On Control Problems Of Delta Operator Formulated Discrete Time Systems

Control problems of the delta operator formulated discrete-time systems have recently drawn considerable attention in the field of systems and control. Delta operator based discretization method can avoid ill-conditioned problems caused by the conventional shift operator when fast sampling is used, and the delta operator model converges to corresponding continuous-time model as the sampling interval goes to zero, thus analysis and synthesis for continuous-time and discrete-time systems can be treated in the unified delta operator framework.This dissertation makes some researchs on the delta operator formulated systems. The main works are stated in brief as follows.Firstly, robust covariance control problems of the delta operator formulated uncertain linear discrete-time systems are studied. This work includes two aspects:①Design approach of robust covariance controllers based on dynamic output feedback is investigated. A sufficient condition for the delta operator system which satisfying the expected covariance constraints is proposed, and then a sufficient condition for the existence of the robust covariance controllers is derived based on the linear matrix inequality(LMI) approach and , a parametrized characterization of the controllers(if they exist) is given in term of the feasible solution to a certain LMIs.②Similarly, a design method of robust covariance output feedback controllers for the delta operator uncertain system with D-stability constraints is also presented. In both processing, the elimination method is introduced to aviod parameter turning when the controllers are solving, and to reduce the conservativeness. The numerical examples solved by MATLAB shows the usefulness of the controller design method.Secondly, reliable control problems of the delta operator systems are studied. The aim is to design reliable controller which can meet the desired performance and tolerate actuator and(or) sensor failure. In this section, a more general failures model, so called continuous failure model is adopted for actuator and(or) sensor failure. Based on the LMI approach, reliable robust stabilization, reliable robust D-stabilization and reliable robust H∞control for the delta operator uncertain system are researched respectively. The conditions of the existence of corresponding controllers are deduced, and then the design methods of corresponding controllers are suggested respectively. A distinctive feature of the proposed method is that the structural information of the continuous failure model is fully utilized, which brings less conservativeness in controller design. This is demonstrated by some numerical examples.Thirdly, non-fragile control problems are studied. The cotrollers to be designed are assumed to have multiplicative(or additive) gain variations. Based on the LMI approach, four problems are discussed.①A criterion of non-fragile quadratic stabilizability is derived, and then the design method of the state feedback cotrollers for non-fragile robust stabilization is presented.②Using the above conclusion, a sufficent and necessary condition of non-fragile quadratic D-stabilizability is obtained via construct an auxiliary delta operator system.③A non-fragile robust guaranteed cost controller design method for uncertain delta operator systems is presented.④An existence condition of the non-fragile robust state feedback control law satisfying disk pole and variance constraints for the delta operator systems is derived, and then the design method of relevant controller is suggested. For all cases, numerical examples are provided to illustrate the corresponding design methods.Finally, robust pole assignment for the delta operator linear time-invariant system is discussed. A method so call pole normal assignment is evolved. The objective is to find a control law, such that the closed-loop system has desired poles and the closed-loop system matrix is a normal matrix, and then robustness of the control system is enhanced. Using the properties of normal matrix and generalized inverse theory of matrix, necessary and sufficient conditions are given to state feedback pole normal assignment and static output feedback pole normal assignment respectively. When the condition holds, the unified expression of the control laws are showed.