Growth rate conditions for uniform asymptotic stability of cascaded time-varying systems

Abstract

In the last decade several results on cascaded autonomous systems have appeared in the literature. The sufficient conditions for the stability and stabilizability of these systems are often related to certain growth rate conditions on the functions which define the dynamics of the system. In this paper we analyze three complementary classes of nonlinear time-varying systems according to these growth rates. For each case, we give further results to guarantee uniform global asymptotic stability of the cascade and relate our contributions to input-to-state stability and other growth rate conditions previously reported. The advantage of this type of analysis is that it is often easier to verify such conditions than to find a Lyapunov function with a negative-definite derivative.

Introduction

Sufficient conditions for uniform global asymptotic stability of nonlinear time-varying (NLTV) systems are typically formulated via a Lyapunov function which is required to have a negative-definite derivative. However, this is often hard to meet for many practical applications. In this paper we present sufficient conditions to guarantee UGAS for cascaded NLTV systems

$$\text{Σ}{}_1\text{:}\text{x}\text{̇}{}_1\text{=f}{}_1\text{(t,x}{}_1\text{)+g(t,x)x}{}_2\text{,}$$

$$\text{Σ}{}_2\text{:}\text{x}\text{̇}{}_2\text{=f}{}_2\text{(t,x}{}_2\text{),}$$

where $\text{x}{}_1\text{∈}\text{R}{}^{\mathrm{n}}\text{,}\text{x}{}_2\text{∈}\text{R}{}^{\mathrm{m}}\text{,}\text{x≔}\text{col}\text{[x}{}_1\text{,x}{}_2\text{]}$. The functions $\text{f}{}_1\text{(t,x}{}_1\text{),}\text{f}{}_2\text{(t,x}{}_2\text{)}$ and g(t,x) are continuous in their arguments, locally Lipschitz in x, uniformly in t, and f₁(t,x₂) is continuously differentiable in both arguments. We also assume that there exists a nondecreasing function G(·) such that,

$$\text{||g(t,x)||≤G(||x||).}$$

We will enunciate sufficient conditions for three classes of cascades: roughly speaking, we consider systems such that, for each fixed x₂, the following hold uniformly in t: (i) the function f₁(t,x₁) grows faster than g(t,x) as ||x₁||→∞ (for each fixed x₂); (ii) both functions f₁(t,x₁) and g(t,x) grow at similar rate as ||x₁||→∞; (iii) the function g(t,x) grows faster than f₁(t,x₁) as functions of x₁.

In each case, we give sufficient conditions to guarantee that a UGAS nonlinear time-varying system

$$\text{x}\text{̇}{}_1\text{=f}{}_1\text{(t,x}{}_1\text{)}$$

remains UGAS when it is perturbed by the output of another UGAS system of the form Σ₂, that is, we establish sufficient conditions to ensure UGAS for system (1) and (2).

Our motivation to consider cascaded systems is that, as shown in Panteley and Lorı́a (1999) and Panteley, Lorı́a, and Sokolov (1999). they may appear in many practical applications. Most remarkably, in some cases a system can be decomposed into two subsystems for which control inputs can be designed with the aim that the closed loop have a cascaded structure. In this direction the results in Panteley and Ortega (1997) suggest that the global stabilization of nonlinear systems which allow a cascades decomposition, may be achieved by ensuring UGAS for both subsystems separately. The question remaining is to know whether the stability properties of both subsystems separately, remains valid under the cascaded interconnection , . See also Lefeber (2000) and references therein, where cascades theorems as those presented in this paper have been successfully used to design simple (even linear) controllers for nonholonomic systems and ships, as opposed to highly nonlinear Lyapunov-based control laws. The latter motivates us to study the stability analysis problem exposed above.

The stability analysis problem for autonomous systems

$$\text{Σ}{}_1\text{′}\text{:}\text{x}\text{̇}{}_1\text{=f}{}_1\text{(x}{}_1\text{,x}{}_2\text{),}$$

$$\text{Σ}{}_2\text{′}\text{:}\text{x}\text{̇}{}_2\text{=f}{}_2\text{(x}{}_2\text{),}$$

where $\text{x}{}_1\text{∈}\text{R}{}^{\mathrm{n}}\text{,}\text{x}{}_2\text{∈}\text{R}{}^{\mathrm{m}}$ and the functions $\text{f}{}_1\text{(·,·),}\text{f}{}_2\text{(·)}$ are sufficiently smooth in their arguments, was addressed for instance in Sontag 1989a, Sontag 1989b, where the author used the “converging input – bounded state” (CIBS) property (i.e., for each input x₂(·) on [0,∞) such that $\text{lim}{}_{\mathrm{t\to \infty }}\text{x}{}_2\text{(t)=0}$, and for each initial state x₁₀, the solution of (5) with x₁₀≔x₁(0) exists for all t≥0 and it is bounded) to prove that the cascaded system Σ₁′, Σ₂′ is GAS if the subsystems $\text{x}\text{̇}{}_1\text{=f}{}_1\text{(x}{}_1\text{,0)}$ and (6), are GAS and CIBS holds. Also, based on Krasovskii–LaSalle's invariance principle, Seibert and Suárez (1990), showed that the composite system is GAS assuming that all solutions are bounded (in short, BS) and that both subsystems, (6) and $\text{x}\text{̇}{}_1\text{=f}{}_1\text{(x}{}_1\text{,0)}$, are GAS.

For autonomous systems this fact is a fundamental result which has been used by many authors to prove GAS of the cascade , . The natural question which arises next, is “how to guarantee boundedness of the solutions”?. One way is to use the now well-known property of input-to-state stability (ISS), introduced in Sontag (1989).

Concerned by the control design problem, i.e., to stabilize the cascaded system Σ₁′, Σ₂′ by using feedback of the state x₂ only, Saberi, Kokotovic, and Sussmann (1990), studied the case when Σ₂′ is a linear controllable system. Assuming f₁(x₁,x₂) in (5) to be continuously differentiable, rewrite (5) as

$$\text{x}\text{̇}{}_1\text{=f}{}_1{}^{\mathrm{\ast }}\text{(x}{}_1\text{)+g(x}{}_1\text{,x}{}_2\text{)x}{}_2\text{.}$$

Saberi et al. (1990), introduced the linear growth condition

$$\text{||g(x}{}_1\text{,x}{}_2\text{)x}{}_2\text{||≤θ(||x}{}_2\text{||)||x}{}_1\text{||,}$$

where θ is $\text{C}{}^1$, nondecreasing and θ(0)=0, together with the assumption that x₂=0 is GES, to prove boundedness of the solutions. Using such a condition one can deal with systems which are not ISS with respect to the input x₂.

From these examples one may conjecture that, in order to prove CIBS for system (7) with decaying input x₂, some growth restrictions should be imposed on the functions $\text{f}{}_1{}^{\mathrm{\ast }}\text{(·)}$ and g(·,·). For instance, for the NL system (7) one may impose a linear growth condition such as (8) or the ISS property with respect to the input x₂. As we will show later, for the latter it is “needed” that the function $\text{f}{}_1{}^{\mathrm{\ast }}\text{(x}{}_1\text{)}$ grows faster than g(x₁,x₂) as ||x₁||→∞, for each x₂.

In the recent papers, Mazenc and Praly (1996) and Janković, Sepulchre, and Kokotović (1996), addressed the problem of global stabilizability of feedforward systems, by a systematic recursive design procedure, which leads to the construction of a Lyapunov function for the complete system. Even though the design procedures differ in both references, a common point is the stability analysis of cascaded systems. In order to prove that all solutions remain bounded under the cascaded interconnection, Janković et al. (1996) used the linear growth restriction

$$\text{||g(x}{}_1\text{,x}{}_2\text{)x}{}_2\text{||≤θ}{}_1\text{(||x}{}_2\text{||)||x}{}_1\text{||+θ}{}_2\text{(||x}{}_2\text{||),}$$

where $\text{θ}{}_1\text{(·),}\text{θ}{}_2\text{(·)}$ are $\text{C}{}^1$ and θ_i(0)=0, together with the growth rate condition on the Lyapunov function V(x₁) for the zero-dynamics $\text{x}\text{̇}{}_1\text{=f}{}_1\text{(x}{}_1\text{,0)}\text{:}\text{||∂V/∂x}{}_1\text{||}\text{||x}{}_1\text{||≤cV}$ for ||x₁||≥c₂ (which holds e.g. for all polynomials V(x₁)) and a condition of exponential stability for Σ₂. Mazenc and Praly (1996) used the assumption on the existence of continuous nonnegative functions $\text{ρ,}\text{κ}\text{:}\text{R}{}_{\mathrm{>0}}\text{→}\text{R}{}_{\mathrm{>0}}$, such that |∂V/∂x₁·g(x)x₂|≤κ(x₂)[1+ρ(V)] and $\text{1/1+ρ(V)∉}\text{L}{}_1$ and $\text{κ(x}{}_2\text{)∈}\text{L}{}_1$. The choice of κ is restricted depending on the type of stability of Σ₂′. In other words, there is a tradeoff between the decay rate of x₂(·) and the growth of g(x).

Unfortunately, all the results mentioned above apply only to autonomous nonlinear systems. Motivated by the practical problem of trajectory tracking control, non-autonomous systems deserve particular attention. Some of the initial efforts made to extend the ideas exposed above for time-varying nonlinear cascaded systems are contained in Jiang and Mareels (1997), Panteley and Lorı́a 1997, Panteley and Lorı́a 1998 and Mazenc and Praly (1997). In Jiang and Mareels (1997), the stabilization problem of a robust (vis-à-vis dynamic uncertainties) controller was considered, while in Panteley and Lorı́a 1997, Panteley and Lorı́a 1998 we established sufficient conditions for UGAS of cascaded nonlinear nonautonomous systems based on a similar linear growth condition as in (9), and an integrability assumption on the input x₂ thereby, relaxing the exponential–decay condition used in other references. In Mazenc and Praly (1997), the results of Mazenc and Praly (1996) are extended to the nonautonomous case.

The rest of this paper is organized as follows. In the next section we enunciate our main results, which address the problem exposed in Section 1.1, i.e., we consider systems for which the ISS, or a linear growth condition (w.r.t. x, for each t) on g(t,x) or none of these can be verified. For clarity of exposition the proofs of our claims are reported in Section 3. Some concluding remarks are given in Section 4.

Notation: In this paper, the solution of a differential equation, $\text{x}\text{̇}\text{=f}\text{(t,x)}$, where $\text{f}\text{:}\text{R}{}_{\mathrm{\ge 0}}\text{×}\text{R}{}^{\mathrm{n}}\text{→}\text{R}{}^{\mathrm{n}}$, with initial conditions $\text{(t}{}_0\text{,x}{}_0\text{)∈}\text{R}{}_{\mathrm{\ge 0}}\text{×}\text{R}{}^{\mathrm{n}}$ and x₀=x(t₀), is denoted by $\text{x(·}\text{;}\text{t}{}_0\text{,x}{}_0\text{)}$ or simply as x(·). We say that the system $\text{x}\text{̇}\text{=f}\text{(t,x)}$, is uniformly globally stable (UGS) if the trivial solution $\text{x(·}\text{;}\text{t}{}_0\text{,x}{}_0\text{)≡0}$ is UGS. Respectively for UGAS. A continuous function $\text{α}\text{:}\text{R}{}_{\mathrm{\ge 0}}\text{→}\text{R}{}_{\mathrm{\ge 0}}$ is said to be of class $\text{K}$ ($\text{α∈}\text{K}$) if α(·) is strictly increasing and α(0)=0; $\text{α∈}\text{K}{}_{\mathrm{\infty }}$ if in addition α(s)→∞ as s→∞. A continuous function $\text{α}\text{:}\text{R}{}_{\mathrm{\ge 0}}\text{→}\text{R}{}_{\mathrm{\ge 0}}$ is said to be of class $\text{L}$ ($\text{α∈}\text{L}$) if α(·) is strictly decreasing and $\text{lim}{}_{\mathrm{s\to \infty }}\text{α(s)=0}$. A continuous function $\text{β}\text{:}\text{R}{}_{\mathrm{\ge 0}}\text{×}\text{R}{}_{\mathrm{\ge 0}}\text{→}\text{R}{}_{\mathrm{\ge 0}}$ is of class $\text{KL}$ if $\text{β(·,t)∈}\text{K}$ for each fixed t≥0 and $\text{β(s,·)∈}\text{L}$ for each s≥0. ||·|| denotes the Euclidean norm. $\text{V}\text{̇}{}_{\mathrm{(\#)}}\text{(t,x)}$ is the time derivative of the Lyapunov function, V(t,x) along the solutions of the differential equation (#). When clear from the context we use the compact notation V(t,x(t))=V(t). We also use L_ψV=∂V/∂x·ψ for a vector field $\text{ψ}\text{:}\text{R}{}_{\mathrm{\ge 0}}\text{×}\text{R}{}^{\mathrm{q}}\text{→}\text{R}{}^{\mathrm{n}}$.