Relative Stability Analysis - Control Systems - Electrical Engineering (EE)

Routh-Hurwitz Stability Criterion

The Routh-Hurwitz stability criterion is a method to determine the stability of a linear time-invariant control system from its characteristic polynomial without explicitly calculating the roots. The criterion provides a necessary condition and a sufficient condition for stability.

Necessary condition

The necessary condition for stability is that all coefficients of the characteristic polynomial must have the same sign (commonly taken as positive). If this condition is violated, the system is certainly unstable.

Consider a characteristic polynomial of order n:

Note that the polynomial should not have missing terms (i.e., coefficients equal to zero) when directly applying the standard Routh procedure; special handling is required if terms are missing.

Sufficient condition

The sufficient condition for stability using the Routh array is that all elements of the first column of the Routh array must have the same sign (all positive or all negative). The number of sign changes in the first column equals the number of characteristic-equation roots in the right half of the s-plane. If there are zero sign changes, all roots lie in the left half and the system is stable.

Routh Array Method

As polynomial order increases, finding roots explicitly becomes impractical. The Routh array gives a tabular procedure to determine how many roots have positive real parts by examining sign changes in the first column.

Procedure to form the Routh array:

• Write the coefficients of the characteristic polynomial across the first two rows: the first row contains coefficients of sⁿ, s^(n-2), s^(n-4), ...; the second row contains coefficients of s^(n-1), s^(n-3), s^(n-5), ...
• Compute subsequent rows using determinants formed from elements of the two rows immediately above, continuing until the s⁰ (or s¹) row is obtained.
• Count sign changes in the first column. Each sign change corresponds to one root in the right half of the s-plane.

Note - If all elements of a row share a common factor, the row may be divided by that factor to simplify arithmetic; this does not change sign pattern or stability conclusions.

The general Routh array for an nth-order polynomial appears as follows:

Interpretation

If the first column contains:

• no sign changes - the system is stable (all roots in left half).
• one or more sign changes - the number of sign changes equals the number of right-half roots (system unstable).

Worked Example 1

Determine the stability of the system whose characteristic equation is

s⁴ + 3s³ + 3s² + 2s + 1 = 0

Solution (stepwise):

Verify the necessary condition by inspecting coefficients.

The coefficients are 1, 3, 3, 2, 1; all are positive.

Form the Routh array for the polynomial. The first two rows are filled from the coefficients and remaining rows computed accordingly.

Inspect the first column of the completed Routh array.

All entries in the first column are positive, so there are no sign changes.

Conclusion: The system is stable (no roots lie in the right half of the s-plane).

Special Cases of the Routh Array

Two special situations may complicate construction of the Routh array:

• The first element of a row becomes zero while other elements in that row are non-zero.
• All elements of a row become zero.

Case 1 - First element of a row is zero

When only the first element of a row is zero but other elements in that row are non-zero, replace the zero with a small positive quantity ε and continue forming the array. After completing the table, examine the sign changes in the first column as ε → 0+. The limiting sign changes give the correct count of right-half roots.

Worked Example 2 (first element zero)

Characteristic equation:

s⁴ + 2s³ + s² + 2s + 1 = 0

Solution (stepwise):

Verify the necessary condition by checking coefficients.

The coefficients 1, 2, 1, 2, 1 are all positive.

Form the Routh array. If a row has a common factor, simplify that row by dividing through (this does not change sign pattern).

During construction, the first element of the s^2 row becomes zero while other elements in that row are non-zero. Replace that zero with ε and continue computing lower rows using algebraic expressions in terms of ε.

Let ε → 0+ and examine the first column sign sequence.

There are two sign changes in the first column as ε → 0+. Therefore there are two roots in the right half of the s-plane and the system is unstable.

Case 2 - Entire row of zeros

If all elements of a row become zero, the corresponding row represents symmetric roots about the origin (pure imaginary roots or reciprocal pairs). To proceed:

• Form the auxiliary polynomial A(s) using the row just above the zero row; A(s) contains only even or only odd powers of s (matching the order of that row).
• Differentiate A(s) with respect to s to obtain A'(s). Replace the zero row by the coefficients of A'(s) and continue the Routh procedure.

Worked Example 3 (row of zeros)

Characteristic equation:

s⁵ + 3s⁴ + s³ + 3s² + s + 3 = 0

Solution (stepwise):

Verify the necessary condition by checking coefficients.

All coefficients 1, 3, 1, 3, 1, 3 are positive.

Form the Routh array; simplify rows if a common factor exists.

If all elements of a row become zero (in this example the s^3 row), write the auxiliary polynomial A(s) from the row above the zero row (the s^4 row). For this example:

A(s) = s⁴ + s² + 1

Differentiate A(s) with respect to s:

dA(s)/ds = 4s³ + 2s

Place the coefficients of dA(s)/ds into the row that was all zeros and continue building the Routh array.

Inspect the first column after completing the array. There are two sign changes in the first column; hence the system has two right-half roots and is unstable.

Practical Notes and Applications

• The Routh method is widely used for preliminary stability checks in control design because it quickly determines the number of unstable poles without root-finding.
• When using numerical tools, rounding errors may produce small zeros; treat such occurrences using ε or by symbolic simplification where possible.
• Routh analysis also helps identify marginal stability (pure imaginary roots) and multiplicity of roots on the imaginary axis through the auxiliary polynomial procedure.
• Keep algebraic simplification stepwise and avoid cancelling sign information when dividing rows by common factors.

Summary

The Routh-Hurwitz criterion provides an efficient tabular method to count right-half zeros of the characteristic polynomial. Verify the necessary condition (all coefficients same sign) first, then construct the Routh array and count sign changes in the first column. Handle special cases-first element zero and full row zero-by using ε substitution and auxiliary polynomials respectively. The result gives a direct stability verdict without computing roots explicitly.