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Characterization of solutions of the discrete-time algebraic Riccati equation based on quadratic difference forms

Characterization of solutions of the discrete-time algebraic Riccati equation based on quadratic difference forms
Characterization of solutions of the discrete-time algebraic Riccati equation based on quadratic difference forms
This paper is concerned with a characterization of all symmetric solutions to the discrete-time algebraic Riccati equation (DARE). Dissipation theory and quadratic difference forms from the behavioral approach play a central role in this paper. Along the line of the continuous-time results due to Trentelman and Rapisarda [H.L. Trentelman, P. Rapisarda, Pick matrix conditions for sign-definite solutions of the algebraic Riccati equation, SIAM J. Contr. Optim. 40 (3) (2001) 969–991], we show that the solvability of the DARE is equivalent to a certain dissipativity of the associated discrete-time state space system. As a main result, we characterize all unmixed solutions of the DARE using the Pick matrix obtained from the quadratic difference forms. This characterization leads to a necessary and sufficient condition for the existence of a non-negative definite solution. It should be noted that, when we study the DARE and the dissipativity of the discrete-time system, there exist two difficulties which are not seen in the continuous-time case. One is the existence of a storage function which is not a quadratic function of state. Another is the cancellation between the zero and infinite singularities of the dipolynomial spectral matrix associated with the DARE, due to the infinite generalized eigenvalues of the associated Hamiltonian pencil. One of the main contributions of this paper is to demonstrate how to resolve these difficulties.
Discrete-time algebraic Riccati equation, Dissipative system, Quadratic difference forms, Behavioral approach, Spectral factorization, Storage function
1060-1082
Kojima, Chiaki
0a50491d-140e-49cd-a257-2816cf504880
Takaba, Kiyotsugu
951ed1dd-9ead-4dad-bb6f-093c68f52052
Kaneko, Osamu
b06e84b6-acad-41ca-9113-91a2be60696c
Rapisarda, Paolo
79efc3b0-a7c6-4ca7-a7f8-de5770a4281b
Kojima, Chiaki
0a50491d-140e-49cd-a257-2816cf504880
Takaba, Kiyotsugu
951ed1dd-9ead-4dad-bb6f-093c68f52052
Kaneko, Osamu
b06e84b6-acad-41ca-9113-91a2be60696c
Rapisarda, Paolo
79efc3b0-a7c6-4ca7-a7f8-de5770a4281b

Kojima, Chiaki, Takaba, Kiyotsugu, Kaneko, Osamu and Rapisarda, Paolo (2006) Characterization of solutions of the discrete-time algebraic Riccati equation based on quadratic difference forms. Linear Algebra and Its Applications, 416 (2-3), 1060-1082.

Record type: Article

Abstract

This paper is concerned with a characterization of all symmetric solutions to the discrete-time algebraic Riccati equation (DARE). Dissipation theory and quadratic difference forms from the behavioral approach play a central role in this paper. Along the line of the continuous-time results due to Trentelman and Rapisarda [H.L. Trentelman, P. Rapisarda, Pick matrix conditions for sign-definite solutions of the algebraic Riccati equation, SIAM J. Contr. Optim. 40 (3) (2001) 969–991], we show that the solvability of the DARE is equivalent to a certain dissipativity of the associated discrete-time state space system. As a main result, we characterize all unmixed solutions of the DARE using the Pick matrix obtained from the quadratic difference forms. This characterization leads to a necessary and sufficient condition for the existence of a non-negative definite solution. It should be noted that, when we study the DARE and the dissipativity of the discrete-time system, there exist two difficulties which are not seen in the continuous-time case. One is the existence of a storage function which is not a quadratic function of state. Another is the cancellation between the zero and infinite singularities of the dipolynomial spectral matrix associated with the DARE, due to the infinite generalized eigenvalues of the associated Hamiltonian pencil. One of the main contributions of this paper is to demonstrate how to resolve these difficulties.

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Published date: 2006
Keywords: Discrete-time algebraic Riccati equation, Dissipative system, Quadratic difference forms, Behavioral approach, Spectral factorization, Storage function
Organisations: Southampton Wireless Group

Identifiers

Local EPrints ID: 262190
URI: http://eprints.soton.ac.uk/id/eprint/262190
PURE UUID: 1ca793bb-e6e5-4ddb-b4ab-e5e72141cd52

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Date deposited: 29 Mar 2006
Last modified: 14 Mar 2024 07:06

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Contributors

Author: Chiaki Kojima
Author: Kiyotsugu Takaba
Author: Osamu Kaneko
Author: Paolo Rapisarda

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