Routh Array Special Case Zero Element

During the construction of a Routh array, a special difficulty arises when a zero appears in the first column of an intermediate row while the rest of the row is not all zero. This creates a division-by-zero problem in the computation of the next row. The epsilon method is the standard technique to resolve this situation and still extract stability information from the array. This special case is a recurring examination topic in GATE control systems.

Core Concept Explanation

When the Routh array is being constructed and a zero appears in the first column of a row where the rest of the row contains nonzero elements, the next row computation requires dividing by zero, which is undefined. This situation cannot be treated as the all-zero row special case because other elements in the row are nonzero. The epsilon method resolves this by replacing the zero with a small positive quantity ε (epsilon), completing the array symbolically, and then evaluating the signs of the first-column elements as ε approaches zero from the positive side.

The key insight is that what matters for stability is only the sign of each first-column element, not its exact value. By keeping ε as a small positive number throughout the computation and then observing whether the first-column elements of subsequent rows are positive or negative in the limit, the sign changes can be correctly counted. The actual value of ε does not affect the sign analysis as long as it remains positive.

Procedure Step by Step

The epsilon method follows a clear sequence. First, identify the row where the zero appears in the first column. Second, replace that zero with ε, where ε is a small positive number. Third, continue computing all remaining rows of the Routh array using normal formulas, treating ε as a symbolic positive constant. Fourth, after the full array is complete, examine each first-column element and determine its sign as ε approaches zero from the positive side (taking the limit). Fifth, count the sign changes in the resulting first-column signs to determine the number of RHP poles.

In evaluating the limit, an expression like 1/ε tends to positive infinity, while -1/ε tends to negative infinity. An expression like ε - 5 approaches -5, which is negative. These sign evaluations are straightforward once the algebra of the row computations is completed with ε in place.

Mathematical Expression

Consider a Routh array row where the computed first-column element is b1 = 0. Replace it with ε. The next row element is c1 = (ε·x - p·q) / ε where x, p, q are elements from the current and previous rows. As ε tends to zero, this expression can evaluate to either +∞ or -∞ depending on the sign of the numerator constant term. The sign of c1 in the limit determines whether a sign change occurs between the ε row and the c1 row.

Practical Understanding

A zero in the first column of a Routh array (with nonzero elements in the rest of the row) indicates that at least one root of the characteristic polynomial has a real part very close to zero, or that the polynomial has a specific algebraic structure causing near-cancellation. In physical terms, such a system is very close to the stability boundary and the Routh test is signaling this near-marginal condition.

It is important to distinguish this special case (zero only in first column) from the zero row special case (entire row of zeros). The epsilon method applies exclusively to the former. In the latter case, the auxiliary polynomial method is used instead. Mixing up these two special cases is a common examination mistake.

Given:
Characteristic polynomial D(s) = s⁴ + s³ + 2s² + 2s + 3
Coefficients: 1, 1, 2, 2, 3

Why this formula applies:
Epsilon method needed because zero will appear in first column.

Routh Array:

  s⁴ |  1     2     3
  s³ |  1     2     0
  s² |  b1    b2

b1 = (1×2 - 1×2)/1 = 0/1 = 0  ← Zero in first column, apply ε
b2 = (1×3 - 1×0)/1 = 3

Replace b1 = 0 with ε:
  s² |  ε     3
  s¹ |  c1

c1 = (ε×2 - 1×3)/ε = (2ε - 3)/ε

Substitution:
As ε → 0⁺: c1 = (0 - 3)/ε = -3/ε → -∞ (negative)

  s⁰ |  d1 = 3 (positive)

First column signs: 1(+), 1(+), ε(+), -3/ε(-), 3(+)
Sign changes: ε→ -3/ε = 1 change, -3/ε → 3 = 1 change

Final Answer:
Two sign changes → 2 poles in the right half plane. System is UNSTABLE.

Exam Tip: When you see a zero appear in the first column but other elements in the same row are nonzero, always use the epsilon method. Never use the auxiliary polynomial here — that is reserved for the all-zero row case. After substituting ε, focus only on the sign of each first column element as ε tends to zero plus, not the magnitude.

• A zero in the first column with nonzero elements remaining in the same row triggers the epsilon method.
• Replace the zero with ε (small positive number) and continue computing remaining rows symbolically.
• After completing the array, evaluate the sign of each first-column element by taking the limit as ε approaches zero from the positive side.
• Count sign changes in the resulting first-column signs to determine the number of right-half-plane poles.
• The epsilon method is different from the auxiliary polynomial method, which applies only when an entire row becomes zero.

Quick Revision

• Zero in first column only (rest of row nonzero): use epsilon method, not auxiliary polynomial.
• Replace zero with ε > 0, complete the array, then take limit ε → 0⁺ for each first-column sign.
• Common sign evaluations: -k/ε → negative, k/ε → positive, ε - c (c>0) → negative.
• Number of sign changes in first column = number of RHP poles = degree of instability.
• GATE trap: confusing zero-first-column case with zero-row case. Always check if other elements in the row are also zero.
• Even if ε causes a first-column element to be very large in magnitude, only the sign matters for stability counting.