Routh-Hurwitz Criterion

Determining whether a closed-loop system is stable without computing the exact pole locations is a significant practical problem, especially for high-order polynomials. The Routh-Hurwitz criterion provides a systematic algebraic procedure to determine the number of closed-loop poles in the right half of the s-plane using a structured tabular computation called the Routh array. It is both a necessary and sufficient condition for stability of linear time-invariant systems.

Core Concept Explanation

The Routh-Hurwitz criterion tests whether the characteristic polynomial of a closed-loop system has any roots in the right half plane (or on the imaginary axis) without computing the roots directly. The characteristic polynomial is obtained from the denominator of the closed-loop transfer function set equal to zero. For a system with denominator D(s) = ansⁿ + an-1sⁿ⁻¹ + ... + a1s + a0, the Routh array is constructed from the coefficients of this polynomial.

Before constructing the array, a necessary (but not sufficient) condition must be checked: all coefficients of the characteristic polynomial must be present and must have the same algebraic sign. If any coefficient is zero or if signs are mixed, the system has at least one pole in the right half plane or on the imaginary axis, and the system is not stable. This necessary condition alone can be used as a quick check in examination problems.

Routh Array Construction

The Routh array is a rectangular table with n+1 rows for an nth-order polynomial. The first two rows are filled directly from the polynomial coefficients by separating alternately into even-indexed and odd-indexed terms. The first row contains the coefficients of the even-powered terms (sⁿ, sⁿ⁻², ...) and the second row contains the coefficients of odd-powered terms (sⁿ⁻¹, sⁿ⁻³, ...).

Each subsequent row is computed using a 2×2 determinant formula. For row 3 element in column j, the formula is:

Using the notation where row 1 has elements r1c1, r1c2, ... and row 2 has elements r2c1, r2c2, ..., the element b1 = (r2c1 × r1c2 - r1c1 × r2c2) / r2c1. Similarly b2 = (r2c1 × r1c3 - r1c1 × r2c3) / r2c1. This pattern continues until the row becomes all zeros or has only one element. The process terminates at the s⁰ row.

Stability Determination Rule

The Routh-Hurwitz stability theorem states that the number of roots of the characteristic polynomial in the right half plane (including the imaginary axis) equals the number of sign changes in the first column of the completed Routh array. The system is BIBO stable if and only if there are no sign changes, meaning all elements of the first column are positive (or all negative if the leading coefficient is negative).

This criterion is both necessary and sufficient for the stability of continuous-time LTI systems. Any sign change in the first column immediately indicates an unstable pole, and counting the sign changes gives the exact number of right-half-plane poles. This information is directly usable in GATE numerical questions asking for the range of gain K for stability.

Mathematical Expression

For the general third-order characteristic polynomial D(s) = a3s³ + a2s² + a1s + a0, the Routh array has four rows. The key determinant element is b1 = (a2·a1 - a3·a0) / a2. For stability, the conditions reduce to: all coefficients positive and a1·a2 > a0·a3. This cross-product inequality is a standard GATE shortcut for third-order systems.

Practical Understanding

The Routh-Hurwitz criterion is particularly useful when a system has a variable gain K in the open-loop transfer function. Substituting the closed-loop characteristic polynomial in terms of K and applying the Routh array, the designer can find the range of K values for which all first-column elements are positive. This range defines the stable operating region and is called the range of stability.

The marginal stability condition, where the system is on the boundary between stable and unstable, occurs when the first-column element of a particular row becomes exactly zero. At this value of K, the system has poles exactly on the imaginary axis. The frequency of these imaginary axis poles can be found using the auxiliary polynomial formed from the row just above the zero row.

Given:
Characteristic polynomial D(s) = s⁴ + 2s³ + 3s² + 4s + 5
Coefficients: a4=1, a3=2, a2=3, a1=4, a0=5

Why this formula applies:
Routh array determines number of RHP poles from sign changes in first column.

Routh Array Construction:

  s⁴ |  1     3     5
  s³ |  2     4     0
  s² |  b1    b2
  s¹ |  c1
  s⁰ |  5

b1 = (2×3 - 1×4)/2 = (6-4)/2 = 1
b2 = (2×5 - 1×0)/2 = 10/2 = 5
c1 = (1×4 - 2×5)/1 = (4-10)/1 = -6

Final Array first column: 1, 2, 1, -6, 5

Calculation:
Sign changes: 1→2 (no), 2→1 (no), 1→-6 (YES), -6→5 (YES)
Number of sign changes = 2

Final Answer:
System has 2 poles in the right half plane. System is UNSTABLE.

Exam Tip: For a third-order polynomial a3s³ + a2s² + a1s + a0, the shortcut stability condition without building the full array is: all coefficients positive AND a1·a2 > a0·a3. If either condition fails, system is unstable. This shortcut saves significant time in GATE numerical problems.

Quick Revision

• Routh-Hurwitz criterion is necessary and sufficient for stability of continuous-time LTI systems.
• Necessary condition: all coefficients of D(s) must be present and have the same sign.
• Number of RHP poles equals the number of sign changes in the first column of the Routh array.
• Third-order stability shortcut: all coefficients positive and a1·a2 > a0·a3.
• To find range of K for stability, express characteristic polynomial in K and set all first-column elements greater than zero.
• GATE trap: a missing intermediate coefficient (zero) automatically means instability even before building the array.
• Marginal K (oscillation frequency) is found from the auxiliary polynomial of the row above the zero-element row.