Substructural Multiobjective H∞ Controller Design for Large Flexible Structures: A Divide-and-Conquer Approach Based on Linear Matrix Inequalities

Abstract

In this paper, a novel substructural approach is proposed and successfully implemented for H∞ robust controller design for large flexible structures. It is assumed that sensors and actuators are discrete and located at some nodal points of the structure. In general, a finite element method (FEM)-based modelling approach results in a matrix differential equation of large dimensions. As the dimension becomes larger and larger controller design algorithms require more and more computation time, and start to have numerical problems. To cope with these difficulties, there are many known techniques in the literature, including the decentralized- and substructural-type methods. In this paper, a substructural-type approach based on the static condensation principle is adopted and the H∞ optimal controller design problem for large flexible structures is studied. The key point in the present approach is that the static condensation is performed in the abstract state space. Geometric information about the flexible structure is utilized in deciding how to do the state decomposition, then H∞ optimal controllers are designed at the substructure level, and finally a global controller is assembled for the whole structure. To improve the convergence of the algorithm, a multi-objective H∞ optimization approach is adopted. More precisely, while forcing the closed-loop poles to be in a given convex region to ensure fast dynamics, and hence improve the convergence of the substructural iterations, the H∞ objective function is minimized to achieve maximum robustness. The main advantage of this approach is that both the H∞ objective and the constraints on closed-loop poles can be expressed as a convex problem and formulated as linear matrix inequalities (LMIs), which can be solved easily, e.g. by LMI Toolbox of MATLAB. Overall, the proposed approach results in a reduction in computation time and improvements in numerical reliability as the problem of large size is decomposed into several smaller-size problems. The accuracy and effectiveness of the substructural H∞ control technique are tested on benchmark problems, and effects of structural non-linearities are studied.