A b s t r a c tIn this report we show how the results and techniques of [33] can be extended t o a very general single-input/single-output mixed sensitivity problem, in which the transfer function of the plant to be controlled has both irrational stable and irrational unstable parts. In the process we show (as an extension of [6]) how t o transform the scalar mixed sensitivity problem into the "standard form" which is the starting point from which [33] computes the optimal norm of the mixed sensitivity. This transformation allows us t o make some new observations about the problem that have design significance. These developments are motivated by the computation of the infimal HM-norm of the mixed sensitivity for the irrational transfer function model developed in [9].
We present a constructive method for 'absorbing' an irrational outer factor of a plant into the 'Q-parameter' in the H"' optimal weighted sensitivity problem for single-inputlsingle-output distributed parameter systems, when the plant has finitely many irrational zeros on the imaginary axis. This problem could not be solved using previous results. We also extend our new results to the mixed sensitivity problem.
KEY WORDS H-infinity control Sensitivity optimization Outer factors
INTRODUCTIONA crucial step in the derivation of the solution to H" optimal control problems is to show that the outer factor of the plant can be ignored at a certain point in the calculation of the infimal norm of the H" control criterion. Specifically, given the expressions po = inf 11 W -FMQ QEH" and with WEL", M E H " inner and F E H " outer, the assumption is made that * = p i , and computation proceeds to find pi.Solutions to H" control problems rely on the equivalence of these two optimization problems, yet the validity of this step has only been demonstrated for rational plants" and for stable plants with irrational inner part but rational outer Our recent study of an example3 has led us to be concerned with plants having irrational outer part. In the present work we present an extension to a class of unstable plants having irrational outer and inner factors.The organization of this paper is as follows:In Section 2 we review the H" optimal weighted sensitivity problem for unstable plants and its reduction to a minimization problem which is affine in a free (functional) parameter, via the so-called 'Q-parametrization'. When the outer and unstable factors of the plant and the weighting function are rational, is known that under additional necessary assumptions on the This paper was recommended for publication by editor M. G. Safonov
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