Belinda B. King

Proper orthogonal decomposition for reduced basis feedback controllers for parabolic equations

In this paper, we present a discussion of the proper orthogonal decomposition (POD) as applied to simulation and feedback control of the one-dimensional heat equation. We provide two examples of input collections to which the POD process is applied. First, we apply POD directly to the finite element basis of linear B-splines. Next, we additionally include time snapshots. We show that although the second case provides better simulations, this POD basis is ill-suited for control problems. We provide a discussion of both the linear quadratic regulator (LQR) problem and the linear quadratic Gaussian (LQG) problem.

Optimal sensor location for robust control of distributed parameter systems

In this paper, we discuss an approach to sensor/estimator design for robust control of distributed parameter systems. This approach involves using minmax compensator design and piecewise approximates to the optimal feedback gains. To illustrate, we present the results for a nonlinear hybrid partial and ordinary differential equation system.<<ETX>>

Reduced Order Controllers for Spatially Distributed Systems via Proper Orthogonal Decomposition

A method for reducing controllers for systems described by partial differential equations (PDEs) is presented. This approach differs from an often used method of reducing the model and then designing the controller. The controller reduction is accomplished by projection of a large scale finite element approximation of the PDE controller onto low order bases that are computed using the proper orthogonal decomposition (POD). Two methods for constructing input collections for POD, and hence low order bases, are discussed and computational results are included. The first uses the method of snapshots found in POD literature. The second is a new idea that uses an integral representation of the feedback control law. Specifically, the kernels, or functional gains, are used as data for POD. A low order controller derived by applying the POD process to functional gains avoids subjective criteria associated with implementing a time snapshot approach and performs favorably.

A fourth-order parabolic equation modeling epitaxial thin film growth

Abstract We study the continuum model for epitaxial thin film growth from Phys. D 132 (1999) 520–542, which is known to simulate experimentally observed dynamics very well. We show existence, uniqueness and regularity of solutions in an appropriate function space, and we characterize the existence of nontrivial equilibria in terms of the size of the underlying domain. In an investigation of asymptotical behavior, we give a weak assumption under which the ω-limit set of the dynamical system consists only of steady states. In the one-dimensional setting we can characterize the set of steady states and determine its unique asymptotically stable element. The article closes with some illustrative numerical examples.

Sensor location in feedback control of partial differential equation systems

The task of placing sensors for purposes of feedback control is vital in order to obtain information necessary for accurate state estimation. We present a method for optimal location of sensors which is motivated by the feedback control law for the distributed parameter system. In particular, we show how feedback functional gains reflect spatial regions over which accurate information is paramount for control. We use this information in an algorithm which computes centroidal Voronoi tesselations, yielding optimal locations for sensors. This placement is compared with three others to show that location can be more important than number of sensors.

Nonlinear dynamic compensator design for flow control in a driven cavity

We discuss minmax compensator design for the driven cavity problem. A nonlinear controller is obtained by using a linear feedback control law with nonlinear compensator equations. A finite difference-Galerkin scheme with divergence free basis is applied to the infinite dimensional system to illustrate the feasibility of this approach.

Semidefinite Programming Techniques for Reduced Order Systems with Guaranteed Stability Margins

In this paper, the compensator based reduced order control design framework of Burns and King (J. Vibrations and Control, vol. 4, pp. 297–323, 1998) is modified to yield low order systems with guaranteed stability margins. This result is achieved through use of a logarithmic barrier function. In addition, a reduced basis method is formulated in which the compensator equations are approximated on uneven grids; guaranteed stability margins are also included. The methods are tested numerically on a one dimensional, nonlinear, damped, hyperbolic structural control problem. Examples are provided to illustrate differences between systems designed by both reduced basis methods.

A NOTE ON THE REGULARITY OF SOLUTIONS OF INFINITE DIMENSIONAL RICCATI EQUATIONS

This note is concerned with the regularity of solutions of algebraic Riccati equations arising from infinite dimensional LQR and LQG control problems. We show that distributed parameter systems described by certain parabolic partial differential equations often have a special structure that smoothes solutions of the corresponding Riccati equation. This analysis is motivated by the need to find specific representations for Riccati operators that can be used in the development of computational schemes for problems where the input and output operators are not Hilbert-Schmidt. This situation occurs in many boundary control problems and in certain distributed control problems associated with optimal sensor/actuator placement.

Reduced order controllers for distributed parameter systems: LQG balanced truncation and an adaptive approach

In this paper, two methods are reviewed and compared for designing reduced order controllers for distributed parameter systems. The first involves a reduction method known as LQG balanced truncation followed by MinMax control design and relies on the theory and properties of the distributed parameter system. The second is a neural network based adaptive output feedback synthesis approach, designed for the large scale discretized system and depends upon the relative degree of the regulated outputs. Both methods are applied to a problem concerning control of vibrations in a nonlinear structure with a bounded disturbance.

Representation of feedback operators for parabolic control problems

In this paper we present results on existence and regularity of integral representations of feedback operators arising from parabolic control problems. The existence of such representations is important for the design of low order compensators and in the placement of sensors. This paper extends earlier results of J. A. Burns and B. B. King to problems with N spatial dimensions.