A comprehensive guide to understanding and using the Routh-Hurwitz Stability Criterion Calculator
The Routh-Hurwitz stability criterion is a fundamental mathematical test used in control systems engineering to determine the stability of linear time-invariant (LTI) systems without explicitly solving for the roots of the characteristic polynomial. Developed independently by Edward John Routh and Adolf Hurwitz in the late 19th century, this criterion provides a systematic method to assess whether all poles of a system lie in the left half of the complex plane, which is the necessary and sufficient condition for stability.
Consider a linear system with a characteristic polynomial of the form:
The system is stable if and only if all roots of D(s) have negative real parts. Rather than solving this polynomial directly (which becomes computationally intensive for high-order systems), the Routh-Hurwitz criterion examines the coefficients through a systematic tabular method.
For a system to be stable, two conditions must be satisfied:
The Routh table is constructed as follows:
The algorithm handles several mathematical edge cases that can occur during the construction of the Routh table:
When the first element of a row becomes zero (but other elements are non-zero), we replace the zero with a small positive number ε (epsilon method). This allows the algorithm to continue and provides meaningful results about the system's stability.
When an entire row becomes zero, it indicates the presence of an auxiliary polynomial. We construct this auxiliary polynomial from the previous row and use its derivative to replace the zero row. This case often occurs in systems with poles on the imaginary axis or symmetric pole placement.
This calculator is implemented in JavaScript and uses a modular approach to handle the various aspects of the Routh-Hurwitz analysis:
computeRH(coefficients) - Main algorithm that constructs the Routh tablecheckStability(matrix) - Analyzes the first column for sign changesmakeRHMatrix(coefficients) - Initializes the table with input coefficientsisEntireRowZero(row) - Detects zero row special casederivativeRow(row) - Handles auxiliary polynomial derivativeThe algorithm begins by creating a matrix with dimensions (n+1) × ⌈(n+1)/2⌉, where n is the order of the polynomial. The first two rows are populated with alternating coefficients from the characteristic polynomial.
Each element in subsequent rows is calculated using a 2×2 determinant operation. The implementation uses the standard formula:
The implementation includes robust handling for mathematical edge cases:
The interface dynamically generates input fields based on the selected system order, provides real-time validation, and displays results with comprehensive error handling. The matrix visualization includes row labels and formatted numerical output for clarity.
Consider the characteristic polynomial: s³ + 6s² + 11s + 6
All elements in the first column [1, 6, 9, 6] are positive, indicating a stable system. This polynomial has poles at s = -1, -2, -3.
Consider: s⁵ + s⁴ - s - 1
This example demonstrates both the zero row special case and sign changes in the first column, indicating instability.
The calculator includes pre-loaded examples that demonstrate:
The reliability of this calculator has been thoroughly validated through a comprehensive test suite that includes 22 different test cases covering various scenarios:
The implementation has been cross-validated against MATLAB's Control System Toolbox to ensure accuracy. Key validation points include:
While the implementation is robust, users should be aware of the following limitations:
routh functionThis calculator was originally developed as a learning tool for ELEC 341 - Systems and Controls. It demonstrates the practical application of classical control theory and provides students with an interactive way to understand the Routh-Hurwitz stability criterion.