Post on 02-Feb-2021
Linear System StabilityChapter 4
Contents
Introduction
System Stability
Routh-Hurwitz Criterion
Construction of Routh Table
Determining System Stability
Routh-Hurwitz Criterion (Special Cases)
1. Zero only in First Column
2. Zero for Entire Row
Stability via Routh Hurwitz
Introduction
Stability is the most important system specification. If a system is unstable, the transient response and steady-state errors are in a moot point.
Definition of stability, for linear, time-invariant system by using natural response:
A system is stable if the natural response approaches zero as time approaches infinity.
A system is unstable if the natural response approaches infinity as time approaches infinity
A system is marginally stable if the natural response neither decays nor grows but remains constant or oscillates.
)()()( tctctc naturalforced
Introduction
Definition of stability using the total response bounded-input,
bounded-output (BIBO):
i. A system is stable if every bounded input yields a bounded output.
ii. A system is unstable if any bounded input yields an unbounded
output.
Absolute & Relative Stability
Absolute Stability:
i. The absolute stability indicates whether the system is stable or not.
ii. This is indicated by the presence of one or more poles in RHP.
Relative Stability:
i. Relative stability refers to the degree of stability of a stable system
described by above.
ii. This depends on the transfer function of the system, which is
represented by both the numerator (that yields the zeros) and
denominator (that yields the poles).
iii. This can then be referred to in the study of system response either in
time or frequency domain.
System Stability
Stable systems have closed-loop transfer functions with poles only
in the left half-plane.
System Stability
Unstable systems have closed-loop transfer functions with at least
one pole in the right half-plane and/or poles of multiplicity
greater than 1 on the imaginary axis.
System Stability
Marginally stable systems have closed-loop transfer functions with
only imaginary axis poles of multiplicity 1 and poles on the
imaginary axis.
Determining System Stability
To determine stability of a given system, we have to consider the
manner in which the system is operating, whether open-loop or
closed-loop.
i. If the system is operating in open-loop, first find the closed loop
transfer function.
ii. Find the closed-loop poles.
iii. If the order of the system is 2 or less, factorise the denominator of
the transfer function. This will provide the roots of the polynomial, or
the closed-loop poles of the system.
iv. If the system order is higher than 2nd-order, use construct Routh
table and apply Routh-Hurwitz Criterion.
v. Any poles that exist on the RHP will indicate that the system is
unstable.
Routh–Hurwitz Criterion
Routh-Hurwitz Stability Criterion:
The number of roots of the polynomial that are in the right half-
plane is equal to the number of changes in the first column.
Systems with the transfer function having all poles in the LHP is
stable.
Hence, we can conclude that a system is stable if there is no change
of sign in the first column of its Routh table.
Routh–Hurwitz Criterion
If a polynomial is given by:
The necessary conditions for stability are:
i. All the coefficients of the polynomial are of the same sign. If not, there are poles on the right hand side of the s-plane.
ii. All the coefficient should exist accept for a0.
0.....)( 011
1
asasasasTn
nn
n
Where,
an, an-1, …, a1, a0 are constants
n = 1, 2, 3,…, ∞
Routh-Hurwitz Criterion
For the sufficient condition, we must form a Routh-array.
1
321
1
31
2
1
n
nnnn
n
nn
nn
a
aaaa
a
aa
aa
b
1
541
1
51
4
2
n
nnnn
n
nn
nn
a
aaaa
a
aa
aa
b
Routh-Hurwitz Criterion
For the sufficient condition, we must formed a Routh-array.
1
2131
1
21
31
1b
baab
b
bb
aa
c nn
nn
1
3151
1
31
51
2b
baab
b
bb
aa
c nn
nn
1
2121
1
21
21
1i
ihhi
i
ii
hh
j
21 ik
Construction of Routh Table
Equivalent closed-loop transfer function
Initial layout for Routh table
Completed Routh table
Determining System Stability
Example: How many roots exist on RHP?
Routh–Hurwitz Criterion (Special Cases)
However, special cases exists when:
1. There exists zero only in the first column.
2. The entire row is zero.
1.Zero only in First Column
If the first element of a row is zero, division by zero would be
required to form the next row.
To avoid this, an epsilon, , is assigned to replace the zero in the first column.
Example: Consider the following closed-loop transfer function T(s).
35632
10)(
2345
ssssssT
Zero only in First Column
To determine the system stability, sign changes were observed
after substituting with a very small positive number or
alternatively a very small negative number.
Zero Only in First Column
Exercise: For the following closed-loop transfer function T(s), determine the number of poles that exist on RHP.
123232
1)(
2345
ssssssT
2. Zero for Entire Row
An entire row of zeros will appear in the Routh table when a
purely even or purely odd polynomial is a factor of the original
polynomial.
Example: s4 + 5s2 + 7 has an even powers of s.
Even polynomials have roots that are symmetrical about the origin.
i. Roots are symmetrical & real
ii. Roots are symmetrical & imaginary
iii. Roots are quadrantal
Zero for Entire Row
Zero for Entire Row
Example:
Differentiate with respect to s:
5684267
10)(
2345
ssssssT
86)( 24 sssP
0124)( 3 ss
ds
sdP
Zero for Entire Row
Example:How many poles are on RHP, LHP and jω-axis for the closed-loop system below?
Zero for Entire Row
Exercise: For the following closed-loop transfer function T(s), determine the number of poles that exist on RHP, LHP and the jω-axis
20384859392212
20)(
2345678
sssssssssT
Stability Via Routh-Hurwitz
Example: Find the range of gain K for the system below that will
cause the system to be stable, unstable and marginally stable,
Assume K > 0.
Closed-loop transfer function:
Ksss
KsT
7718)(
23
Stability Via Routh-Hurwitz
Example: Find the range of gain K for the system below that will
cause the system to be stable, unstable and marginally stable,
Assume K > 0.
Forming the Routh table:
Stability Via Routh-Hurwitz
Example: Find the range of gain K for the system below that will
cause the system to be stable, unstable and marginally stable,
Assume K > 0.
If K < 1386:
All the terms in 1st column will be positive and since there are no
sign changes, the system will have 3 poles in the left-half plane
and are stable.
If K > 1386:
The s1 in the first column is negative. There are 2 sign changes,
indicating that the system has two right-half-plane poles and one
left-half plane pole, which make the system unstable.
Stability Via Routh-Hurwitz
Example: Find the range of gain K for the system below that will
cause the system to be stable, unstable and marginally stable,
Assume K > 0.
If K = 1386:
The entire row of zeros, which signify the existence of jω poles. Returning to the s2 row and replacing K with 1386, so we have:
P(s)=18s2 +1386