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Matlab and Designing Controller System

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Consider the following system whose state space representation is as follows: (see attached file for better representation)

x'1 0 1 0 x1 0

x'2 = 0 0 1 x2 + 0 u

x'3 24 14 -1 x3 1

y = [2 -1 1]x

a) Design a controller for the system, place the poles at s=-10 and s = -15± 5j.

b) Design (if possible) a full-order observer for the system. Place the observer poles at s= -20,-20,-20.

c) Using MATLAB, repeat part (a) and (b). Also check that the matrices A-BK and A-LC have the desired eigenvalues.

d) Using MATLAB, plot x1,x2,x3, x1,x2 ,x3 in the same graph. Note that the initial state of the system is (-3,-3,-3); the initial state of the observer is 0.

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Consider the following system whose state space representation is as follows:

x'1 0 1 0 x1 0

x'2 = 0 0 1 x2 + 0 u

x'3 24 14 -1 x3 1

y = [2 -1 1]x

a) Design a controller for the system, place the poles at s=-10 and s = -15± 5j.

b) Design (if possible) a full-order observer for the system. Place the observer poles at s= -20,-20,-20.

c) Using MATLAB, repeat part (a) and (b). Also check that the matrices A-BK and A-LC have the desired eigenvalues.

d) Using MATLAB, plot x1,x2,x3, x1,x2 ,x3 in the same graph. Note that the initial state of the system is (-3,-3,-3); the ...

Solution Summary

This solution has full explanations and step-by-step calculations to design a controller and full-order observer for the system using Matlab.

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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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