On analytical approach to generalized mixed $H_2$/H∞ control problem

Abstract

To improve the reliability of control systems, certain robustness to plant uncertainties and disturbance inputs is required in terms of well founded mathematical basis. Robust control theory was set up and developed until now from this motivation. In this field, $H_2$ or $H_{\infty}$-norm performance measures are frequently used nowadays. Moreover a mixed $H_2/H_{\infty}$ control problem is introduced to combine the merits of each measure since $H_2$ control usually makes more sense for performance while $H_{\infty}$ control is better for robustness to plant perturbations. However there exists no complete analytic solution to this problem at this time. In this paper, the mixed $H_2/H_{\infty}$ control problem is considered. The basic two-input, two-output system structure is transformed to the equivalent auxiliary system structure of one input and two outputs. Then analytic solutions of (sub)optimal mixed $H_2/H_{\infty}$ state-feedback controller are derived on the one-input, two-output structure for the scalar plant case and the multivariable plant case, respectively. An illustrative example is given to compare the proposed analytic solution with the existing numerical one. Finally, the results are extended to output feedback problems by reducing it to the equivalent state-feedback problems.