Routh–Hurwitz Criterion Calculator

Mathematical background

Consider a real-coefficient polynomial of degree n with positive leading term

with all coefficients real and a_n > 0. The polynomial is called Hurwitz stable if all its roots satisfy Re(s) < 0. In control systems, these roots are the poles of the closed-loop transfer function, and Hurwitz stability corresponds to an asymptotically stable response to perturbations.

The Routh–Hurwitz criterion gives necessary and sufficient conditions for Hurwitz stability using only the coefficients a_0, a_1, …, a_n. One standard form of the test is the Routh table, an array whose first column encodes the number of roots in the right half-plane. The number of sign changes in that column equals the number of roots with positive real part.

How the Routh table is constructed

For a polynomial of degree n, the Routh table has n + 1 rows. Denote the entries in row i and column j by r[i, j], starting from row 0 at the top for convenience.

Top two rows

The first row contains coefficients of the highest power in steps of 2, starting at a_n:

Row 0: [a_n, a_{n-2}, a_{n-4}, …]

The second row begins with a_{n-1} and also steps down by 2 in the power of s:

Row 1: [a_{n-1}, a_{n-3}, a_{n-5}, …]

If the degree is odd or even, some of these trailing coefficients may be absent; missing entries are treated as zeros when constructing the table.

Recursive formula for lower rows

For row indices i ≥ 2, entries are computed from the two rows above using a determinant-like formula. Let r[i, 0] be the first element of row i. For column index j ≥ 0, the general recurrence used by this calculator is:

r[i, j] = ( r[i-1, 0] * r[i-2, j+1] - r[i-2, 0] * r[i-1, j+1] ) / r[i-1, 0]

This is algebraically equivalent to the standard Routh construction and yields the same sequence in the first column, which is the part used for stability decisions.

Handling zero first-column entries

A practical difficulty arises if r[i-1, 0] = 0, because that value appears in the denominator. In analytic treatments, this is addressed by replacing the zero with a small positive parameter ε and taking limits. Numerically, the usual workaround is:

• If a first-column pivot is exactly zero, replace it by a small positive number ε.
• Compute the rest of the row using the recurrence.

This calculator uses a fixed value ε = 10−6 whenever it encounters a zero pivot in the first column. This avoids division by zero and keeps the sign pattern meaningful for most engineering problems, though it slightly perturbs borderline cases.

Interpreting the results

Once the Routh table is constructed, focus on the first column. Let that column be

[r[0,0], r[1,0], r[2,0], …, r[n,0]].

Count the number of sign changes when moving from top to bottom. Each sign change corresponds to one root of the polynomial with positive real part. For example:

• No sign changes → no right-half-plane roots → all roots are in the left half-plane or on the imaginary axis.
• One sign change → one root (or a conjugate pair plus compensating left-half-plane roots) in the right half-plane.
• k sign changes → exactly k roots with Re(s) > 0.

For standard control applications, the main classification is:

• Asymptotically stable: no sign changes and no zero entries in the first column.
• Unstable: at least one sign change, meaning at least one pole in the right half-plane.
• Borderline / marginal: rows of zeros or exact zeros in the first column; additional analysis is needed (typically auxiliary polynomials) to determine repeated or purely imaginary roots.

Worked example

Consider the polynomial

P(s) = s^4 + 3s^3 + 3s^2 + 3s + 1.

The coefficients, from highest degree to constant term, are

[1, 3, 3, 3, 1].

Enter these as

1,3,3,3,1

in the calculator input field.

The Routh table begins with the top two rows:

• Row 0 (powers s^4, s^2, s^0): [1, 3, 1]
• Row 1 (powers s^3, s^1): [3, 3, 0]

Using the recurrence, the lower rows are calculated step by step. The first column of the completed table is

[1, 3, 2, 1, 1].

All entries are positive, so there are no sign changes. Therefore, the polynomial has no roots in the right half-plane and the corresponding LTI system is asymptotically stable.

If instead you enter coefficients

1,0,-2,0,1

corresponding to

P(s) = s^4 - 2s^2 + 1,

the constructed Routh table yields a first column with two sign changes. The calculator reports that there are two roots in the right half-plane, so the polynomial (and associated system) is unstable.

Summary of the algorithm

1. Read the coefficients. Ensure they are ordered from the highest power of s down to the constant term, and that the leading coefficient is positive. If necessary, multiply the polynomial by −1 to make the leading coefficient positive.
2. Form the first two rows. Place a_n, a_{n-2}, a_{n-4}, … in the first row, and a_{n-1}, a_{n-3}, a_{n-5}, … in the second row. Treat missing coefficients as zeros.
3. Compute subsequent rows. Use the recursive formula for r[i, j] involving the two rows above, replacing any zero first-column pivot by ε = 10−6 before dividing.
4. Handle special patterns. If an entire row becomes zero, construct the corresponding auxiliary polynomial from the row above and differentiate it to build the replacement row (standard Routh–Hurwitz procedure).
5. Count sign changes. Inspect the first column of the completed table. The number of sign changes from top to bottom equals the number of roots of P(s) in the right half-plane. Use this to classify the polynomial as stable, unstable, or borderline.

Comparison with direct root finding

In practice, engineers often combine both approaches: the Routh–Hurwitz test quickly indicates whether a tentative design is stable, and then numerical root finding or simulation refines the understanding of transient and frequency-domain behavior.

Assumptions and limitations

• Real coefficients only. The method, and this calculator, assume that all polynomial coefficients are real numbers. Polynomials with complex coefficients require more general techniques.
• Positive leading coefficient. The classic formulation assumes a_n > 0. If your polynomial has a negative leading coefficient, multiply all coefficients by −1 before entering them. This does not change the roots but simplifies the sign interpretation.
• Linear time-invariant systems. The stability interpretation corresponds to LTI systems. For nonlinear, time-varying, or discrete-time systems, additional or different tools are required.
• Epsilon substitution. When a first-column pivot is zero, this calculator replaces it by ε = 10−6 to avoid division by zero. For genuinely marginally stable systems with roots on the imaginary axis, this small perturbation can slightly alter the sign pattern. Use the results as a guide and, if the system appears borderline, confirm using an independent method (e.g., direct root computation or simulation).
• Rows of zeros. If an entire row becomes zero, the underlying theory calls for forming an auxiliary polynomial from the row above and differentiating it to continue the construction. The calculator follows the standard procedure, but such cases signal repeated or imaginary-axis roots, so conclusions should be interpreted with care.
• Input validation. The tool expects a comma-separated list of real numbers, highest power first (for example, 1, 5, 6 for s^2 + 5s + 6). Extra commas, missing coefficients, or non-numeric text may lead to errors or to coefficients being treated as zero.
• Not a complete design review. The Routh–Hurwitz criterion is an important step in control design, but it does not replace a full analysis of performance, robustness, or implementation details. Use it alongside root locus, Bode plots, simulation, and other control system stability tools.