laboratory in automatic control lab6
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8/2/2019 Laboratory in Automatic Control Lab6
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8/2/2019 Laboratory in Automatic Control Lab6
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The Stability of Linear Feedback Systems
Routh-Hurwitz Stability (1/10)
Compute the closed-loop control system poles of
the system G(s)
Solutions such that this equation
equals to zero are poles.
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8/2/2019 Laboratory in Automatic Control Lab6
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The Stability of Linear Feedback Systems
Routh-Hurwitz Stability (2/10)
Matlab code
numg=[1]; deng=[1 1 2 23];
sysg=tf(numg,deng);sys=feedback(sysg,[1]);pole(sys)
Result
Negative polemeans the system isstable.
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8/2/2019 Laboratory in Automatic Control Lab6
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The Stability of Linear Feedback Systems
Routh-Hurwitz Stability (3/10)
Compute the closed-loop control system poles of
the system G(s) for
3 2
1( )
42G s
ss s
3 2
1( )
42G s
ss s
+
R(s) K
3 21 2 4
GK Kfeedback
GK s s s K
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8/2/2019 Laboratory in Automatic Control Lab6
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The Stability of Linear Feedback Systems
Routh-Hurwitz Stability (4/10)
Method 1
Matlab code
K=[0:0.5:20];for i=1:length(K)
q=[1 2 4 K(i)];P(:,i)=roots(q);
end
plot(real(P),imag(P),'x'),gridxlabel('Real axis'),ylabel('Imaginary axis')
Result
Right half plane means it isunstable.
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The Stability of Linear Feedback Systems
Routh-Hurwitz Stability (5/9)
Method 2
Matlab code
K=[0:0.5:20];for i=1:length(K)num=[K(i)]; den=[1 2 4 0];sys=feedback(tf(num,den),[1]);P(:,i)=pole(sys);
endplot(real(P),imag(P),'x'),gridxlabel('Real axis'),ylabel('Imaginary axis')
Result
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8/2/2019 Laboratory in Automatic Control Lab6
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The Stability of Linear Feedback Systems
Routh-Hurwitz Stability (6/10)
3 2
1( )
42G s
ss s
+
R(s)
Compute the closed-loop control system poles of
the system G(s) for , plot the value ofKsuch that the system is stable.
K
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8/2/2019 Laboratory in Automatic Control Lab6
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The Stability of Linear Feedback Systems
Routh-Hurwitz Stability (7/9)
Matlab code - Method 1
K=[0:0.5:20];for i=1:length(K)
q=[1 2 4 K(i)];P(:,i)=roots(q);if(real(P(1,i))
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8/2/2019 Laboratory in Automatic Control Lab6
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The Stability of Linear Feedback Systems
Routh-Hurwitz Stability (8/10)
Matlab code - Method 2
K=[0:0.5:20];for i=1:length(K)
num=[K(i)]; den=[1 2 4 0];sys=feedback(tf(num,den),[1]);P(:,i)=pole(sys);if(real(P(1,i))
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8/2/2019 Laboratory in Automatic Control Lab6
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The Stability of Linear Feedback Systems
Routh-Hurwitz Stability (9/10)
Result
-3 -2.5 -2 -1.5 -1 -0.5 0 0.5-3
-2
-1
0
1
2
3
Real axis
Imaginaryaxis
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The Stability of Linear Feedback Systems
Routh-Hurwitz Stability (10/10)
Check: Step response
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Lab Assignments
Lab 6:
Solve Problems MP6.6, and MP6.8(b). For
MP6.6, you don't have to use the Routh-Hurwitz method.
Lab report should at least contain:
1. The MATLAB code and plot for MP6.6. 2. The MATLAB code and plot for MP6.8(b) with
comments.
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Lab Assignments:MP6.6
Consider the feedback control system in FigureMP6.6. Using the function , develop a MATLAB
script to compute the closed-loop transfer function
poles for 0 5 and plot the results denoting the
poles
for
K
with the " " symbol. Determine the maximum
range of for stability with the Routh-Hurwitz
method. Compute the roots of the characteristic
equation when is the minimum value allowed for
stability.
K
K
3 21
5 3s s K s K R s Y s
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Lab Assignments:MP6.8(b)
1
Consider the feedback control system in Figure
MP6.8. (b) Using MATLAB, plot the pole locations
as a function of 0<
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