杭州电子科技大学现代控制理论基础课件第一章 状态空间描述（4）.pptVIP

Modern Control Theory Lecture 4, written by Xiaodong Zhao Modern Control Theory Letures Notes Coryrighr@Xiaodong zhao 1.6.3 Diagonal canonical realization Theorem 1: For a LTI system, if all eigenvalues are mutually different, there exists nonsingular matrix P and linear transform such that Proof: Let be eigenvector corresponding to . We have Let Theorem 2: If matrix A is a companion matrix with mutually different eigenvalues and let P be the Vandermonde matrix Then ,we have Proof: Let be eigenvector corresponding to . We have We then end the proof by Theorem 1. 1.6.4 Jodan canonical realization When A has multiple eigenvalues, there exists nonsingular matrix P and linear transform such that Assume and the independent eigenvector corresponding to is unique, then we have How to determine P? From , we have and are independent eigenvectors are generalized eigenvectors If matrix A is a companion matrix with eigenvalues and the independent eigenvector corresponding to is unique, then we have Example 2: consider Let , we get . Since A is a companion matrix , we have From , we can get and 1.7 State Space of Discrete Time System are the sampled data, determined by ,where T is the sampling period Unit Time delay : T second time delay 1.7.1 State Space Realization Case I: No input derivative Let Then, we can get Case II: Include input derivative Let Where Let Then we can get Or in matrix form Example 1: The system equation is We