440 Asian Journal of Control, Vol. 7, No. 4, pp. 440-447, December 2005 Manuscript received November 25, 2003; revised June 16, 2004; accepted February 12, 2005. M.P. Lazarević is with the Department of Mechanics, Faculty of Mechanical Engineering (e-mail: mlazarevic@mas.bg. ac.yu). D.Lj. Debeljković is with the Department of Control Engineering, Faculty of Mechanical Engineering (e-mail: ddebeljkovic@ mas.bg.ac.yu). －Brief Paper－ FINITE TIME STABILITY ANALYSIS OF LINEAR AUTONOMOUS FRACTIONAL ORDER SYSTEMS WITH DELAYED STATE M.P. Lazarević and D.Lj. Debeljković ABSTRACT For the first time, in this paper, a stability test procedure is proposed for linear time-invariant fractional order systems (LTI FOS). Paper extends some basic results from the area of finite time and practical stability to linear, continuous, fractional order time invariant time-delay systems given in state space form. Sufficient conditions of this kind of stability, for particular class of fractional time-delay systems is derived. KeyWords: Linear analysis, stability criteria, time delay, fractional system. I. INTRODUCTION The question of stability is of main interest in control theory. Also, the problem of investigation of time delay system has been exploited over many years. Delay is very often encountered in different technical systems, such as electric, pneumatic and hydraulic networks, chemical processes, long transmission lines, etc. The existence of pure time delay, regardless if it present in the control or/and state, may cause undesirable system transient response, or generally, even an instability. Numerous reports have been published on this matter, with particular emphasis on the application of Lyapunov’s second method, or on using idea of matrix measure [1-4]. Here, we present another approach, i.e. we investigate system stability from the non-Lyapunov point of view. In practice one is not only interested in system stability (e.g. in the sense of Lyapunov), but also in bounds of system trajectories. A system could be stable but still completely useless because it possesses undesirable transient performances. Thus, it may be useful to consider the stability of such systems with respect to certain subsets of state-space which are defined a priori in a given problem. Besides that, it is of particular significance to concern the behaviour of dynamical systems only over a finite time interval. These bounded ness properties of system responses, i.e. the solution of system models, are very important from the engineering point of view. Realizing this fact, numerous definitions of the so-called technical and practical stability were introduced. Roughly speaking, these definitions are essentially based on the predefined boundaries for the perturbation of initial conditions and allowable perturbation of system response. Thus, the analysis of these particular boundedness properties of solutions is an important step, which precedes the design of control signals, when finite time or practical stability control is concern. Motivated by “brief discussion” on practical stability in the monograph of LaSalle and Lefschet [5], Weiss and Infante [6] have introduced various notations of stability over finite time interval for continuous-time systems and constant set trajectory bounds. A more general type of stability (“practical stability with settling time”, practical exponential stability, etc.) which includes many previous definitions of finite stability was introduced and considered by Grujić [7,8]. Concept of finite-time stability, called “final stability”, was introduced by Lashirer and Story [9] and further development of these results was due to Lam and Weiss [10]. Also, analysis of linear time-delay systems in the context of finite and practical stability was introduced and considered by Debeljković and Lazarević [11-13]. Recently, there has been some advances in control theory of fractional differential systems for stability questions [14]. Fractional-order means that the delay differential equation order is non-integer. However, for fractional order dynamic systems, it is difficult to evaluate the stability by simply examining its characteristic equation