Determine the stability of your linear time-invariant (LTI) system instantly using our Routh Criterion Calculator. Simply input your characteristic equation coefficients to analyze closed-loop stability without finding actual roots.
Routh Criterion Calculator
Routh Criterion Calculator Formula
$b_1 = \frac{a_{n-1}a_{n-2} – a_n a_{n-3}}{a_{n-1}}$
The Routh-Hurwitz stability criterion is a mathematical test that is a necessary and sufficient condition for the stability of a linear time-invariant control system. Source: Wikipedia – Routh Criterion | Source: Wolfram MathWorld
Variables:
- Coefficients ($a_n, a_{n-1}, …$): The numerical constants of the characteristic equation $D(s)$.
- $n$: The degree of the polynomial.
- First Column: The critical column used to determine sign changes.
What is Routh Criterion Calculator?
A Routh Criterion Calculator is a specialized tool used by control engineers to determine if a system is stable without solving for the roots of the characteristic equation. By constructing a “Routh Table,” it identifies if any roots reside in the Right-Half Plane (RHP).
If all elements in the first column of the Routh table have the same sign, the system is stable. If there are sign changes, the number of sign changes equals the number of roots in the RHP, indicating an unstable system.
How to Calculate Routh Criterion (Example)
Example: $s^3 + 10s^2 + 31s + 30 = 0$
- Step 1: List coefficients: $a_3=1, a_2=10, a_1=31, a_0=30$.
- Step 2: Arrange Row 1 ($s^3$): 1, 31.
- Step 3: Arrange Row 2 ($s^2$): 10, 30.
- Step 4: Calculate $b_1$ for Row 3 ($s^1$): $(10 \times 31 – 1 \times 30) / 10 = 28$.
- Step 5: Calculate Row 4 ($s^0$): $(28 \times 30 – 10 \times 0) / 28 = 30$.
- Result: First column [1, 10, 28, 30]. No sign changes = Stable.