On finite time stability for fractional order time delay systems

M.P.Lazarević

University of Belgrade

Kraljice Marije 16, 11070 Belgrade, Serbia

 

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., Zavarei M., Jamshidi M. For an example, Williams described a typical problem occurring in the chemical and petroleum industries. Pipelines recycle certain distillation column by products back to a chemical reactor. This recycle transport lag can be as much 10 min. In his paper Ross applied an optimal control of time delay system i.e. refining plant-chemical reactor with recycle loop. On the other side, 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, Lee and Diant, Mori, Hmamed, Chen et al.

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 La Salle and Lefschetz, Weiss and Infante 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. Concept of finite-time stability, called "final stability", was introduced by Lashirer and Story and further development of these results was due to Lam and Weiss. Also, analysis of linear time-delay systems in the context of finite and practical stability was introduced and considered by Debeljković and Lazarević.

Recently, there has been some advances in control theory of fractional differential systems for stability questions Matignon. 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 either by finding its dominant roots or by using other algebraic methods. At the moment, direct check of the stability of fractional order systems using polynomial criteria (e.g., Routh's or Jury's type) is not possible, because the characteristic equation of the system is, in general, not a polynomial but a pseudopolynomial function of fractional powers of the complex variable s. Thus there remain only geometrical methods of complex analysis based on the so called argument principle (e. g. Nyquist type) which can be used for the stability check in the BIBO sense (bounded-input bounded-output). Also, for linear fractional differential systems of finite dimensions in state-space form, both internal and external stabilities are investigated by Matignon. Analytical approach is suggested by Chen and Moore, where it is considered the analytical stability bound using Lambert function W for a class of second-order ordinary delay differential equations (DDE) and case of the linear fractional-order DDE.On the other side, our approach doesn't demand any solving of delay differential equation (DDE) but it is based on forming the corresponding criteria (criterion of practical stability and finite time stability) in which basis matrices of system A₀, A₁ exclusively appear, where basis matrices may contain tuning parameters which inflence the stability of system more obviously than in both papers Chen and Moore.

For the first time, a finite time stability test procedure is proposed for (non)linear nonautonomous time-invariant delay fractional order systems (LTID FOS). Here, we examine the problem of sufficient conditions that enable system trajectories to stay within the a priori given sets for the particular class of (non)linear nonautonomous fractional order time-delay systems. To the best knowledge of author, these problems are not yet analysed for the fractional order time-delay systems and this class of systems.

In this paper, stability results for finite-dimensional (non)linear (non)autonomous time delay fractional differential systems given in state space form. To the best knowledge of author, these problems have not yet been analysed for this class of nonlinear time-delay fractional order systems. Sufficient conditions of this kind of stability is derived by applying generalized Bellman-Gronwall theorem. In that way, one can check system stability over finite time interval.