The Routh–Hurwitz stability criterion is a method used to determine the stability of a linear time-invariant (LTI) system without explicitly calculating the roots of the characteristic equation. It helps determine the number of poles located in the left half-plane (LHP), right half-plane (RHP), and on the imaginary axis.
For a system to be stable, all poles of the characteristic equation must lie in the left half of the s-plane.
Table of Contents
Given Closed-Loop Transfer Function
\[
T(s)=\frac{200}{s^4+6s^3+11s^2+6s+200}
\]
The characteristic equation is obtained from the denominator:
\[
s^4+6s^3+11s^2+6s+200=0
\]
Step 1: Construct the Routh Array
The first two rows are formed using the coefficients of the characteristic equation.
| Power of s | Coefficients |
|---|---|
| \(s^4\) | 1 11 200 |
| \(s^3\) | 6 6 0 |
Step 2: Compute Row \(s^2\)
\[
a = \frac{(6)(11) – (1)(6)}{6}
\]
\[
a = \frac{66 – 6}{6} = 10
\]
Second element:
\[
b = \frac{(6)(200) – (1)(0)}{6}
\]
\[
b = 200
\]
| Power | Elements |
|---|---|
| \(s^2\) | 10 200 |
Step 3: Compute Row \(s^1\)
\[
\frac{10\times6 – 6\times200}{10}
\]
\[
= \frac{60 – 1200}{10}
\]
\[
= -114
\]
| Power | Elements |
|---|---|
| \(s^1\) | -114 |
Step 4: Row \(s^0\)
| Power | Element |
|---|---|
| \(s^0\) | 200 |
Complete Routh Array
| Power | First Column |
|---|---|
| \(s^4\) | 1 |
| \(s^3\) | 6 |
| \(s^2\) | 10 |
| \(s^1\) | -114 |
| \(s^0\) | 200 |
Step 5: Check Sign Changes
First column:
\[
1,\;6,\;10,\;-114,\;200
\]
Sign pattern:
\[
+,\; +,\; +,\; -,\; +
\]
Number of sign changes = 2.
Interpretation
- Right Half Plane poles = 2
- Left Half Plane poles = 2
- Imaginary axis poles = 0
System Stability
Since there are poles in the right half-plane, the system is unstable with two right-half plane poles and two left-half plane poles.