Matlab and Designing Controller System
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.
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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