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

선형 시변 시스템에 대한 모드 및 총가제어성/가관측성 척도

  • 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 )
  • 최재원 (부산대학교 기계공학부 및 기계기술연구소) ;
  • 이호철 (부산대학교 기계공학부) ;
  • 이달호 (경원대학교 전자공학과)
  • Published : 1999.08.01


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.


  1. Linear System Theory and Design C T. Chen
  2. Linear System Theory W. J. Rough
  3. Linear Systems T. Kailath
  4. Proc. of Sceond VPI & SU AIAA Symposium on Dynamics and Control of Large Flexible Spacecraft A definition of the degree of controllability - A criterion for actuator placement C. N. Viswanathan;R. W. Longman;P. W. Linkins
  5. Journal of Guidance, Control, and Dynamics v.12 no.3 Measures of modal controllability and observability for first- and second-order linear systems A. M. A. Hamdan;A. H. Nayfeh
  6. Ph. D. Dissertation, Seoul National University Control Design Methodologies Using Left and Right Eigenstructure with Applications to Flight Systems J. W. Choi
  7. Journal of Guidance, Control, and Dynamics v.18 no.2 Design of an effective controller via accommodating left eigenstructure assignment J. W. Choi;J. G. Lee;Y. Kim;T. Kang
  8. Annales Scientifiques de l'Ecole Normale Superieure v.13 no.2 Equation differentielles lineaires a coefficients periodiques G. Floquet
  9. Ph. D. Dissertation, ECE Dept., UAH A Unified Eigenvalue Theory for Linear Dynamical Systems J. Zhu
  10. Linear Algebra and Its Applications v.147 Unified canonical forms for matrices over a differential ring J. Zhu;C. D. Johnson
  11. Proc. of the 34th IEEE Conference on Decision and Control A unified spectral theory for linaear time-varying systems - progress and challenges J. Zhu
  12. Proc. of the 1991 American Control Conference New spectral canonical realizations for time-varying linear dynamical systems using a unified eigenvalues concept J. Zhu;C. D. Johnson
  13. Advanced Mathematical Methods for Scientists and Engineering C. M. Bender
  14. SIAM Review v.29 no.1 Decoupling and order reduction via the Riccati transformation D. R. Smith
  15. Annales Scientifiques de l'Ecole Normale Superieure v.8 no.2 Sur la theorie des equations differentielles lineaires G. Floquet
  16. Ordinary Differential Equations E. L. Ince
  17. 제어·자동화·시스템공학회 논문지 v.5 no.5 선형 시변 시스템에 대한 잘 정의된 직렬 및 병렬 D-스펙트럼 J. Zhu;이호철;최재원
  18. Linear Time-Varying Systems: Analysis and Synthesis H. D'angelo
  19. Proc. of the 28th IEEE Southeastern Symposium on Systems Theory A necessary and sufficient stability criterion for linear time-varying systems J. Zhu
  20. Final Report for the 1995 AFOSR Summer Research Extension Program PD-eigenstructure assignment control for multivariable nonlinear tracking and decoupling J. Zhu