PPT: Routh-Hurwitz Stability | Control Systems - Electrical Engineering (EE) PDF Download

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Routh Harwitz Criterion
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Routh Harwitz Criterion
Stability
• A linear, time-invariant system is stable if the natural response 
approaches zero as time approaches infinity.
• A linear, time-invariant system is unstable if the natural 
response grows without bound as time approaches infinity.
• A linear, time-invariant system is marginally stable if the 
natural response neither decays nor grows but remains 
constant or oscillates as time approaches infinity.
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Routh Harwitz Criterion
Stability
• A linear, time-invariant system is stable if the natural response 
approaches zero as time approaches infinity.
• A linear, time-invariant system is unstable if the natural 
response grows without bound as time approaches infinity.
• A linear, time-invariant system is marginally stable if the 
natural response neither decays nor grows but remains 
constant or oscillates as time approaches infinity.
How to Determine System Stability?
Let us focus on the natural response definitions of stability
• Stable systems have closed-loop transfer functions with poles 
only on the left half-plane.
• If the closed-loop system poles are on the right half of the s-
plane and hence have a positive real part, the system is 
unstable.
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Routh Harwitz Criterion
Stability
• A linear, time-invariant system is stable if the natural response 
approaches zero as time approaches infinity.
• A linear, time-invariant system is unstable if the natural 
response grows without bound as time approaches infinity.
• A linear, time-invariant system is marginally stable if the 
natural response neither decays nor grows but remains 
constant or oscillates as time approaches infinity.
How to Determine System Stability?
Let us focus on the natural response definitions of stability
• Stable systems have closed-loop transfer functions with poles 
only on the left half-plane.
• If the closed-loop system poles are on the right half of the s-
plane and hence have a positive real part, the system is 
unstable.
Stability Determined by TF Poles 
Page 5


Routh Harwitz Criterion
Stability
• A linear, time-invariant system is stable if the natural response 
approaches zero as time approaches infinity.
• A linear, time-invariant system is unstable if the natural 
response grows without bound as time approaches infinity.
• A linear, time-invariant system is marginally stable if the 
natural response neither decays nor grows but remains 
constant or oscillates as time approaches infinity.
How to Determine System Stability?
Let us focus on the natural response definitions of stability
• Stable systems have closed-loop transfer functions with poles 
only on the left half-plane.
• If the closed-loop system poles are on the right half of the s-
plane and hence have a positive real part, the system is 
unstable.
Stability Determined by TF Poles 
Routh-Hurwitz Stability Criterion
• It is a method for determining continuous system stability.
• The Routh-Hurwitz criterion states that “the number of 
roots of the characteristic equation with positive real parts 
is equal to the number of changes in sign of the first 
column of the Routh array ”.