Stability of distributed parameter systems with finite-dimensional controller-compensators using singular perturbations

Using a singular perturbation formulation of the linear time-invariant distributed parameter system, we develop a method to design finite-dimensional feedback compensators of any fixed order which will stabilize the infinite-dimensional distributed parameter system. The synthesis conditions are given entirely in terms of a finite-dimensional reduced-order model; the stability results depend on an infinite-dimensional version of the Klimushchev-Krasovskii lemma presented here.     

Mobile control of distributed parameter systems

Book Review

Mobile control of distributed parameter systemsA. G. Butkovskiy and L. M. Pustyl'nikov: translated by L. W. Longdon, Ellis Horwood, Halsted Press, New York, Chichester, Brisbane, Toronto, 1987, 310 pp. US$45.00. ISBN 0-85312-507-4 (Ellis Horwood Limited), ISBN 0-470-20817-1 (Halsted Press).     

Optimal control of parameter distributed systems with impulses

This paper concerns optimal control problems with impulses. The optimal magnitude of impulses and the spatial position of impulses are studied. We obtain maximum principles for these problems.     

The galerkin method and feedback control of linear distributed parameter systems

In the development of feedback control theory for distributed parameter systems (DPS), i.e., systems described by partial differential equations, it is important to maintain the finite dimensionality of the controller even though the DPS is infinite dimensional. Since this dimension is directly related to the available on-line computer capacity, it must be finite (and not very large) in order to make the controller implementable from an engineering standpoint. In previous work, it has been our intention to investigate what can be accomplished by finite-dimensional control of infinite-dimensional systems; in particular, we have concentrated on controller design and closed-loop stability. The starting point for all of this is some means for producing a finite-dimensional approximation—a reduced-order model—of the actual DPS. When the “modes” of the DPS are known, the natural candidate for model reduction is projection onto the modal subspace spanned by a finite number of critical modes. Unfortunately, in real engineering systems, these modes are never known exactly and some other reasonable approximation must be used. In this paper, the model reduction is based on the well-known Galerkin procedure. We generate the Galerkin reduced-order model and develop a finite-dimensional controller from it; then we analyze the stability of this controller in closed loop with the actual DPS. Our results indicate conditions under which model reduction based on consistent Galerkin approximations will lead to stable finite-dimensional control.     

Exponentially stabilizing finite-dimensional controllers for linear distributed parameter systems: Galerkin approximation of infinite dimensional controllers

The Galerkin method is presented as a way to develop finite-dimensional controllers for linear distributed parameter systems (DPS). The direct approach approximates the open-loop DPS and then generates the controller from this approximation; the indirect approach approximates the infinite-dimensional stabilizing controller. The indirect approach is shown to converge to the stable closed-loop system consisting of DPS and infinite-dimensional controller; conditions are presented on the behavior of the Galerkin method for the open-loop DPS which guarantee closed-loop stability for large enough finite-dimensional approximations.     

Dissipativity for Lur’e distributed parameter control systems

In this work, dissipativity of Lur’e distributed parameter control systems has been addressed. Delay-dependent sufficient conditions for the dissipativity with respect to the infinite-dimensional version of energy supply rate (Q₁,S₁,R₁) characterized exclusively by unbounded operator Q₁ are established in terms of linear operator inequalities (LOIs). Finally, the heat equation illustrates our result.     

Abstract theory of the optimal control of distributed parameter systems

Leningrad. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 29, No. 1, pp. 94–107, January–February, 1988.     

Nonnegative control of finite-dimensional linear systems

We consider the controllability problem for finite-dimensional linear autonomous control systems with nonnegative controls. Despite the Kalman condition, the unilateral nonnegativity control constraint may cause a positive minimal controllability time. When this happens, we prove that, if the matrix of the system has a real eigenvalue, then there is a minimal time control in the space of Radon measures, which consists of a finite sum of Dirac impulses. When all eigenvalues are real, this control is unique and the number of impulses is less than half the dimension of the space. We also focus on the control system corresponding to a finite-difference spatial discretization of the one-dimensional heat equation with Dirichlet boundary controls, and we provide numerical simulations.     

Optimal control problems for distributed parameter systems in Banach spaces

We consider the infinite-dimensional nonlinear programming problem of minimizing a real-valued functionf ₀ (u) defined in a metric spaceV subject to the constraintf(u) Y, wheref(u) is defined inV and takes values in a Banach spaceE and Y is a subset ofE. We derive and use a theorem of Kuhn-Tucker type to obtain Pontryagin's maximum principle for certain semilinear parabolic distributed parameter systems. The results apply to systems described by nonlinear heat equations and reaction-diffusion equations inL ¹ andL spaces.This work was supported in part by the National Science Foundation under Grant DMS-9001793.     

Stabilization of nonlinear discrete-time dynamic control systems with a parameter and state-dependent coefficients

A numerical-analytical algorithm for designing nonlinear stabilizing regulators for the class of nonlinear discrete-time control systems is proposed that significantly reduces computational costs. The resulting regulator is suboptimal with respect to the constructed quadratic functional with state-dependent coefficients. The conditions for the stability of the closed-loop system are established, and a stability result is stated. Numerical results are presented showing that the nonlinear regulator designed is superior to the linear one with respect to both nonlinear and standard time-invariant cost functionals. An example demonstrates that the closed-loop system is uniformly asymptotically stable.     

Reduced-order feedback control of distributed parameter systems via singular perturbation methods

Implementable feedback control of distributed parameter systems must often be based on reduced-order models due to the infinite dimensional nature of the actual open-loop system. The behavior of controllers designed via reduced-order models obtained with singular-perturbation techniques is analyzed. When such controllers are used in the actual distributed parameter system, the closed-loop stability is in question. The results presented here provide bounds on the smallness of the singular perturbation parameter to ensure stable operation; such a priori bounds may be used to evaluate potential reduced-order controllers for distributed parameter systems.     

Absolutely exponential stability of Lur’e distributed parameter control systems

In this work, absolutely exponential stability of Lur’e distributed parameter control systems with delayed state has been addressed. Delay-dependent sufficient conditions for the absolutely exponential stability in Hilbert spaces are established in terms of linear operator inequalities (LOIs). Finally, the wave equation is given to illustrate our result.     

Linear quadratic control problem with a terminal convex constraint for discrete-time distributed systems

The present work deals with the linear quadratic control problemfor a discrete distributed system with terminal convex constraint.Using techniques of perturbation by feedback, it is shown thatthe resolution of the considered problem is equivalent to thatof a controllability, one so-called Extended Exact Controllabilitywith time-varying operators. The Hilbert uniqueness method approachis then extended to this case to provide an explicit form forthe optimal control. In the same framework, the inequality constraintcase is examined for which a practical numerical resolutionis given. Finally, the results obtained are used to treat aminimum-time reachability problem.     

Smooth finite-dimensional approximations of distributed optimization problems via control discretization

Approximating finite-dimensional mathematical programming problems are studied that arise from piecewise constant discretization of controls in the optimization of distributed systems of a fairly broad class. The smoothness of the approximating problems is established. Gradient formulas are derived that make use of the analytical solution of the original control system and its adjoint, thus providing an opportunity for algorithmic separation of numerical optimization and the task of solving a controlled initial-boundary value problem. The approximating problems are proved to converge to the original optimization problem with respect to the functional as the discretization is refined. The application of the approach to optimization problems is illustrated by solving the semilinear wave equation controlled by applying an integral criterion. The results of numerical experiments are analyzed.     

Input-to-state stability for Lur’e stochastic distributed parameter control systems

In this work, the stochastic input-to-state stability (SISS) of Lur’e distributed parameter control systems has been addressed. Using a comparison principle, delay-dependent sufficient conditions for the stochastic input-to-state stability in Hilbert spaces are established in terms of linear operator inequalities (LOIs). Finally, the stochastic wave equation illustrates our result.     

The spectral factorization problem for multivariable distributed parameter systems

This paper studies the solution of the spectral factorization problem for multivariable distributed parameter systems with an impulse response having an infinite number of delayed impulses. A coercivity criterion for the existence of an invertible spectral factor is given for the cases that the delays are a) arbitrary (not necessarily commensurate) and b) equally spaced (commensurate); for the latter case the criterion is applied to a system consisting of two parallel transmission lines without distortion. In all cases, it is essentially shown that, under the given criterion, the spectral density matrix has a spectral factor whenever this is true for its singular atomic part, i.e. its series of delayed impulses (with almost periodic symbol). Finally, a small-gain type sufficient condition is studied for the existence of spectral factors with arbitrary delays. The latter condition is meaningful from the system theoretic point of view, since it guarantees feedback stability robustness with respect to small delays in the feedback loop. Moreover its proof contains constructive elements.     

Distributed parameter control of nonlinear flexible structures with linear finite-dimensional controllers

In many applications, mechanically flexible structures must be actively controlled to improve their performance. These structures are distributed parameter systems but they must be controlled by on-line computers and a few control actuators and sensors. A variety of controllers based on reduced-order linearized models of the structure may be designed to satisfy a given set of performance requirements. In actual operation, any such controller operates on the total structure and not the model. This paper determines bounds on the controller interaction with the unmodeled part of the structure; such bounds can be used to guarantee the successful operation of the linearly controlled structure even in the presence of nonlinear interactions.     

Open-loop control of parameter-dependent discrete-time systems

We consider parameter-dependent linear time-invariant discrete-time single input systems, where the system matrix and the input vector are assumed to depend continuously on a parameter varying over a compact interval. We face the problem of steering the zero state simultaneously arbitrarily close towards a given continuous family of desired terminal states with a finite parameter independent open-loop input sequence. Starting from existing sufficient conditions, which include simplicity of the eigenvalues of the system matrices, we examine the case of multiple eigenvalues. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)