Question

Consider an open-loop unstable system with the transfer function \( G(s)H(s) = \frac{s+2}{(s+1)(s-1)} \) when the feedback path is closed, then the system is

• A. unstable
• B. stable
• C. If two poles are added in left half s-plane then the system is stable
• D. cannot be determined

Hint:

For negative feedback, closed-loop poles are roots of \(1+G(s)H(s)=0\).
System is stable if all closed-loop poles are in the LHP (negative real parts).
Open-loop instability (RHP poles in G(s)H(s)) does not necessarily mean closed-loop instability.
If positive feedback is used, characteristic equation is \(1-G(s)H(s)=0\).

Answer 1

The Correct Option is A

The transfer function of the open-loop system is given by:

\(G(s)H(s) = \frac{s+2}{(s+1)(s-1)}\)

This indicates there is a pole at \(s=1\) and another pole at \(s=-1\). The pole at \(s=1\) is located in the right half of the s-plane, which makes the open-loop system unstable.

When the feedback path is closed, we need to analyze the closed-loop transfer function, which can be defined as:

\(T(s) = \frac{G(s)H(s)}{1 + G(s)H(s)} = \frac{\frac{s+2}{(s+1)(s-1)}}{1 + \frac{s+2}{(s+1)(s-1)}}\)

This can be further simplified for analysis, but the key point here is to understand that the presence of a pole in the right half of the s-plane in the open-loop transfer function will lead to instability in the closed-loop system as well.

Therefore, the closed-loop system will be unstable due to the right half s-plane pole.