LMI Relaxations for Reduced-Order Robust ${cal H}_{\infty }$ Control of Continuous-Time Uncertain Linear Systems

This technical note is concerned with the problem of reduced order robust H_∞ dynamic output feedback control design for uncertain continuous-time linear systems. The uncertain time-invariant parameters belong to a polytopic domain and affect all the system matrices. The search for a reduced-order controller is converted in a problem of static output feedback control design for an augmented system. To solve the problem, a two-stage linear matrix inequality (LMI) procedure is proposed. At the first step, a stabilizing state feedback scheduled controller with polynomial or rational dependence on the parameters is determined. This parameter-dependent state feedback controller is used at the second stage, which synthesizes the robust (parameter-independent) output feedback H_∞ dynamic controller. A homogeneous polynomially parameter-dependent Lyapunov function of arbitrary degree is used to assess closed-loop stability with a prescribed H_∞ attenuation level. As illustrated by numerical examples, the proposed method provides better results than other LMI based conditions from the literature.