H∞ Tracking Control of Uncertain Markovian Hybrid Switching Systems: A Fuzzy Switching Dynamic Adaptive Control Approach

This article investigates the $H_{\infty }$ stochastic tracking control problem for uncertain fuzzy Markovian hybrid switching systems by using a fuzzy switching dynamic adaptive control approach. The long and the short is to construct multiple piecewise stochastic Lyapunov functions which provide an effective tool for designing hybrid switching law and fuzzy switching dynamic adaptive law. A hybrid switching law, including both stochastic switching and deterministic switching, is designed to represent more general switching scenarios, which can improve the $H_{\infty }$ adaptive tracking performance through offering a running time before stochastic switching for the adaptive control strategy to work well. A fuzzy switching dynamic adaptive control technique is developed such that all signals of the tracking error equation are bounded, and the system state trajectory tracks the reference model state trajectory under a disturbance attenuation level as closely as possible. Finally, an application study verifies the effectiveness of the acquired methods.

I. Introduction

Markovian switching exhibits great advantages in modeling unexpected abrupt changes [1]–[3]. Over the past decades, a large number of research results on Markovian switching systems have been reported from science to engineering [4], [5]. A pioneer work is presented in [6] for the control synthesis of semi-Markovian uncertain fuzzy systems with hidden modes, which can efficiently balance the conservatism caused by the mismatching between system modes and observed ones. Due to the sojourn time [7], [8], the intrinsical conservativeness limits to a large extent the application of Markovian switching for describing the actual switching behaviors. In fact, Markovian switching is classical stochastic switching [9]. Besides, the deterministic switching has been extensively investigated in switched system fields [10]–[12]. Since the early 2000s, some exciting progress has been obtained for the deterministic switching design [13], [14]. With the introduction of deterministic switching, switched systems can not only inherit the original nature of the individual subsystem but also exhibit complicated behaviors [15]–[17]. Several significant results on deterministic switching have been reported concerning dissipativity, stabilization, and adaptive control [18], [19]. It is noted that the above switching is either stochastic switching or deterministic switching. However, the switching is usually influenced by a variety of characteristics, and these two switching characteristics interrelate with each other [20]. Practically, the switching maybe not merely Markovian or deterministic, but a mix of them [21]. Therefore, how to study the interaction of these switching characteristics is of great theoretical value and practical significance. This motivates us to further investigate the hybrid switching law carried with determinacy and randomness for Markovian switching systems.