Pole Assignment And Observers Design For Linear Discrete-time Periodic Systems

As a bridge between the linear time invariant systems and time varying systems,linear periodic time varying systems are a class of systems of great importance. On theone hand, many problems in real world can be reduced into periodic models; on the otherhand, in the control process of some systems, when the constant feedback control lawsdo not work, the periodic ones do perfectly. Due to the rapid development of computertechnology, many continuous systems need to be discretized. Thus, the research on lineardiscrete-time periodic systems is especially important. Up to now, there are already agreat deal of results in the field of discrete-time periodic systems, but there are still manyproblems to be discussed.Aiming at the basic problem in linear discrete periodic systems, employing paramet-ric design approaches, this thesis investigates pole assignment via periodic control lawsand design of periodic observers. The main contributions given by this thesis are list asfollows.1. Pole assignment for linear discrete periodic systems via periodic state feedback isdiscussed. The monodromy matrix of the closed-loop system is represented in a specialform. By combining this special form with the recent result on solutions to a class ofgeneralized Sylvester matrix equations, a complete parametric approach for pole assign-ment via periodic state feedback is proposed. The free parameters existing in the solutionsto pole assignment are further used to achieve robustness performance. The robust poleassignment problem is converted into a static optimization problem. The algorithm forrobust pole assignment is proposed.2. Pole assignment for linear discrete periodic systems via periodic output feedbackis investigated. Under the condition that the periodic system is completely reachable andcompletely observable, parametric solutions to the pole assignment problem for lineardiscrete periodic systems via periodic output feedback are presented. Further, robust poleassignment via periodic output feedback are solved. Both parametric pole assignmentalgorithm and robust pole assignment algorithm are obtained. It is shown that using peri-odic output feedback control laws can not only realize arbitrary pole assignment but alsoprovide many degree of freedom. 3. Pole assignment for linear discrete periodic systems via periodic dynamical feed-back is investigated. Similar to time invariant systems, the design of a periodic dynamicalcompensator can be reduced into the design of a periodic output feedback controller fora augmented periodic system. By employing the parametric approach of periodic outputfeedback controllers, a parametric design method for periodic dynamical compensators isproposed. For the system with several pertubation variables, a robustness index is pro-posed and robust dynamical compensator design problem is solved.4. Periodic observers design problem for linear discrete periodic systems is re-searched. Both solving forward-time discrete periodic Sylvester matrix equations andsolving reverse-time discrete-time discrete periodic Sylvester matrix equations are re-duced into solving a class of generalized Sylvester matrix equations. The explicit, com-plete and parametric solutions for the two class of matrix equations are presented. Basedon the solutions to reverse-time periodic Sylvester matrix equation, problem for peri-odic Luenberger function observer design is solved. Further, it is shown that there existsseparation principle of observer-based controllers design in this type of systems. An algo-righm for design a periodic state feedback controller based on a full order state observeris provided.5. Stabilization of satellite attitude by magnetorquer is considered. Under the ac-tion of geomagnetic field, the roll and yaw dynamics result in a linear periodic model.By setting an objective function of robust performance and optimization, a set of robustperiodic state feedback controllers is obtained. A numerical simulation through the zero-order-hold is conducted. The result shows that the periodic controllers are very effectivein regulating roll angle and yaw angle to zero.