Controle e Observador de Sistema

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Problem 7.48PP
A certain process has the transfer function G(i) = . .
(a) Find A. B, and C for this system in observer canonicai fonn.
(b) If u = -Kx, compute K so that the closed-loop control poles are located at s = -2 ± 2j.
(c) Compute L so that the estimator error poles are located at s = -10±10y.
(d) Give the transfer function of the resulting controller [for example, using Eq.].
(d) Give the transfer function of the resulting controller [for example, using Eq.].
(e) What are the gain and phase margins of the controller and the given open-loop system? 
Eq.
V(j)
D .(j) “ 7 ^ - - “ ,( x )— K ( x l - A + B K 4 L C ) ' L
‘■ • o - c g
-•w -i’ g :])■'&]
• ^ [ 2 1 2 , + 4]
, , - 2 6 4 J - 6 9 2
j* + 2 4 s + 2 9 2
Thus, the transfer function of the compensator is D , ( i)=
-2 6 4 J-6 9 2
5^ + 24^ + 292
Step 6 of 6
(e)
Consider the MATLAB code to determine the phase and gain margin of the controller using 
nyquist plot.
H=tf([-264-133924].[1 24 292]) 
nyquist{H)
The output after executing the MATLAB code is shown in Figure 1.
NyiliiistDiagnB
From Figure 1. the phase margin is —137® and gain margin is —5 3 ^ dB 
Thus, the phase margin is |-137®| and gain margin is 1-53.2 d s l •