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# Control Systems Complex Controller

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Problem 1:
Consider the closed-loop transfer function

T(s) = 10K/(s2 + 20s + K)

1. Obtain the family of step responses for K=10, 100, and 500. Co-plot the responses and develop a table that includes the following:

a. Percent overshoot
b. Settling time

Figure 1
Problem 2:
A negative feedback control system is depicted in Figure 1. Suppose that your design is to find a controller, Gc(s), of minimal complexity such that your closed-loop system can track a unit step input with steady-state error of zero.

1. As a first try, consider a simple proportional controller: Gc (s) = K,
where K is a fixed gain. Let K = 2. Using MATLAB, plot the unit step response.

2. Determine the steady-state error from the plot above. Calculate the theoretical steady-state error and compare your results.

3. Now consider a more complex controller: Gc (s) = K0 + K1/s

where K0 = 2, and K1 = 20. This type of controller is known as a proportional, integral (PI) controller. Plot the unit step response using this controller.

4. Determine the steady-state error from the plot. Calculate the theoretical steady-state error and compare your results.

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