SGER: New Approaches to the Design of Fixed Order Controllers

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Control theory is at a eculiar crossroad.Much of the elegant theory developed over the last 43 years hasrun its course and now appeals to a dwindling number of academics,while it has generated little interestfrom industry and is mostly irrelevant for engineering design.On the other hand the real design roblemsfaced by control engineers remain unresearched,lack theoretical foundation,are of a fundamental natureand theoretically extremely challenging,and there is a crying need for analytical and computer aideddesign ap roaches.Our roposal attem ts to address this gap and to work towards its resolution.The exploratory research described here deals with the develo ment of new approaches to the design of.xed order and structure controllers.The need for this research stems from the following:a)The o timalcontrol theory developed intensively since 1960 by control theorists result in high order controllers thatare model-based and the theory cannot handle the constraint of .xed order,which is ever present inindustrial applications;this makes modern theory inapplicable to even such widely used controllers asPID and lead/lag,for instance.b)O timal and high order controllers can have severe structural andsensitivity roblems as shown dramatically in the 1997 paper "Robust,Fragile or Optimal?"h(IEEE T-ACAugust 1997)by Keel and Bhattacharyya.This a er showed that controllers designed by H2,H ",l1and techniques could be acutely sensitive to erturbations in the controller coe .cients rendering themunusable.Since then no systematic solution to the fragility roblem has emerged despite signi .cantactivity in this area.c)An important need in practical design roblems is to have sets of feasiblesolutions meeting various objectives such as stability,robustness,time-delay tolerance etc so that designtradeo .s can be intelligently compared.The resent theory of .xed order controllers is far from answeringsuch questions-indeed it cannot in most cases determine if even nominal stabilization is ossible at all.With the above background as motivation our roposed research sets out to explore some new ap-proaches to solve the .xed order controller design roblem.The .rst step in this,necessarily,is thedetermination of the stabilizing set.This is a di .cult and longstanding unsolved roblem in controltheory.We have recently obtained a breakthrough in two s ecial cases.Speci .cally we obtained a mixedanalytical/com utational solution to the roblem of determining whether a .rst order or PID controllercan stabilize a given lant.Our solution can determine the complete set of stabilizing controllers andshows that this set is not convex or even connected.In view of this it is not clear how erformanceconstraints can be e .ectively imposed.Moreover the solution technique used by us in these cases doesnot seem to extend to the general case.We intend to explore two approaches for the general case.The .rst will be an attempt to approximatethe .xed order controller stabilizing sets,which are disconnected in general,through unions of convexsets.We will explore the use of the interlacing conditions for stability to generate these and it appearsossible that we can get a good and tight approximation to the stabilizing sets as unions of convexsets.This can open the door to the im osition of erformance constraints and their satisfaction throughcomputationally tractable methods such as linear rogramming and convex optimization,despite thecomplicated to ology of the stabilizing sets.The second approach that we intend to explore is theuse of a hybrid Nyquist/D-Decomposition technique that we have found to be useful in simple cases.This method has the additional otential advantage that it may be ossible to determine stabilizing setsfrom frequency domain data only thus making the design based exclusively on experimental data andessentially "gmodel-free "h.In this setting achieving robustness of the design to the data should also bedevelo ed.In summary the exploratory research pro osed here addresses the design of fixed but arbitrary or-der,robust,non-fragile controllers based on classical as well as modern erformance criteria.This is alongstanding and di .cult open control roblem.Nevertheless it needs to be solved in general for thetheory and ractice of the .eld to rogress.In the roposal we describe some seemingly romising newapproaches which are signi .cant departures from the optimization and model based approaches thatdominate the .eld at resent.We intend to explore how far these approaches can be develo ed intodesign e .ective rocedures.We believe that such procedures will bene .t control engineering ractice aswell as academic research and hel close the looming theory-practice gap in our .eld.