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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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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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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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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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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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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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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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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<