Page - (000056) - in Control Theory Tutorial - Basic Concepts Illustrated by Software Examples

48 6 Regulation C(s)= √ 2ρμ (√ ρ+√μ)s+ρμ s2+√2(√ρ+√μ)s+(√ρ+√μ)2 , withassociatedminimizedcost, J∗ =√2[μ2√ρ+ρ2√μ+2ρμ(√μ+√ρ)]. Forρ=1, thecontrollerbecomes C(s)= √ 2μ ( 1+√μ)s+μ s2+√2(1+√μ)s+(1+√μ)2 , (6.7) withassociatedminimizedcost, J∗ =√2[μ2+√μ+2μ(√μ+1)]. Wecanseethetradeoffsindesignmostclearlyfromthecontrollerwithρ=1.When μ is small, loaddisturbance inputsare smaller thansensornoise inputs.Anoptimal systemshould therefore tolerategreater sensitivity to loaddisturbances in returnfor reducedsensitivity to sensornoise. In the optimal controller described byEq. 6.7, a small value ofμproduces low gain, becauseC(s)becomes smaller asμdeclines.Wecan see fromEq. 6.1 that a small gain for the controller,C, reduces the sensitivity to noise inputs by lowering Gun andGηn.Similarly,asmallgainforC raisesthesensitivityofthesystemoutput, η, todisturbance inputsby raisingGηd. The optimal system achieves the prescribed rise in sensitivity to disturbance in order toachieve lower sensitivity tonoise. 6.3 ResonancePeakExample This sectionapplies theprevious section’sH2 optimizationmethod to theprocess P(s)= 1 s2+0.1s+1. (6.8) Thisprocesshasa resonancepeaknearω=1. MysupplementalMathematicacodederives theoptimalcontrollerof theformin Eq.6.6.Theoptimalcontroller isexpressedin termsof thecostweightingsμandρ. Thesolutionhasmany terms, so there isnobenefit in showing it here.