# Measures of modal and gross controllability/observability for linear time-varying systems

• 최재원 (부산대학교 기계공학부 및 기계기술연구소) ;
• 이호철 (부산대학교 기계공학부) ;
• 이달호 (경원대학교 전자공학과)
• Choe, Jae-Won (Mechanical Technology Research Center, Dept.of Mechanical Engineering, Busan National University) ;
• Lee, Ho-Chul (Busan National University) ;
• Lee, Dal-Ho (Dept.of Electrical Information Engineering, Kyungwon University )
• 발행 : 1999.08.01

#### 초록

For linear time-varying systems described by the triple (A(t),B(t),C(t)) where A(t),B(t),C(t) are the system, the input, and the output matrices, respectively, we propose concepts for measures of modal and gross controllability /observability. We introduce a differential algebraic eigenbvalue theory for linear time-varying systems to calculate the PD-eigenvalues and left and right PD-eigenvectors of the system matrix A(t) which will be used to derive the concepts for the measures. The time-dependent angle between the left PD-eigenvectors of the system matrix A(t) and the columns of the input matrix B(t), and the magnitude of the each element of the input matrix B(t) are used to propose the modal controllability measure. Similarly, the time-dependent angle between the right PD-eigenvectors of the system matrix A(t) and the rows of the output matrix C(t) are used to propose the madal observability measure. Gross measure of controllability of a mode from all inputs and its gross measure of observability in all outputs for the linear time-varying systems are also proposed. Numerical examples are presented to illustrate the proposed concepts.

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