You will examine behavior of a control system with different controllers. Your goal is to select...

  

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You will examine behavior of a control system with different controllers. Your goal is to select controller parameters so that closed loop system has critically damped dynamics. The plant (P) is given is as (P) + ay = bu+bd where a,b and disturbance d (t) are constant. The control structure has unity feedback as shown in Figure 1. The transfer functions of P, PI, PD, and PID controller are assumed as: P controller k C(s)=k,; PI controller C(s) = k₁+; PD controller C(s) = k₁, +kas; PID controller C(s)=k, ++ ks. k S S d -ODY C P Figure 1 у C1 u Figure 2 d P у Problem 1: For system shown in Figure 1: a) Find closed loop transfer functions G, (s) = y(s)/r(s), G (s) = y(s)/d(s), G (s) = e(s)/r(s). Gyd (s) = e(s)/d(s) where e(s)=r(s)-y(s) for each of the controller (P, PI, PD, and PID). b) Discuss the difference between closed loop systems if controllers are selected as (a) P and PD, and (b) PI and PID. c) For given controllers determine if you can place poles and zeroes of G, (s) = y(s)/r(s) independently d) If the desired pole(s) of G, (s)= y(s)/r(s) are given can you uniquely determine parameters for each yr controller? e) Compare the system selection of the controller parameters if (a) plant is determined by a > 0,5> 0 and (b) plant is determined by a <0, b>0. Discuss the constraints that you may have in selection of the parameters in both cases. You will examine behavior of a control system with different controllers. Your goal is to select controller parameters so that closed loop system has critically damped dynamics. The plant (P) is given is as (P) + ay = bu+bd where a,b and disturbance d (t) are constant. The control structure has unity feedback as shown in Figure 1. The transfer functions of P, PI, PD, and PID controller are assumed as: P controller k C(s)=k,; PI controller C(s) = k₁+; PD controller C(s) = k₁, +kas; PID controller C(s)=k, ++ ks. k S S d -ODY C P Figure 1 у C1 u Figure 2 d P у Problem 1: For system shown in Figure 1: a) Find closed loop transfer functions G, (s) = y(s)/r(s), G (s) = y(s)/d(s), G (s) = e(s)/r(s). Gyd (s) = e(s)/d(s) where e(s)=r(s)-y(s) for each of the controller (P, PI, PD, and PID). b) Discuss the difference between closed loop systems if controllers are selected as (a) P and PD, and (b) PI and PID. c) For given controllers determine if you can place poles and zeroes of G, (s) = y(s)/r(s) independently d) If the desired pole(s) of G, (s)= y(s)/r(s) are given can you uniquely determine parameters for each yr controller? e) Compare the system selection of the controller parameters if (a) plant is determined by a > 0,5> 0 and (b) plant is determined by a <0, b>0. Discuss the constraints that you may have in selection of the parameters in both cases.