Optimal control of stable weakly regular linear systems

The paper extends quadratic optimal control theory to weakly regular linear systems, a rather broad class of infinite-dimensional systems with unbounded control and observation operators. We assume that the system is stable (in a sense to be defined) and that the associated Popov function is bounded from below. We study the properties of the optimally controlled system, of the optimal cost operatorX, and the various Riccati equations which are satisfied byX. We introduce the concept of an optimal state feedback operator, which is an observation operator for the open-loop system, and which produces the optimal feedback system when its output is connected to the input of the system. We show that if the spectral factors of the Popov function are regular, then a (unique) optimal state feedback operator exists, and we give its formula in terms ofX. Most of the formulas are quite reminiscent of the classical formulas from the finite-dimensional theory. However, an unexpected factor appears both in the formula of the optimal state feedback operator as well as in the main Riccati equation. We apply our theory to an extensive example.

[1]  O. Staffans.  Quadratic Optimal Control Through Coprime And Spectral Factorizations , 1996 .

[2]  O. Staffans.  On the discrete and continuous time infinite-dimensional algebraic Riccati equations , 1996 .

[3]  Piotr Grabowski.  Admissibility of observation functionals , 1995 .

[4]  H. Zwart,et al.  An example in linear quadratic optimal control , 1998 .

[5]  Olof J. Staffans,et al.  Coprime factorizations and well-posed linear systems , 1998, Proceedings of the 37th IEEE Conference on Decision and Control (Cat. No.98CH36171).

[6]  O. Staffans.  Quadratic Optimal Control of Well-Posed Linear Systems , 1999 .

[7]  Vlad Ionescu,et al.  Continuous and discrete-time Riccati theory: A Popov-function approach , 1993 .

[8]  Vladimír Kucera.  New results in state estimation and regulation , 1981, Autom..

[9]  George Weiss,et al.  Regular linear systems with feedback , 1994, Math. Control. Signals Syst..

[10]  D. Salamon.  Infinite Dimensional Linear Systems with Unbounded Control and Observation: A Functional Analytic Approach. , 1987 .

[11]  Linear quadratic optimal control for abstract linear systems , 1996 .

[12]  Irena Lasiecka,et al.  Algebraic Riccati equations with non-smoothing observation arising in hyperbolic and Euler-Bernoulli boundary control problems , 1988 .

[13]  George Weiss,et al.  Admissible observation operators for linear semigroups , 1989 .

[14]  Hans Zwart,et al.  An Introduction to Infinite-Dimensional Linear Systems Theory , 1995, Texts in Applied Mathematics.

[15]  Piotr Grabowski.  The LQ controller synthesis problem , 1993 .

[16]  Joseph J. Winkin,et al.  Spectral factorization and LQ-optimal regulation for multivariable distributed systems , 1990 .

[17]  Piotr Grabowski.  The LQ controller problem: an example , 1994 .

[18]  George Weiss,et al.  Transfer Functions of Regular Linear Systems. Part I: Characterizations of Regularity , 1994 .

[19]  Olof J. Staffans,et al.  Quadratic optimal control of stable systems through spectral factorization , 1995, Math. Control. Signals Syst..

[20]  S. Mitter,et al.  Representation and Control of Infinite Dimensional Systems , 1992 .

[21]  Vladimír Kuera,et al.  Brief papers: New results in state estimation and regulation , 1981 .

[22]  Frank M. Callier,et al.  LQ-optimal control of infinite-dimensional systems by spectral factorization , 1992, Autom..

[23]  Martin Weiss.  Riccati equations in Hilbert spaces : a Popov function approach , 1994 .

[24]  Olof J. Staffans,et al.  Quadratic optimal control of well-posed linear systems , 1998, Proceedings of the 37th IEEE Conference on Decision and Control (Cat. No.98CH36171).

[25]  O. Staffans.  Coprime Factorizations and Well-Posed Linear Systems , 1998 .

[26]  R. Rebarber.  Conditions for the equivalence of internal and external stability for distributed parameter systems , 1993, IEEE Trans. Autom. Control..

[27]  O. Staffans.  Quadratic optimal control of stable well-posed linear systems , 1997 .

[28]  George Weiss,et al.  Weak Lp-stability of a linear semigroup on a Hilbert space implies exponential stability , 1988 .

[29]  R. Triggiani,et al.  Riccati equations for hyperbolic partial differential equations with L2(O,T; L2(T)) - Dirichlet boundary terms , 1985, 1985 24th IEEE Conference on Decision and Control.

[30]  Scott W. Hansen,et al.  New results on the operator Carleson measure criterion , 1997 .

[31]  B. Molinari.  Equivalence Relations for the Algebraic Riccati Equation , 1973 .

[32]  Irena Lasiecka,et al.  Differential and Algebraic Riccati Equations With Application to Boundary/Point Control Problems: Continuous Theory and Approximation Theory , 1991 .