On the asymptotically optimal tuning of robust controllers for systems in the CD-algebra

The authors previously (2000) showed that a low-gain controller of the form C_ε(s)=Σ_k=-nⁿ εK_k/(s-iω_k) is able to track and reject constant and sinusoidal reference and disturbance signal for a stable plant in the Callier-Desoer (CD) algebra. In this note, we investigate the optimal tuning of the matrix gains K_k of the controller C_ε(s) as the scalar gain ε↓0. The cost function is the maximum error between the reference signal and the measured output signal over all frequencies and bounded reference and disturbance signal amplitudes. Closed forms for asymptotically globally optimal solutions are given. The optimal matrix gains K_k are expressed in terms of the values of the plant transfer matrix at the reference and disturbance signal frequencies. Thus the matrices K_k can be tuned with input-output measurements made from the open loop plant without knowledge of the plant model. Although the analysis is in the CD-algebra, to the authors´ knowledge the main results are new even for finite-dimensional systems