Module 9
Routh’s Method,Root Locus,
Magnitude and Phase Equations
(4 hours)
9.1 Routh’s Stability Criterion
About Stability,( P145,Section 1)
9.1.1 Define on the Stability of Closed-loop System:
When the transfer function of system is Ф(s),the output
is:
)(
)(
)(
)(
)()(1
)(
)()()(
1
sR
ss
sKM
sR
sHsG
sG
sRsΦsC
i
n
i
Suppose
1)(),()( sRttr
n
i i
i
i
n
i
ss
c
ss
sKM
sC
1
1
)(
)(
)(
ts
n
i
i
iecsCLtC?
1
1 )]([)(
)(0,0Re tesif tsi i
The system is stable;
)(,0Re tet h e nsanyif tsj j
ts
n
i
i
iectC
1
)(
Then total response
The system is unstable.
Conclusion
The system is stable only when all the closed-
loop poles are located in the left-hand half of the
s complex plane (LHP);
The system become unstable as soon as one
closed-loop pole is located in the right-hand half
of the s complex plane (RHP).
跨越华盛顿州 Puget Sound
的 塔 科 马 峡 谷 的 首 座 大桥 ——开始晃动时灾难发生时
9.2 Algebra Stability Criterion
(代数稳定判据）
And its characteristic roots are
nsss?,,21
On the basis of the relationships between roots
and coefficients of the equation,we know that:
i
n
i
n
n
n
kji
kji
kji
n
ji
ji
ji
n
i
i
s
a
a
sss
a
a
ss
a
a
s
a
a
1
0
1,,
0
3
1,
0
2
1
0
1
)1(
must all be
If we request that these roots s1-sn
have all negative real-part,those
values
00
2
0
1,,aaaaaa n?
positive,and must be no zero.
Otherwise there is one positive
real-part root at least.
The necessary condition of
stable system.
The Sufficient and Necessary Condition on
Stable System —— The Routh’s Method
(系统稳定的充要条件） ( P146 )
Original data
Calculated data
Routh Criterion (A) – The sufficient
and necessary condition is,
All the data of the first column of
Routh’s array must be positive.
(B)
06K
5.7430,0304 KK
0?K
5.7?K
SP,The open-loop transfer function is
)1(
)1(')(
2?
Tss
sKsG?
Determine the range of value of K,τ,T for which
the system is stable.
Solution,The characteristic equation is:
0)1()1(2 sKTss? 023 KsKsTs?
Ks
KTKs
Ks
KTs
0
1
2
3
0
1
0
0
KTK
K
Ka n dT 0,?
9.3 Theorems about Routh Criterion
Theorem 1,Division of row.
The coefficients of any row may be multiplied or divided
by a positive number without changing the signs of the first
column,The labor of evaluating the coefficients in Routh’s
array can be reduced by multiplying or dividing any row by
a constant,This may result,for example,in reducing the
size of the coefficients and therefore simplifying the
evaluation of the remaining coefficients.
020125923 23456 ssssss
SP.
When the first term in a row is zero but not all the other
terms are zero,the following methods can be used:
Substitute s = 1 / x in the original equation,then solve for
the roots of x with positive real parts,The number of
roots x with positive real parts will be the same as the
number of s roots with positive real parts.
Multiply the original polynomial by the factor (s +1),
which introduces an additional negative root,Then form
the Routh’s array for the new polynomial.
Substitute a small variable ε (ε >0,and ε 0) for this
zero element.
Theorem 2,A zero coefficient in the column.
There are two roots with positive real parts.
When all the coefficients of one row are zero,the
procedure is as follows:
The auxiliary equation can be formed from the preceding
row,as shown below.
The Routh’s array can be completed by replacing the all-
zero row with the coefficients obtained by differentiating
the auxiliary equation.
The roots of the auxiliary equation are also roots of the
original equation,These roots occur in pair and are the
negative of each other,Therefore,these roots may be
imaginary (complex conjugates) or real (one positive and
one negative),may lie in quadruplets (two pairs of
complex-conjugate roots),etc.
Theorem 3,A zero row.
In fact,all the roots of closed-loop system are:
11;11;3;3 4321 jsjsjsjs
Routh Criterion A criterion for determining the
stability of a system by examining the characteristic
equation of the transfer function,The criterion states that
the number of roots of the characteristic equation with
positive real parts is equal to the number of changes of
sign of the coefficients in the first column of the Routh’s
array.
Stability A performance measure of a system,A system
is stable if all the poles of the transfer function have
negative real parts.
Stable System A dynamic system with a bounded
system response to a bounded input.
Concept and Term
Problem 1
Problem 2
9.4.1 Definition of Root Locus
–– is a plot of the roots of characteristic equation
of the closed-loop system as a function of the
gain of the open-loop transfer function,
9.4 Root Locus Method:
Magnitude and Phase Equation
)()(1
)(
)(
)(
sHsG
sG
sR
sC
0)()(1 sHsG 1)()(sHsG
1
)(
)(
)()(
1
1
i
n
i
j
m
j
ps
zsK
sHsG )( Kfs
SP,A second-order system:
22
2
2 22)(
)(;1)(;
)2()( nn
n
ssKss
K
sR
sCsH
ss
KsG
Ks 112,1
Location of
roots for the
characteristic
equation:
K S1 S2
0 0 -2
0.5 -0.3 -1.7
0.75 -0.5 -1.5
1.0 -1.0 -1.0
2.0 -1.0+j1.0 -1.0-j1.0
3.0 -1.0+j1.4 -1.0-j1.4
50.0 -1.0+j7.0 -1.0-j7.0;t a n;;; tc o n sdn
The root locus is a vertical
line for K > Ka,
Analysis:
9.4.2 Magnitude and Phase
Equation (P148)
Note:
When plotting the root locus,only the argument
(phase) equation is the sufficient and necessary
condition (because s is vary with K ),The
magnitude equation can be used only when a
root si corresponding some K is needed to be
determined.
Instructional objectives:
At the end of this lecture students
should be able to
Determine the stability of a system using
Routh’s criterion
Plot approximate root locus
Routh’s Method,Root Locus,
Magnitude and Phase Equations
(4 hours)
9.1 Routh’s Stability Criterion
About Stability,( P145,Section 1)
9.1.1 Define on the Stability of Closed-loop System:
When the transfer function of system is Ф(s),the output
is:
)(
)(
)(
)(
)()(1
)(
)()()(
1
sR
ss
sKM
sR
sHsG
sG
sRsΦsC
i
n
i
Suppose
1)(),()( sRttr
n
i i
i
i
n
i
ss
c
ss
sKM
sC
1
1
)(
)(
)(
ts
n
i
i
iecsCLtC?
1
1 )]([)(
)(0,0Re tesif tsi i
The system is stable;
)(,0Re tet h e nsanyif tsj j
ts
n
i
i
iectC
1
)(
Then total response
The system is unstable.
Conclusion
The system is stable only when all the closed-
loop poles are located in the left-hand half of the
s complex plane (LHP);
The system become unstable as soon as one
closed-loop pole is located in the right-hand half
of the s complex plane (RHP).
跨越华盛顿州 Puget Sound
的 塔 科 马 峡 谷 的 首 座 大桥 ——开始晃动时灾难发生时
9.2 Algebra Stability Criterion
(代数稳定判据）
And its characteristic roots are
nsss?,,21
On the basis of the relationships between roots
and coefficients of the equation,we know that:
i
n
i
n
n
n
kji
kji
kji
n
ji
ji
ji
n
i
i
s
a
a
sss
a
a
ss
a
a
s
a
a
1
0
1,,
0
3
1,
0
2
1
0
1
)1(
must all be
If we request that these roots s1-sn
have all negative real-part,those
values
00
2
0
1,,aaaaaa n?
positive,and must be no zero.
Otherwise there is one positive
real-part root at least.
The necessary condition of
stable system.
The Sufficient and Necessary Condition on
Stable System —— The Routh’s Method
(系统稳定的充要条件） ( P146 )
Original data
Calculated data
Routh Criterion (A) – The sufficient
and necessary condition is,
All the data of the first column of
Routh’s array must be positive.
(B)
06K
5.7430,0304 KK
0?K
5.7?K
SP,The open-loop transfer function is
)1(
)1(')(
2?
Tss
sKsG?
Determine the range of value of K,τ,T for which
the system is stable.
Solution,The characteristic equation is:
0)1()1(2 sKTss? 023 KsKsTs?
Ks
KTKs
Ks
KTs
0
1
2
3
0
1
0
0
KTK
K
Ka n dT 0,?
9.3 Theorems about Routh Criterion
Theorem 1,Division of row.
The coefficients of any row may be multiplied or divided
by a positive number without changing the signs of the first
column,The labor of evaluating the coefficients in Routh’s
array can be reduced by multiplying or dividing any row by
a constant,This may result,for example,in reducing the
size of the coefficients and therefore simplifying the
evaluation of the remaining coefficients.
020125923 23456 ssssss
SP.
When the first term in a row is zero but not all the other
terms are zero,the following methods can be used:
Substitute s = 1 / x in the original equation,then solve for
the roots of x with positive real parts,The number of
roots x with positive real parts will be the same as the
number of s roots with positive real parts.
Multiply the original polynomial by the factor (s +1),
which introduces an additional negative root,Then form
the Routh’s array for the new polynomial.
Substitute a small variable ε (ε >0,and ε 0) for this
zero element.
Theorem 2,A zero coefficient in the column.
There are two roots with positive real parts.
When all the coefficients of one row are zero,the
procedure is as follows:
The auxiliary equation can be formed from the preceding
row,as shown below.
The Routh’s array can be completed by replacing the all-
zero row with the coefficients obtained by differentiating
the auxiliary equation.
The roots of the auxiliary equation are also roots of the
original equation,These roots occur in pair and are the
negative of each other,Therefore,these roots may be
imaginary (complex conjugates) or real (one positive and
one negative),may lie in quadruplets (two pairs of
complex-conjugate roots),etc.
Theorem 3,A zero row.
In fact,all the roots of closed-loop system are:
11;11;3;3 4321 jsjsjsjs
Routh Criterion A criterion for determining the
stability of a system by examining the characteristic
equation of the transfer function,The criterion states that
the number of roots of the characteristic equation with
positive real parts is equal to the number of changes of
sign of the coefficients in the first column of the Routh’s
array.
Stability A performance measure of a system,A system
is stable if all the poles of the transfer function have
negative real parts.
Stable System A dynamic system with a bounded
system response to a bounded input.
Concept and Term
Problem 1
Problem 2
9.4.1 Definition of Root Locus
–– is a plot of the roots of characteristic equation
of the closed-loop system as a function of the
gain of the open-loop transfer function,
9.4 Root Locus Method:
Magnitude and Phase Equation
)()(1
)(
)(
)(
sHsG
sG
sR
sC
0)()(1 sHsG 1)()(sHsG
1
)(
)(
)()(
1
1
i
n
i
j
m
j
ps
zsK
sHsG )( Kfs
SP,A second-order system:
22
2
2 22)(
)(;1)(;
)2()( nn
n
ssKss
K
sR
sCsH
ss
KsG
Ks 112,1
Location of
roots for the
characteristic
equation:
K S1 S2
0 0 -2
0.5 -0.3 -1.7
0.75 -0.5 -1.5
1.0 -1.0 -1.0
2.0 -1.0+j1.0 -1.0-j1.0
3.0 -1.0+j1.4 -1.0-j1.4
50.0 -1.0+j7.0 -1.0-j7.0;t a n;;; tc o n sdn
The root locus is a vertical
line for K > Ka,
Analysis:
9.4.2 Magnitude and Phase
Equation (P148)
Note:
When plotting the root locus,only the argument
(phase) equation is the sufficient and necessary
condition (because s is vary with K ),The
magnitude equation can be used only when a
root si corresponding some K is needed to be
determined.
Instructional objectives:
At the end of this lecture students
should be able to
Determine the stability of a system using
Routh’s criterion
Plot approximate root locus
