FEEDBACK CONTROL SYSTEMS

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ECM2105 - Control Engineering Dr Mustafa M Aziz (2013)________________________________________________________________________________3.2. Disturbance rejectionSuppose that there is an input disturbance D(s) applied between the controller and plant:D(s)Open-loop:U(s) = G c (s)R(s)Y(s) = G c (s)G(s)R(s)+G(s)D(s)R(s)G c(s)+U(s) +G(s)Y(s)Closed-loop: U(s) = G c (s)[R(s) – H(s)Y(s)]Gc(s)G(s)Y(s) =R(s)1+G (s)G(s)H(s)cG(s)+D(s)1+G (s)G(s)H(s)cR(s)D(s)+U(s) +G c(s)+ –H(s)G(s)Y(s)Exercise: For the systems shown above, the process transfer function G(s) = 1/(0.5s+1), thecontroller G c (s) = 99, R = 0, a step-unit disturbance D(s) = 1/s, and H(s) = 1. Determine theresponse to the disturbance for both the open-loop and closed-loop systems and sketch the outputs.4
ECM2105 - Control Engineering Dr Mustafa M Aziz (2013)________________________________________________________________________________3.3. Transient responseThe transient response is the response of a system as a function of time.A motor speed control system:− Objective: the actual speed y approaches the desired speed r quickly− Open-loop: G c (s) = k, G(s) = k 1 /(Ts+1)Assume, for example, kk 1 = 1, T = 10− Closed-loop with a proportional control term k, H(s) = 1Assume, for example, kk 1 = 10, T = 10.1Unit-step response of open-loop control1Unit-step response of closed-loop control0.90.90.80.80.70.7Amplitude0.60.50.4Amplitude0.60.50.40.30.30.20.20.10.100 5 10 15Time (sec.)00 5 10 15Time (sec.)3.4. Steady-state errorThe steady-state error is the error after the transient response has decayed, leaving only thecontinuous response, that is for a unity-feedback system:e( ∞)= lim[r(t) − y(t)]t→∞E(s) = R(s) - Y(s). Considering a unit-step input, R(s) = 1/s:− Open-loop: Y(s) = G(s)R(s), E(s) = [1-G(s)]R(s).By the final-value theorem,e( ∞)= lime(t) = limsE(s) = 1−G(0)t→∞− Closed-loop: Y(s) = G(s)R(s)/[1+G(s)H(s)]. For H(s) = 1, E(s) = R(s)/[1+G(s)]By the final-value theorem,s→0e( ∞)= lime(t) = limsE(s) = 1/[1 + G(0)]t→∞s→05
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