The Routh Table Calculator is a specialized tool used in control system engineering to determine the stability of a linear time-invariant (LTI) system by analyzing the coefficients of its characteristic equation. This calculator automatically generates the Routh Array and identifies the number of roots in the right-half of the s-plane.
Routh Table Calculator
Example: 1, 10, 31, 1030 (for s³ + 10s² + 31s + 1030)
Routh Table Calculator Formula
The Routh Array is constructed as follows:
For $a_n s^n + a_{n-1} s^{n-1} + … + a_0 = 0$:
sⁿ | aₙ aₙ₋₂ aₙ₋₄ ...
sⁿ⁻¹ | aₙ₋₁ aₙ₋₃ aₙ₋₅ ...
sⁿ⁻² | b₁ b₂ b₃ ...
Where $b_1 = \frac{a_{n-1} \cdot a_{n-2} – a_n \cdot a_{n-3}}{a_{n-1}}$
Source: Wikipedia – Routh-Hurwitz Stability | Control Tutorials (UMich)
Variables:
- Coefficients ($a_n$): Real numbers representing the constants in front of each power of $s$.
- s-plane: The complex plane where the stability is analyzed.
- First Column: The critical column used to determine system stability.
What is a Routh Table Calculator?
A Routh Table Calculator helps engineers determine the stability of a control system without solving for the actual roots of the characteristic equation. By creating a mathematical array (the Routh Array), it checks if any roots of the polynomial lie in the right-half of the complex s-plane (RHP).
According to the Routh-Hurwitz criterion, a system is stable if and only if all coefficients of the characteristic equation are positive and there are no sign changes in the first column of the Routh Array.
How to Calculate Routh Table (Example)
Example: $s^3 + 2s^2 + 4s + 2 = 0$
- List coefficients: $a_3=1, a_2=2, a_1=4, a_0=2$.
- Row 1 ($s^3$): 1, 4.
- Row 2 ($s^2$): 2, 2.
- Calculate Row 3 ($s^1$): $b_1 = ((2 \times 4) – (1 \times 2)) / 2 = 3$.
- Calculate Row 4 ($s^0$): $c_1 = ((3 \times 2) – (2 \times 0)) / 3 = 2$.
- Check first column: 1, 2, 3, 2. No sign changes → System is Stable.
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Frequently Asked Questions (FAQ)
What does a sign change in the first column mean?
The number of sign changes in the first column equals the number of roots in the right-half s-plane, indicating an unstable system.
What if a zero appears in the first column?
If a zero appears, we replace it with a small positive value $\epsilon$ and continue the calculation to check for sign changes.
Can it handle complex coefficients?
Standard Routh-Hurwitz criterion applies to polynomials with real coefficients.
Is this relevant for discrete systems?
No, for discrete systems, the Jury Stability Criterion or Bilinear Transformation is typically used.