I. G. Rosen

Model Reference Adaptive Control of Distributed Parameter Systems

A model reference adaptive control law is defined for nonlinear distributed parameter systems. The reference model is assumed to be governed by a strongly coercive linear operator defined with respect to a Gelfand triple of reflexive Banach and Hilbert spaces. The resulting nonlinear closed-loop system is shown to be well posed. The tracking error is shown to converge to zero, and regularity results for the control input and the output are established. With an additional richness, or persistence of excitation assumption, the parameter error is shown to converge to zero as well. A finite-dimensional approximation theory is developed. Examples involving both first- and second-order, parabolic and hyperbolic, and linear and nonlinear systems are discussed, and numerical simulation results are presented.

On-Line Parameter Estimation for Infinite-Dimensional Dynamical Systems

The on-line or adaptive identification of parameters in abstract linear and nonlinear infinite-dimensional dynamical systems is considered. An estimator in the form of an infinite-dimensional linear evolution system having the state and parameter estimates as its states is defined. Convergence of the state estimator is established via a Lyapunov estimate. The finite-dimensional notion of a plant being sufficiently rich or persistently excited is extended to infinite dimensions. Convergence of the parameter estimates is established under the additional assumption that the plant is persistently excited. A finite-dimensional approximation theory is developed, and convergence results are established. Numerical results for examples involving the estimation of both constant and functional parameters in one-dimensional linear and nonlinear heat or diffusion equations and the estimation of stiffness and damping parameters in a one-dimensional wave equation with Kelvin--Voigt viscoelastic damping are presented.

Adaptive identification of second-order distributed parameter systems

The adaptive (on-line) estimation of parameters for a class of second-order distributed parameter systems is considered. This class of systems, which includes abstract wave and beam equations with a variety of forms of damping, is frequently used to model the vibration of large flexible structures. A combined state and parameter estimator is constructed as an initial-value problem for an infinite-dimensional evolution equation in weak or variational form. State convergence is established via a Lyapunov-like estimate. The finite-dimensional notion of persistence of excitation is extended to the infinite-dimensional case and used to establish parameter convergence. A finite-dimensional approximation theory is presented and a convergence result is proven. An example involving the identification of a damped one-dimensional wave equation is discussed and results of a numerical study are presented.

An approximation theory for the identification of nonlinear distributed parameter systems

An abstract approximation framework for the identification of nonlinear, distributed parameter systems is developed. Inverse problems for nonlinear systems governed by strongly maximal monotone operators (satisfying a mild continuous dependence condition with respect to the unknown parameters to be identified) are treated. Convergence of Galerkin approximations and the corresponding solutions of finite-dimensional approximating identification problems to a solution of the original infinite-dimensional identification problem is demonstrated, using the theory of nonlinear evolution systems and a nonlinear analogue of the Trotter–Kato approximation result for semigroups of bounded linear operators. The nonlinear theory developed here is shown to subsume an existing linear theory as a special case. It is also shown to be applicable to a broad class of nonlinear elliptic operators and the corresponding nonlinear parabolic partial differential equations to which they lead. An application of the theory to a quas...

Numerical approximation for the infinite-dimensional discrete-time optimal linear-quadratic regulator problem

An abstract approximation framework is developed for the finite and infinite horizon discrete-time linear-quadratic regulator problems for systems whose state dynamics are described by a linear semigroup of operators on an infinite-dimensional Hilbert space. The schemes included in the framework yield finite-dimensional approximations to the linear state feedback gains which determine the optimal control law. Convergence arguments are given. Examples involving hereditary and parabolic systems and the vibration of a flexible beam are considered. Spline-based finite element schemes for these classes of problems, together with numerical results, are presented and discussed.

The identification of a distributed parameter model for a flexible structure

We develop a computational method for the estimation of parameters in a distributed model for a flexible structure. The structure we consider (part of the “RPL experiment”) consists of a cantilevered beam with a thruster and linear accelerometer at the free end. The thruster is fed by a pressurized hose whose horizontal motion effects the transverse vibration of the beam. We use the Euler-Bernoulli theory to model the vibration of the beam and treat the hose-thruster assembly as a lumped or point-mass-dashpotspring system at the tip. Using measurements of linear acceleration at the tip, we estimate the hose parameters (mass, stiffness, damping) and a Voigt-Kelvin viscoelastic structural damping parameter for the beam using a least-squares fit to the data.We consider spline based approximations to the hybrid (coupled ordinary and partial differential equations) system; theoretical convergence results and numerical studies with both simulation and actual experimental data obtained from the structure are pre...

Approximation in control of thermoelastic systems

This paper develops an abstract framework for analysis and approximation of linear thermoelastic control systems, and for design of finite-dimensional compensators. The thermoelastic systems in this paper consist of abstract wave and diffusion equations coupled in a skew self-adjoint fashion. Linear semigroup theory is used to establish that the abstract thermoelastic models are well posed and to prove convergence of generic approximation schemes. Open-loop uniform exponential stability for a subclass of thermoelastic systems is proved via a Lyapunov function. An example involving the design of an optimal linear-quadratic-Gaussian (LQG) compensator for a thermoelastic rod illustrates the application of the abstract theory. Results of an extensive numerical study, including a comparison of the closed-loop performance of different compensator designs, are presented and discussed.

Convergence of Galerkin approximations for operator Riccati equations—A nonlinear evolution equation approach

Abstract We develop an approximation and convergence theory for Galerkin approximations to infinite dimensional operator Riccati differential equations formulated in the space of Hilbert-Schmidt operators on a separable Hilbert space. We treat the Riccati equation as a nonlinear evolution equation with dynamics described by a nonlinear monotone perturbation of a strongly coercive linear operator. We prove a generic approximation result for quasi-autonomous nonlinear evolution systems involving accretive operators which we then use to demonstrate the Hilbert-Schmidt norm convergence of Galerkin approximations to the solution of the Riccati equation. We illustrate the application of our results in the context of a linear quadratic optimal control problem for a one dimensional heat equation.

Galerkin approximation for inverse problems for nonautonomous nonlinear distributed systems

We develop an abstract framework and convergence theory for Galerkin approximation for inverse problems involving the identification of nonautonomous, in general nonlinear, distributed parameter systems. We provide a set of relatively easily verified conditions which are sufficient to guarantee the existence of optimal solutions and their approximation by a sequence of solutions to a sequence of approximating finite-dimensional identification problems. Our approach is based upon the theory of monotone operators in Banach spaces and is applicable to a reasonably broad class of nonlinear distributed systems. Operator theoretic and variational techniques are used to establish a fundamental convergence result. An example involving evolution systems with dynamics described by nonstationary quasi-linear elliptic operators along with some applications and numerical results are presented and discussed.

Variable structure model reference adaptive control of parabolic distributed parameter systems

The objective of this note is to extend results of variable structure model reference adaptive control (VS-MRAC) for finite dimensional systems to a class of parabolic distributed parameter systems. Using a priori bounds on the plants unknown parameters, a switching law along with a control law is proposed which guarantees exponential convergence of the model reference state error to zero. In a similar fashion as in the finite dimensional case, the presence of unmodelled dynamics yields exponential convergence of the state error to a residual set. Numerical studies of a 1D parabolic system are included to provide insight into the applicability of VS-MRAC to distributed parameter systems.