For the control system given, (a) Sketch, by hand, its root locus for Ge(s) = K....

 

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For the control system given, (a) Sketch, by hand, its root locus for Ge(s) = K. When sketching the root locus, if necessary, make use of the asymptotes finding σa and a that are the intersecting point and C(s) R(S) + 1 Ge(s) (s+2)(s+5)(s +12) angles with the real axis, respectively using the following formula, Σfinite poles-Σfinite zeros σα and a #finite poles-# finite pzeros (2k+1)π #finite poles-#finite pzeros' , where k = 0, +1, +2,... (b) If the root locus intersects the jo-axis, find the values of poles 51,2 and gain K at the crossing points. Then write the range of gain K making the system stable. (c) Determine the type of the system (type 0, type 1, type 2, etc.). (d) Find the steady-state error for this P-controlled system for Ge(s) = 162, when a unit step and then ramp inputs are introduced. (e) It is determined that the root locus intersects the 10% overshoot line (which corresponds to a damping ratio of <= 0.59) for K = 162, at s = -2.62 + j3.67. Now, justify the second order approximation by finding the third closed-loop pole at this gain, then give an estimation of the settling time using Ts ≈ 4/(<wn). (f) Design a PID controller to substitute for G,(s) so that the system's response becomes twice as faster with the same maximum overshoot and produces no steady-state error to a step input. Hint: For a simple PID controller, design a PD controller to substitute for Ge(s) so that the settling time is reduced 2 times while keeping the same maximum overshoot. Then add a PI controller to meet the steady-state error requirement. For simplicity, you may use the same gain you found for PD controller, also you may select the zero of PI controller at -0.1. (g) Plot the responses of the P, PD and PID controlled system on the same plane for comparison. (h) Determine the constant of proportionality, K₂, the reset (integral) time T; and the rate (derivative) time Ta values of the PID controller assuming that the controller has the following transfer function, GPID (S)= 80(s + 0.4) (s + 10) S (i) Realize the same PID-controller given in (h) i. as a block diagram where three blocks for P, I, D actions are connected in parallel. Remember to determine and write the constants of proportion, integral and derivative (Kp, Ki and Ka) ii. physically with a circuit having two OPAMPs. Select the first capacitor, i.e. the one at the input, as C₁ = 10μF, also select the second OPAMP's impedances as Z3 Z4 = 10k For the control system given, (a) Sketch, by hand, its root locus for Ge(s) = K. When sketching the root locus, if necessary, make use of the asymptotes finding σa and a that are the intersecting point and C(s) R(S) + 1 Ge(s) (s+2)(s+5)(s +12) angles with the real axis, respectively using the following formula, Σfinite poles-Σfinite zeros σα and a #finite poles-# finite pzeros (2k+1)π #finite poles-#finite pzeros' , where k = 0, +1, +2,... (b) If the root locus intersects the jo-axis, find the values of poles 51,2 and gain K at the crossing points. Then write the range of gain K making the system stable. (c) Determine the type of the system (type 0, type 1, type 2, etc.). (d) Find the steady-state error for this P-controlled system for Ge(s) = 162, when a unit step and then ramp inputs are introduced. (e) It is determined that the root locus intersects the 10% overshoot line (which corresponds to a damping ratio of <= 0.59) for K = 162, at s = -2.62 + j3.67. Now, justify the second order approximation by finding the third closed-loop pole at this gain, then give an estimation of the settling time using Ts ≈ 4/(<wn). (f) Design a PID controller to substitute for G,(s) so that the system's response becomes twice as faster with the same maximum overshoot and produces no steady-state error to a step input. Hint: For a simple PID controller, design a PD controller to substitute for Ge(s) so that the settling time is reduced 2 times while keeping the same maximum overshoot. Then add a PI controller to meet the steady-state error requirement. For simplicity, you may use the same gain you found for PD controller, also you may select the zero of PI controller at -0.1. (g) Plot the responses of the P, PD and PID controlled system on the same plane for comparison. (h) Determine the constant of proportionality, K₂, the reset (integral) time T; and the rate (derivative) time Ta values of the PID controller assuming that the controller has the following transfer function, GPID (S)= 80(s + 0.4) (s + 10) S (i) Realize the same PID-controller given in (h) i. as a block diagram where three blocks for P, I, D actions are connected in parallel. Remember to determine and write the constants of proportion, integral and derivative (Kp, Ki and Ka) ii. physically with a circuit having two OPAMPs. Select the first capacitor, i.e. the one at the input, as C₁ = 10μF, also select the second OPAMP's impedances as Z3 Z4 = 10k

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a Root locus for Ges K The transfer function for the given control system is Gs s5s2s3s12 The number of finite poles is 2 and the number of finite zeros is also 2 Therefore the root locus will start f... View the full answer