The Nyquist plot of a strictly stable $G(s)$ having the numerator polynomial as $(s – 3)$ encircles the critical point –1 once in the anti-clockwise direction. Which one of the following statements on the closed-loop system shown in figure, is correct?
To determine the stability of the closed-loop system from the given Nyquist plot, we need to apply the Nyquist stability criterion. Here are the steps:
- According to the Nyquist stability criterion, the number of clockwise encirclements of the point -1 by the Nyquist plot is related to the difference between the number of poles and zeros of 1 + G(s)H(s) in the right half of the complex plane.
- The system function is G(s) = \frac{s-3}{\text{...}}. The Nyquist plot encircles the point -1 once in the anti-clockwise direction.
- Given that G(s) is strictly stable, it means there are no poles in the right half-plane.
- The number of anti-clockwise encirclements \(N\) is given as -1 because encirclement is made in the anti-clockwise direction.
- Nyquist criterion states:
N = P - Z
- \(P\) = number of poles of \(G(s)H(s)\) in the right half-plane.
- \(Z\) = number of zeros of \(1 + G(s)H(s)\) in the right half-plane.
- Since \(P = 0\) (because \(G(s)\) is strictly stable): -1 = 0 - Z
- Solving the above equation gives \(Z = 1\).
- Having Z = 1 means there is one zero of 1 + G(s)H(s) in the right half-plane, indicating that the system is unstable.
The correct statement is: The system is unstable.
Important Questions from Nyquist Plot
______indicates not only whether a system is stable, but also its degree of stability and how stability may be imposed if necessary.
In Nyquist plot of a system on adding a pole at s = 0, then plot will -
The Nyquist plot of the transfer function \(G\left( s \right) = \frac{K}{{\left( {{s^2} + 2s + 2} \right)\left( {s + 2} \right)}}\)
Does not encircle the point (–1 + j0) for K = 10 but does encircle the point (-1 + j0) for K = 100 . Then the closed-loop system (having unity gain feedback) is
A closed-loop control system is stable if the Nyquist plot of the corresponding open-loop transfer function
The Nyquist stability criterion and the Routh criterion both are powerful analysis tools for determining the stability of feedback controllers. Identify which of the following statements is FALSE:
