such that for all $$0<\varepsilon< \varepsilon^*,$$ there exist function $$\beta_{\varepsilon}(.,.) \in \mathcal{KL}$$ and positive scalars $$\rho(\varepsilon),$$ such that for each initial condition $$x_0$$ at any initial time $$t_0$$, the solution $$x(.)$$ of the system (2.1) exists on $$\mathbb{R}_+$$ and satisfies \begin{equation} \label{dp2} \|x(t)\| \leq \beta_{\varepsilon}(\|x_0\|,t-t_0)+ \rho(\varepsilon), \ \forall t \geq t_0, \end{equation} (2.3)with$$\rho(\varepsilon)\to 0$$as$$\varepsilon \to 0.$$ We also recall that the system (2.1) is said to be $$0-$$GPUAS if the origin of $$\dot{x} = f(t,x,0)$$ is GPUAS. Remark 1 Systems of the form (2.1) can result from the design of high gain observers (see Benabdallah et al., 2011) or closed loop control systems. $$\rho(\varepsilon)$$ can be regarded as a measure of the distance of the system behavior from that of exponential stability (see Benabdallah et al., 2011). Since $$\rho(\varepsilon)$$ can be made arbitrarily small, the stability is called practical. Definition 2.3 A smooth function $$ V_{\varepsilon}:\mathbb{R}_+ \times \mathbb{R}^n \longrightarrow \mathbb{R}_+$$ is said to be a PUISS-Lyapunov function for the system (2.1), if there exist $$\mathcal{K}_{\infty}-$$functions $$\alpha_{1\varepsilon}(.), \ \alpha_{2\varepsilon}(.)$$, $$\mathcal{K}-$$functions $$\alpha_{3\varepsilon}(.)$$, $$\gamma_{\varepsilon}(.)$$ and a positive scalars $$\kappa(\varepsilon)> 0,$$ where $$\kappa(\varepsilon)\to 0$$ as $$\varepsilon \to 0$$, such that $$\forall x\in \mathbb{R}^n,$$$$\forall u \in \mathbb{R}^m$$ and $$\forall t\geq t_0:$$ \begin{equation} \label{eth0} \alpha_{1\varepsilon}(\|x\|)\leq V_{\varepsilon}(t,x) \leq \alpha_{2\varepsilon}(\|x\|) \end{equation} (2.4) and \begin{equation} \label{eth1}V_{\varepsilon}(t,x) \geq \gamma_{\varepsilon}(\|u\|_{\infty}) \ \Longrightarrow \ {\dot{V}_{\varepsilon}(t,x) \leq -\alpha_{3\varepsilon}(V_{\varepsilon})+\kappa(\varepsilon)}. \end{equation} (2.5) When (2.5) holds with $$\kappa(\varepsilon)=0$$, $$V_{\varepsilon}$$ is called an ISS-Lyapunou function for the system (2.1). It was shown in Sontag & Wang (1995) that for a time-invariant system $$\dot{x}= f(x,u)$$, the ISS property is equivalent to the existence of an ISS-Lyapunov function $$V$$ which is independent of $$t$$ and property (2.5) is equivalent to the existence of a $$\mathcal{K}$$-function $$\alpha_{3\varepsilon}(.)$$ and $$\chi_{\varepsilon}(.)$$ such that \begin{equation} \label{eth2} \dot{V}_{\varepsilon} (t,x) \leq -\alpha_{3\varepsilon}(V_{\varepsilon} )+\chi_{\varepsilon}(\|u\|_{\infty})+ \kappa(\varepsilon). \end{equation} (2.6) Remark 2 Observe that this definition is slightly different from the original definition proposed by Sontag & Wang (1995) in that $$\alpha_{3\varepsilon}$$ depending on a parameters $$\varepsilon>0$$ rather than independent on parameters as in Sontag & Wang (1995). Remark 3 It is immediate that the system (2.1) admits an iSpS- (respectively, ISS-) Lyapunov function satisfying (2.4) and (2.5) if and only if it admits an iSpS- (respectively, ISS-) Lyapunov function satisfying (2.4) and (2.6) in Sontag & Wang (1995). Recently, the equivalence between the iSpS property and the existence of an iSpS-Lyapunov function was shown by Sontag & Wang (1995), i.e. the following was proved. Proposition 2.2 The system (2.1) is PUISS (respectively, ISS) if and only if it has an PUISS- (respectively, ISS-) Lyapunov function. 3. PUISS of perturbed systems Now we will study the PUISS of two classes of perturbed parameter-dependent systems. Stability of perturbed systems is investigated, for instance, in Khalil (1996) but the considered systems do not depend on parameters. The first one concerns systems with a nominal part that depends on a parameter $$\varepsilon>0.$$ Let $$\mathcal{H}=\{\omega\in \mathcal{K} \ \text{such that } \ (id-\omega)\in \mathcal{K} \}.$$ Theorem 3.1 Consider the perturbed system \begin{equation} \dot{x}=F^{\varepsilon}(t,x) +G(t,x,u), \end{equation} (3.1) where $$F^{\varepsilon}: {\mathbb{R}}_+ \times {\mathbb{R}}^n \rightarrow {\mathbb{R}}^n $$ and $$G: {\mathbb{R}}_+ \times {\mathbb{R}}^n \times {\mathbb{R}}^m \rightarrow {\mathbb{R}}^n$$ are continuous and locally Lipschitz in $$x$$ with $$F^{\varepsilon}$$ known and $$G$$ unknown ‘the perturbed term’. Suppose that the nominal system $$\dot{x} = F^{\varepsilon}(t,x)$$ is uniformly globally practically asymptotically stable, with a Lyapunov function $$V_{\varepsilon}:{\mathbb{R}}_+\times {\mathbb{R}}^n \to {\mathbb{R}}_+,$$ which satisfies \begin{gather*} \alpha_{1_{\varepsilon}}(\|x\|)\leq V_{\varepsilon}(t,x) \leq \alpha_{2_{\varepsilon}}(\|x\|) \\ \frac{\partial V_{\varepsilon}}{\partial t}(t,x)+\frac{\partial V_{\varepsilon}}{\partial x}(t,x)F^{\varepsilon}(t,x)\leq -\alpha_{3_{\varepsilon}}(V_{\varepsilon})+\delta(\varepsilon), \end{gather*} where $$\alpha_{1_{\varepsilon}}(.)$$ and $$\alpha_{{2}_{ \varepsilon}}(.)$$$$\in{ \mathcal{K}}_{\infty}$$, $$\alpha_{3_{\varepsilon}}(.)$$ is a $$\mathcal{K}-$$ function and $$\delta(\varepsilon)$$ is a positive constant with $$\delta(\varepsilon)\to 0$$ as $$\varepsilon \to 0).$$ We suppose also that there exist positive constants $$\delta_1(\varepsilon)$$, $$\delta_2(\varepsilon)$$$$>0$$ and $$0< \theta< 1,$$ such that: \[ \|\frac{\partial V_{\varepsilon}}{\partial x}G(t,x,u)\| \leq \delta_1(\varepsilon) \|x\|+ \delta_2(\varepsilon) \|u\|_{\infty}, \quad \forall (t,x,u)\in {\mathbb{R}}_+ \times {\mathbb{R}}^ n \times {\mathbb{R}}^m \] and \begin{equation} \label{gain} \displaystyle\frac{\delta_1(\varepsilon)} {\theta}(\alpha_{3_{\varepsilon}} \circ \alpha_{1_{\varepsilon}})^{-1} \in \mathcal{H}. \end{equation} (3.2) Then, the system (3.1) is practically uniformly input-to-state stable. Proof. The derivative of $$V_{\varepsilon}$$ along the trajectories of system (3.1) is given by $$\dot{V}_{\varepsilon}(t,x(t))\leq -\alpha_{3_{\varepsilon}}(V_{\varepsilon})+\delta(\varepsilon)+\delta_1(\varepsilon) \|x\|+ \delta_2(\varepsilon) \|u\|_{\infty}.$$ Then, $$\dot{V}_{\varepsilon}(t,x(t))\leq -\alpha_{3_{\varepsilon}}(V_{\varepsilon})+\delta(\varepsilon)+\delta_1(\varepsilon) \alpha^{-1}_{1_{\varepsilon}}(V_{\varepsilon})+ \delta_2(\varepsilon) \|u\|_{\infty}.$$ Thus, there exists $$0<\theta<1$$ such that $$\dot{V}_{\varepsilon}(t,x(t))\leq -(1-\theta) \alpha_{3_{\varepsilon}}(V_{\varepsilon})-\theta \alpha_{3_{\varepsilon}}(V_{\varepsilon})+\delta(\varepsilon)+\delta_1(\varepsilon) \alpha^{-1}_{1_{\varepsilon}}(V_{\varepsilon})+ \delta_2(\varepsilon) \|u\|_{\infty},$$ which implies that $$\dot{V}_{\varepsilon}(t,x(t))\leq -(1-\theta) \alpha_{3_{\varepsilon}}(V_{\varepsilon})+\delta(\varepsilon), \ \forall \ V_{\varepsilon}(t,x) \geq \gamma_{\varepsilon}(\|u\|_{\infty}), $$ with $$\gamma_{\varepsilon}(.)$$ defined by: $$\gamma_{\varepsilon}(s):= (\theta\alpha_{3_{\varepsilon}})^{-1}\circ \left( id-\displaystyle\frac{\delta_1(\varepsilon)} {\theta}(\alpha_{3_{\varepsilon}} \circ \alpha_{1_{\varepsilon}})^{-1}\right) ^{-1}(\delta_2(\varepsilon) s).$$ According to the hypothesis (3.2), $$\gamma_{\varepsilon}(.)$$ is a $$\mathcal{K}$$-function. Thus, \begin{equation} V_{\varepsilon}(t,x) \geq \gamma_{\varepsilon}(\|u\|_{\infty}) \ \Longrightarrow \ \dot{V}(t,x) \leq -(1-\theta) \alpha_{3\varepsilon}(V_{\varepsilon})+\delta(\varepsilon) . \end{equation} (3.3) We conclude that the perturbed system (3.1) is practically uniformly input-to-state stable. □ 4. PUISS of perturbed triangular systems The first result we recall here is concerned with cascade connection of two PUISS systems, in the case when an PUISS-Lyapunov function is explicitly known for each of them. For the sake of generality, it is allowed that the driven subsystem depends also on the external input. We therefore deal with systems of the form: \begin{eqnarray} {\dot x}_1&=& f_1^{\varepsilon}(t,x_1,x_2,u) \label{Pertcas3}\\ \end{eqnarray} (4.1) \begin{eqnarray} {\dot x}_2&=& f_2^{\varepsilon}(t,x_2,u), \label{Pertcas4} \end{eqnarray} (4.2) where $$t\geq 0,$$$$x_1\in \mathbb{R}^n,$$$$x_2\in \mathbb{R}^p,$$$$u: \mathbb{R}_+\longrightarrow \mathbb{R}^m,$$ is measurable locally essentially bounded, $$f_1^{\varepsilon},$$$$f_2^{\varepsilon}$$ are both assumed to be locally Lipschitz. We also assume that $$f_1^{\varepsilon}(t,0,0,0) = 0$$ and $$f_2^{\varepsilon}(t,0,0) = 0$$. We present here a cascade of a PUISS subsystem controlled by another subsystem PUISS to be PUISS. Theorem 4.1 (PUISS+PUISS) Let $$V_{1\varepsilon}$$ and $$V_{2\varepsilon}$$ be continuous positive definite radially unbounded functions, differentiable on $$\mathbb{R}^n$$ and $$\mathbb{R}^p,$$ respectively. Suppose that $$\forall x_1\in \mathbb{R}^n,$$$$\forall x_2\in \mathbb{R}^p,$$$$\forall u \in L^{m}_{\infty}$$ and $$\forall t\geq t_0,$$ one has \begin{gather*} \frac{\partial V_{1\varepsilon}}{\partial t}+\frac{\partial V_{1\varepsilon}}{\partial x} f_1^{\varepsilon}(t,x_1,x_2,u)\leq -\alpha_{1\varepsilon}(V_{1\varepsilon})+\alpha_{2\varepsilon}(\|x_2\|)+\gamma_{1\varepsilon}(\|u\|_{\infty})+ \rho_1(\varepsilon),\\ \delta_{1_{\varepsilon}}(\|x_2\|)\leq V_{2\varepsilon}(t,x_2) \leq \delta_{2_{\varepsilon}}(\|x_2\|) \end{gather*} and \[ \frac{\partial V_{2\varepsilon}}{\partial t}+\frac{\partial V_{2\varepsilon}}{\partial x} f_2^{\varepsilon}(t,x_2,u)\leq -\delta_{3\varepsilon}(V_{2\varepsilon})+\gamma_{2\varepsilon}(\|u\|_{\infty})+ \rho_2(\varepsilon).\] where $$\delta_{{1\varepsilon}}(.) $$ and $$\delta_{{2\varepsilon}}(.) $$ are $$\mathcal{K}_{\infty}-$$functions, $$\alpha_{1\varepsilon}(.)$$, $$\alpha_{{2\varepsilon}}(.), $$$$\gamma_{{1\varepsilon}}(.), $$$$\gamma_{{2\varepsilon}}(.)$$ and $$\delta_{{3\varepsilon}}(.)$$ are $$\mathcal{K}-$$functions and $$\rho_i(\varepsilon) \rightarrow 0$$ as $$\varepsilon \rightarrow 0$$, $$\forall i=1,2.$$ If \begin{equation} \label{casc5}\displaystyle\frac{1}{\theta}\alpha_{2\varepsilon}\circ (\delta_{3\varepsilon} \circ \delta_{{1\varepsilon}})^{-1} \in \mathcal{H}, \end{equation} (4.3) the cascade system (4.1)–(4.2) is PUISS. Proof. The derivative of $$V_{\varepsilon}:=V_{1\varepsilon}+V_{2\varepsilon}$$ along the trajectories of system (4.1)–(4.2) is given by $$\dot{V_{\varepsilon}}(t,x)\leq -\alpha_{1\varepsilon}(V_{1\varepsilon})-\delta_{3\varepsilon}(V_{2\varepsilon}) +\alpha_{2\varepsilon}(\|x_2\|)+\gamma_{1\varepsilon}(\|u\|_{\infty})+\gamma_{2\varepsilon}(\|u\|_{\infty})+ \rho_1(\varepsilon)+ \rho_2(\varepsilon) .$$ Thus, there exist $$0<\theta<1$$ such that: $$\dot{V_{\varepsilon}}(t,x)\,{\leq }\,{-}\,\alpha_{1\varepsilon}(V_{1\varepsilon})-(1-\theta) \delta_{3\varepsilon}(V_{2\varepsilon}) -\theta \delta_{3\varepsilon}(V_{2\varepsilon}) +\alpha_{2\varepsilon}\circ \delta_{1_{\varepsilon}}^{-1}(V_{2\varepsilon})+\gamma_{1\varepsilon}(\|u\|_{\infty})+\gamma_{2\varepsilon}(\|u\|_{\infty})+ \rho_1(\varepsilon)+ \rho_2(\varepsilon)\!.$$ Then, using Lemma 2.1, there exists $$\alpha_{\varepsilon}\in \mathcal{K}$$ such that: $$\dot{V_{\varepsilon}}(t,x)\leq -\alpha_{\varepsilon}(V_{\varepsilon})+[\alpha_{2\varepsilon}\circ \delta_{1_{\varepsilon}}^{-1}-\theta \delta_{3\varepsilon}](V_{2\varepsilon}) +\gamma_{\varepsilon}(\|u\|_{\infty})+ \rho(\varepsilon),$$ where $$\gamma_{\varepsilon}(s):= \gamma_{1\varepsilon}(s)+\gamma_{2\varepsilon}(s) \in \mathcal{K}$$ and $$\rho(\varepsilon):= \rho_1(\varepsilon)+ \rho_2(\varepsilon) \rightarrow 0$$ as $$\varepsilon \rightarrow 0,$$ which implies that $$\dot{V_{\varepsilon}}(t,x)\leq -\alpha_{\varepsilon}(V_{\varepsilon})+\rho(\varepsilon), \ \ \text{if} \ [\alpha_{2\varepsilon}\circ \delta_{1_{\varepsilon}}^{-1}-\theta \delta_{3\varepsilon}](V_{2\varepsilon}) +\gamma_{\varepsilon}(\|u\|_{\infty})\leq 0. $$ Then, using (4.3), there exists $$\chi_{\varepsilon}\in \mathcal{K}$$ defined by: $$\chi_{\varepsilon}(s):= (\theta \delta_{3\varepsilon})^{-1}\circ \left( id-\displaystyle\frac{1}{\theta}\alpha_{2\varepsilon}\circ (\delta_{3\varepsilon} \circ \delta_{{1\varepsilon}})^{-1}\right)^{-1}\circ \gamma_{\varepsilon}( s),$$ such that \begin{equation} V_{\varepsilon}(t,x) \geq \chi_{\varepsilon}(\|u\|_{\infty}) \ \Longrightarrow \ \dot{V_{\varepsilon}}(t,x)\leq -\alpha_{\varepsilon}(V_{\varepsilon})+\rho(\varepsilon). \end{equation} (4.4) We conclude that the cascade system (4.1)–(4.2) is practically uniformly input-to-state stable. □ Remark 4 While theorem 4.1 is stated for the cascade interconnection of two nonlinear subsystems, it easily extends to multiple cascades. 4.1. Perturbation terms depending on a parameter $$\varepsilon$$ Now we consider cascaded systems of the form with perturbations depending on a parameter $$\varepsilon>0$$. \begin{eqnarray} {\dot x}_1&=& f_1(t,x_1,x_2,u)+g_1^{\varepsilon}(t,x_1,x_2) \label{Pert3}\\ \end{eqnarray} (4.5) \begin{eqnarray} {\dot x}_2&=& f_2(t,x_2,u)+g_2^{\varepsilon}(t,x_2) \label{Pert4} \end{eqnarray} (4.6) where $$t\geq 0,$$$$x_1\in \mathbb{R}^n,$$$$x_2\in \mathbb{R}^p,$$$$u: \mathbb{R}_+\longrightarrow \mathbb{R}^m,$$ is measurable locally essentially bounded and both $$f_1,$$$$f_2$$ are assumed to be locally Lipschitz in the state and piecewise continuous in the time. We also assume that $$f_1^{\varepsilon}(t,0,0,0) = 0$$ and $$f_2^{\varepsilon}(t,0,0) = 0$$. $$g^{\varepsilon}_1$$ and $$g^{\varepsilon}_2$$ are locally Lipschitz in the state and piecewise continuous in the time for all parameters $${\varepsilon}$$ under consideration. ($$\mathcal{A}_3$$) The system \[{\dot x}_1= f_1(t,x_1,0,u)\] is PUISS, with a Lyapunov PUISS function $$V_{1}$$ be continuous positive definite radially unbounded function, differentiable on $$\mathbb{R}^n$$. Suppose that $$\forall x_1\in \mathbb{R}^n,$$$$\forall u \in L^{m}_{\infty}$$ and $$\forall t\geq t_0,$$ the following holds \begin{gather} \label{a1} a_1 \|x_1\|^2\leq V_1(t,x_1)\leq a_2 \|x_1\|^2\\ \frac{\partial V_1}{\partial t}+\frac{\partial V_1}{\partial x_1}f_1(t,x_1,0,u) \leq -\alpha_1(V_1)+\chi_1(\|u\|_{\infty})+\delta_1\notag\\ \|\frac{\partial V_1}{\partial x_1}\|\leq a_3 \|x_1\|\notag \end{gather} (4.7) and \[\|\frac{\partial V_1}{\partial x_1}g_1^{\varepsilon}(t,x_1,x_2)\| \leq r_1(\varepsilon),\] where $$a_1$$, $$a_2,$$$$a_3$$, $$\delta_1$$ and $$r_1(\varepsilon)$$ are positives constants with $$r_1(\varepsilon)\rightarrow 0$$ as $$\varepsilon \rightarrow 0$$, $$\alpha_1(.) \in \mathcal{K}$$ and $$\chi_1(.) \in \mathcal{K}$$. In addition, there exists a positive constant $$M<\infty$$ such that \[\|\frac {\partial f_1 }{\partial x_2}\|\leq M , \quad \forall (t,x_1,x_2,u)\in \mathbb{R}_+\times \mathbb{R}^n \times\mathbb{R}^p\times L^{m}_{\infty}\] ($$\mathcal{A}_4$$) The system \[\dot x _2 = f_2 (t,x_2,u) \] is PUISS, with a Lyapunov PUISS function $$V_{2}$$ be continuous positive definite radially unbounded function, differentiable on $$\mathbb{R}^p$$. Suppose that $$\forall x_2\in \mathbb{R}^p,$$$$\forall u \in L^{m}_{\infty}$$ and $$\forall t\geq t_0:$$ \begin{gather} \label{b1} b_1 \|x_2\|^2\leq V_2(t,x_2)\leq b_2\|x_2\|^2\\ \frac{\partial V_2}{\partial t}+\frac{\partial V_2}{\partial x_2}f_2(t,x_2,u) \leq -\alpha_2(V_2)+\chi_2(\|u\|_{\infty})+\delta_2,\notag\\ \|\frac{\partial V_2}{\partial x_2}g_2^{\varepsilon}(t,x_2)\| \leq r_2(\varepsilon),\notag \end{gather} (4.8) where $$b_1$$, $$b_2$$, $$\delta_2$$ and $$r_2(\varepsilon)$$$$\geq 0$$ with $$r_2(\varepsilon) \rightarrow 0$$ as $$\varepsilon \rightarrow 0$$, $$\alpha_2(.)\in \mathcal{K}, \ \chi_2(.)\in \mathcal{K}$$. We suppose also there exists a parameter $$0<\theta<1$$ such that: \begin{equation} \label{gain5} \displaystyle\frac{M a_3}{2\theta \min{(a_1,b_1)}} {{\alpha}}^{-1}\in \mathcal{H}, \end{equation} (4.9) where $$\alpha(s):=\min{\{\alpha_1(\frac{s}{2}), \ \alpha_2(\frac{s}{2})\}}.$$ The following result establishes PUISS of triangular systems under some assumption and provided the knowledge of a Lyapunov function for the driven subsystem. Theorem 4.2 Under the assumptions $$(\mathcal{A}_3)$$ and $$(\mathcal{A}_4)$$, the cascaded system (4.5)–(4.6) is PUISS. Proof. The derivative of $$V:=V_{1}+V_{2}$$ along the trajectories of system (4.5)–(4.6) is given by $$\dot{V}(t,x)\leq -\alpha_{1}(V_{1})-\alpha_{2}(V_{2}) +\chi_1(\|u\|_{\infty})+\delta_1+\chi_2(\|u\|_{\infty})+\delta_2+a_3 M\|x_1\|\|x_2\|+r_1(\varepsilon)+r_2(\varepsilon). $$ Then, using lemma 2.1, there exist $$\alpha\in \mathcal{K}$$ such that: $$\dot{V}(t,x)\leq -\alpha(V) + \chi(\|u\|_{\infty})+ r(\varepsilon)+\displaystyle\frac{a_3M}{2a_1}V_1+\displaystyle\frac{a_3M}{2b_1}V_2,$$ where $$\chi(s):= \chi_{1}(s)+\chi_{2}(s) \in \mathcal{K}$$, $$\delta= \delta_1+\delta_2$$ and $$r(\varepsilon):= r_1(\varepsilon)+ r_2(\varepsilon) \rightarrow 0$$ as $$\varepsilon \rightarrow 0.$$ Then, $$\dot{V}(t,x)\leq -\alpha(V) + \chi(\|u\|_{\infty})+ r(\varepsilon)+\delta+\displaystyle\frac{a_3M}{2 \min{(a_1,b_1)}}V.$$ Thus, there exists $$0<\theta<1$$ such that $$\dot{V}(t,x)\leq -(1-\theta)\alpha(V)- \theta \alpha(V)+\displaystyle\frac{a_3M}{2 \min{(a_1,b_1)}}V+ \chi(\|u\|_{\infty})+ r(\varepsilon)+\delta,$$ which implies that $$\dot{V}(t,x)\leq -(1-\theta)\alpha(V)+r(\varepsilon)+\delta, \ \ \text{if} \ [ - \theta \alpha+\displaystyle\frac{a_3M}{2 \min{(a_1,b_1)}}id](V)+ \chi(\|u\|_{\infty})\leq 0. $$ Then, using 4.9, there exist $$\gamma\in \mathcal{K}$$ defined by $$\gamma(s):= (\theta \alpha)^{-1}\circ \left( id- \displaystyle\frac{M a_3}{2\theta \min{(a_1,b_1)}} {{\alpha}}^{-1}\right)^{-1}\circ \chi( s),$$ such that \begin{equation} V(t,x) \geq \gamma(\|u\|_{\infty}) \ \Longrightarrow \ \dot{V}(t,x)\leq -(1-\theta)\alpha(V)+r(\varepsilon)+\delta. \end{equation} (4.10) We conclude that the cascade system (4.5)–(4.6) is practically uniformly input-to-state stable. □ 4.2. Nominal system depending on a parameter $$\varepsilon$$ Now, we consider triangular perturbed systems of the following form with a nominal systems depending on a parameter $$\varepsilon>0$$: \begin{eqnarray} \dot{x}_1&=&f_1^{\varepsilon}(t,x_1)+g(t,x,u) \label{R11} \\ \end{eqnarray} (4.11) \begin{eqnarray} \dot{x}_2&=&f_2^{\varepsilon}(t,x_2), \label{R12} \ \end{eqnarray} (4.12) where $$x_1\in\mathbb{R}^n, \ x_2\in\mathbb{R}^p$$, $$u: \mathbb{R}_+\longrightarrow \mathbb{R}^m,$$ is measurable locally essentially bounded, $$x:=(x_1,x_2)^T$$, $$f_1^{\varepsilon}(t,x_1)$$ and $$f_2^{\varepsilon}(t,x_2) $$ are differentiable functions for all parameters $${\varepsilon}$$ under consideration and $$g(t,x,u)$$ are locally Lipschitz in the state and piecewise continuous in the time. Introduce the following assumption: $$(H_1)$$ Suppose that \begin{equation} \label{cascade1} \dot{x}_1=f_1^{\varepsilon}(t,x_1) \end{equation} (4.13) is globally practically uniformly asymptotically stable, with a Lyapunov function $$V_{1 \varepsilon},$$$$\alpha_{1_{\varepsilon}}(.)$$ and $$\alpha_{{2}_{ \varepsilon}}(.)$$$$\in{ \mathcal{K}}_{\infty}$$, a $$\mathcal{K}-$$ function $$\alpha_{3_{\varepsilon}}(.)$$ and a positive constant $$\delta(\varepsilon)$$ with $$\delta(\varepsilon)\to 0$$ as $$\varepsilon \to 0).$$ We suppose also that there exist positive constants $$\rho_1(\varepsilon) \geq 0,$$ with $$ \rho_1(\varepsilon) \rightarrow 0$$ as $$\varepsilon \rightarrow 0$$, such that: \begin{gather}\label{R:35}\alpha_{1\varepsilon} (\|x_1\|)\leq V_{1\varepsilon}(t,x_1)\leq \alpha_{2\varepsilon}(\|x_1\|)\\ \end{gather} (4.14) \begin{gather} \label{R:36}\displaystyle\frac{\partial V_{1\varepsilon}}{\partial t} +\displaystyle\frac{\partial V_{1\varepsilon}}{\partial x_1} f_1^{\varepsilon}(t,x_1)\leq \ -\alpha_{3\varepsilon} (V_{1\varepsilon})+\rho_1(\varepsilon) \end{gather} (4.15) $$(H_2)$$ Suppose that \begin{equation} \label{cascade2} \dot{x}_2=f_2^{\varepsilon}(t,x_2) \end{equation} (4.16) is GPUAS, there exists a Lyapunov function $$V_{2\varepsilon}(t,x_1),$$$$\gamma_{1\varepsilon},$$$$\gamma_{2\varepsilon}$$$$\in{ \mathcal{K}}_{\infty}$$, a $$\mathcal{K}-$$ function $$\gamma_{3\varepsilon}$$ and $$\rho_2(\varepsilon)\geq 0,$$ with $$ \rho_2(\varepsilon) \rightarrow 0$$ as $$\varepsilon \rightarrow 0$$ such that: \begin{gather}\label{R:37}\gamma_{1\varepsilon} (\|x_2\|)\leq V_{2\varepsilon}(t,x_1)\leq \gamma_{2\varepsilon}(\|x_2\|)\\ \end{gather} (4.17) \begin{gather} \label{R:38}\displaystyle\frac{\partial V_{2\varepsilon}}{\partial t} +\displaystyle\frac{\partial V_{2\varepsilon}}{\partial x_2} f_2^{\varepsilon}(t,x_1)\leq \ -\gamma_{3\varepsilon} (V_{2\varepsilon})+\rho_2(\varepsilon) \end{gather} (4.18) $$(H_3)$$ We suppose the existence for scalars $$\delta_1(\varepsilon)$$, $$\delta_2(\varepsilon)$$$$\geq 0$$, a $$\mathcal{K}$$-function $$\alpha_{4\varepsilon}(.)$$ a constant $$\theta,$$$$0< \theta< 1$$ such that: \[\|\displaystyle\frac{\partial V_{1\varepsilon}}{\partial x_1} g(t,x,u)\| \leq \delta_1(\varepsilon) \|x_1\|+ \delta_2(\varepsilon) \|x_2\| +\alpha_{4\varepsilon}(\|u\|_{\infty}), \quad \forall (t,x_1,x_2,u)\in {\mathbb{R}}_+\times {\mathbb{R}}^ n\times {\mathbb{R}}^p \times {\mathbb{R}}^m .\] and \begin{equation} \label{gain4} \displaystyle\frac{1}{\theta} {\bar{\beta}}_{\varepsilon}\circ \sigma_{\varepsilon}^{-1} \in \mathcal{H}, \end{equation} (4.19) where \begin{align*} \bar{\beta}_{\varepsilon}(s)&:=\max{\{\delta_1(\varepsilon)\alpha_{1_{\varepsilon}}^{-1}(s), \ \delta_2(\varepsilon) \gamma_{1_{\varepsilon}}^{-1}(s)\}},\\ \sigma_{\varepsilon}(s)&:=\min{\{\alpha_{3_{\varepsilon}}(\frac{s}{2}), \ \gamma_{3_{\varepsilon}}(\frac{s}{2})\}}. \end{align*} Theorem 4.3 Under assumptions $$(H_1)$$, $$(H_2)$$ and $$(H_3),$$ the triangular perturbed system (4.11)–(4.12) is (PUISS). Proof. The derivative of $$V_{\varepsilon}:=V_{1\varepsilon}+V_{2\varepsilon}$$ along the trajectories of system (4.11)–(4.12) is given by $$\dot{V_{\varepsilon}}(t,x)\leq -\alpha_{3\varepsilon}(V_{1\varepsilon})-\gamma_{3\varepsilon}(V_{2\varepsilon}) + \delta_1(\varepsilon) \|x_1\|+ \delta_2(\varepsilon) \|x_2\| +\alpha_{4\varepsilon}(\|u\|_{\infty})+\rho_1(\varepsilon)+\rho_2(\varepsilon)$$ then, using lemma 2.1, there exist $$\sigma_{\varepsilon}(s):=\min{\{\alpha_{3_{\varepsilon}}(\frac{s}{2}), \ \gamma_{3_{\varepsilon}}(\frac{s}{2})\}}\in \mathcal{K}$$, such that: $$\dot{V_{\varepsilon}}(t,x)\leq -\sigma_{\varepsilon}(V_{\varepsilon})+ \delta_1(\varepsilon) \alpha_{1_{\varepsilon}}^{-1}(V_{1\varepsilon})+\delta_2(\varepsilon) \gamma_{1_{\varepsilon}}^{-1}(V_{2\varepsilon}) +\alpha_{4\varepsilon}(\|u\|_{\infty})+\rho_1(\varepsilon)+\rho_2(\varepsilon),$$ which implies that $$\dot{V_{\varepsilon}}(t,x)\leq -\sigma_{\varepsilon}(V_{\varepsilon})+ {\bar{\beta}}_{\varepsilon}(V_{\varepsilon}) +\alpha_{4\varepsilon}(\|u\|_{\infty})+\rho(\varepsilon),$$ where $${\bar{\beta}}_{\varepsilon}(s):=\max{\{\delta_1(\varepsilon)\alpha_{1_{\varepsilon}}^{-1}(s), \ \delta_2(\varepsilon) \gamma_{1_{\varepsilon}}^{-1}(s)\}}\in \mathcal{K}$$ and $$\rho(\varepsilon):= \rho_1(\varepsilon)+ \rho_2(\varepsilon) \rightarrow 0$$ as $$\varepsilon \rightarrow 0,$$ Thus, there exist $$0<\theta<1$$ such that: $$\dot{V_{\varepsilon}}(t,x)\leq -(1-\theta)\sigma_{\varepsilon}(V_{\varepsilon})- \theta \sigma_{\varepsilon}(V_{\varepsilon})+ {\bar{\beta}}_{\varepsilon}(V_{\varepsilon}) +\alpha_{4\varepsilon}(\|u\|_{\infty})+\rho(\varepsilon),$$ which implies that $$\dot{V_{\varepsilon}}(t,x)\leq -(1-\theta)\sigma_{\varepsilon}(V_{\varepsilon})+\rho(\varepsilon), \ \ \text{if} \ [ {\bar{\beta}}_{\varepsilon}-\theta \sigma_{\varepsilon}](V_{\varepsilon}) +\alpha_{4\varepsilon}(\|u\|_{\infty})\leq 0. $$ Then, using 4.19 there exist $$\chi_{\varepsilon}\in \mathcal{K}$$ defined by: $$\chi_{\varepsilon}(s):= (\theta \sigma_{\varepsilon})^{-1}\circ \left( id-\displaystyle\frac{1}{\theta} {\bar{\beta}}_{\varepsilon}\circ \sigma_{\varepsilon}^{-1}\right)^{-1}\circ \alpha_{4\varepsilon}( s),$$ such that \begin{equation} V_{\varepsilon}(t,x) \geq \chi_{\varepsilon}(\|u\|_{\infty}) \ \Longrightarrow \ \dot{V_{\varepsilon}}(t,x)\leq -(1-\theta)\sigma_{\varepsilon}(V_{\varepsilon})+\rho(\varepsilon). \end{equation} (4.20) We conclude that the cascade system (4.11)–(4.12) is practically uniformly input-to-state stable. □ Example 1 Consider the system \begin{eqnarray} {\dot x}_1&=& -\varepsilon x_1+ \displaystyle\frac{x_1x_2}{1+x_1^2}+ \displaystyle e^{-t} u \end{eqnarray} (4.21) \begin{eqnarray} {\dot x}_2&=& -x_2^3+\varepsilon \displaystyle\frac{x_2e^{-2t}}{1+x_2^2}. \label{exple3} \end{eqnarray} (4.22) Setting \begin{gather*} V_{1{\varepsilon}}:= \frac{\varepsilon}{2}x_1^2, \ V_{2{\varepsilon}}:= \frac{\varepsilon}{2}x_2^2\\ f_1^{\varepsilon}(t,x_1)=-\varepsilon x_1, \ \ f_2^{\varepsilon}(t,x_2)=-x_2^3+ \varepsilon \displaystyle\frac{x_2e^{-2t}}{1+x_2^2} \ \text{and} \ \ g(t,x,u)=\displaystyle\frac{x_1 x_2}{1+x_1^2}+ \displaystyle e^{-t} u. \end{gather*} We deduce that assumptions $$(H_1)$$, $$(H_2)$$ and $$(H_3)$$ are satisfied. 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Published: Feb 12, 2018

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