Test: Stability Analysis of Linear Control Systems - 1 - Electronics and Communication Engineering (ECE) MCQ

Number of sign changes in the first column of Routh’s array indicates the number of closed loop poles or roots of characteristic equation lying in the right half of s-plane.

Both assertion and reason are individually true. The correct reason for assertion is BIBO stability criteria,

The characteristic equation is,
1 + G(s)H(s) = 0 

For absolute stability, we have:

The settling time is inversely proportional to the real part of the dominant roots. If the settling time is less compared to other system, then the system will be relatively more stable

Given, step response:

∴  Transfer function is

Since one of the closed loop pole (s = 1) lies in RH s-plane, therefore given system is unstable.

The Rouths array is formed as follows:

For the system to be stable, there should not be any sign change in the first column of Routh’s array.

Hence, above conditions must hold for the given system to be stable.

The characteristic equation of given system is: 1 + G(s) = 0


Given system will be stable when,
200 + K > 0 or K > -200

System will oscillate when K = K_mar = 666.25
Auxiliary equation is : 52.5 s² + (200 + K) = 0
or, 52.5 s² = - 866.25 or, s² = -16.5

Routh’s array is formed as shown below, 

Replacing ε in place of 0 in 1st column, we have:

To check for sign change:

Thus, there are two sign changes in 1st column of Routh’s array (+0 to -∞ and -∞ to + 2).
Hence, number of roots in RH s-plane = 2.
Number of roots in LH s-plane = 1.
Number of roots on the jω-axis = 0.