Finite-time stability analysis of fractional order time delay systems: Bellman-Gronwall`s approach

The paper extends some basic results from the area of finite time and practical stability to nonlinear , perturbed , fractional order time-delay systems where a robust stability test procedure is proposed. The problem of sufficient conditions that enable system trajectories to stay within the a priori given sets for the particular class of nonlinear fractional order time delay systems is examined. Introduction HE question of stability is of particular interest in the control theory. Also , the problem of investigation of time delay systems has been explored over many years. Delay is very often encountered in different technical systems , such as electric , pneumatic , hydraulic ; in networks , chemical processes , long transmission lines , etc. , [ 1 ]. On the other hand , the existence of pure time delay , whether it is present in control or / and state , may cause undesirable system transient response or even instability. Numerous reports have been published on this matter , with particular emphasis on the application of Lyapunov`s second method or using the idea of matrix measure , [ 2 , 3 , 4 , 5 ]. Another approach will be presented here i. e. , system stability from the non-Lyapunov point of view will be investigated. In practice , system stability (e. g. in the sense of Lyapunov) , is investigated along with the bounds of system trajectories. A system could be stable but still completely useless if 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. For the more it is of particular significance to consider the behavior of dynamical systems only over a finite time interval. These boundedness properties of system responses , i. e. , the solution of system models , are very important from the engineering point of view. Bearing this in mind , numerous definitions of the so-called technical and practical stability were introduced. 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 concerned. Motivated by " brief discussion " on practical stability the monograph of LaSalle and Lefschet [ 6 ] , Weiss and Infante [ 7 ] have introduced various notations …