On stability of linear time-invariant fractional order systems with multiple delays

Abstract

An intriguing perspective is presented in studying the stability robustness of linear time invariant (LTI) frac- tional order systems with multiple time delays against delay uncertainties (this means that the time delays are constant but their true values are not exactly known). The new approach is based on a holographic mapping, which is implemented over the parametric space of the delays. This mapping considerably allays the problem, which is otherwise known to be notoriously complex. We show that this procedure analytically reveals all possible stability regions precisely in the space of multiple time delays. For simplicity of conveyance and without loss of generality, the number of time delays is taken as two in this document. As an added strength, it does not require the system under consideration to be stable for zero delays. An example case study is also provided to approve the proposed method results, which is not possible to analyze using any other methodology known to the authors. I. INTRODUCTION Delay can appear in the input, output or in the state variables. The existence of delay in input has no effect on system stability as long as the system has an open-loop form. However, if such systems are controlled via feedback, the delay will be transferred to the characteristic equation of the closed loop system and will cause it to have infinite dimensional system and, therefore, it will have a considerable impact on the stability of the system. It is a known fact that the existence of numerous delays in a dynamic system can lead to the systems poor performance and even its instability. The presence of delay in the characteristic equation results in an infinite number of poles in the system, and this turns the stability analysis of time-delay systems into a challenging problem. To overcome this problem in the linear time- invariant integer order systems, some useful methods (1- 7) have been presented with the purpose of evaluating the stability of these dynamics. Regarding the fractional order time-delay systems, due to the involvement of fractional mathematics in these problems, the stability analysis of these systems will be much more complicated than the integer order systems. The researchers of (8) and (9) are probably the pioneer to consider stability analysis of the fractional delay system with single-delay. They have extended the Ruth- Hurwitz criteria for analyzing the stability of some special