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
Routh’s Stability Criterion - ECE 680 Modern Automatic Control
Explanation:
Let's take 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 and bi 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 plane.
Algorithm for applying Routh’s stability criterion:
The algorithm requires the order of A(s) to be finite.
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.
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 as:
sn = a0 a2 a4 a6 · · ·
sn−1 = a1 a3 a5 a7 · · ·
sn−2 = b1 b2 b3 b4 · · ·
sn−3 = c1 c2 c3 c4 · · ·
sn−4 = d1 d2 d3 d4 · · ·
........
s2 = e1 e2
s1 = f1
s0 = g0
where the coefficients bi are b1 = a1a2 − a0a3.a1(4)
b2 = a1a4 − a0a5. a1(5)
b3 = a1a6 − a0a7. a1(6)
....
The Routh’s Stability Criterion will be 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)