# The Stability of a Class of Fractional Order Switching System with Time-Delay Actuator

DOI: 10.4018/IJSDA.2018010105

## Abstract

Since switching systems are important in research and industry, the article is concerned about the stabilization of fractional order switching systems with the order of 1 < q < 2 and a time delay actuator. To this end, the so-called system was initially converted to a system with no delay using a trick, such that the impact of delay was considered in the state matrix of the system in form of a coefficient. In the following, the switching rule was obtained based on the variable structure control with the sliding section. The necessary stability condition for the fractional order switching system with the order of 1 < q < 2 and time delay actuator is presented and approved based on the convex analysis and linear matrix inequalities. Then, a Lyapunov function was introduced with its negative derivative. By defining the Lyapunov function, the system that can be chosen at any time by the switching rule would be stable. Finally, the simulation results were expressed to show the impact of the proposed method.
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## 1. Introduction

The hybrid dynamic systems play an important role in the control science, which is due to a need for high-reliability control in the hybrid systems with a complicated dynamic (Cavaleri et al., 2012).

Switching system is one of the most prevalent models of hybrid systems, which is expressed by a combination of differential state equations and a switching logic. In fact, a switching system is a dynamic system, which includes a number of sub-systems coordinating the switch of sub-systems based on a logical law.

During the recent years, the switching systems gained more importance in modeling and design of controllers in many processes, involving airplane control, power network, network control, and biological systems. (Lin & Antsaklis, 2009; Azar & Vaidyanathan, 2015).

The switching rule analysis is an important factor in the study of switching systems and plays a leading role in issues, such as the design of control systems and stability analysis (Zhai & Xu, 2010). One of the common methods for analyzing the stability of switched systems is finding a Lyapunov function, such that if there is a common Lyapunov function for all the sub-systems, the stability of switching system is guaranteed (Bao et al., 2014; Jungers, 2008; Zhai & Xu, 2010).

Equations with derivatives and fractional order integrals have gained importance for about three centuries. The significance of these derivatives is due to the fact that the behavior of many real-world systems can be raised with a better fractional order dynamics than the integer order dynamics (Gorenflo, 1997; Lin & Kuo, 2012; Ortigueira, 2008; Petras, 2000).

Generally, in the context of describing the model of the systems, the fractional order differential equations are capable of describing more complex systems (Azar et al., 2017). Moreover, in the field of control systems, fractional order controllers are more flexible than classic controllers and would increase the ability to control different systems (Arara et al., 2010; Hosseinnia et al., 2010; Monje et al., 2010; Ortigueira, 2008; Rida & Arafa, 2011; Boulkroune et al., 2016; Ghoudelbourk et al., 2016).

So far, many studies have been conducted on the stability of fractional order systems (Balochian & Sedigh, 2012; Moze et al., 2007; Tavazoei & Haeri, 2009), but less attention has been paid to the stability of fractional order switching systems. In particular, in (Balochian & Sedigh, 2012) the stability of a linear time invariant switching system with a fractional order of 1 < q < 2 was confirmed by a Lyapunov function that is valid for all sub-systems.

In some systems, the rate of change in mode only not relies on the current state of the system, but also depends on sometimes in the past, especially in control systems in which the actuators and sensors may have a flashback (Mahmoud & Sunni, 2012; Pham et al., 2017). Thus, investigating time delay is of particular importance that (Boukas & Al-Muthairi, 2005; Han, 2004; Han & Gu, 2001; Liu et al., 2007; Xu & Lam, 2005) are concerned about this issue.

In (Kim et al., 2006; Sun et al., 2008; Sun et al, 2006; Xie & Wang, 2004, 2005), the issue of time delay was investigated at different classes of integer order switching systems. In (Balochian & Sedigh, 2012) the fractional order switching system is evaluated without time delay. However, given the importance of fractional order systems with time delay actuator, the stability of these switching systems is discussed in this paper.

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