# Control Systems Comparison

Problem 1:

Using the riocus function, obtain the root locus for the following transfer function shown in Figure 1 when 0 < k < ∞ and G(s) is defined as the following:

G(s) = (s5+4s4+ 6s3+8s2+6s+4)/( s6+2 s5+2s4+s3+s2+10 -1)

1. Comment on the stability of the system as k varies.

Figure 2

Problem 2:

Consider the feedback system shown in Figure 2 above. There are three potential controllers for your system:

• Gc (s) = K (proportional controller)

• Gc (s) = K/s (integral controller)

• Gc (s) = K(1+1/s) (proportional, integral (PI) controller)

The design specifications are T(s) ≤ 10 seconds P.O. ≤ 10% for a unit step input.

1. For the proportional controller, sketch the root locus using MATLAB with 0 < K < ∞. Determine the range of K which results in stability. Determine the value of K, K/s, and K(1+1/s) so that the design specifications are satisfied

2. Co-plot the unit step responses for the closed-loop systems with each controller designed.

3. Compare and contrast the three controllers, concentrating on the steady-state errors and transient performance.

4. Recommend a controller and justify your choice.

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#### Solution Preview

The solution is provided in the attached files. The MATLAB script is also included in the zip file.

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#### Solution Summary

The solution examines control system comparisons. Three different controllers are compared and contracted concentration on the steady-state errors and transient performance.