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SIAM J. on Control and Optimization

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1981

Volume 19, Issue 6, pp. 711-853


Control Canonical Forms and Eigenvalue Assignment by Feedback for a Class of Linear Hyperbolic Systems

B. M. N. Clarke and D. Williamson

SIAM J. Control Optim. 19, pp. 711-729 (19 pages) | Cited 7 times

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Canonical forms are developed for a class of linear hyperbolic systems. They are then applied to solve the problem of eigenvalue assignment by distributed feedback and boundary control. The duality of this problem is demonstrated to one of eigenvalue assignment by boundary feedback of an adjoint system subject to distributed control. For both systems it is shown that by feedback, the set $\{ \rho _j \} $, $j \in \mathbb{Z}$, can be assigned as eigenvalues of the closed loop system, subject to an asymptotic condition on the set $\{ \rho _j \} $. The feedback control is explicitly characterized.
Analogous results are obtained for the problem of eigenvalue assignment by distributed feedback and distributed control.

On Spectrum Distribution of Completely Controllable Linear Systems

Sun Shun-Hua

SIAM J. Control Optim. 19, pp. 730-743 (14 pages) | Cited 9 times

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This paper is concerned with the placement of the spectrum of the closed-loop operator $A + BK$ resulting from use of a linear feedback control law $u = Kx$ in the infinite dimensional linear control system $x' = Ax + Bu$. For a class of systems in Hilbert space with certain assumptions on the spectrum of the operator $A$, a complete characterization of the achievable spectra is obtained. The proofs are carried out in an operator-theoretic context.

Finite Elements and Terminal Penalization for Quadratic Cost Optimal Control Problems Governed by Ordinary Differential Equations

Goong Chen and Wendell H. Mills, Jr.

SIAM J. Control Optim. 19, pp. 744-764 (21 pages) | Cited 4 times

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We use the finite element method to compute optimal controls of systems governed by linear ordinary differential equations with a quadratic performance index. As an application we use the penalty technique to solve terminal state optimal controllability problems. Numerical instabilities, which are common in the use of penalty, are minimized when the finite element method is applied to solve this problem. Convergence theorems are given and error and penalty parameter estimates are presented. Concrete examples for various situations are given to illustrate the theory.

Exact Controllability Theorems and Numerical Simulations for Some Nonlinear Differential Equations

Goong Chen, Wendell H. Mills, Jr., and Giovanni Crosta

SIAM J. Control Optim. 19, pp. 765-790 (26 pages) | Cited 2 times

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We study exact controllability problems for some nonlinear systems with linear controls. Our tools are contraction fixed point theorems and nonlinear semigroup properties. We show that under the assumptions of low order nonlinearity, reversibility and the existence of certain feedback controls, the nonlinear system is exactly controllable. The constructive aspect of the theory allows the application of numerical simulation. An analog-digital realization diagram is discussed. Accurate numerical schemes are developed and error estimates are presented with concrete examples to illustrate the theory.

Parameter Estimation and Identification for Systems with Delays

H. T. Banks, J. A. Burns, and E. M. Cliff

SIAM J. Control Optim. 19, pp. 791-828 (38 pages) | Cited 20 times

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Parameter identification problems for delay systems motivated by examples from aerody- namics and biochemistry are considered. The problem of estimation of the delays is included. Using approximation results from semigroup theory, a class of theoretical approximation schemes is developed and two specific cases (“averaging” and “spline” methods) are shown to be included in this treatment. Convergence results, error estimates, and a sample of numerical findings are given.

Discrete Time Stochastic Adaptive Control

Graham C. Goodwin, Peter J. Ramadge, and Peter E. Caines

SIAM J. Control Optim. 19, pp. 829-853 (25 pages) | Cited 90 times

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This paper establishes global convergence of a stochastic adaptive control algorithm for discrete time linear systems. It is shown that, with probability one, the algorithm will ensure the system inputs and outputs are sample mean square bounded and the conditional mean square output tracking error achieves its global minimum possible value for linear feedback control. Thus, asymptotically, the adaptive control algorithm achieves the same performance as could be achieved if the system parameters were known.
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