Harvey Thomas Banks

Hereditary Control Problems: Numerical Methods Based on Averaging Approximations

An approximation scheme involving approximation of linear functional differential equations by systems of high order ordinary differential equations is formulated and convergence is established in the context of known results from linear semigroup theory. Applications to optimal control problems are discussed and a summary of numerical results is given. The paper is concluded with a brief survey of previous literature on this class of approximations for systems with delays.

On damping mechanisms in beams

A partial differential equation model of a cantilevered beam with a tip mass at its free end is used to study damping in a composite. Four separate damping mechanisms consisting of air damping, strain rate damping, spatial hysteresis and time hysteresis are considered experimentally. Dynamic tests were performed to produce time histories. The time history data is then used along with an approximate model to form a sequence of least squares problems. The solution of the least squares problem yields the estimated damping coefficients. The resulting experimentally determined analytical model is compared with the time histories via numerical simulation of the dynamic response. The procedure suggested here is compared with a standard modal damping ratio model commonly used in experimental modal analysis.

Spline approximations for functional differential equations

Abstract We develop an approximation framework for linear hereditary systems which includes as special cases approximation schemes employing splines of arbitrary order. Numerical results for first- and third-order spline-based methods are presented and compared with results obtained using a previously developed scheme based on averaging ideas.

Optimal control of linear time-delay systems

A method is presented whereby an optimal control may be obtained for a linear time-varying system with time delay. The performance criterion is quadratic with a fixed, finite upper limit, and results in a set of differential equations with boundary conditions whose solution yields an optimal feedback control. A numerical technique is developed for the solution of the differential equations, and two examples are worked.

The Linear Regulator Problem for Parabolic Systems

We present an approximation framework for computation (in finite dimensional spaces) of Riccati operators that can be guaranteed to converge to the Riccati operator in feedback controls for abstract evolution systems in a Hilbert space. It is shown how these results may be used in the linear optimal regulator problem for a large class of parabolic systems.

Nonlinear feedback controllers and compensators: a state-dependent Riccati equation approach

Abstract State-dependent Riccati equation (SDRE) techniques are rapidly emerging as general design and synthesis methods of nonlinear feedback controllers and estimators for a broad class of nonlinear regulator problems. In essence, the SDRE approach involves mimicking standard linear quadratic regulator (LQR) formulation for linear systems. In particular, the technique consists of using direct parameterization to bring the nonlinear system to a linear structure having state-dependent coefficient matrices. Theoretical advances have been made regarding the nonlinear regulator problem and the asymptotic stability properties of the system with full state feedback. However, there have not been any attempts at the theory regarding the asymptotic convergence of the estimator and the compensated system. This paper addresses these two issues as well as discussing numerical methods for approximating the solution to the SDRE. The Taylor series numerical methods works only for a certain class of systems, namely with constant control coefficient matrices, and only in small regions. The interpolation numerical method can be applied globally to a much larger class of systems. Examples will be provided to illustrate the effectiveness and potential of the SDRE technique for the design of nonlinear compensator-based feedback controllers.

Parameter Estimation and Identification for Systems with Delays

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.

Modeling and Estimating Uncertainty in Parameter Estimation

In this paper we discuss questions related to reliability or variability of estimated parameters in deterministic least-squares problems. By viewing the parameters for the inverse problem as realizations for a random variable we are able to use standard results from probability theory to formulate a tractable probabilistic framework to treat this uncertainty. We discuss method stability and approximate problems and are able to show convergence of solutions of the approximate problems to those of the original problem. The efficacy of our approach is demonstrated in numerical examples involving estimation of constant parameters in differential equations.

Feedback control methodologies for nonlinear systems

A number of computational methods have been proposed in the literature to design and synthesize feedback controls when the plant is modeled by nonlinear dynamics. However, it is not immediately clear which is the best method for a given problem; this may depend on the nature of the nonlinearities, size of the system, whether the amount of control used or time needed for the method is a concern, and other factors. In this paper, a comprehensive comparison study of five methods for the synthesis of nonlinear control systems is carried out. The performance of the methods on several test problems are studied, and some recommendations are made as to which feedback control method is best to use under various conditions.

Reduced-order model feedback control design: numerical implementation in a thin shell model

Reduced-order models employing the Lagrange and popular proper orthogonal decomposition (POD) reduced-basis methods in numerical approximation and feedback control of systems are presented and numerically tested. The system under consideration is a thin cylindrical shell with surface-mounted piezoceramic actuators. Donnell-Mushtari equations, modified to include Kelvin-Voigt damping, are used to model the system dynamics. Basis functions constructed from Fourier polynomials tensored with cubic splines are employed in the Galerkin expansion of the full-order model. Reduced-basis elements are then formed from full order approximations of the exogenously excited shell taken at different time instances. Numerical examples illustrating the features of the reduced-basis methods are presented. As a first step toward investigating the behavior of the methods when implemented in physical systems, the use of reduced-order model feedback control gains in the full order model is considered and numerical examples are presented.