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The standard regulator problem for systems with input delays. An approach through singular control theory. (English) Zbl 0815.49006
For the finite-dimensional state space system \[ \dot x= Ax+ \left\{\sum^ N_{i= 1} B_ i u(t- h_ i)+ \int^ 0_{-h} B(s) u(t+ s) ds\right\}, \] where the novelty is in the introduction of “delays” in the control (modelling perhaps actuator-dynamics), the author considers the “infinite-horizon” LQR problem: \[ \min_ u \int^ \infty_ 0 [x(t), Qx(t)] dt+ \int^ \infty_ 0 [Ru(t), u(t)] dt,\quad R> 0,\quad Q\geq 0, \] and shows the existence of solution under appropriate “stabilizability” conditions, involving a generalized algebraic Riccati equation. The novelty in the technique of solution is the use of the so-called “singular” LQR problem, where \(R\) as above is singular.

MSC:
49J25 Optimal control problems with equations with ret. arguments (exist.) (MSC2000)
93C25 Control/observation systems in abstract spaces
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[1] Balakrishnan AV (1978) Boundary control of parabolic equations: L-Q-R theory. In: Theory of Nonlinear Operators, Akademie-Verlag, Berlin · Zbl 0427.49005
[2] Balakrishnan AV (1981) Applied Functional Analysis, 2nd edn. Springer-Verlag, New York · Zbl 0459.46014
[3] Bittanti S, Laub AJ, Willems JC (1991) The Riccati Equations. Springer-Verlag, Berlin
[4] Delfour MC (1986) The linear quadratic optimal control problem with delays in state and control variables: a state space approach. SIAM J Control Optim 24:835-883 · Zbl 0606.93037
[5] Doyle J, Glover K, Khargonekar PP, Francis BA (1989) State space solutions to standardH 2 andH ? control problems. IEEE Trans Automat Control 34:831-847 · Zbl 0698.93031
[6] Fattorini O (1968) Boundary control systems. SIAM J Control Optim 6:349-385 · Zbl 0164.10902
[7] Folland GB (1984) Real Analysis. Wiley, New York
[8] Ichikawa A (1982) Quadratic control of evolution equations with delays in control. SIAM J Control Optim 20:645-668 · Zbl 0495.49006
[9] Kalo T (1984) Perturbation Theory of Linear Operators. Springer-Verlag, Berlin
[10] Khargonekar PP (1981) Canonical forms for linear-quadratic optimal control problems. PhD Thesis, University of Florida
[11] Lasiecka I, Triggiani R (1991) Differential and Algebraic Riccati Equations with Applications to Boundary/Point Control Problems: Continuous Theory and Approximation Theory. Springer-Verlag, Berlin · Zbl 0754.93038
[12] Lee EB, You Y (1989) Optimal syntheses for infinite-dimensional linear delayed state output systems: a semicausal approach. Appl Math Optim 19:113-136 · Zbl 0674.49025
[13] Louis J-Cl (1985) The regulator problems in Hilbert spaces and some applications to stability of nonlinear control systems. Thesis, Départment de Mathématiques, Facultés Universitaires Notre-Dame de la Paix, Namur, Belgie
[14] Louis J-Cl, Wexler D (1982) On the regulator problem and the operational Riccati equation in Hilbert spaces. In: Dynamical Systems II (Bednarek AR, Cesari L, eds), Academic Press, New York, pp 605-611 · Zbl 0599.34084
[15] Louis J-Cl, Wexler D (1991) The Hilbert space regulator problem and operator Riccati equation under stabilizability. Ann Soc Sci Bruxelles 105:137-165 · Zbl 0771.47026
[16] Manitius A, Olbrot AW (1979) Finite spectrum assignment problem for systems with delays. IEEE Trans Automat Control 24:541-553 · Zbl 0425.93029
[17] Pandolfi L (1990) Generalized control systems, boundary control systems, and delayed control systems. Math Control Signals Systems 3:165-181 · Zbl 0694.93047
[18] Pandolfi L (1991) Dynamic stabilization of systems with input delays. Automatica 27:1047-1050
[19] Salamon D (1984) On control and observation of neutral systems. Pitman, Boston · Zbl 0546.93041
[20] Vinter RB, Kwong RH (1981) The finite time quadratic control problem for linear systems with state and control delays: an evolution equation approach. SIAM J Control Optim 19:139-153 · Zbl 0465.93043
[21] Willems JC (1971) Least squares stationary optimal control and the algebraic Riccati equation. IEEE Trans Automat Control 16:621-634
[22] Yakubovich VA (1973) The frequency theorem in control theory. Siberian J Math 14:384-419
[23] Yakubovich VA (1974) A frequency theorem for the case in which the state and control spaces are Hilbert spaces, with applications to some problems in the synthesis of optimal controls, I. Siberian J Math 15:457-476 · Zbl 0302.49017
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