General optimal attenuation of harmonic disturbance with unknown frequencies

• Published in: International journal of control 2012-03, Vol.85 (3), p.260-279
• Main Author: Stefanovski, Jovan D.
• Format: Article
• Language: English
• Subjects: Applied sciences; Attenuation; Computer science; control theory; systems; Control theory. Systems; Controllers; Disturbances; Exact sciences and technology; Harmonics; inner and co-inner matrices; Invariants; Mathematical models; Modelling and identification; Norms; Optimization; parametrisation of stabilising controllers; SOF control; Studies
• ISSN: 0020-7179; 1366-5820
• DOI: 10.1080/00207179.2011.645167

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.