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

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Control S ystems F ormula Sheet:
Routh-Hurwitz Stability
1. Routh-Hurwitz Stability Criterion
• Purpose : Determines the stability of a linear time-invariant system b y analyz-
ing the char acteristic equation without solving for roots.
• Char acteristic Equation : F or a system with tr ansfer function denominator:
a
n
s
n
+a
n-1
s
n-1
+···+a
1
s+a
0
=0
where a
n
?= 0 is the coefficient of the highest power of s .
• Necessary Condition : All coefficients a
n
,a
n-1
,...,a
0
must be non-zero and have
the same sign (all positive or all negative) for stability .
• Sufficient Condition : All elements in the first column of the Routh arr a y must
be positive for the system to be stable.
2. Routh Arr a y Construction
• Gener al F orm of Char acteristic Equation :
a
n
s
n
+a
n-1
s
n-1
+a
n-2
s
n-2
+···+a
1
s+a
0
=0
• Routh Arr a y Structure : F or ann -th order polynomial, the arr a y is constructed
as follows:
s
n
a
n
a
n-2
a
n-4
···
s
n-1
a
n-1
a
n-3
a
n-5
···
s
n-2
b
1
b
2
b
3
···
s
n-3
c
1
c
2
c
3
···
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
s
0
Last row
• Calculation of Elements :
b
1
=
a
n-1
a
n-2
-a
n
a
n-3
a
n-1
, b
2
=
a
n-1
a
n-4
-a
n
a
n-5
a
n-1
, ...
c
1
=
b
1
a
n-3
-a
n-1
b
2
b
1
, c
2
=
b
1
a
n-5
-a
n-1
b
3
b
1
, ...
1
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Control S ystems F ormula Sheet:
Routh-Hurwitz Stability
1. Routh-Hurwitz Stability Criterion
• Purpose : Determines the stability of a linear time-invariant system b y analyz-
ing the char acteristic equation without solving for roots.
• Char acteristic Equation : F or a system with tr ansfer function denominator:
a
n
s
n
+a
n-1
s
n-1
+···+a
1
s+a
0
=0
where a
n
?= 0 is the coefficient of the highest power of s .
• Necessary Condition : All coefficients a
n
,a
n-1
,...,a
0
must be non-zero and have
the same sign (all positive or all negative) for stability .
• Sufficient Condition : All elements in the first column of the Routh arr a y must
be positive for the system to be stable.
2. Routh Arr a y Construction
• Gener al F orm of Char acteristic Equation :
a
n
s
n
+a
n-1
s
n-1
+a
n-2
s
n-2
+···+a
1
s+a
0
=0
• Routh Arr a y Structure : F or ann -th order polynomial, the arr a y is constructed
as follows:
s
n
a
n
a
n-2
a
n-4
···
s
n-1
a
n-1
a
n-3
a
n-5
···
s
n-2
b
1
b
2
b
3
···
s
n-3
c
1
c
2
c
3
···
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
s
0
Last row
• Calculation of Elements :
b
1
=
a
n-1
a
n-2
-a
n
a
n-3
a
n-1
, b
2
=
a
n-1
a
n-4
-a
n
a
n-5
a
n-1
, ...
c
1
=
b
1
a
n-3
-a
n-1
b
2
b
1
, c
2
=
b
1
a
n-5
-a
n-1
b
3
b
1
, ...
1
• Stability Condition : S ystem is stable if all elements in the first column ( a
n
,a
n-1
,b
1
,c
1
,... )
are positive.
• Number of Unstable Poles : Number of sign changes in the first column of the
Routh arr a y equals the number of poles in the right half-plane (RHP).
3. Special Cases in Routh Arr a y
• Zero in First Column:
– Replace the zero with a small positive number ? and proceed with arr a y
construction.
– Analyze the first column as ?? 0
+
to determine sign c hanges.
– Indicates potential poles on the imaginary axis (margin al stability).
• Entire Row of Zeros :
– Indicates a symmetric polynomial or repeated po les on the imaginary axis.
– F orm the auxiliary polynomial from the row above the zero ro w:
A(s) = (coefficient of s
k
)s
k
+ (coefficient of s
k-2
)s
k-2
+···
– T ak e the derivative of the auxiliary polynomial:
dA(s)
ds
– Use coefficients of
dA(s)
ds
to replace the zero row and continue arr a y con-
struction.
– A uxiliary polynomial roots lie on the imaginary axis, indicating oscillatory
behavior (marginal stability).
4. Applications
• Stability Analysis : Determine if all roots of the char acteristic equation have
negative real parts (stable system).
• Range of Par ameter for Stability : F or systems with a par ameter (e.g., gain
K ), solve for the r ange of K such that all first-column elements ar e positive.
• Marginal Stability : Identify conditions where poles lie on the imaginary axis
(zeros in first column or entire row of zeros).
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