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

 Page 1


Routh-Hurwitz Stabilit y
The Routh-Hurwitz stabilit y criterion is a mathematical to ol used in con trol systems and signal
pro cessing to determine the stabilit y of a linear time-in v arian t (L TI) system without explicitly computing
the ro ots of its c haracteristic p olynomial. It is widely applied to assess whether a system’s resp onse
remains b ounded o v er time.
1. In tro duction to Routh-Hurwitz Stabilit y
A system is stable if all the ro ots of its c haracteristic p olynomial lie in the left half of the complex plane
(for con tin uous-time systems) or inside the unit circle (for discrete-time systems). The Routh-Hurwitz
criterion pro vides a tabular metho d to c hec k stabilit y b y analyzing the p olynomial co e?icien ts, a v oiding
the need to solv e for ro ots directly .
2. Characteristic P olynomial
F or a con tin uous-time L TI system, the c haracteristic p olynomial is deriv ed from the system’s transfer
function denominator or state-space mo del, t ypically expressed as:
P(s) = a
n
s
n
+a
n-1
s
n-1
+···+a
1
s+a
0
where a
n
?= 0 is the leading co e?icien t, and n is the p olynomial degree. The system is stable if all ro ots
of P(s) = 0 ha v e negativ e real parts.
3. Ro uth-Hurwitz Criterion
The Routh-Hurwitz criterion constructs a Routh arra y to test stabilit y . The arra y is formed as follo ws:
• Step 1 : Arrange the p olynomial co e?icien ts in t w o ro ws for s
n
and s
n-1
:
s
n
a
n
a
n-2
a
n-4
···
s
n-1
a
n-1
a
n-3
a
n-5
···
• Step 2 : Compute subsequen t ro ws using the form ula for ro w s
k
:
b
i
=
a
k+1,1
a
k,i+1
-a
k,1
a
k+1,i+1
a
k+1,1
where a
k,j
is the eleme n t in ro w s
k
, column j . Con tin ue un til the r o w for s
0
.
• Step 3 : Analyze the first column of the Routh arra y . The system is stable if all elemen ts in the
first column ha v e the same sign (t ypically p ositiv e if a
n
> 0 ).
The n um b er of sign c hanges in the first column equals the n um b er of ro ots with p ositiv e real parts
(indicating instabilit y).
4. Exa mple
Consider the p olynomial:
P(s) = s
3
+3s
2
+3s+1
1
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Routh-Hurwitz Stabilit y
The Routh-Hurwitz stabilit y criterion is a mathematical to ol used in con trol systems and signal
pro cessing to determine the stabilit y of a linear time-in v arian t (L TI) system without explicitly computing
the ro ots of its c haracteristic p olynomial. It is widely applied to assess whether a system’s resp onse
remains b ounded o v er time.
1. In tro duction to Routh-Hurwitz Stabilit y
A system is stable if all the ro ots of its c haracteristic p olynomial lie in the left half of the complex plane
(for con tin uous-time systems) or inside the unit circle (for discrete-time systems). The Routh-Hurwitz
criterion pro vides a tabular metho d to c hec k stabilit y b y analyzing the p olynomial co e?icien ts, a v oiding
the need to solv e for ro ots directly .
2. Characteristic P olynomial
F or a con tin uous-time L TI system, the c haracteristic p olynomial is deriv ed from the system’s transfer
function denominator or state-space mo del, t ypically expressed as:
P(s) = a
n
s
n
+a
n-1
s
n-1
+···+a
1
s+a
0
where a
n
?= 0 is the leading co e?icien t, and n is the p olynomial degree. The system is stable if all ro ots
of P(s) = 0 ha v e negativ e real parts.
3. Ro uth-Hurwitz Criterion
The Routh-Hurwitz criterion constructs a Routh arra y to test stabilit y . The arra y is formed as follo ws:
• Step 1 : Arrange the p olynomial co e?icien ts in t w o ro ws for s
n
and s
n-1
:
s
n
a
n
a
n-2
a
n-4
···
s
n-1
a
n-1
a
n-3
a
n-5
···
• Step 2 : Compute subsequen t ro ws using the form ula for ro w s
k
:
b
i
=
a
k+1,1
a
k,i+1
-a
k,1
a
k+1,i+1
a
k+1,1
where a
k,j
is the eleme n t in ro w s
k
, column j . Con tin ue un til the r o w for s
0
.
• Step 3 : Analyze the first column of the Routh arra y . The system is stable if all elemen ts in the
first column ha v e the same sign (t ypically p ositiv e if a
n
> 0 ).
The n um b er of sign c hanges in the first column equals the n um b er of ro ots with p ositiv e real parts
(indicating instabilit y).
4. Exa mple
Consider the p olynomial:
P(s) = s
3
+3s
2
+3s+1
1
Construct the Routh arra y:
s
3
1 3 0
s
2
3 1 0
s
1 3·3-1·1
3
=
8
3
0 0
s
0
8
3
·1-3·0
8
3
= 1 0 0
First column: [1,3,
8
3
,1] . All elemen ts are p ositiv e, so the system is stable (no ro ots with p ositiv e real
parts).
5. Sp e cial Cases
• Zero in First Column : If an elemen t in the first column is zero but the ro w is non-zero, replace
the zero with a small p ositiv e n um b er ? and pro ceed, taking the limit as ? ? 0 to c hec k sign
c hanges.
• En tire Ro w Zero : Indicates p ossible ro ots on the imaginary axis or rep eated ro ots. F orm an
auxiliary p olynomial from the previous ro w:
A(s) = a
k,1
s
k
+a
k,2
s
k-2
+···
Differen tiate A(s) to obtain co e?icien ts for the next ro w, then con tin ue. Ro ots of A(s) = 0 lie on
the imaginary axis, indicating marginal stabilit y if no sign c hanges o ccur.
6. Appli cations
The Routh-Hurwitz criterion is used in:
• Con trol System Design : T o ensure stabilit y of feedbac k systems (e.g., PID con trollers).
• Signal Pro cessing : T o analyze stabilit y of digital filters.
• A erospace and Rob otics : T o v erify stabilit y of dynamic systems.
• Electrical Circuits : T o assess s tabilit y of amplifiers and oscillators.
7. Pr actical Considerations
• Co e?icien t Sensitivit y : Small c hanges in p olynomial co e?icien ts can affect stabilit y , requiring
careful design.
• Numerical Precision : Computations m ust b e accurate to a v oid errors in the Routh arra y , esp e-
cially for high-degree p olynomials.
• Marginal Stabilit y : Systems with ro ots on the imaginary axis ma y oscillate, requiring additional
analysis (e.g., Nyquist criterion).
• Discrete-Time Systems : F or z-domain p olynomials, stabilit y requires ro ots inside the unit circle,
but the Routh-Hurwitz criterion can b e adapted via bilinear transformation.
2
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Routh-Hurwitz Stabilit y
The Routh-Hurwitz stabilit y criterion is a mathematical to ol used in con trol systems and signal
pro cessing to determine the stabilit y of a linear time-in v arian t (L TI) system without explicitly computing
the ro ots of its c haracteristic p olynomial. It is widely applied to assess whether a system’s resp onse
remains b ounded o v er time.
1. In tro duction to Routh-Hurwitz Stabilit y
A system is stable if all the ro ots of its c haracteristic p olynomial lie in the left half of the complex plane
(for con tin uous-time systems) or inside the unit circle (for discrete-time systems). The Routh-Hurwitz
criterion pro vides a tabular metho d to c hec k stabilit y b y analyzing the p olynomial co e?icien ts, a v oiding
the need to solv e for ro ots directly .
2. Characteristic P olynomial
F or a con tin uous-time L TI system, the c haracteristic p olynomial is deriv ed from the system’s transfer
function denominator or state-space mo del, t ypically expressed as:
P(s) = a
n
s
n
+a
n-1
s
n-1
+···+a
1
s+a
0
where a
n
?= 0 is the leading co e?icien t, and n is the p olynomial degree. The system is stable if all ro ots
of P(s) = 0 ha v e negativ e real parts.
3. Ro uth-Hurwitz Criterion
The Routh-Hurwitz criterion constructs a Routh arra y to test stabilit y . The arra y is formed as follo ws:
• Step 1 : Arrange the p olynomial co e?icien ts in t w o ro ws for s
n
and s
n-1
:
s
n
a
n
a
n-2
a
n-4
···
s
n-1
a
n-1
a
n-3
a
n-5
···
• Step 2 : Compute subsequen t ro ws using the form ula for ro w s
k
:
b
i
=
a
k+1,1
a
k,i+1
-a
k,1
a
k+1,i+1
a
k+1,1
where a
k,j
is the eleme n t in ro w s
k
, column j . Con tin ue un til the r o w for s
0
.
• Step 3 : Analyze the first column of the Routh arra y . The system is stable if all elemen ts in the
first column ha v e the same sign (t ypically p ositiv e if a
n
> 0 ).
The n um b er of sign c hanges in the first column equals the n um b er of ro ots with p ositiv e real parts
(indicating instabilit y).
4. Exa mple
Consider the p olynomial:
P(s) = s
3
+3s
2
+3s+1
1
Construct the Routh arra y:
s
3
1 3 0
s
2
3 1 0
s
1 3·3-1·1
3
=
8
3
0 0
s
0
8
3
·1-3·0
8
3
= 1 0 0
First column: [1,3,
8
3
,1] . All elemen ts are p ositiv e, so the system is stable (no ro ots with p ositiv e real
parts).
5. Sp e cial Cases
• Zero in First Column : If an elemen t in the first column is zero but the ro w is non-zero, replace
the zero with a small p ositiv e n um b er ? and pro ceed, taking the limit as ? ? 0 to c hec k sign
c hanges.
• En tire Ro w Zero : Indicates p ossible ro ots on the imaginary axis or rep eated ro ots. F orm an
auxiliary p olynomial from the previous ro w:
A(s) = a
k,1
s
k
+a
k,2
s
k-2
+···
Differen tiate A(s) to obtain co e?icien ts for the next ro w, then con tin ue. Ro ots of A(s) = 0 lie on
the imaginary axis, indicating marginal stabilit y if no sign c hanges o ccur.
6. Appli cations
The Routh-Hurwitz criterion is used in:
• Con trol System Design : T o ensure stabilit y of feedbac k systems (e.g., PID con trollers).
• Signal Pro cessing : T o analyze stabilit y of digital filters.
• A erospace and Rob otics : T o v erify stabilit y of dynamic systems.
• Electrical Circuits : T o assess s tabilit y of amplifiers and oscillators.
7. Pr actical Considerations
• Co e?icien t Sensitivit y : Small c hanges in p olynomial co e?icien ts can affect stabilit y , requiring
careful design.
• Numerical Precision : Computations m ust b e accurate to a v oid errors in the Routh arra y , esp e-
cially for high-degree p olynomials.
• Marginal Stabilit y : Systems with ro ots on the imaginary axis ma y oscillate, requiring additional
analysis (e.g., Nyquist criterion).
• Discrete-Time Systems : F or z-domain p olynomials, stabilit y requires ro ots inside the unit circle,
but the Routh-Hurwitz criterion can b e adapted via bilinear transformation.
2
8. Limitations
• Only determines stabilit y , not the exact lo cation of ro ots.
• Less practical for v ery high-degree p olynomials due to computational complexit y .
• Sp ecial cases (zero ro ws or elemen ts) require careful handling to a v oid misin terpretation.
9. Conclusion
The Routh-Hurwitz stabilit y criterion is a p o w erful and e?icien t metho d for assessing the stabilit y of
L TI systems without solving for p olynomial ro ots. By constructing a Routh arra y and analyzing its first
column, engineers can quic kly determine whether a system is stable, marginally stable, or unstable. Its
simplicit y and applicabilit y mak e it indisp ensable in con trol systems and signal pro cessing design.
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