Find the transfer function Tdy from the disturbance d to the output y (assume r = 0). Find the transfer function Try from the reference r to the output y (assume d = 0).
Figure 1. A
closed loop control system with disturbance
where
· P(s) is the transfer function of the plant
· C(s) is the transfer function of the controller
· r is the reference signal
· e is the error signal
· u is the output signal from the controller
· d is the disturbance signal
· y is the output signal
1. Find the transfer function Tdy from the disturbance d to the output y (assume r = 0). Find the transfer function Try from the reference r to the output y (assume d = 0).
2. If the transfer functions of the plant and controller are given as
· Plant: , (e.g., a normalized description of car velocity with force as the input; this can represent a cruise control system.)
· Controller: , where K and z are constants.
a) Verify that this is a PI controller. Show your work and state the expression for the P and I gains in terms of K and z.
b) Show that if the input d is a step disturbance and , the effect of d on y approaches zero as and (Hint: Assume K,z > 0. Use the final value theorem.) Demonstrate your result by choosing the following different sets of values for K and z (varied by orders of magnitude) and plotting their closed-loop step response. Include MATLAB code and plots. Discuss your observation.
1) K = 0.1, z = 0.1
2) K = 1, z = 1
3) K = 10, z = 10
c) Design a controller (select values of K and z) such that the following specifications are met:
· Target crossover frequency = 1 rad/s (approximately equal to the closed-loop bandwidth).
· Target phase margin is 60⁰.
1) Show all your work and demonstrate your design meets these specifications by the use of “allmargin” command on the open-loop transfer function L(s) = P(s)C(s). (hint: “help allmargin” in Matlab). Include MATLAB code and results.
2) Plot the closed-loop frequency and step responses from the reference input r to the output y. Show the achieved bandwidth by marking on the Bode plot with a “data cursor”. Comment on how different, if so, it is from the specified bandwidth. Include MATLAB code and plots. (Read http://www.mathworks.com/help/matlab/creating_plots/data-cursor-displaying-data-values-interactively.html on how to use the data cursor.)
3) Plot the closed-loop frequency and step responses from the disturbance d to the output y. Does the steady state error e become 0 when a step disturbance d is applied to this closed loop transfer function? Include MATLAB code and plots.
3. Now design a proportional-only controller (i.e., C(s) = K) to have the same closed loop crossover frequency (bandwidth) = 1 rad/s. Show your work. You can ignore the phase margin requirement, since it cannot be modified without a zero in the controller. Repeat problem 2 c) part 2) and 3). Is an integrator needed in the controller? Explain your observation.
NOTE: in order to form a closed loop transfer function you need to utilize the “feedback” command. Assuming you have created transfer function objects for both the plant and controller, i.e., P and C, the closed-loop transfer function from r to y, Try, will be computed as
>> T_ry = feedback(P*C,1)
Subsequently the bode plot and step response for Try can be found with
>> bode(T_ry)
>> step(T_ry)
In order to compute the transfer function from the disturbance d to the output y, use
>> T_dy = feedback(P, C)
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