Solutions To Periodic Sylvester Matrix Equations And Its Several Applications

As bridges between the time-varying systems and the time-invariant systems,the periodic systems occupy a pivotal position in the analysis and research of modern control theory.In the field of periodic systems control,the study of discrete linear periodic matrix equations is an important topic.Especially,solving the periodic Sylvester matrix equations is a key step in the study of classical problems in the control domain such as robust pole assignment,state observer design,observer-based robust control and fault diagnosis in discrete-time systems.In addition,the study of the solution to the periodic Lyapunov matrix equation also involves the solution to the periodic Sylvester matrix equation,which makes it have an important role in the system analysis.At present,some valuable research results have been obtained on solving periodic Sylvester matrix equations,but at the same time there are still many problems to be further studied.In this thesis,solutions to the periodic Sylvester matrix equations and their applications are investigated.Iterative approaches based on conjugate gradient are adopt to generate the solutions to the objective equations.Furthermore,the problems of the robust periodic pole assignment and the robust periodic observer design for linear discrete-time periodic systems are discussed.The main contributions of this thesis include the following points:First,the finite-iterative solutions to forward/backward periodic Sylvester matrix equations and generalized periodic Sylvester matrix equations are studied.By designing new iteration steps,based on the conjugate gradient method,the algorithm for solving time-invariant equations is extended to the time-varying domain,thus a new finite-iteration algorithm is given.After theoretical derivation and numerical example,what can be concluded is that the proposed algorithm can achieve error-free solution to the objective equations in a finite step with any initial conditions.Second,the problem of solving generalized coupled periodic Sylvester matrix equations is studied.Firstly,the sufficient and necessary conditions for the existence of the solution to this type of coupled equations are studied.If the solutions exist,starting from the case of only two coupled equations,based on the idea of conjugate gradient,an iterative algorithm for solving such matrix equations is given.Further,the proposed algorithm is extended to the case of multi-coupled equations.Finally,numerical examples illustrate the correctness and efficiency of the proposed algorithms.Third,the applications of the periodic Sylvester matrix equations in control theory are studied.The problems of poles assignment via periodic state feedback,periodic full-dimensional state observer design and poles assignment for second-order linear discrete-time periodic systems via periodic PD feedback are investigated,respectively.Through theoretical derivation,these problems are put in touch with the corresponding matrix equations.Then,by using the iterative method,the corresponding periodic controllers are solved.In addition,to reduce the influence of unknown disturbance on the control systems,robust periodic controllers are designed.