Model reduction and control of flexible structures using Krylov vectors

Abstract

Krylov vectors and the concept of parameter matching are combined together to develop a model reduction algorithm for a damped structural dynamics system. The obtained reduced-order model matches a certain number of low-frequency moments of the full-order system. The major application of the present method is to the control of flexible structures. It is shown that, in the control of flexible structures, three types of control energy spillover generally exist: control, observation, and dynamic. The formulation based on Krylov vectors can eliminate both the control and observation spillovers while leaving only the dynamic spillover to be considered. Two examples are used to illustrate the efficacy of the Krylov method. I. Introduction A MAJOR difficulty in the control of flexible structures or any other large-scale system is, in the words of Bellman, the curse of dimensionality. A flexible structure is, by nature, a distributed-pa rameter system, and, hence, it has infinitely many degrees of freedom. Even approximate structural models obtained by some discretization approach are generally still too large for use in control design applications. Therefore, model order reduction plays an indispensable role in the control of flexible structures. Usually, model reduction of a structural dynamics system is performed by the RayleighRitz method, which transforms the system equation to a smaller scale by using a projection subspace. It is indisputable that the choice of projection subspace is important to the accuracy of the reduced model. The eigensubspace, or the normal mode subspace, is frequently used for projection because it has a clear physical meaning and can preserve the system natural frequencies. However, with regard to the accuracy of system response, numerical experience has shown that preservation of the natural frequencies is usually not the only • concern. Other than normal modes, there are other static modes, e.g., constraint modes, attachment modes, and inertiarelief modes, which are frequently used in component mode synthesis.1 In this paper, Krylov vectors, which can be considered as static modes, are used for model reduction. There has been quite a bit of research concerning the convergence and efficiency of Krylov vectors in application to eigenvalue analysis and to the structural dynamics model reduction problem.2'5 Krylov vectors are also efficient when employed in general linear system and controller reduction problems.6'7 The major purpose of this paper is to discuss the possible application of Krylov vectors to controller design for flexible structures. The structural dynamics system studied here is described by a second-order matrix differential equation together with an output measurement equation. To perform model reduction to a structural dynamics system in this input-output configuration, the concept of parameter matching for general linear system model reduction is adopted. Parameter matching constitutes a class of efficient methods for model order reduction