Linear Quadratic Optimization for Systems in the Behavioral Approach

In this paper the following formulation of the linear quadratic optimal control problem for dynamical systems in the behavioral setting is proposed: given a linear, time-invariant, and complete system, find the set of trajectories of the system that optimize a quadratic-type cost function and that satisfy some linear static constraints. This formulation provides a unifying framework, where several generalized versions of the classical LQ optimal control can be stated and solved. The existence of solutions is first discussed. It is shown that a necessary and sufficient condition for the existence of solutions may be obtained as a by-product of a reduction procedure translating the problem into an equivalent one of minimum complexity. Such a procedure is based on the theory of $\ell^2$-systems in the behavioral setting. Once the complexity is reduced, a parametrization of the set of optimal solutions is obtained by employing a behavioral realization technique and a two-step optimization procedure.

[1]  J. Magnus,et al.  Matrix Differential Calculus with Applications in Statistics and Econometrics (Revised Edition) , 1999 .

[2]  Jan C. Willems,et al.  Models for Dynamics , 1989 .

[3]  S. Sastry,et al.  Optimal path planning on matrix Lie groups , 1994, Proceedings of 1994 33rd IEEE Conference on Decision and Control.

[4]  Alan J. Laub,et al.  The linear-quadratic optimal regulator for descriptor systems , 1985, 1985 24th IEEE Conference on Decision and Control.

[5]  Jan C. Willems,et al.  LQ-control: a behavioral approach , 1993, Proceedings of 32nd IEEE Conference on Decision and Control.

[6]  Jan C. Willems,et al.  From time series to linear system - Part III: Approximate modelling , 1987, Autom..

[7]  J. C. Willems,et al.  Controllability of l2-systems , 1992 .

[8]  D. Cobb,et al.  Descriptor variable systems and optimal state regulation , 1983 .

[9]  J. Willems.  On interconnections, control, and feedback , 1997, IEEE Trans. Autom. Control..

[10]  Tryphon T. Georgiou,et al.  Graphs, causality, and stabilizability: Linear, shift-invariant systems on ℒ2[0, ∞) , 1993, Math. Control. Signals Syst..

[11]  F. Lewis.  A survey of linear singular systems , 1986 .

[12]  J. Willems.  Paradigms and puzzles in the theory of dynamical systems , 1991 .

[13]  Jan C. Willems,et al.  Controllability of l 2 -systems , 1992 .

[14]  Siep Weiland,et al.  Rational representations of behaviors: Interconnectability and stabilizability , 1997, Math. Control. Signals Syst..

[15]  L. Silverman.  Discrete Riccati Equations: Alternative Algorithms, Asymptotic Properties, and System Theory Interpretations , 1976 .

[16]  Jan C. Willems,et al.  From time series to linear system - Part I. Finite dimensional linear time invariant systems , 1986, Autom..

[17]  Christiaan Heij,et al.  Deterministic Identification of Dynamical Systems , 1989 .

[18]  D. Whittaker,et al.  A Course in Functional Analysis , 1991, The Mathematical Gazette.

[19]  B. Anderson,et al.  Optimal control: linear quadratic methods , 1990 .

[20]  Ahw Ton Geerts,et al.  The algebraic Riccati equation and singular optimal control: The discrete-time case , 1988 .

[21]  A. Krall,et al.  Modeling stabilization and control of serially connected beams , 1987 .

[22]  Tryphon T. Georgiou,et al.  Graphs, causality and stabilizability: linear, shift-invariant systems on L/sub 2/(0, infinity ) , 1992, [1992] Proceedings of the 31st IEEE Conference on Decision and Control.

[23]  J. Willems,et al.  On Quadratic Differential Forms , 1998 .

[24]  Yasuo Murata.  Optimal control methods for linear discrete-time economic systems , 1982 .

[25]  C. Heij,et al.  Global total least squares modeling of multivariable time series , 1995, IEEE Trans. Autom. Control..

[26]  J. Nieuwenhuis.  Another look at linear-quadratic optimization problems over time , 1995 .