Research On The Lmi-based Control Of Discrete-time Piecewise Affine Systems

Piecewise affine (PWA) systems have been receiving increasing attention by thecontrol and engineering communities because they provide a powerful means of analysisand design for non-linear control systems. Thus, the control issues of PWA systems haveboth theoretical significance and engineering value.This thesis is composed of 6 chapters. In the framework of linear matrix inequality(LMI), the state-feedback control and observer-based output-feedback control problemsof discrete-time PWA systems are investigated deeply. During controller designs, theperformances of robust stability, disturbance attenuation, time-domain satisfaction, etc.,are considered. Accordingly, the feasibility of applying the methods to vehicle anti-lockbrake system (ABS) is discussed.In Chapter 2, the disturbance attenuation problem of discrete-time PWA systems isresearched. Two different state-feedback H_∞control schemes are proposed. The basicidea of them is to construct piecewise quadratic Lyapunov function and introduce a dis-sipation inequality to guarantee the system energy dissipation. The designed controllersnot only guarantee the stability of the closed-loop systems, but also obtain the disturbanceattenuation ability. During one controller design, the regions are described by ellipsoidsand the results can be formulated as LMIs, while during the other controller design, theregions are described by polyhedra and the results can be formulated as bilinear matrix in-equalities (BMIs). Finally, the advantages and disadvantages of the two control schemesare compared.In practice, control systems always suffer uncertainties and time-domain constraints.If their in?uences on control performances are not taken into account, the expected con-trol objectives can not be realized. Motivated by this situation, in Chapter 3, the resultsobtained in Chapter 2 are extended, and a robust H_∞controller is designed for discrete-time PWA systems in the presence of time-varying uncertainties, external disturbance andtime-domain constraints. If the time-varying parameters are unknown in future, but mea-surable at current sampling time, they can be incorporated in controller design to reduceconservatism, and then a less conservative parameter-dependent controller can be ob-tained. In addition, the concepts of reachable set and state-space ellipsoid are introduced to capture the time-domain constraints. The designed robust controller and parameter-dependent controller can guarantee closed-loop properties, including stability, H_∞per-formance and the satisfaction of constraints.Usually, not all system states can be directly measured. So, how to design controllerby using the measured output to guarantee the stability of closed-loop system is an im-portant control problem. In Chapter 4, an observer-based output feedback controller isdesigned for discrete-time PWA systems. In the chapter, it is supposed that the PWA sys-tems are partitioned based on output space. The assumption implies that the system andthe controller can be guaranteed to operate in the same region at the same time. Controllerdesign methods are carried out for nominal systems and uncertain systems respectively.For nominal systems, a state-feedback controller and an observer are designed firstly, thenit is proved that the output-feedback controller constructed by the resulting state-feedbackcontroller and observer gains can guarantee the stability of the closed-loop systems. Theidea is similar to the separation principle of linear systems. For uncertain systems, theidea of designing the controller gains separatively is no longer satisfied. Another controlmethod is proposed for this case, where the controller gains should be designed simul-taneously. A matrix equality will be encountered during controller design. The singularvalue decomposition technique is used to treat the constraint of matrix equality, then thesuggested control method can be formulated as LMIs.If the PWA systems are partitioned based on state space, or there exists measurednoise, it is impossible to infer the currently active region from the measured output, andthe transitions of system and controller can not be guaranteed to be synchronized. InChapter 5, for the discrete-time PWA systems which are partitioned based on state space,the non-synchronized H_∞estimation problem is discussed firstly. During the estimatordesign, the issue that the system and the estimator may stay in different regions fromtime to time is explicitly considered. By estimating the system state and the currently ac-tive region simultaneously, the estimation error systems can be guaranteed to be globallyasymptotically stable with optimized H_∞performance. At last, the results are extendedto design non-synchronized observer-based output-feedback controller.In Chapter 6, the control theory of PWA systems is applied to vehicle ABS controllerdesign. By approximating the slip-related nonlinear characteristics using PWA function,and considering the vehicle speed as the slow time-varying parameter which is unknown in future, but measurable at current sampling time, the vehicle braking model is treatedas PWA system with measurable time-varying parameter. Then a parameter-dependentABS controller is designed according to the method proposed in Chapter 3. At last, theeffectiveness of the ABS controller is assessed on a detailed full-car simulator (veDYNA,tuned to fit the characteristics of Red-?ag Mingshi CA7180A3E). Simulation results showthat the vehicle with the proposed ABS controller obtains a good braking performance.Thus, the methods presented in this thesis have a good prospect of application.