Question

The number of times the Nyquist plot of \(G(s)H(s)=\dfrac{1}{2}\,\dfrac{(s-1)(s-2)}{(s+1)(s+2)}\) encircles the origin is ____.

Hint:

For encirclements of the {origin} by the Nyquist of \(G\), use \(N=Z_{\text{RHP}}-P_{\text{RHP}}\) for \(G(s)\).

Answer 1

Correct Answer: 2

Step 1: Count RHP poles and zeros of \(G(s)H(s)\).
Zeros at \(s=1,2\) \(\Rightarrow\) two RHP zeros.
Poles at \(s=-1,-2\) \(\Rightarrow\) no RHP poles.
Step 2: Argument principle for encirclements of the origin.
Number of encirclements of the origin by the Nyquist plot of \(G(j\omega)\) is \(N = Z_{\text{RHP}} - P_{\text{RHP}} = 2-0 = 2\).
Final Answer: 2