A MARTINGALE KRONECKER LEMMA AND PARAMETER-ESTIMATION FOR LINEAR-SYSTEMS - COMMENT

Recently the issue of fragile controllers (high sensitivity of closed- loop stability and/or performance to small changes in controller coeff icients) produced by using popular robust and optimal control synthesi s methods was raised by the above-mentioned paper.(1) This paper had a t least three serious flaws. First, the authors did not provide any re ferences to the wide earlier work in which methods are given to analyz e and solve fragility and related robustness issues. Second, Keel and Bhattacharyya used mostly simple textbook examples in which the optimi zation criteria for controller synthesis were so simple that they do n ot incorporate realistic design considerations. This also resulted in the third big flaw of the paper by Keel and Bhattacharyya; namely, the possible explanation given to the cause of these problems via holes i n parametric stability space for high-order controllers is misplaced. A more direct explanation is in the badly chosen optimization criteria and controller parameterizations which make the controller synthesis and realization problems considered rather unrealistic and (mathematic ally) ill-posed. In the present paper we comment on the paper by Keel and Bhattacharyya and discuss fragility and other robustness issues th rough the well-established tools of coprime factorizations and robustn ess optimization. The main conclusion is that by adopting sensible opt imization criteria, which take into account enough of the important de sign considerations directly, and numerically robust controller parame terizations, controller fragility should not be a big problem in appli cations of modern robust and optimal control theory.