Research On Control System Symmetrization Based On Feedback

Symmetric systems is a class of systems which have special structure and comprehensive application background,such as electrical and power networks,viscoelastic materials,chemical reaction,structural systems and so on.At present,there are many achievements in theory of symmetric systems.Descriptor systems are more general systems.They appear in many practical systems,which are natural mathematical representation for those systems.Examples of such systems include economical systems,chemical systems and micro-electro-mechanical systems.Therefore descriptor systems have received considerable attentions of scholars and made great progress.Based on the current research situation of symmetric system and descriptor system,this thesis focuses on the problem of the problem,that an asymmetric general or descriptor system is transformed into a state-space symmetric system through feedback control.First,the problem of solution to matrix equation A~TX±X~T=B is discussed.We propose a method to solve the matrix equation A~TX±X~T=B through singular value decomposition.Firstly,the necessary and sufficient condition for the existence of the solution to the matrix equation A~TX±X~T=B and its solution expression is given.Secondly,the necessary and sufficient condition for the existence of the solution to the matrix equation and its solution expression is given when matrix A is real,row full rank,column full rank and nonsingular respectively.Second,the problem that the normal and descriptor asymmetric systems are symmetrized through state feedback is investigated.Firstly,the problem that the normal asymmetric systems are transformed into state-space symmetric systems through state feedback control is concerned.The necessary and sufficient condition for the existence of state feedback control which meets the condition is given,and the expressions of state feedback control is obtained.The necessary and sufficient condition for stability of closed-loop systems is also given.Secondly,the problem that the descriptor asymmetric systems are transformed into state-space symmetric systems through state feedback control and transformation matrix is discussed.The necessary and sufficient condition for the existence of state feedback control and transformation matrix which meet the condition is given.And the expressions of state feedback control and transformation matrix are obtained.The necessary and sufficient condition for admissibility of closed-loop systems is also given.Third,the problem of that the asymmetric systems are symmetrized through output feedback and dynamic compensator is studied.Firstly,two methods to solve the problem that the asymmetric systems are symmetrized through output feedback control are given.The necessary and sufficient condition for the existence of output feedback control which meets the condition is given with the two methods,and the expressions of output feedback control is obtained.For the first method,the necessary and sufficient condition for stability of closed-loop systems is also given.Afterwards,dynamic feedback control is equivalent to output feedback control.Corresponding conclusions about that the asymmetric systems are symmetrized through dynamic compensation are obtained.