Routh- Hurwitz Criterion

Table of Contents
1. Statement of the Routh-Hurwitz Criterion
2. Necessary but not Sufficient Conditions for Stability
3. Advantages of the Routh-Hurwitz Criterion
4. Limitations of the Routh-Hurwitz Criterion
5. The Routh Array and Procedure
6. Special Cases in the Routh Procedure
7. Worked Example
8. Applications of Routh-Hurwitz Criterion
9. PYQs: Competitive Exams
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Before discussing the Routh-Hurwitz Criterion, let us study the three basic types of stability for linear time-invariant systems.

• Stable system: All the roots (poles) of the characteristic equation lie in the left half of the s-plane (i.e. have negative real parts). The response to any bounded input is bounded and the natural response decays to zero as t → ∞.
• Marginally stable system: One or more roots lie on the imaginary axis (purely imaginary), and the remaining roots lie in the left half of the s-plane. The system neither diverges nor does the natural response necessarily decay to zero; sustained oscillations may occur.
• Unstable system: One or more roots lie in the right half of the s-plane (i.e. have positive real parts). The natural response grows without bound as t → ∞.

Statement of the Routh-Hurwitz Criterion

The Routh-Hurwitz criterion provides a simple algebraic test to determine the number of roots of a real-coefficient polynomial that lie in the right half of the s-plane, without explicitly computing the roots.

• The Routh-Hurwitz criterion states that a necessary and sufficient condition for a polynomial to have no roots with positive real parts is that all the entries in the first column of the Routh array have the same sign.
• If the first column of the Routh array has sign changes, the number of sign changes equals the number of roots of the characteristic equation in the right half of the s-plane (the number of roots with positive real parts).

Necessary but not Sufficient Conditions for Stability

• For a polynomial characteristic equation with real coefficients, the following are necessary conditions for all roots to lie in the left half s-plane:
• All coefficients of the polynomial must have the same sign (for a standard form with descending powers and a positive leading coefficient).
• There should be no missing terms; i.e., no coefficient in the sequence should be zero unless handled specially by the Routh procedure.

Even when the coefficients are all positive and no terms are missing, the polynomial may still have roots in the right half plane. Therefore these conditions are necessary but not sufficient. The Routh-Hurwitz procedure (developed by A. Hurwitz and E. J. Routh) is used to conclusively determine stability.

Advantages of the Routh-Hurwitz Criterion

• It determines stability without explicitly solving for the polynomial roots.
• It provides information about relative stability by indicating how many roots lie in the right half plane.
• It can be used to determine the range of a parameter (for example gain K) for which the closed-loop system is stable.
• It helps to find points where the root locus crosses the imaginary axis (to locate marginal stability boundaries).

Limitations of the Routh-Hurwitz Criterion

• It applies only to linear systems described by polynomials with real coefficients.
• It does not give the exact locations of poles in the s-plane; it only counts how many are in each half-plane.
• Special cases (zeros in the first column or an entire row of zeros) require careful handling (auxiliary polynomial, ε-method), which must be applied correctly.

The Routh Array and Procedure

Consider the characteristic polynomial

Assume coefficients a0, a1, ..., an are real and non-zero and arranged for descending powers of s. The Routh array is constructed as follows.

Step 1: Arrange coefficients in the first two rows of the Routh array. The first row contains coefficients of highest power descending by two, and the second row contains the next set of coefficients.

Step 2: Form the third row entries by using determinants of the two rows above. Each element in a lower row is computed from elements of the two rows immediately above it.

Step 3: Form the fourth row in the same manner using the second and third rows.

Step 4: Continue this process until a final row with constants is obtained. The first column values of this array are then checked for sign changes to conclude stability.

Special Cases in the Routh Procedure

• Zero in the first column element: If an element in the first column becomes zero while other elements in the row are non-zero, substitute a small positive number ε in place of zero and proceed with the array. Evaluate the limit as ε → 0+ to determine the sign pattern.
• Entire row of zeros: If an entire row becomes zero, it indicates the presence of symmetrical root pairs about the origin (even or odd polynomial factors). Form the auxiliary polynomial from the row above the zero row and differentiate the auxiliary polynomial to replace the zero row; then continue the array.
• Repeated or purely imaginary roots: These appear when the auxiliary polynomial method produces factors that correspond to imaginary axis roots; such cases indicate marginal stability or boundary between stability and instability.

Worked Example

Example: Check the stability of the system whose characteristic equation is

s⁴ + 2s³ + 6s² + 4s + 1 = 0

Sol:

Form the Routh array for the polynomial.

Write the first two rows with coefficients:

First row: coefficients of s⁴, s², s⁰.

Second row: coefficients of s³, s¹.

Compute the third row entries using the determinant formula from the two rows above.

Examine the first column of the completed Routh array. Since all entries in the first column are positive (same sign), there are no sign changes.

Therefore, the given characteristic equation has no roots with positive real parts and the system is stable.

Applications of Routh-Hurwitz Criterion

• Determining closed-loop stability without root computation.
• Finding ranges of controller gains (K) that ensure closed-loop stability.
• Determining marginal stability boundaries where poles cross the imaginary axis.
• Analysing effect of parameter variations on stability (robustness checks).

PYQs: Competitive Exams

Q1: The open loop transfer function of a unity gain negative feedback system is given by

The range of k for which the system is stable, is
(a) K > 3
(b) K < 3
(c) K > 5
(d) K < 5

Ans: (c)
Sol:

Given

Routh-Hurwitz criterion can be used to determine the range of k for a stable system.
Characteristic equation: 1 + G(s) = 0

Construct the Routh table.

For a stable system, the elements of the first column must not change sign.
For that to happen
k - 5 > 0
or, k > 5
Therefore, option (c) is correct.

Q2: A single-input single-output feedback system has forward transfer function G(s) and feedback transfer function H(s). It is given that |G(s)H(s)| < 1. Which of the following is true about the stability of the system?

(a) The system is always stable
(b) The system is stable if all zeros of G(s)H(s) are in left half of the s-plane
(c) The system is stable if all poles of G(s)H(s) are in left half of the s-plane
(d) It is not possible to say whether or not the system is stable from the information given

Ans: (c)
Sol:
The closed-loop characteristic equation is
D(s) = 1 + G(s)H(s)
The roots of D(s) are the closed-loop poles.
Nyquist stability criterion relates the number of unstable closed-loop poles (Z) to the number of unstable open-loop poles (P) and the number of encirclements (N) of the point (-1, j0) by the Nyquist plot of G(s)H(s) via the relation
N = P - Z
Given |G(s)H(s)| < 1, the Nyquist plot of G(s)H(s) about the origin does not encircle the point (-1, 0); therefore, N = 0.
For Z = 0 (no unstable closed-loop poles) we require P = 0.
Thus the system is stable if all poles of G(s)H(s) are in the left half of the s-plane (i.e. P = 0).