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).

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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 T_dy from the disturbance d to the output y (assume r = 0). Find the transfer function T_ry 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, T_ry, will be computed as

>> T_ry = feedback(P*C,1)

Subsequently the bode plot and step response for T_ry 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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