Control System (ECE411) Lectures 13 & 14

Time-Domain AnalysisAnalyzing Simple ControllersControl System (ECE411) Lectures 13 & Azimi, ProfessorDepartment of Electrical and Computer EngineeringColorado State UniversityFall AzimiControl SystemsTime-Domain AnalysisAnalyzing Simple ControllersSteady-State Error AnalysisRemark: For aunity feedback System (H(s) = 1):e(t) =r(t) c(t)E(s) =R(s) C(s) =R(s) R(s)M(s) =E(s) = [1 M(s)]R(s)whereM(s)is the closed loop transfer ,ess= lims 0sE(s) = lims 0s[1 M(s)]R(s)For a unit stepR(s) =1s, we getess= [1 M(0)]Note:The above results could sometimes be used for cases whenH(s)6= 1(tracking error).ExampleGiven a unity feedback System shown below with closed loop transfer functionM(s) =K(s2+2s+2)(s+a), AzimiControl SystemsTime-Domain AnalysisAnalyzing Simple ControllersSteady-State Error Analysis-Cont.(a) findKandasuch thatess= unit ramp input,(b) findessfor unit step (a): First, we find the open-loop transfer functionG(s)fromM(s)using,M(s) =G(s)1 +G(s)=K(s2+ 2s+ 2)(s+a)= G(s) =K(s2+ 2s+ 2)(s+a) K=Ks3+ (a+ 2)s2+ (2a+ 2)s+ (2a K)Now, in order to avoid a Type 0 System which yieldsess for ramp input,2a K= 0 = K= a unit ramp input:ess=1 KvwhereKv= lims sG(s)Thus,ess=1Kv= = Kv= AzimiControl SystemsTime-Domain AnalysisAnalyzing Simple Cont

A typical time-domain response of a second order system (closed loop) to a unit step input is shown. M.R. Azimi Control Systems. Time-Domain Analysis Analyzing Simple Controllers Transient Analysis-Cont. Key De nitions: 1 Max Overshoot (M p) M p= c ... (repeated real roots) Characteristics: No overshoot, slow/large rise time. M.R. Azimi Control ...

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