Input–output finite time stabilization of linear systems

Abstract

Bounded Input Bounded Output (BIBO) stability is usually studied when only the input–output behavior of a dynamical system is of concern. The present paper investigates the analogous concept in the framework of Finite Time Stability (FTS), namely the Input–Output FTS (IO-FTS). FTS has been already investigated in several papers in terms of state boundedness, whereas in this work we deal with the characterization of the input–output behavior. Sufficient conditions are given, concerning the class of ${\mathcal{L}}_2$ and ${\mathcal{L}}_{\infty }$ input signals, for the analysis of IO-FTS and for the design of a static state feedback controller, guaranteeing IO-FTS of the closed-loop system. The effectiveness of the proposed results is eventually illustrated by means of some numerical examples.

Introduction

The concept of finite time stability (FTS) dates back to the sixties, when this idea was introduced in the control literature (Dorato, 1961). A system is said to be finite time stable if, given a bound on the initial condition, its state does not exceed a certain threshold during a specified time interval. It is important to recall that FTS and Lyapunov Asymptotic Stability (LAS) are independent concepts; indeed a system can be FTS but not LAS, and vice versa. While LAS deals with the behavior of a system within a sufficiently long (in principle infinite) time interval, FTS is a more practical concept, useful to study the behavior of the system within a finite (possibly short) interval, and therefore it finds application whenever it is desired that the state variables do not exceed a given threshold (for example to avoid saturations or the excitation of nonlinear dynamics) during the transients. Sufficient conditions for FTS and finite time stabilization (the corresponding design problem) have been provided in Amato, Ariola, and Cosentino (2006) and Shen (2008) in the context of linear systems, and in Amato et al. (2009), Ambrosino et al. (2009) and Zhao, Sun, and Liu (2008) in the context of impulsive and hybrid systems.

In this work we focus on the input–output behavior of linear systems over a finite time interval. Roughly speaking, and consistently with the definition of FTS given in Dorato (1961), a system is defined IO-FTS if, given a class of norm bounded input signals over a specified time interval $T$, the outputs of the system do not exceed an assigned threshold during $T$.

In order to correctly frame our work in the current literature, we recall that a system is said to be IO ${\mathcal{L}}_p$-stable (see Khalil (1992, Ch. 4)) if for any input of class ${\mathcal{L}}_p$, the system exhibits a corresponding output which belongs to the same class. The main differences between classic IO stability and IO-FTS are that the latter involves signals defined over a finite time interval, does not necessarily require the inputs and outputs to belong to the same class, and that quantitative bounds on both inputs and outputs must be specified. Therefore, IO stability and IO-FTS are independent concepts.

For the sake of completeness, it should be mentioned that, with respect to the one given in this paper, a different concept of IO-FTS for nonlinear systems has been given in Hong, Jiang, and Feng (2008) extending the definition of finite time stability given in Bhat and Bernstein (2000) to nonautonomous systems. In the latter works, the authors focus on the Lyapunov stability analysis of nonlinear systems whose trajectories converge to an equilibrium point in finite time and on the characterization of the associated settling time. According to this definition of FTS, in Hong et al. (2008) a different concept of finite time input–output stability is introduced. In particular, the authors consider the case of nonautonomous system with a norm bounded input signal over the interval $[0+\infty ]$ and an initial condition $x\left(0\right)=x_0$. The finite time input–output stability is related to the property of a system to have a norm bounded output whose bound, after a finite time interval $T$, does not depend anymore on the initial state. Hence, we can conclude that the concept of IO-FTS introduced in this paper and the one in Hong et al. (2008) are different.

Input–output stabilization of time-varying system on finite time horizon is tackled also in Shaked and Suplin (2001). However, as for classic IO stability, their concept of IO stability over a finite time horizon does not give explicit bounds on input and output signals, and does not allow the input and output to belong to different classes.

The main contributions of this paper are two sufficient conditions which guarantee that a given system is IO-FTS over a specified time interval, for two different input classes. Furthermore the problem of IO finite time stabilization via state feedback is also tackled.

Our work is organized as follows. In Section 2 the problem we deal with is precisely stated, and some preliminary definitions are provided. Two sufficient conditions which guarantee IO-FTS of a given linear system are introduced in Section 3. These two conditions concern with two different class of input signals. Sufficient conditions to solve the IO finite time stabilization problem via static state feedback are also provided. In Section 4 some examples illustrating the applicability of the devised results are discussed. Some concluding remarks are eventually provided.

Notation

The symbol ${\mathcal{L}}_p$ denotes the space of vector-valued signals whose $p$-th power is absolutely integrable over $[0,+\infty )$. The restriction of ${\mathcal{L}}_p$ to $\Omega \subset [0,+\infty )$ is denoted by ${\mathcal{L}}_{p,\Omega }$. Given a set $\Omega \subseteq \mathbb{R}$, a symmetric positive definite matrix $R$ and a signal $\sigma (\cdot ):\Omega \mapsto {\mathbb{R}}^m$, the weighted norm ${\left({\int }_{\Omega }\sigma {\left(\tau \right)}^{T}R\sigma \left(\tau \right)d\tau \right)}^{1/2}$ will be denoted by ${\Vert \sigma (\cdot )\Vert }_{\Omega ,R}$.