Abstract
In this paper we present results about the algebraic Riccati equation (ARE) and a weaker version of the ARE, the algebraic Riccati system (ARS), for infinite-dimensional, discrete-time systems. We introduce an operator pencil, associated with these equations, the so-called extended symplectic pencil (ESP). We present a general form for all linear bounded solutions of the ARS in terms of the deflating subspaces of the ESP. This relation is analogous to the results of the Hamiltonian approach for the continuous-time ARE and to the symplectic pencil approach for the finite-dimensional discrete-time ARE. In particular, we show that there is a one-to-one relation between deflating subspaces with a special structure and the solutions of the ARS.
Using the relation between the solutions of the ARS and the deflating subspaces of the ESP, we give characterizations of self-adjoint, nonnegative, and stabilizing solutions. In addition we give criteria for the discrete-time, infinite-dimensional ARE to have a maximal self-adjoint solution. Furthermore, we consider under which conditions a solution of the ARS satisfies the ARE as well.
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References
R. F. Curtain and L. Rodman. Comparison theorems for infinite-dimensional Riccati equations.System Control Lett., 15: 153–159, 1990.
R. F. Curtain and H. J. Zwart.An Introduction to Infinite-Dimensional Linear Systems Theory. Springer-Verlag, Berlin, 1995.
I. Gohberg, S. Goldberg, and M. A. Kaashoek.Classes of Linear Operators, vol. 1. Operator Theory, Advances and Applications, no. 49. Birkhäuser, Basel, 1990.
A. Halanay and V. Ionescu.Time Varying Discrete Linear Systems. Operator Theory, Advances and Applications, no. 68. Birkhäuser, Basel, 1994.
J. W. Helton. A spectral factorization approach to the distributed stable regulator problem; the algebraic Riccati equation.SIAM J. Control Optim., 14(4): 639–661, 1976.
V. Ionescu and M. Weiss. On computing the stabilizing solution of the discrete-time Riccati equation.Linear Algebra Appl., 174: 249–238, 1992.
T. Kato.Perturbation Theory of Linear Operators. Springer-Verlag, New York, 1966.
V. Kučera. Algebraic Riccati equations: Hermitian and definite solutions. In s. Bittanti, A. J. Laub, and J. C. Willems, editors,The Riccati Equation, pp. 54–88. Springer-Verlag, Berlin, 1991
C. R. Kuiper and H. J. Zwart. Relations between the algebraic Riccati equation and the Hamiltonian for Riesz-spectral systems. Technical Report 1054, Faculty of Applied Mathematics, University of Twente, 1994.
A. J. Laub. Invariant subspace methods for the numerical solution of Riccati equations. In S. Bittanti, A. J. Laub, and J. C. Willems, editors,The Riccati Equation, pp. 163–196. Springer-Verlag, Berlin, 1991.
K. Y. Lee, S. N. Chow, and R. O. Barr. On the control of discrete-time distributed parameter systems.SIAM J. Control, 10 (2): 361–376, 1972.
H. Logemann. Stability and stabilizability of linear infinite-dimensional discrete-time systems.IMA J. Math. Control Inform, 9: 255–263, 1992.
J. C. Louis and D. Wexler. The Hilbert space regulator problem and operator Riccati equation under stabilizability.Ann. Soc. Sci. Bruxelles, 105: 101–121, 1991.
J. C. Oostveen. Relations between the discrete-time algebraic Riccati equation and the extended symplectic pencil. Master's thesis, University of Twente, November 1994.
J. C. Oostveen and H. J. Zwart. Solving the infinite-dimensional discrete-time algebraic Riccati equation using the extended sympletic pencil. Technical Report W-9511, Department of Mathematics, University of Groningen, 1995.
S. A. Pohjolainen. On the discrete-time regulator problem in infinite-dimensional spaces.J. Optim. Theory Appl., 30: 319–327, 1980.
K. M. Przyłuski. The Lyapunov equation and the problem of stability for linear bounded discrete-time systems in Hilbert space.Appl. Math. Optim., 6: 97–112, 1980.
K. M. Przyłuski. Stability of linear infinite-dimensional systems revisited.Internat J. Control, 48(2): 513–523, 1988.
A. C. M. Ran and R. Vreugdenhil. Existence and comparison theorems for algebraic Riccati equations for continuous-and discrete-time systems.Linear Algebra Appl., 99: 63–83, 1988.
G. W. Stewart. Error and perturbation bounds for subspaces associated with certain eigenvalue problems.SIAM Rev., 12: 727–764, 1976.
A. A. Stoorvogel and A. Saberi. The discrete Riccati equation and linear matrix inequality. Preprint, 1994.
R. M. Young.An Introduction to Non-Harmonic Fourier Series. Academic Press, New York, 1980.
J. Zabczyk. Remarks on the control of discrete-time distributed parameter systems.SIAM J. Control, 12(4): 721–735, 1974.
J. Zabczyk. On optimal stochastic control of discrete-time systems in Hilbert space.SIAM J. Control, 13(6): 1217–1234, 1975.
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Oostveen, J., Zwart, H. Solving the infinite-dimensional discrete-time algebraic riccati equation using the extended symplectic pencil. Math. Control Signal Systems 9, 242–265 (1996). https://doi.org/10.1007/BF02551329
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DOI: https://doi.org/10.1007/BF02551329