2 lec Eng
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Hurwitz Stability Criterion
Let we have characteristic equation for a system:
a0 sn a1sn 1 ... an 1s an 0
The Hurwitz criterion gives the necessary and sufficient condition for all roots of the characteristic equation to lie in the left half of the s-plane.
Necessary (but not sufficient) condition:
1. All the coefficients of the equation have the positive sign.
2. None of the coefficients vanish.
Hurwitz Stability Criterion
Sufficient condition:
The criterion requires that the equation's n Hurwitz determinants must all be positive.
a |
a |
a ... |
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0 |
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1 |
3 |
5 |
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a0 |
a2 |
a4 ... |
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0 |
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0 |
a1 |
a3 ... |
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0 |
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H |
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... ... ... ... ... |
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0 |
0 |
0 ... |
a |
n 1 |
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0 |
0 |
0 ... |
a |
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n 2 |
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0
0
0
...
0 an
H 2 |
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a1 |
a0 |
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a1 |
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H3 |
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a0 |
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0 |
... |
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H n |
an |
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a3 a2 a3 a2 a1
H n 1
0
a5
a4 0 a3
0
H – Hurwitz matrix |
Hurwitz determinants |
Nyquist Stability Criterion
The Nyquist stability criterion determines the stability of a closed-loop system from its open-loop frequency response and open-loop poles.
The Nyquist criterion possesses the following basic features that make it desirable for the analysis as well as the design of control systems:
1.It provides the same amount of information on the absolute stability of a control system as the Hurwitz criterion
2.In addition to absolute system stability, the Nyquist criterion indicates the degree of stability and gives an indication of how the system stability may be improved.
Nyquist Stability Criterion
If the open-loop transfer function W(s) has m poles in the right-half s-plane, then for stability, the Nyquist diagram W j as ω varies from 0 to infinity, must encircle the –1+j0 point m/2 times in the counterclockwise (positive) direction.