Page - (000055) - in Control Theory Tutorial - Basic Concepts Illustrated by Software Examples

6.1 CostFunction 47 Thetransferfunctionforanimpulseisequaltoone.Thus,thetransferfunctionsfor disturbanceandnoise inputsare, respectively,D(s)=μandN(s)=1.Asystem’s responsetoaninputissimplytheproductoftheinputandthesystemtransferfunction. Forexample, thefirst terminEq.6.3becomes ||D(s)Gud(s)||22 =μ2||Gud(s)||22, and the full cost functionbecomes J = μ2||Gud(s)||22+μ2ρ2||Gηd(s)||22 +||Gun(s)||22+ρ2||Gηn(s)||22. (6.4) UsingthesensitivityexpressionsinEq.6.2,wecanwritethisexpressionmoresimply as J =||CS||22+(μ2+ρ2)||T||22+μ2ρ2||PS||22. (6.5) 6.2 OptimizationMethod ThissectionfollowsQiuandZhou’s(2013)optimizationalgorithm.Theircostfunc- tion in thefinal equationonpage31of their book is equivalent tomycost function inEq.6.4. Optimization finds the controller,C(s), that minimizes the cost function. We search for optimal controllers subject to the constraint that all transfer functions in Eq. 6.1 are stable. Stability requires that the real component be negative for all eigenvaluesofeach transfer function. Atransferfunction’seigenvaluesaretherootsofthedenominator’spolynomial in s. For each transfer function inEq.6.1, theeigenvalues, s, areobtainedbysolution of1+P(s)C(s)=0. Weassumeafixedprocess, P, andweighting coefficients,μ andρ. Tofind the optimal controller,webeginwithageneral formfor thecontroller, suchas C(s)= q1s+q2 p0s2+ p1s+ p2 . (6.6) Weseek thecoefficients p andq thatminimize thecost function. QiuandZhou(2013)solvetheexampleinwhichP(s)=1/s2,forarbitraryvalues ofμandρ.TheaccompanyingMathematicacodedescribes thesteps in thesolution algorithm. Here, I simply state the solution. Check the article by Qiu and Zhou (2013)andmyMathematicacodefor thedetailsandforastartingpoint toapply the optimizationalgorithmtootherproblems.Thefollowingsectionapplies thismethod toanotherexampleandillustrates theoptimizedsystem’sresponsetovariousinputs. ForP =1/s2,QiuandZhou(2013)give theoptimal controller