Question

Routh test

• A. Criterion provides information about the actual location of roots
• B. Cannot be used to test the stability of a control system containing transportation lag
• C. Criterion is not applicable to systems with polynomial characteristic equation
• D. Cannot determine as to how many roots of the characteristic equation have positive real roots

Hint:

The Routh-Hurwitz criterion requires a polynomial characteristic equation and cannot be directly applied to systems with time delays (transportation lag), which result in transcendental equations.

Answer 1

The Correct Option is B

Step 1: Understand the Routh-Hurwitz criterion. 
The Routh-Hurwitz criterion (or Routh test) is a method to determine the stability of a linear time-invariant system by analyzing the characteristic equation of the system, which is typically a polynomial of the form: \[ a_n s^n + a_{n-1} s^{n-1} + \cdots + a_1 s + a_0 = 0. \] The criterion checks whether all roots of this polynomial have negative real parts (indicating stability) by constructing the Routh array and examining the signs of the elements in the first column. Key features:
It determines the number of roots with positive real parts (unstable roots).
It does not provide the exact location (values) of the roots, only their distribution relative to the imaginary axis.
It applies to systems with polynomial characteristic equations.
Step 2: Analyze the applicability to transportation lag. 
A transportation lag (or time delay) introduces a term like \( e^{-\tau s} \) into the system’s transfer function, where \( \tau \) is the delay time. For example, a system with a transfer function \( G(s) = \frac{K e^{-\tau s}}{1 + Ts} \) has a time delay. The characteristic equation becomes: \[ 1 + G(s) = 0 \quad \Rightarrow \quad 1 + \frac{K e^{-\tau s}}{1 + Ts} = 0, \] \[ 1 + Ts + K e^{-\tau s} = 0. \] This equation is transcendental (not a polynomial) due to the \( e^{-\tau s} \) term, which has an infinite series expansion: \[ e^{-\tau s} = 1 - \tau s + \frac{(\tau s)^2}{2!} - \frac{(\tau s)^3}{3!} + \cdots. \] The Routh-Hurwitz criterion requires a polynomial characteristic equation to construct the Routh array. Since a transportation lag results in a non-polynomial equation, the Routh test cannot be directly applied. Alternative methods, like the Nyquist criterion or Padé approximation (approximating \( e^{-\tau s} \) as a rational function), are used for systems with time delays. 
Step 3: Evaluate the options. 
(1) Criterion provides information about the actual location of roots:
Incorrect, as the Routh test only determines the number of roots with positive real parts, not their exact locations (e.g., specific values). Incorrect.
(2) Cannot be used to test the stability of a control system containing transportation lag: Correct, as the transportation lag introduces a transcendental term, making the characteristic equation non-polynomial, so the Routh test cannot be applied directly. Correct.
(3) Criterion is not applicable to systems with polynomial characteristic equation: Incorrect, as the Routh test is specifically designed for polynomial characteristic equations. Incorrect.
(4) Cannot determine as to how many roots of the characteristic equation have positive real roots: Incorrect, as the Routh test explicitly determines the number of roots with positive real parts by counting sign changes in the first column of the Routh array. Incorrect. 
Step 4: Select the correct answer. 
The Routh test cannot be used to test the stability of a control system containing transportation lag, matching option (2).