International Journal of Control, Automation, and Systems (2012) 10(6):1096-1101 DOI 10.1007/s12555-012-0603-2 ISSN:1598-6446 eISSN:2005-4092 http://www.springer.com/12555 An Integral Function Approach to the Exponential Stability of Linear Time-Varying Systems Yu Yao, Kai Liu*, Dengfeng Sun, Venkataramanan Balakrishnan, and Jian Guo Abstract: This paper studies the exponential stability of linear time-varying (LTV) systems using the recent proposed integral function. By showing the properties of the integral function and applying the Bellman-Gronwall Lemma, a sufficient and necessary condition for the exponential stability of LTV systems is derived. Furthermore, the exponential decay rate of the system trajectories can be obtained by computing the radii of convergence of integral function. The algorithm for computing the integral function is also developed and two classical examples are given to illustrate the proposed approach. Keywords: Bellman-Gronwall Lemma, exponential stability, integral function, linear time-varying systems. 1. INTRODUCTION The linear continuous time-varying (LTV) systems has been receiving increasing attention by system and control community, since they appear frequently in practical engineering areas such as aerospace control systems [1,2]. While important, LTV systems are very hard to investigate despite of the fundamental stability problem. It is well known that, even when the eigenvalues of the system have strictly negative real parts for all instants of time, the time-varying system may be unstable. However, numerous important progresses, including but not limited to [3-10] have been achieved through the effort of researchers. They more or less all rely on the use of a linear time-invariant plant as an approximation of the LTV system and ensuring that the influence of the approximation is not excessive. The main advantage of this frozen time method is the possibility of exploiting the great deal of tools which have been developed for linear time-invariant (LTI) systems. This paper tries to present a novel approach to investigate the exponential stability of LTV systems. In the previous work [11], an integral function approach was proposed to analyze the exponential stability of a class of piecewise-linear systems, and a computational sufficient and necessary criterion was provided in terms of the integral function. In this paper, an improved integral function is introduced, which has some nice properties including homogeneity, sub-additivity, convexity, common-bound and vertex-bound. Based on the properties and Bellman- Gronwall lemma, a sufficient and necessary condition for the exponential stability of LTV systems is derived, and the exponential decay rate of the LTV systems is characterized by the radius of convergence of integral function without conservatism. As our best knowledge of LTV systems, it is the first time that such a rate has been characterized exactly. 2. PROBLEM FORMULATION We consider a class of continuous-time PPLS repre- sented by 0 () ( ) ( ), , xt Atxt t t = ≥  (1) where () xt ∈ R n and () R nn At × ∈ are. It is assumed that A(t) is continuous in t, and bounded for all 0 . t t ≥ Definition 1: Let 0 (, ;) xtt z denotes the solution of LTV system with initial time t 0 and initial state z. The system (1) is called exponentially stable if there exist (0,1) r ∈ and r κ > 0 such that 0 0 (, ;) t t r xtt z r z κ - ≤ for all t ≥ t 0 . Definition 2: Define the exponential decay rate of LTV system (1) as 0 * 0 0 inf{ | (, ;) , R, } t t n r r r xtt z r z z t t κ - = ≤ ∈ ≥ (2) to characterize the convergence rate of the “most unstable” trajectories of LTV system (1). The objectives are: (i) present a computable sufficient and necessary criterion of exponential stability for LTV systems; (ii) compute the exponential decay rate without conservatism. © ICROS, KIEE and Springer 2012 __________ Manuscript received February 27, 2012; revised May 28, 2012; accepted July 11, 2012. Recommended by Editorial Board mem- ber Nam H. Jo under the direction of Editor Hyungbo Shim. This work was partially supported by National Natural Science Foundation of China under grants NSFC 61074160, 61021002 and 61104193. Yu Yao, Kai Liu, and Jian Guo are with the Control and Simu- lation Center, Harbin Institute of Technology, China (e-mails: yaoyu@hit.edu.cn, carsonliu.hit@gmail.com, guojianhit@163.com). Dengfeng Sun is with the School of Aeronautics & Astronau- tics Engineering, Purdue University, USA (e-mail: dsun@purdue. edu). Venkataramanan Balakrishnan is with the School of Electrical and Computer Engineering, Purdue University, USA (e-mail: ragu @purdue.edu). * Corresponding author.