Routh-Hurwitz Table Calculator

Determine system stability by analyzing the characteristic polynomial.

Routh-Hurwitz Table

The generated Routh Array. The first column is used to determine stability.

First Column Values Chart

A visual representation of the first column values to highlight sign changes.

What is the Routh-Hurwitz Table Calculator?

A Routh-Hurwitz Table Calculator is an engineering tool used to determine the stability of a linear time-invariant (LTI) system without having to compute the roots of its characteristic equation. The Routh-Hurwitz stability criterion is a mathematical test that provides a necessary and sufficient condition for system stability by examining the coefficients of the system’s characteristic polynomial. The “table,” more formally known as the Routh Array, is a systematic arrangement of these coefficients. The primary insight comes from analyzing the first column of this array: if there are no sign changes in the first column, all the roots of the polynomial have negative real parts, and the system is stable.

This calculator is essential for control systems engineers, students, and researchers who need to quickly assess system stability. Manually creating the Routh array can be tedious and prone to errors, especially for higher-order systems or when dealing with special cases. This routh hurwitz table calculator automates the process, handling both standard and special cases, such as zeros in the first column or an entire row of zeros.

The Routh-Hurwitz Formula and Explanation

The Routh-Hurwitz criterion doesn’t involve a single formula but a recursive algorithm to construct the Routh Array. Given a characteristic polynomial of degree ‘n’:

ansn + an-1sn-1 + ... + a1s + a0 = 0

The array construction begins by populating the first two rows with the polynomial’s coefficients:

• Row 1 (sⁿ): Consists of the first, third, fifth, … coefficients (a_n, a_n-2, a_n-4, …).
• Row 2 (s^n-1): Consists of the second, fourth, sixth, … coefficients (a_n-1, a_n-3, a_n-5, …).

Subsequent rows are calculated using determinants from the two preceding rows. For instance, the first element of the third row (b₁) is calculated as:

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

This process continues until all rows down to s⁰ are completed. The stability is then determined by the number of sign changes in the first column of the generated table.

Variables Table

Variable	Meaning	Unit	Typical Range
an, an-1, …, a0	Coefficients of the characteristic polynomial.	Unitless	Any real number. For stability, a necessary (but not sufficient) condition is that all coefficients have the same sign.
Sign Changes	The number of times the sign (+/-) changes when reading down the first column of the Routh Array.	Count	An integer from 0 to n.

A great resource for understanding stability analysis can be found at our Pole-Zero Plot Analyzer page.

Practical Examples

Example 1: A Stable System

Consider the characteristic equation: s3 + 6s2 + 11s + 6 = 0.

• Inputs: 1, 6, 11, 6
• Routh Table:
  
  s³: 1, 11
  
  s²: 6, 6
  
  s¹: 10
  
  s⁰: 6
• Result: The first column is. There are no sign changes. Therefore, the system is **stable**.

Example 2: An Unstable System

Consider the characteristic equation: s4 + s3 + 2s2 + 10s + 8 = 0.

• Inputs: 1, 1, 2, 10, 8
• Routh Table:
  
  s⁴: 1, 2, 8
  
  s³: 1, 10
  
  s²: -8, 8
  
  s¹: 11
  
  s⁰: 8
• Result: The first column is [1, 1, -8, 11, 8]. There are two sign changes (from 1 to -8, and from -8 to 11). This indicates there are **two roots in the right-half plane**, and the system is **unstable**.

For more on system behavior, check out the Bode Plot Calculator.

How to Use This Routh-Hurwitz Table Calculator

1. Identify Coefficients: Start with the characteristic equation of your control system. List the coefficients in order, from the highest power of ‘s’ down to the constant term.
2. Enter Coefficients: Type the coefficients into the input field, separated by commas. For example, for s³ + 2s² + 3s + 4, you would enter 1,2,3,4.
3. Calculate: Click the “Calculate Stability” button to generate the Routh Array.
4. Interpret Results: The calculator will immediately tell you if the system is stable or unstable. It will also state the number of right-half plane roots based on the sign changes in the first column of the table.
5. Analyze the Table and Chart: Review the generated table to understand the calculation. The chart provides a quick visual reference of the first column, making it easy to spot the sign changes that lead to instability.

Key Factors That Affect Routh-Hurwitz Analysis

• Missing Terms: If any coefficient (other than a₀) is zero or negative, and not all coefficients are negative, the system is unstable or at best marginally stable. This is a quick preliminary check.
• Zero in the First Column: This is a special case. The calculator handles this by substituting the zero with a small positive number (epsilon, ε) to continue the calculation, which reveals the underlying stability.
• A Row of Zeros: This indicates the presence of roots on the imaginary axis (marginally stable) or symmetrically located roots. The calculator manages this by using an auxiliary polynomial derived from the row above the zero row.
• Polynomial Order: Higher-order systems lead to larger Routh arrays, increasing the complexity and chance of manual error. This is where a routh hurwitz table calculator becomes invaluable.
• Coefficient Magnitude: The relative magnitudes of the coefficients determine the values in the array and, consequently, the presence of any sign changes.
• Numerical Precision: In manual calculations, floating-point errors can accumulate. The calculator uses precise floating-point arithmetic to ensure accurate results. You can find related information at the Nyquist Plot Calculator.

Frequently Asked Questions (FAQ)

What does it mean if a system is stable?

A stable system is one whose output remains bounded (does not go to infinity) in response to a bounded input. In terms of the characteristic equation, it means all its roots (poles) lie in the left-half of the complex s-plane.

What does a sign change in the first column mean?

The number of sign changes in the first column of the Routh array is exactly equal to the number of roots of the polynomial that have a positive real part (i.e., the number of poles in the right-half of the s-plane).

What if the first element of a row is zero?

This is a special case. The standard procedure, which this calculator implements, is to replace the zero with a very small positive number (epsilon) and proceed with the calculations. The sign of the elements in the first column is then evaluated as epsilon approaches zero.

What if an entire row becomes zero?

This indicates that the polynomial has roots that are symmetric about the origin, such as roots on the imaginary axis (e.g., ±jω) or real roots in opposite locations (e.g., ±σ). The system is not strictly stable. The calculator forms an auxiliary polynomial from the preceding row to complete the analysis. You might find our Root Locus Plotter tool useful here.

Can this calculator handle symbolic coefficients (e.g., ‘K’)?

This routh hurwitz table calculator is designed for numerical coefficients. For stability analysis with a variable gain ‘K’, you would typically solve the inequalities derived from the first column by hand to find the range of K for stability.

Why are all my coefficients positive, but the system is unstable?

Having all positive coefficients is a necessary condition for stability, but it is not sufficient for systems of order 3 or higher. An unstable system can still have all positive coefficients, which is why the full Routh-Hurwitz test is required.

Are the coefficient values unitless?

Yes, the coefficients of the characteristic polynomial are treated as dimensionless or unitless numbers for the purpose of the Routh-Hurwitz analysis.

Where can I learn about the underlying math?

Control system theory textbooks provide in-depth derivations. You can also visit our PID Controller Tuning Guide for related control theory concepts.