Periodic Lyapunov equations: Some applications and new algorithms

The discrete-time periodic Lyapunov equation has several important applications in the analysis and design of linear periodic control systems. Specific applications considered are the solution of state- and output-feedback optimal periodic control problems, the stabilization by periodic state feedback and the square-root balancing of discrete-time periodic systems. Efficient and numerically reliable algorithms based on the periodic Schur decomposition are proposed for the solution of periodic Lyapunov equations. The proposed algorithms are extensions of the direct solution methods for standard discrete-time Lyapunov equations for the case of indefinite as well as non-negative definite solution.

[1]  J. H. Wilkinson.  The algebraic eigenvalue problem , 1966 .

[2]  G. Hewer.  An iterative technique for the computation of the steady state gains for the discrete optimal regulator , 1971 .

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

[4]  Genshiro Kitagawa,et al.  An algorithm for solving the matrix equation X = FXF T + S , 1977 .

[5]  A. Barraud.  A numerical algorithm to solve AT X A - X = Q , 1977 .

[6]  A. Barraud.  ' An algorithm for solving the matrix equation X=FXFT + S' , 1978 .

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

[8]  S. Hammarling.  Numerical Solution of the Stable, Non-negative Definite Lyapunov Equation , 1982 .

[9]  Solution of the matrix equation AX−XB=C , 1986 .

[10]  I. Postlethwaite,et al.  Truncated balanced realization of a stable non-minimal state-space system , 1987 .

[11]  S. Bittanti,et al.  The difference periodic Ricati equation for the periodic prediction problem , 1988 .

[12]  Nicholas J. Higham,et al.  FORTRAN codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation , 1988, TOMS.

[13]  Alan J. Laub,et al.  Sensitivity of the stable discrete-time Lyapunov equation , 1990 .

[14]  Nicholas J. Higham,et al.  INVERSE PROBLEMS NEWSLETTER , 1991 .

[15]  S. Bittanti,et al.  The Periodic Riccati Equation , 1991 .

[16]  Adam W. Bojanczyk,et al.  Periodic Schur decomposition: algorithms and applications , 1992, Optics & Photonics.

[17]  M. Pittelkau.  Optimal periodic control for spacecraft pointing and attitude determination , 1993 .

[18]  A. Laub,et al.  Numerical solution of the discrete-time periodic Riccati equation , 1994, IEEE Trans. Autom. Control..

[19]  V. Ionescu,et al.  Time-Varying Discrete Linear Systems , 1994 .

[20]  J. Sreedhar,et al.  On finding stabilizing state feedback gains for a discrete-time periodic system , 1994, Proceedings of 1994 American Control Conference - ACC '94.

[21]  P. Dooren,et al.  Periodic Schur forms and some matrix equations , 1994 .

[22]  Alan J. Laub,et al.  Residual bounds for discrete-time Lyapunov equations , 1995, IEEE Trans. Autom. Control..

[23]  Noah H. Rhee,et al.  Cyclic Schur and Hessenberg-Schur Numerical Methods for Solving Periodic Lyapunov and Sylvester Equa , 1995 .

[24]  S. Pieters,et al.  A computational approach for optimal periodic output feedback control , 1996, Proceedings of Joint Conference on Control Applications Intelligent Control and Computer Aided Control System Design.

[25]  Thilo Penzl,et al.  Numerical solution of generalized Lyapunov equations , 1998, Adv. Comput. Math..