Finite Dimensional Approximate Modeling with Error Bounds of Flexible Vibrating Systems Based on Partial Eigenstructures

An approach to control-oriented uncertainty modeling is presented for a class of elastic vibrating systems such as flexible structures, beams and strings, described by partial differential equations. Uncertainty bounding techniques are developed using upper and lower bounds of the unknown eigenparameters. The result forms a basis for a finite-dimensional controller design in which closed loop stability and performance are guaranteed. A feasible set of systems is defined of all systems governed by a class of differential equations with certain norm bounds of unknown input and output operators and with partially known bounds of eigenparameters. Then the perturbation magnitude covering the feasible set is evaluated in frequency domain where a standard truncated modal model is chosen as the nominal one. An upper bound to the truncated error magnitude is proposed which is calculated using linear programming. It is demonstrated that all the parameters formulating a feasible set are derived using finite element analysis for a flexible beam example, and feasibility of the proposed scheme is also illustrated by numerical bounding results.