Routh Array Calculator

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Expert Review by David Chen, PE • Control Systems Specialist
Verified for mathematical accuracy and stability analysis standards.

The Routh-Hurwitz stability criterion is a mathematical test that is a necessary and sufficient condition for the stability of a linear time-invariant (LTI) control system. Use our Routh Array Calculator to quickly determine system stability by analyzing the coefficients of your characteristic equation.

Routh Array Calculator

Separate by commas: $a_n, a_{n-1}, \dots, a_0$
System Stability Result
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Routh Array Calculator Formula

The Routh-Hurwitz criterion determines stability without solving the differential equation. For a polynomial:

P(s) = ansn + an-1sn-1 + … + a1s + a0

Elements in the third row ($b_i$) are calculated as:

b1 = ( (an-1 * an-2) – (an * an-3) ) / an-1

Formula Sources: Wolfram MathWorld | Wikipedia

Variables:

  • an: The coefficient of the highest power of $s$.
  • Coefficients: All real numbers in the characteristic equation.
  • Sign Changes: The number of sign changes in the first column equals the number of roots in the Right-Half Plane (RHP).

Related Calculators:

What is Routh Array Calculator?

A Routh Array Calculator is a specialized engineering tool used in control theory to determine the stability of a system based on its characteristic polynomial. By organizing coefficients into a tabular format known as the Routh Table, engineers can check if any roots of the system’s characteristic equation have positive real parts.

If any sign change occurs in the first column of the generated Routh Table, the system is considered unstable. This tool is essential for students and professionals working with feedback control systems, aerospace engineering, and robotics.

How to Calculate Routh Array (Example)

Consider the polynomial: $s^3 + 2s^2 + 3s + 10 = 0$

  1. List coefficients: $a_3=1, a_2=2, a_1=3, a_0=10$.
  2. Row 1 ($s^3$): $a_3, a_1 \rightarrow 1, 3$.
  3. Row 2 ($s^2$): $a_2, a_0 \rightarrow 2, 10$.
  4. Row 3 ($s^1$): Calculated as $( (2*3) – (1*10) ) / 2 = -2$.
  5. Row 4 ($s^0$): Calculated as $( (-2*10) – (2*0) ) / -2 = 10$.
  6. Check column 1: $1, 2, -2, 10$. There are 2 sign changes ($2 \rightarrow -2$ and $-2 \rightarrow 10$). The system is unstable.

Frequently Asked Questions (FAQ)

What does a zero in the first column mean? If a zero appears, it indicates a marginal stability or a specific root configuration. A small value $\epsilon$ is typically used to continue the calculation.

Can I use this for non-linear systems? No, the Routh-Hurwitz criterion is strictly for Linear Time-Invariant (LTI) systems defined by polynomial characteristic equations.

Is the system stable if there are no sign changes? Yes, if all elements in the first column are of the same sign (and non-zero), all roots are in the Left-Half Plane (LHP), meaning the system is stable.

What happens if a whole row is zero? This indicates roots that are symmetric about the origin (e.g., purely imaginary roots). You must use an auxiliary polynomial to proceed.

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