Motion Control Final Exam

Part 1: Concept problems. Briefly answer the following questions.

1. What pole locations determine (1) a stable system, (2) an unstable system? Use diagrams.

2. What information is contained in the specification K_p = 10,000?

3. What kind of compensation (controller) improves both steady-state error and transient response?

Part 2: System modeling and evaluation problems. Show your steps.

4. For the rotational mechanical system shown below, a step torque is applied at θ₁(t). K = 1.5 N-m/rad.

a) Write two equations of motion, and find the transfer function Tc(s) = θ₂(s)/T(s).

b) Tell the stability of the system and explain why.

c) Find the system output θ₂(t), if all initial conditions are zero.

d) Plot the system response with appropriate label and title.

e) Find the damping ratio ζ, natural frequency ω_n, and state the nature of system response.

f) Calculate the percent overshoot %OS, settling time T_s, and peak time T_p., and mark those specifications in plotd).

g) Find the steady-state error of the system and mark the value in plot d).

h) Find the final value of the system and mark the value in plot d).

i) If the steady-state error in g) is not zero, determine the new spring constant value K to achieve zero steady-state error.

Part 3: Design problem. Show steps.

5. The engine, body, and tires of a racing vehicle affect the acceleration and speed attainable. The engine, body, and tires of a racing vehicle affect the acceleration and speed attainable. The speed control of the car is represented by the model shown in the figure below.

(a) Find the speed control system close loop transfer function, T(s).

(b) Calculate the settling time of the speed to a step command.

(c) Calculate the steady-state error of the car speed to a step command.

(d) If less than or equal to 2% for steady-state error and 1.0 second less for the settling time are required for the system, can this system meet the requirement? If not, choose a compensator (controller) from the list below to go with the system to improve the system performance and explain why certain compensator is chosen.

(e) Draw block diagram to show the compensated car speed control system and find the single block transfer function, T_c(s). Use K₁ = 1.25, K₂ = 2.30, and K₃ = 0.1.

(f) Plot compensated system and uncompensated system to step input in the same figure and label which is which. Title the figures.

(g) Find the steady-state error and estimate settling time for compensated system, and mark them on the plot. Also mark the steady-state error and settling time for uncompensated system (from (b) and (c)) on the plot.

(h) Verity or describe whether or not compensated system meets the design requirements.

Part 4: Simulink

6. Proceeding from Problem 4 above, use Simulink to tune PID controlled system.

(a) Make a Simulink model below with PID controller parameters P(K1) = 1.25, I(K2) = 2.30, D(K3) = 0.1.

(b) Tune PID controlled system to achieve the same settling time Ts with tolerance +-0.05 seconds but OS% less than 2%. Record the control parameters P, I, and D and print out the step plot.

(c) Tune PID controlled system to achieve the same OS% with tolerance +-1% but settling time Ts less than 0.2 seconds. Record the control parameters P, I, and D and print out the step plot.

(d) Tune PID controlled system to achieve OS% less than 7% and the settling time Ts less than 0.4 seconds. Record the control parameters P, I, and D and print out the step plot.