Robustness of the independent modal-space control method

Abstract

The effect of parameter uncertainties on the control system performance of distributed-p arameter systems is examined. Because in general, the parameters contained in the equations of motion of the actual distributed system are, not known accurately, control forces designed on the basis of a postulated model will not control the actual distributed system effectively. In this paper it is shown by means of a stability theorem that, when the independent modal-space control (IMSC) method is used in conjunction with modal filters, any errors in the system parameters cannot lead to instability of the closed-loop system, so that the control system is very robust. A perturbation analysis is proposed for the computation of the closed-loop poles of large-order systems in the presence of parameter changes. I. Introduction T HE motion of a distributed-parameter system is governed generally by a set of simultaneous partial differential equations of motion. 1 The parameters contained in the equations of motion are, in general, continuous functions of the spatial variables. For flexible structures, these parameters represent mass, stiffness, and damping distributions. To control the distributed system, one must construct a mathematical model of the distributed system. The control forces then are designed on the basis of the mathematical model. A common approach to modeling is to convert the partial differential equations of the distributed system into an infinite set of ordinary differential equations.1'3 Then, a limited number of modes (generally the lowest) are retained for control. In designing the control system, one assumes that the eigensolution associated with the controlled modes is known with sufficient accuracy, which, in turn, assumes that the system parameters are known accurately. Errors in the eigensolution produce errors in the design and implementation of the controls. Hence, the question arises whether the control system designed on the basis of system parameters that are in error can control the actual system effectively; i.e., whether the control system is robust. The answer clearly depends on the degree of inaccuracy in the estimated state of the distributed system. For cases when this error is not very large, one intuitively expects very small deviations from the control system performance. In general, one should make some allowance in the control system design for parameter uncertainties. For cases when the parameters contained in the equations of motion, such as the mass and stiffness distributions, are known to within a multiplicative constant, only the system eigenvalues change and the eigenfunctions retain the same shape.4 A sensitivity study, based on a perturbation analysis treating the parameter errors as perturbations reveals that if the independent modal-space control (IMSC) method is used in conjunction with modal filters, the control system is relatively insensitive to parameter errors.4 When the spatial distributions of the system parameters are not known, however, both the estimated eigenvalues and eigenfunctions tend to differ from their actual values, so that the sensitivity analysis of Ref. 4 is not applicable.