# Nyquist Stability Criterion

The rational function $b(s) = (s+3)(s-4)^{-1}$ is the frequency response function (FRF) of a system $B$. Is $B$ stable?

I understand that a system is unstable if there are poles in the closed right half plane. Would we say this system is not stable because of the pole at s = 4? Or does the zero at s = -3 interfere with the notion of stability? If someone could explain what is correct, with a bit more conceptual explanation it would be appreciated.

• You have the correct understanding: the system is unstable because of the pole at $s = 4$. – Omnomnomnom Jun 9 '15 at 10:38
• Also, where does nyquist stability feature? Are we describing some feedback system? – Omnomnomnom Jun 9 '15 at 10:45

The system is unstable if any bounded input gives unbounded outout.

Physically, an unstable system whose natural response grows without bound can cause damage to system or property or human life. Poles in the left half plane yield either exponential decay or sinusoidal responses, thus lhp poles indicate stability.

The system is stable if it doesn't have any right half plane poles. We say that the system is unstable because for values of s close to 4, the system tends to infinity and such large values make systems unstable. The zero at s=-3 has nothing to do with the notion of stability. You can read control theory - a complete course on this. Being more specific - Read Nyquist Stability criterion and Root locus plots for better visualisation.

• The Nyquist stability criterion has little to do with the question as presented – Omnomnomnom Jun 9 '15 at 11:03
• Also, it's unclear what you mean when you say for values of s close to 4, the system tends to infinity and such large values make systems unstable. – Omnomnomnom Jun 9 '15 at 11:03
• If we draw Nyquist plot of the given function, then we can say about the stability by noticing the enclosement of point s = -1. – Yash Jun 10 '15 at 4:44
• Also, we want our system to work efficiently in normal conditions, if it goes to infinity or large values, then the system can behave unexpectedly. – Yash Jun 10 '15 at 4:45
• Yes, but what does "for values of $s$ close to $4$" actually mean? – Omnomnomnom Jun 10 '15 at 13:20

Plot the Root locus diagram for the system and then analyze the stability.