Fuzzy of Continuous-Time Nonlinear Two-Dimensional Systems

Chapter 2 – Chapter 6 have fully investigated the T-S fuzzy model-based state feedback and output feedback sliding mode control problems for some nonlinear systems, which include general nonlinear systems and multiple time-scale nonlinear systems. It should be also noted that the system information of the aforementioned nonlinear systems evolves along only one direction, and these systems are actually known as one-dimensional (1-D) systems. Recent decades have witnessed an increasing development on multi-dimensional (m-D) dynamical systems as many practical processes or systems intrinsically contain m-D properties, such as image deblurring, long transmission line, and so on [115, 116, 118, 119, 120, 121]. Differing from 1-D systems, m-D systems tend to be more complex in system structure, modelling and design. Among various m-D systems, two-dimensional (2-D) systems are recognized as a typical class of m-D systems, and also contain highly complex nonlinearities generally. Consequently, many researchers have devoted efforts to 2-D systems and a substantial body of valuable results have been reported [122, 123, 124, 125, 126, 127]. For instance, [126] addressed the robust Hinf control problem for the gas absorption, water stream heating and air drying processes based on Roesser model. In view of the theoretical advancement and engineering application of SMC , great efforts have been also concentrated on SMC for 2-D systems [122, 132, 133, 134, 135, 136, 137]. The authors in [132] addressed the SMC problem for the gas absorption, water stream heating and air drying processes via discrete-time 2-D Roesser model without considering external disturbance. In [134], a direct state feedback SMC law was developed for 2-D systems through the Fornasini-Marchesini model. When matched exogenous disturbance exists, the authors in [135] investigated the SMC issue for discrete-time linear 2-D systems. For the unavailable full system states situation, in [122], an observer-based SMC method was proposed for differential linear repetitive processes, where the control input matrix was not allowed to have parameter uncertainties. Notice that the existing SMC results [132, 133, 134, 135] for 2-D systems were basically obtained through the full state feedback method. Unfortunately, the full system state variables are not accessible in most complex engineering applications, especially for some nonlinear 2-D systems. Furthermore, all those results in 165 [132, 122, 133, 134, 135] were mainly derived through a common quadratic Lyapunov function, which also tends to remain conservative. In addition, note that T-S fuzzy affine models considered in this monograph inherently have much improved function approximation capability, while there have been few fuzzy SMC results for nonlinear 2-D systems. To our best knowledge, few efforts have been devoted to the SMC problems for both continuous-time and discrete-time nonlinear 2-D processes through T-S fuzzy affine models and piecewise quadratic Lyapunov functions (PQLFs), which inspires the study in Chapter 7 and Chapter 8 . In this chapter, the dynamic SMC design problem for continuous-time nonlinear 2-D systems with exogenous disturbance will be firstly investigated by T-S fuzzy affine models. Associated with sliding surface, a singular 2-D system is first constructed to describe the sliding mode dynamical features. Based on PQLFs , some novel stability and robust performance analysis results on the sliding mode dynamical system are proposed, and the analysis conservatism can be further reduced. The gain of the sliding surface can be attained by solving a set of LMIs . In sequel, a robust state feedback dynamic sliding mode controller and an output feedback dynamic sliding mode controller are designed, respectively, such that the finite-time reachability of the sliding surface can be guaranteed.