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GATE 2014 EE (EE1) – Question 27

Control Systems · Stability analysis using Routh-Hurwitz and Nyquist criteria, Bode plots, Root loci · 1 mark · Multiple choice

All elements in a row of a polynomial’s Routh–Hurwitz array are zero. This premature termination indicates the presence of ______.

  1. only one root at the origin
  2. imaginary roots
  3. only positive real roots
  4. only negative real roots

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Show answer and explanation

Correct answer: (B) imaginary roots

Explanation

If every element of a row of the Routh array is zero, the array cannot be continued, because the next row would need a division by zero.

The row above the zero row gives an **auxiliary polynomial** $A(s)$. It is an even polynomial, and its roots are the part of the characteristic equation that is symmetric about the origin. The zero row is replaced by the coefficients of $dA(s)/ds$ and the array continues.

In the standard case these symmetric roots lie on the imaginary axis, so the zero row indicates **imaginary roots, B**.

Note: a symmetric pair can also be real. For example $s^2-1$ gives a zero row and has roots $\pm1$. Of the options offered, imaginary roots is the one the zero row is used to indicate.