Research On Analysis And Feedback Control Problems For Polynomial Matrix Systems

Polynomial matrix systems are more general description form of state space systems,descriptor systems,and high-order differential systems.It is a new research perspective and approach to investigate the structures of polynomial matrix systems and their relative feedback control problems by polynomial matrix theory.The main contributions of this work are as follows.The impulse-freeness conditions for polynomial matrix systems are given by using strong linearization method.The infinite zero structures of linear matrix pencil are analyzing,and the relationship for infinite zero structures and impulsive modes of linear matrix pencil are derived.The results are extended to polynomial matrix systems.The algorithms are proposed for arbitrary given polynomial matrix to obtain infinite zero structures.The Lyapunov equations for stability of second-order systems are established by using Lyapunov method.The existence conditions for solutions of Lyapunov equations are given.Based on given Lyapunov candidate function,a linear matrix inequality condition for stability of second-order systems is obtained.Based on derived linear matrix inequality condition,all the parametric solutions of Lyapunov equations are constructed.The Lyapunov equations and linear matrix inequality condition for stability of second-order systems are extended to polynomial matrix systems.Necessary and sufficient condition of diagonalization under strict equivalence transformation for cubic polynomial matrix are presented.Necessary and sufficient conditions of diagonalization for nonsingular cubic polynomial matrix under isospectral transformation are given by using unimodular matrix transformation,and the results are extended to singular cubic polynomial matrix.The algorithms are proposed for solving isospectral diagonal cubic polynomial matrix.The parametric expressions of feedback controller gains for eigenstructure assignment ensuring normalization in second-order systems are derived.The eignstructure assignment problems are investigated in polynomial matrix systems by using the same research technique as second-order systems case.The solving algorithms of feedback controller gains for eigenstructure assignment are proposed.The parametric expressions of feedback controller gains for impulse elimination in second-order systems are formulated by eigenstructure assignment method.The impulse elimination problems of polynomial matrix systems are investigated.The solving algorithms of feedback controller gains for impulse elimination in polynomial matrix systems are proposed by eigenstructure assignment and singular value decomposition methods.