Disturbance attenuation by a frequency-shaped linear-quadratic-regulator method

Abstract

A linear-quadrat ic-regulator method for frequency-dependent weighting matrices is applied to the design of a feedback control system such that a specified level of disturbance attenuation can be achieved while the stability of the system is ensured. Some inequalities relating the frequency-dependent weighting matrices to the return- difference matrix of the system are derived. They provide a straightforward design method for the return-difference matrix or, equivalently, the disturbance attenuation property. The result is applied to the design of an aircraft gust-alleviation system. IRCRAFT control systems should be designed to al- leviate the effect of gusts or external disturbances, be- cause the vibration due to gusts significantly degrades aircraft flying qualities. The effect of disturbances can be reduced by designing the controller so that the sensitivity function of the resultant feedback control system is small over the frequency range where the power spectrum of gusts is significant. This can be achieved by making the loop gain of the closed-loop system large over that frequency range. A high loop gain, however, significantly includes the stability and/or stability robustness of the closed-loop system.2 A control system must be designed to alleviate the effect of disturbances without losing its stability. Therefore, a linear-quadrat ic-regulator (LQR) method with frequency-dependent weighting matrices3'4 has been adopted for use herein. The I*QR method ensures the stability of the closed-loop system, while the use of frequency-dep endent weighting matrices adds more flexibil- ity in designing the sensitivity matrix. The problem of the LQR method with frequency-dependent weighting matrices has been studied by several authors. Gupta4 solved the problem by state-space augmentation. Safonov et al. 3 developed some connections between the weighting matrices and the feedback properties of the resulting optimal system. Anderson and Mingori5 discussed the robustness of the resulting system when high frequencies are weighted more heavily than low frequencies in the penalty on control effort. The purpose of mis paper is to develop a method of designing the return-difference matrix, the inverse of the sensi- tivity matrix, by means of adjusting the frequency-dependent weighting matrices.