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（3）.pptVIP

Modern Control Theory Lecture 3, written by Xiaodong Zhao Modern Control Theory Letures Notes Coryrighr@Xiaodong zhao 1.5.4 Combination system System I: System II: Transfer functions: Case I: parallel connection Case II: series connection We can get Or in matrix form Since We get Case III: feedback connection (assume ) We have In matrix form From We get and 1.6 Canonical Realization 1.6.1 Linear transform Let be two different state variables of a system. We have , where transformation of coordinates (equivalent transformation) is similar matrix of A Same fundamental characteristics (rank, trace, eigenvalues, …) Example 1: For a given system Let We have and 1.6.2 Eigenvalues and eigenvectors For a LTI system Characteristic polynomial Characteristic equation Eigenvalues : roots of characteristic equation Eigenvector: let be an eigenvalue, if there exists non-zero vector such that then is an eigenvector corresponding to . Equivalent transformation does not change eigenvalues Example 2: Calculate the eigenvalues and eigenvectors of Let , we get Let , from , we have Therefore . Let we can get 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 Example 3: Consider Let , we get From example 2, we have Then, Theorem 2: If matrix A is a companion matrix with mutually diffe