Analysis of singular systems using orthogonal functions

The use of orthogonal functions to analyze singular systems is investigated. It is shown that the differential-algebraic system equation may be converted to an algebraic generalized Lyapunov equation that can be solved for the coefficients of x(t) in terms of the orthogonal basis functions. This generalized Lyapunov equation may be considered as a "discrete" equation on the slow subspace of the system, and as a "continuous" equation on its fast subspace. Necessary and sufficient conditions for the existence of a unique solution are given in terms of the relative spectrum of the system. A generalized Bartels/Stewart algorithm based on the QZ algorithm is presented for its efficient solution. Relations are drawn with the invariant subspaces of the system.

[1]  R. F. Sincovec,et al.  Solvability, controllability, and observability of continuous descriptor systems , 1981 .

[2]  P. Paraskevopoulos.  Analysis of singular systems using orthogonal functions , 1984 .

[3]  K. Wong.  The eigenvalue problem λTx + Sx , 1974 .

[4]  Stephen L. Campbell,et al.  On using orthogonal functions with singular systems , 1984 .

[5]  G. Golub,et al.  A Hessenberg-Schur method for the problem AX + XB= C , 1979 .

[6]  A. Laub.  A schur method for solving algebraic Riccati equations , 1978, 1978 IEEE Conference on Decision and Control including the 17th Symposium on Adaptive Processes.

[7]  C. F. Chen,et al.  A state-space approach to Walsh series solution of linear systems , 1975 .

[8]  Richard H. Bartels,et al.  Algorithm 432 [C2]: Solution of the matrix equation AX + XB = C [F4] , 1972, Commun. ACM.

[9]  H. Rosenbrock,et al.  Contributions to a hierarchical theory of systems , 1974 .

[10]  G. Stewart,et al.  An Algorithm for Generalized Matrix Eigenvalue Problems. , 1973 .

[11]  Yen-Ping Shih,et al.  Analysis and optimal control of time-varying linear systems via Walsh functions , 1978 .

[12]  B. Stott,et al.  Power system dynamic response calculations , 1979, Proceedings of the IEEE.

[13]  W. Vetter.  Vector structures and solutions of linear matrix equations , 1975 .

[14]  R. Newcomb.  The semistate description of nonlinear time-variable circuits , 1981 .

[15]  F. Lewis.  A survey of linear singular systems , 1986 .

[16]  P. Lancaster.  Explicit Solutions of Linear Matrix Equations , 1970 .

[17]  D. Cobb.  Feedback and pole placement in descriptor variable Systems , 1981 .

[18]  G. Kekkeris.  On the analysis of singular systems using orthogonal functions , 1986 .

[19]  D. Luenberger.  Dynamic equations in descriptor form , 1977 .