Robust H_∞ control design for best mean performance over an uncertain-parameters box

This paper deals with robust, polytopic, probabilistic H∞ analysis and state-feedback synthesis of linear systems and focuses on the performance distribution over the uncertainty region (rather than on the performance bound). The proposed approach allows different disturbance attenuation levels (DALs) at the vertices of the uncertainty polytope. It is shown that the mean disturbance attenuation level (DAL) over an uncertain parameters-box is the average of the DALs at the vertices, if each parameter has an independent, symmetrical, centered probability density function. In such a (most common) case, the mean DAL over the uncertain parameters-box can be optimized by minimizing the sum of the DALs at the vertices. The standard deviation of the DAL over the uncertain parameters-box is also addressed, and a method to minimize this standard deviation is shown. A new robust H∞ state-feedback synthesis theorem is given; it is based on a recent, most efficient analysis method and applies the proposed multiple-vertex-DALs approach. A state-feedback design example utilizing the latter theorem shows that a control design which minimizes the sum of the vertex-DALs leads to a better actual closed-loop performance than a similar design which minimizes only the bound of the DAL over the uncertainty polytope. The comparison is based on the statistics of a population of closed-loop 'point-wise' H∞-norms created by a Monte-Carlo mechanism.