2 lec Eng
.pdfDirect method of Stability analysis
u(t)
Input signal
System
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Output signal
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The equation |
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characteristic equation.
The roots of this equation 1 , 2 , ... , n are called poles of the system (or the transfer function)
Direct method of Stability analysis
Let |
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The poles: |
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The impulse response for this system is
y t 2e 1 t 32 e 12 t
The exponential terms e i t control the response y t as t
The real parts of the poles must be negative
Direct method of Stability analysis
For asymptotic stability the roots of the characteristic equation must all be located in the left-half s-plane.
Direct method of Stability analysis
If there is at least one pole located in the right-half s-plane, the system is unstable:
If there is no one pole located in the right-half s-plane, but there are one or more poles directly on the imaginary axis, the system is neutral stable:
Direct method of Stability analysis
There is an exception.
In the case of multiple poles in the imaginary axis, the system is unstable:
Examples
1. |
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s-plane
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Equilibrium |
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1.5
zero-input
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s-plane
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Equilibrium |
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zero-input
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Examples
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s-plane
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Examples |
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position |
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zero-input
zero-state
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time, sec |
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