Q1: Determine the stability of a closed loop control system whose characteristic equation is S6+s5+3s4+3s3+2s2+s+1=0. Investigate the stability using Routh Hurwitz criterion. Write a C/C++ program to determine the stability
Answers
Answer:ECE 680 Modern Automatic Control Routh’s Stability Criterion June 13, 2007 1
ROUTH’S STABILITY CRITERION
Consider a closed-loop transfer function
H(s) = b0s
m + b1s
m−1 + · · · + bm−1s + bm
a0s
n + a1s
n−1 + · · · + an−1s + an
=
B(s)
A(s)
(1)
where the ai
’s and bi
’s are real constants and m ≤ n. An alternative to factoring the
denominator polynomial, Routh’s stability criterion, determines the number of closedloop poles in the right-half s plane.
Algorithm for applying Routh’s stability criterion
The algorithm described below, like the stability criterion, requires the order of A(s) to
be finite.
1. Factor out any roots at the origin to obtain the polynomial, and multiply by −1 if
necessary, to obtain
a0s
n + a1s
n−1 + · · · + an−1s + an = 0 (2)
where a0 6= 0 and an > 0.
2. If the order of the resulting polynomial is at least two and any coefficient ai
is zero
or negative, the polynomial has at least one root with nonnegative real part. To
obtain the precise number of roots with nonnegative real part, proceed as follows.
Arrange the coefficients of the polynomial, and values subsequently calculated from
them as shown below:
s
n a0 a2 a4 a6 · · ·
s
n−1 a1 a3 a5 a7 · · ·
s
n−2
b1 b2 b3 b4 · · ·
s
n−3
c1 c2 c3 c4 · · ·
s
n−4 d1 d2 d3 d4 · · ·
.
.
.
.
.
.
.
.
.
s
2
e1 e2
s
1
f1
s
0
g0
(3)
where the coefficients bi are
b1 =
a1a2 − a0a3
a1
(4)
b2 =
a1a4 − a0a5
a1
(5)
b3 =
a1a6 − a0a7
a1
(6)
.
.
.
ECE 680 Modern Automatic Control Routh’s Stability Criterion June 13, 2007 2
generated until all subsequent coefficients are zero. Similarly, cross multiply the
coefficients of the two previous rows to obtain the ci
, di
, etc.
c1 =
b1a3 − a1b2
b1
(7)
c2 =
b1a5 − a1b3
b1
(8)
c3 =
b1a7 − a1b4
b1
(9)
.
.
.
d1 =
c1b2 − b1c2
c1
(10)
d2 =
c1b3 − b1c3
c1
(11)
.
Explanation: