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

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Routh-Hurwitz Criterion
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Routh-Hurwitz Criterion
Understanding System Stability
Stable System
All roots of the 
characteristic equation lie 
on the left half of the 'S' 
plane.
Marginally Stable 
System
All roots of the 
characteristic equation lie 
on the imaginary axis of 
the 'S' plane.
Unstable System
All roots of the 
characteristic equation lie 
on the right half of the 'S' 
plane.
Before diving into the Routh-Hurwitz Criterion, it's crucial to understand the three fundamental 
stability states of a system. The location of the characteristic equation's roots in the 'S' plane 
determines whether a system is stable, marginally stable, or unstable.
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Routh-Hurwitz Criterion
Understanding System Stability
Stable System
All roots of the 
characteristic equation lie 
on the left half of the 'S' 
plane.
Marginally Stable 
System
All roots of the 
characteristic equation lie 
on the imaginary axis of 
the 'S' plane.
Unstable System
All roots of the 
characteristic equation lie 
on the right half of the 'S' 
plane.
Before diving into the Routh-Hurwitz Criterion, it's crucial to understand the three fundamental 
stability states of a system. The location of the characteristic equation's roots in the 'S' plane 
determines whether a system is stable, marginally stable, or unstable.
Routh-Hurwitz Criterion: Statement and Conditions
Statement
A system is stable if and only if 
all the elements in the first 
column of the Routh array have 
the same sign. The number of 
sign changes equals the 
number of roots with positive 
real parts.
Necessary Conditions
All coefficients of the 
characteristic equation must 
have the same sign, and there 
should be no missing terms.
Important Note
Meeting the necessary 
conditions doesn't guarantee 
stability. The Routh-Hurwitz 
Criterion must be applied to 
confirm stability.
This criterion, developed by Alfred Hurwitz and Edward Routh, provides a systematic approach to determine 
system stability without solving the characteristic equation directly. If the necessary conditions aren't met, the 
system is immediately classified as unstable.
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Routh-Hurwitz Criterion
Understanding System Stability
Stable System
All roots of the 
characteristic equation lie 
on the left half of the 'S' 
plane.
Marginally Stable 
System
All roots of the 
characteristic equation lie 
on the imaginary axis of 
the 'S' plane.
Unstable System
All roots of the 
characteristic equation lie 
on the right half of the 'S' 
plane.
Before diving into the Routh-Hurwitz Criterion, it's crucial to understand the three fundamental 
stability states of a system. The location of the characteristic equation's roots in the 'S' plane 
determines whether a system is stable, marginally stable, or unstable.
Routh-Hurwitz Criterion: Statement and Conditions
Statement
A system is stable if and only if 
all the elements in the first 
column of the Routh array have 
the same sign. The number of 
sign changes equals the 
number of roots with positive 
real parts.
Necessary Conditions
All coefficients of the 
characteristic equation must 
have the same sign, and there 
should be no missing terms.
Important Note
Meeting the necessary 
conditions doesn't guarantee 
stability. The Routh-Hurwitz 
Criterion must be applied to 
confirm stability.
This criterion, developed by Alfred Hurwitz and Edward Routh, provides a systematic approach to determine 
system stability without solving the characteristic equation directly. If the necessary conditions aren't met, the 
system is immediately classified as unstable.
Advantages and Limitations
Advantages
Determines stability without solving 
equations
Easily determines relative stability
Can determine the range of K for stability
Identifies root locus intersection points 
with the imaginary axis
Limitations
Only applicable for linear systems
Doesn't provide exact pole locations
Valid only for characteristic equations with 
real coefficients
The Routh-Hurwitz Criterion helps determine control system stability without complex math. 
However, it's limited to certain systems and doesn't pinpoint pole locations.
Page 5


Routh-Hurwitz Criterion
Understanding System Stability
Stable System
All roots of the 
characteristic equation lie 
on the left half of the 'S' 
plane.
Marginally Stable 
System
All roots of the 
characteristic equation lie 
on the imaginary axis of 
the 'S' plane.
Unstable System
All roots of the 
characteristic equation lie 
on the right half of the 'S' 
plane.
Before diving into the Routh-Hurwitz Criterion, it's crucial to understand the three fundamental 
stability states of a system. The location of the characteristic equation's roots in the 'S' plane 
determines whether a system is stable, marginally stable, or unstable.
Routh-Hurwitz Criterion: Statement and Conditions
Statement
A system is stable if and only if 
all the elements in the first 
column of the Routh array have 
the same sign. The number of 
sign changes equals the 
number of roots with positive 
real parts.
Necessary Conditions
All coefficients of the 
characteristic equation must 
have the same sign, and there 
should be no missing terms.
Important Note
Meeting the necessary 
conditions doesn't guarantee 
stability. The Routh-Hurwitz 
Criterion must be applied to 
confirm stability.
This criterion, developed by Alfred Hurwitz and Edward Routh, provides a systematic approach to determine 
system stability without solving the characteristic equation directly. If the necessary conditions aren't met, the 
system is immediately classified as unstable.
Advantages and Limitations
Advantages
Determines stability without solving 
equations
Easily determines relative stability
Can determine the range of K for stability
Identifies root locus intersection points 
with the imaginary axis
Limitations
Only applicable for linear systems
Doesn't provide exact pole locations
Valid only for characteristic equations with 
real coefficients
The Routh-Hurwitz Criterion helps determine control system stability without complex math. 
However, it's limited to certain systems and doesn't pinpoint pole locations.
Procedure 
Step 1: Arrange all the coefficients of the above equation in two rows:
Step 2: From these two rows we will form the third row: Step 3:
Step 4: We shall continue this procedure of forming a new rows.