Page - (000065) - in Control Theory Tutorial - Basic Concepts Illustrated by Software Examples

58 7 Stabilization a set of alternatives could be specified as all processes, P′, forwhich the distance fromthenominalprocess,P, is less thansomeupperbound. Wewill write the distance between two processeswhenmeasured at input fre- quencyω as δ[P(jω),P′(jω)]=distanceat frequencyω, (7.4) for which δ is defined below. Themaximum distance between processes over all frequencies is δν(P,P ′)=max ω δ[P(jω),P′(jω)], (7.5) subjecttoconditionsthatdefinewhetherPandP′arecomparable(Vinnicombe2001; QiuandZhou2013).Thisdistancehasvalues0≤ δν ≤1,providingastandardized measureof separation. Todevelopmeasuresofdistance,wefocusonhowperturbationsmayaltersystem stability.Supposewestartwithaprocess,P,andcontroller,C, inafeedbacksystem. Howfar cananalternativeprocess,P′, be fromP andstillmaintain stability in the feedback loopwithC? In otherwords,what is the stabilitymargin of safety for a feedbacksystemwithP andC? Robust control theory provides an extensive analysis of the distances between systemswith respect tostabilitymargins (Vinnicombe2001;ZhouandDoyle1998; Qiu andZhou2010, 2013).Here, I present a rough intuitive descriptionof thekey ideas. For a negative feedback loopwith P andC, the various input–output pathways all have transfer functionswithdenominator 1+PC, as inEq. 6.1.These systems becomeunstablewhen thedenominatorgoes tozero,whichhappens ifP =−1/C. Thus, the stabilitymargin is thedistancebetweenP and−1/C. The values of these transfer functions, P(jω) andC(jω), varywith frequency, ω.Theworstcasewithregard tostabilityoccurswhenP and−1/C areclosest; that is,when thedistancebetween these functions is aminimumwith respect tovarying frequency.Thus,wemaydefine the stabilitymargin as theminimumdistanceover frequency bP,C =min ω δ[P(jω),−1/C(jω)]. (7.6) Here is thekey idea. Startwith anominal process,P1, and a controller,C. If an alternative or perturbedprocess, P2, is close to P1, then the stabilitymargin for P2 shouldnotbemuchworse than forP1. In otherwords, a controller that stabilizes P1 should also stabilize all processes that are reasonably close to P1. Thus, by designing a good stabilitymargin for P1, weguarantee robust stabilization for all processes sufficientlynearP1. Wecan express these ideas quantitatively, allowing the potential to design for a targeted levelof robustness.Forexample, bP2,C ≥bP1,C −δν(P1,P2).