# Control Systems: Root-Locus Problems

Question 1:

a) Construct the root-locus for the K > 0 for the transfer function

GH = K / [s(s+1)(s^2+7s+12)

b) If the design value for the gain is K = 6, calculate the gain margin.

c) Determine the value of the gain factor K for which the system with the above open loop transfer has closed loop poles with a damping ratio of 0.5.

Question 2:

a) Figure 2 models a pressure control system with a plant G = 1{(s+1)(s+3)] consisting of twwo lags, and PI control Gc = K(s+2)/s.

i) Sketch the loci of the closed-loop system poles for varying K.

ii) Find, reasonably accurately, the value of K for a damping ratio 0.5 for the dominating poles.

b) Plot the root loci for a system with the loop gain function (i.e. open loop transfer function).

K/[(s+2)(s+6)(s^2 + 8s + 20)

and find the limiting value of K for stability.

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#### Solution Summary

This solution provides steps to answer each of the root-locus problems.

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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