A Twofold Spline Approximation for Finite Horizon LQG Control of Hereditary Systems

In this paper an approximation scheme is developed for the solution of the linear quadratic Gaussian (LQG) control on a finite time interval for hereditary systems with multiple noncommensurate delays and distributed delay. The solution here proposed is achieved by means of two approximating subspaces: the first one to approximate the Riccati equation for control and the second one to approximate the filtering equations. Since the approximating subspaces have finite dimension, the resulting equations can be implemented. The convergence of the approximated control law to the optimal one is proved. Simulation results are reported on a wind tunnel model, showing the high performance of the method.

[1]  I. G. Rosen,et al.  A Spline Based Technique for Computing Riccati Operators and Feedback Controls in Regulator Problems for Delay Equations , 1984 .

[2]  Fritz Colonius.  Stable and Regular Reachability for Relaxed Hereditary Differential Systems , 1982 .

[3]  Michel C. Delfour,et al.  The linear quadratic optimal control problem for hereditary differential systems: Theory and numerical solution , 1976 .

[4]  Fritz Colonius.  Addendum: Stable and Regular Reachability of Relaxed Hereditary Differential Systems , 1985 .

[5]  A. Manitius,et al.  Numerical Approximations for Hereditary Systems with Input and OutputDelays: Convergence Results and Convergence Rates , 1994 .

[6]  H. Banks,et al.  Spline approximations for functional differential equations , 1979 .

[7]  Michel C. Delfour,et al.  Controllability, Observability and Optimal Feedback Control of Affine Hereditary Differential Systems , 1972 .

[8]  D. Salamon.  Structure and Stability of Finite Dimensional Approximations for Functional Differential Equations , 1985 .

[9]  Georg Propst,et al.  Piecewise linear approximation for hereditary control problems , 1990 .

[10]  Irena Lasiecka,et al.  Differentiability and Convergence Rates of Approximating Semigroups for Retarded Functional Differential Equations , 1988 .

[11]  Alfredo Germani,et al.  Galerkin approximation of optimal linear filtering of infinite-dimensional linear systems , 1988 .

[12]  R. B. VlNTER.  On the Evolution of the State of Linear Differential Delay Equations in M2: Properties of the Generator , 1978 .

[13]  A. Feliachi,et al.  Memoryless stabilization of linear delay-differential systems , 1981 .

[14]  R. Curtain.  Infinite-Dimensional Linear Systems Theory , 1978 .

[15]  Fritz Colonius.  The Maximum Principle for Relaxed Hereditary Differential Systems with Function Space end Condition , 1982 .

[16]  R. H. Fabiano.  Stability preserving spline approximations for scalar functional differential equations , 1995 .

[17]  F. Kappel,et al.  A uniformly differentiable approximation scheme for delay systems using splines , 1987, 26th IEEE Conference on Decision and Control.

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

[19]  Karl Kunisch.  Approximation Schemes for the Linear-Quadratic Optimal Control Problem Associated with Delay-Equations. , 1982 .

[20]  K. A. Morrisy.  Design of Finite-dimensional Controllers for Innnite-dimensional Systems by Approximation , 1994 .

[21]  J. Gibson.  Linear-Quadratic Optimal Control of Hereditary Differential Systems: Infinite Dimensional Riccati Equations and Numerical Approximations , 1983 .

[22]  L. Pandolfi.  The standard regulator problem for systems with input delays. An approach through singular control theory , 1995 .

[23]  Kazufumi Ito,et al.  On nonconvergence of adjoint semigroups for control systems with delays , 1988 .

[24]  Franz Kappel,et al.  An approximation theorem for the algebraic Riccati equation , 1990 .

[25]  L. Jetto,et al.  A Twofold Spline Scheme Approximation Method for the LQG Control Problem on Hilbert Spaces , 1993 .

[26]  Franz Kappel,et al.  Spline approximation for retarded systems and the Riccati equation , 1987 .

[27]  M. Delfour.  The linear quadratic optimal control problem with delays in state and control variables: a state space approach , 1986 .

[28]  Kazumfumi Lto,et al.  Finite-dimensional compensators for infinite-dimensional systems via Galerkin-type approximation , 1990 .

[29]  H. Banks,et al.  Hereditary Control Problems: Numerical Methods Based on Averaging Approximations , 1978 .

[30]  Alfredo Germani,et al.  The LQG control problem for a class of hereditary systems: a method for computing its approximate solution , 1994, Proceedings of 1994 33rd IEEE Conference on Decision and Control.

[31]  F. Kappel,et al.  On the stability properties of spline approximates for retarded systems , 1989 .

[32]  J. S. Gibson,et al.  The Riccati Integral Equations for Optimal Control Problems on Hilbert Spaces , 1979 .

[33]  Kirsten Morris,et al.  Convergence of controllers designed using state-space techniques , 1994, IEEE Trans. Autom. Control..

[34]  R. Kwong,et al.  A stability theory for the linear-quadratic-Gaussian problem for systems with delays in the state, control, and observations , 1979, 1979 18th IEEE Conference on Decision and Control including the Symposium on Adaptive Processes.

[35]  Alfredo Germani,et al.  Approximation of the algebraic Riccati equation in the Hilbert space of Hilbert-Schmidt operators , 1993 .

[36]  Alfredo Germani,et al.  Numerical solution for optimal regulation of stochastic hereditary systems with multiple discrete delays , 1995, Proceedings of 1995 34th IEEE Conference on Decision and Control.

[37]  Raymond Y. Kwong.  A stability theory for the linear-quadratic-Gaussian problem for systems with delays in the state, control, and observations , 1979 .

[38]  T. Mori.  Criteria for asymptotic stability of linear time-delay systems , 1985, IEEE Transactions on Automatic Control.

[39]  Kazufumi Ito,et al.  Semigroup theory and numerical approximation for equations in linear viscoelasticity , 1990 .