Page - (000063) - in Control Theory Tutorial - Basic Concepts Illustrated by Software Examples

56 7 Stabilization 7.1 SmallGainTheorem Supposewehaveastable systemtransfer function,G.That systemmayrepresent a process, acontroller, or acomplexcascadewithvarious feedback loops.Toexpress themathematical formofG,wemustknowexactly thedynamical processesof the system. Howmuchmay the system deviate from our assumptions about dynamics and still remainstable?Forexample, if theuncertaintiesmaybeexpressedbyapositive feedback loop,as inFig.7.1, thenwecananalyzewhetheraparticular system,G, is stably robust against thoseuncertainties. In Fig.7.1, the stable transfer function,Δ, may represent the upper bound on ouruncertainty.The feedback loopshowshowthenominalunperturbedsystem,G, responds to an input andbecomes anewsystem, G˜, that accounts for theperturba- tions.Thesystem, G˜, represents theentire loopshown inFig.7.1. The small gain theoremstates that thenewsystem, G˜, is stable if theproduct of theH∞normsof theoriginal system,G, and theperturbations,Δ, is less thanone ||G||∞||Δ||∞ <1. (7.1) Here, we interpretG as a given systemwith a knownH∞ norm. By contrast, we assume thatΔ represents the set of all stable systems that have anH∞ normbelow someupperbound, ||Δ||∞.Fortheperturbedsystem,G˜, tobestable,theupperbound for theH∞normofΔmust satisfy ||Δ||∞ < 1||G||∞ . (7.2) IfG isasystemthatwecandesignorcontrol, then thesmallerwecanmake ||G||∞, the greater the upper bound on uncertainty, ||Δ||∞, that can be tolerated by the perturbed system. Put another way, smaller ||G||∞ corresponds to greater robust stability. Gw Fig. 7.1 Systemuncertainty represented by a feedback loop.The transfer function,Δ, describes an upper bound on the extent to which the actual system, G˜=G/(1−GΔ), deviates from the nominal system,G.Here,Gmayrepresent aprocess, a controller, or anentire feedbacksystem