Controller Design And Model Predictive Control Approaches For Piecewise Affine System

In engineering practice, there are many hybrid systems described by piecewise affine systems (PWA) which are composed of linear subsystems and convex polytopic regions. Hybrid systems are composed of discrete event dynamic systems and continuous time dynamic systems or discrete time dynamic systems, which interact on each other. The hybrid system theory, which is proposed for the demand of the economic development, is the result of the development of computer science and control theory. Piecewise affine system is one of the most important branches of hybrid system, and is also straightforward. It consists of some subsystems that integrate the logical and continuous dynamics by switching. Theoretically, any nonlinear system can be approximated as piecewise affine system.For decades, scientists have conducted much research work in PWA system and its applications in many engineering problems, and make a series of satisfied achievements. But, because of the difficulty and the popularity of the PWA system (i.e. PWA is essentially a non-smooth and even non- continuous nonlinear system and hybrid system has a extensive engineering background), up to today, the topic of PWA system and its applications is one of the most difficult research work and also one of the hot spots of studies in the field of system and control science.On the basis of summarizing and reviewing previous research work, this dissertation makes some further research work and exploration in PWA and its applications in nonlinear systems. The research work of this thesis consists of following five aspects:The concept and the background of the development of PWA, the importance and necessity of studies of PWA are explained. Various widely used modeling approaches for hybrid systems, their characteristics and the equivalences of typical hybrid models are summarized. The work focuses mainly on important subclass of hybrid systems: piecewise affine systems (PWA). Typical examples of the applications of PWA in engineering practice are also illustrated.Using the convex polytopes technique, the corresponding operating region of the PWA systems in state space is described as ellipsoid which can be characterized by a set of vector inequalities. With these ellipsoid partitions, each model is valid in a given region of a certain space, and the PWA systems switches between different partitions. Based on the common Lyapunov and multiple Lyapunov functions, the stable controller is designed for the PWA systems involving mode switching between different operating regions. Furthermore, the design methods of the robust controllers are proposed for the PWA with sector uncertainty or parameter perturbation respectively.In practice, the PWA systems with certain switching order exist universally. Simultaneously, the analysis and synthesis of control systems with actuator saturation are met frequently. To decrease the conservation of the controller which designed for the PWA systems with constrained control input based on certain switching order, a multi-model predictive control (MPC) algorithm is developed by a sequence of piecewise models along the transition trajectory. The control algorithm is a receding horizon scheme with a quasi-infinite horizon objective function that has finite and infinite horizon cost components. The finite horizon cost consists of free input variables that direct the system towards a terminal region that contains the desired operating point. The infinite horizon cost has an upper bound and takes the system to the final operating point. The control problem is formulated as a convex optimization in terms of Linear Matrix Inequalities (LMIs). Furthermore, on the basis of former algorithm, an additional terminal ellipsoid is introduced to ensure the states converging to the equilibrium faster. The problem of constrained control input is solved by convex optimization involving LMIs. Lastly, a simulation involving Continuous Stirred Tank Reactor (CSTR) model results verify the effectiveness of these proposed methods.In practice, systems are generally uncertain and time-varying. A robust MPC is proposed for PWA systems with polytopic uncertainties. The problem of minimizing an upper bound on the worst-case objective function is reduced to a convex optimization involving LMIs. The feasible free input variables guarantee the PWA systems switching in certain order, and the receding horizon state-feedback control guarantees closed-loop robust asymptotic stability and input constraints. The approach allows explicit incorporation of the description of plant uncertainty. Furthermore, the closed-loop robust stability and feasibility of receding horizon implementation are proved. The simulation results verify the effectiveness of the proposed method.A model predictive control (MPC) is proposed for a time-delay PWA systems with constrained input and certain switching order. The corresponding operating region of the considered systems in state space is described as ellipsoid which can be characterized by a set of vector inequalities. The control law is obtained by convex optimization based on MPC involving linear matrix inequalities (LMIs). And the constrained control input of the considered systems is solved in terms of LMIs. The simulation results verify the effectiveness of the proposed method. It is shown that, based on LMI constraints, it is easy to get the MPC for the PWA systems with time-delay and a simulation involving Autonomous Land Vehicle (ALV) model results verify the effectiveness of these proposed methods.