Abstract
An algorithm for computing proper deflating subspaces with specified spectrum for an arbitrary matrix pencil is presented. The method uses refined algorithms for computing the generalized Schur form of a matrix pencil and enlightens the connection that exists between reducing and proper deflating subspaces. The proposed algorithm can be applied for computing the stabilizing solution of the generalized algebraic Riccati equation, a recently introduced concept which extends the usual algebraic Riccati equation.
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References
Th. Beelen, New algorithms for computing the Kronecker structure of a pencil with applications to systems and control theory, Ph.D. Dissertation, Eindhoven (1987).
P. Van Dooren, A generalized eigenvalue appoach for solving Riccati equations SIAM Sci. Statist. Comp. 2 (1981) 121–135.
P. Van Dooren, Reducing subspaces: definitions properties and algorithms, in:Lecture Notes in Mathematics, vol. 973 (Springer, 1983) pp. 58–73.
P. Van Dooren, The computation of Kronecker canonical form of a singular pencil, Lin. Alg. Appl. 27 (1979) 103–141.
P. Van Dooren, Factorization of a rational matrix: The singular case, Integral Equations and Operator Theory 7 (1984) 704–741.
G.D. Forney, Jr., Minimal bases of rational vector spaces, with applications to multivariable linear systems, SIAM J. Contr. (1975) 493–520.
F.R. Gantmacher,Theory of Matrices, vols. 1 and 2 (Chelsea, New York, 1959).
G.H. Golub and C.F. Van Loan,Matrix Computations (Johns Hopkins University Press, 1989).
V. Ionescu and C. Oara, Generalized continuous-time Riccati theory, Lin. Alg. Appl. (1994), to appear.
V. Ionescu and M. Weiss, The constrained continuous-time algebraic Riccati equation,Proc. 12th World Congress, IFAC (1993).
V. Ionescu and M. Weiss, The constrained discrete-time algebraic Riccati equation,Proc. 2nd European Control Conf. ECC 93 (1993).
B. Kagstrom, RGSVD—An algorithm for computing the Kronecker structure and reducing subspaces of singularA—λB pencils, SIAM J. Sci. Statist. Comp. 7 (1986) 185–211.
V. Kublanovskaya, On an algorithm for the solution of spectral problems of linear matrix pencils, LOMI preprint E-1-82, USSR, Academy of Sciences (1982).
F.L. Lewis and K. Ozcaldiran, Geometric structure and feedback in singular systems, IEEE Trans. Auto. Contr. AC 34 (1989) 450–455.
B.P. Molinari, A strong controllability and observability in linear multivariable control, IEEE Trans. Auto. Contr. AC 21 (1976) 761–764.
A.S. Morse, Structural invariants of linear multivariable systems, SIAM J. Contr. Optim. 11 (1973) 446–465.
D. Rappaport and L.M. Silverman, Structure and stability of discrete-time optimal systems, IEEE Trans. Auto. Contr. AC 16 (1971) 227–232.
A Varga, A pole assignment algorithm for systems in generalized state-space form,Proc. 5th Int. Conf. on Control and Informational Systems in Industry, Bucharest, Romania (1983).
A Varga, Computational methods for stabilization of descriptor systems,Proc. MTNS'93, Regensburg, Germany (1993).
M.H. Verhaegen and P. Van Dooren, A reduced-order observer for descriptor systems, Syst. Contr. Lett. 8 (1986) 9–37.
M. Weiss, Spectral and inner—outer factorization in the general case through the constrained Riccati equation, IEEE Trans. Auto. Contr., to appear.
W.M. Wonham, Linear multivariable control. A geometric approach, in:Lecture Notes in Economic and Mathematical Systems, vol. 101 (Springer, 1974).
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Oara, C. Proper deflating subspaces: properties, algorithms and applications. Numer Algor 7, 355–373 (1994). https://doi.org/10.1007/BF02140690
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DOI: https://doi.org/10.1007/BF02140690