Abstract
We consider the infinite horizon quadratic cost minimization problem for a linear system with finitely many inputs and outputs. A common approach to treat a problem of this type is to construct a semigroup in an abstract state space, and to use infinite-dimensional control theory. However, this approach is less appealing in the case where there are discrete time delays in the impulse response, because such time delays force both the control operator and the observation operator to be unbounded at the same time. In order to be able to include this case we take an alternative approach. We work in an input-output framework, and reduce the problem to a symmetric Wiener-Hopf problem, that can be solved by means of a canonical factorization of the symbol. In a standard shift semigroup realization this amounts to factorizations of the Riccati operator and the feedback operator into convolution operators and projections. Our approach leads to a new significant discovery: in the case where the impulse response of the system contains discrete time delays, the standard Riccati equation is incorrect; to get the correct Riccati equation the feed-through matrix of the system must be partially replaced by the feed-through matrix of the spectral factor. This means that, before it is even possible to write down the correct Riccati equation, a spectral factorization problem must first be solved to find one of the weighting matrices in this equation.
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References
F. M. Callier and C. A. Desoer, An algebra for transfer functions for distributed linear timeinvariant systems,IEEE Trans. Circuits Systems,25 (1978), 651–662.
F. M. Callier and J. Winkin, The spectral factorization problem for SISO distributed systems, inModelling, Robustness and Sensitivity Reduction in Control Theory (Ruth F. Curtain, ed.), pp. 463–489, NATO ASI Series, vol. F34, Springer-Verlag, Berlin, 1987.
F. M. Callier and J. Winkin, Spectral factorization and LQ-optimal regulation for multivariable distributed systems,Internat. J. Control,52 (1990), 55–75.
M. C. Delfour, Linear optimal control of systems with state and control variable delays,Automatica,20 (1984), 69–77.
M. C. Delfour, The linear-quadratic optimal control problem with delays in state and control variables: a state space approach,SIAM J. Control Optim.,24 (1986), 835–883.
C. A. Desoer and M. Vidyasagar,Feedback Systems: Input-Output Properties, Electrical Science Series, Academic Press, New York, 1975.
I. C. Gohberg and I. A. Fel'dman,Convolution Equations and Projection Methods for Their Solutions, Translations of Mathematical Monographs, vol. 41, American Mathematical Society, Providence, RI, 1974.
I. C. Gohberg and M. G. Krein, Systems of integral equations on a half line with kernels depending on the difference of arguments,Uspehi Mat. Nauk SSSR,13 (1958), 3–72. Translated as: American Mathematical Society Translations, series 2, vol. 14, pp. 217–287, 1960.
P. Grabowski, The LQ controller synthesis problem,IMA J. Math. Control Inform.,10 (1993), 131–148.
P. Grabowski, The LQ controller problem: an example,IMA J. Math. Control Inform.,11 (1994), 384–386.
P. Grabowski, The LQ controller synthesis problem: operator case, Manuscript, 1994.
G. Gripenberg, S.-O. Londen, and O. J. Staffans,Volterra Integral and Functional Equations, Cambridge University Press, Cambridge, 1990.
M. G. Krein, Integral equations on a half line with kernels depending on the difference of arguments,Uspehi Mat. Nauk SSSR,13 (1958), 3–120. Translated as: American Mathematical Society Translations, series 2, vol. 22, pp. 163–288, 1962.
A. J. Pritchard and D. Salamon, The linear-quadratic control problem for retarded systems with delays in control and observation.IM A J. Math. Control Inform.,2 (1985), 335–362.
A. J. Pritchard and D. Salamon, The linear quadratic control problem for infinite dimensional systems with unbounded input and output operators,SIAM J. Control Optim.,25 (1987), 121–144.
D. Salamon, Infinite dimensional linear systems with unbounded control and observation: a functional analytic approach,Trans. Amer. Math. Soc.,300 (1987), 383–431.
D. Salamon, Realization theory in Hilbert space,Math. Systems Theory,21 (1989), 147–164.
O. J. Staffans, Extended initial and forcing function semigroups generated by a functional equation,SIAM J. Math. Anal.,16 (1985), 1034–1048.
O. J. Staffans, Quadratic optimal control of unstable systems through coprime factorization, Submitted, December 1994.
O. J. Staffans, A neutral FDE is similar to the product of an ODE and a shift,J. Math. Anal. Appl.,192 (1985), 627–654.
O. J. Staffans, Quadratic optimal control of stable abstract linear systems, Submitted, January 1995.
O. J. Staffans, Coprime factorizations and quadratic optimal control of abstract linear systems, Submitted, April 1995.
O. J. Staffans, The nonstandard quadratic cost minimization problem for abstract linear systems, Submitted, May 1995.
B. van Keulen,H ∞-Control for Distributed Parameter Systems: A State Space Approach, Birkhäuser Verlag, Basel, 1993.
G. Weiss, Admissibility of unbounded control operators,SIAM J. Control Optim.,27 (1989), 527–545.
G. Weiss, Admissibile observation operators for linear semigroups,Israel J. Math.,65 (1989), 17–43.
G. Weiss, Transfer functions of regular linear systems, Part I: Characterizations of regularity,Trans. Amer. Math. Soc.,342 (1994), 827–854.
G. Weiss, Regular linear systems with feedback,Math. Control Signals Systems,7 (1994), 23–57.
J. Winkin, Spectral factorization and feedback control for infinite-dimensional control systems, Doctoral dissertation, Faciliteés Universitaires Notre-Dame de la Paix á Namur, 1989.
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Staffans, O.J. Quadratic optimal control of stable systems through spectral factorization. Math. Control Signal Systems 8, 167–197 (1995). https://doi.org/10.1007/BF01210206
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DOI: https://doi.org/10.1007/BF01210206