General optimal attenuation of harmonic disturbance with unknown frequencies

Abstract

We present algorithms for optimal harmonic disturbance attenuation in standard discrete-time control structure, based on a parametrisation of (marginally) stabilising controllers. The Frobenius norm and the spectral norm of the closed-loop transfer matrix at the disturbance frequencies are minimised. If there is only one frequency of the disturbance, the controller has an observer–based form, which we obtain by solving a static output feedback (SOF) stabilisation control problem. Although the SOF stabilisation problem is hard, the generical case of nonsquare matrix G 22 is solved by linear algebra methods. Numerical simulation results are presented. As a corollary, we transform the control problem with unit circle invariant zeros into a ℋ∞ control problem without such zeros. The elimination of the unit circle invariant zeros is based on the fact that matrix Y(zI − A + BF)−1 is stable, where (Y, F) with Y ≥ 0 is a solution of a discrete-time algebraic Riccati system.