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?

Option 3 : The system is stable if all poles of G(s)H(s) are in left half of the s-plane

Concept:

D(s) = 1 + G(s)H(s)

D(s) gives the roots of characteristic equation i.e. closed-loop poles.

Nyquist stability criteria state that the number of unstable closed-loop poles is equal to the number of unstable open-loop poles plus the number of encirclements of the origin of the Nyquist plot of the complex function D(s).

It can be slightly simplified if instead of plotting the function D(s) = 1 + G(s)H(s), we plot only the function G(s)H(s) around the point and count encirclement of the Nyquist plot of around the point (-1, j0).

From the principal of argument theorem, the number of encirclements about (-1, j0) is

N = P - Z

Where

Where P = Number of open-loop poles on the right half of s plane

Z = Number of closed-loop poles on the right half of s plane

Calculation:

For the given system, we have |G(s)H(s)| < 1

So, we may easily conclude that the Nyquist-plot intersect the negative real axis between 0 and -1 point, i.e. the Nyquist plot does not enclose the point (-1, 0) or in other words number of encirclements is zero.

⇒ N = 0

For a stable closed loop system, we must have Z = 0.

To get Z = 0, from the Nyquist criteria, P must be equal to zero.

Therefore, the system is stable if all poles of G(s)H(s) are in left half of the s-plane.