Page - (000064) - in Control Theory Tutorial - Basic Concepts Illustrated by Software Examples

7.1 SmallGainTheorem 57 Afulldiscussionof thesmallgain theoremcanbefoundin textbooks(e.g.,Zhou andDoyle1998;LiuandYao2016). I present abrief intuitive summary. Thepositive feedback loop inFig.7.1has transfer function G˜= G 1−GΔ. (7.3) Wederive that result by the following steps.Assume that the input toG isw+ν, whichis thesumoftheexternal input,w,andthefeedbackinput,ν.Thus, thesystem output isη=G(w+ν). Wecanwritethefeedbackinputastheoutputof theuncertaintyprocess,ν=Δη. Substituting into the systemoutput expression,wehave η=G(w+ν)=Gw+GΔη. The new system transfer function is the ratio of its output to its external input, G˜=η/w,whichwecansolve for toobtainEq.7.3. The new system, G˜, is unstable if any eigenvalue has real part greater than or equal to zero, inwhich the eigenvalues are the roots of s of the denominator, 1− G(s)Δ(s)=0. Intuitively,wecan see that G˜(s)blowsupunstably if thedenominator becomes zero at some input frequency,ω, for s= jω. Thedenominatorwill begreater than zero as long as the product of themaximumvalues ofG(jω) andΔ(jω) are less thanone,as inEq.7.1.Thatconditionexpresses thekeyidea.Themathematicalpre- sentations in the textbooksshowthatEq.7.1 isnecessaryandsufficient forstability. Reducing theH∞ normofG increases its robustnesswith respect to stability. In Eq. 7.2, a smaller ||G||∞ corresponds to a larger upper boundon the perturbations that canbe tolerated. A lowermaximumgainalsoassociateswitha smaller response toperturbations, improving the robust performance of the systemwith respect to disturbances and noise.Thus, robustdesignmethodsoftenconsider reductionof theH∞norm. 7.2 Uncertainty:DistanceBetweenSystems Supposeweassumeanominal formforaprocess,P.Wecandesignacontroller,C, in a feedback loop to improve system stability and performance. Ifwe design our controller for theprocess,P, thenhowrobust is the feedback system to alternative formsofP? The real process, P′, may differ from P because of inherent stochasticity, or becauseofour simplemodel forPmisspecified the trueunderlyingprocess. Whatistheappropriatesetofalternativeformstodescribeuncertaintywithrespect toP?SupposewedefinedadistancebetweenP andanalternativeprocess,P′.Then