A closed-loop control system is stable if the Nyquist plot of the corresponding open-loop transfer function
Encircles the s-plane point (−1 + j0) in the counterclockwise direction as many times as the number of right-half s-plane poles.
Nyquist Stability Criterion Explained
The Nyquist stability criterion is a powerful graphical method used to determine the stability of a closed-loop control system from its open-loop frequency response, represented by the Nyquist plot. It helps engineers assess system stability without explicitly calculating the closed-loop poles, which can be complex for higher-order systems. The criterion is based on the Argument Principle from complex analysis.
Open-Loop Transfer Function and Nyquist Plot
- Open-Loop Transfer Function: For a unity feedback system, this is typically denoted as \(G(s)H(s)\), where \(G(s)\) is the forward path transfer function and \(H(s)\) is the feedback path transfer function.
- Nyquist Plot: This is a polar plot of the open-loop transfer function \(G(j\omega)H(j\omega)\) as the frequency \(\omega\) varies from \(0\) to \(\infty\). It includes the mapping of the entire Nyquist contour in the s-plane, which encloses the entire right-half of the s-plane.
Critical Point and Encirclements
The stability of the closed-loop system is determined by the behavior of the Nyquist plot with respect to a specific point in the complex plane, known as the critical point. This point is \((-1 + j0)\).
The Nyquist stability criterion establishes a relationship between:
- The number of open-loop poles located in the right-half s-plane (RHS-plane), denoted as \(P\). These are the unstable poles of the open-loop system.
- The number of closed-loop poles located in the right-half s-plane (RHS-plane), denoted as \(Z\). For a stable closed-loop system, \(Z\) must be zero.
- The number of encirclements (\(N\)) of the critical point \((-1 + j0)\) by the Nyquist plot.
Nyquist Stability Criterion Formulation
The most common mathematical formulation of the Nyquist stability criterion is given by the equation:
\(N = P - Z\)
Where:
- \(N\) = Number of clockwise encirclements of the critical point \((-1 + j0)\) by the Nyquist plot of \(G(s)H(s)\).
- \(P\) = Number of open-loop poles (poles of \(G(s)H(s)\)) in the right-half s-plane.
- \(Z\) = Number of closed-loop poles (zeros of \(1 + G(s)H(s)\)) in the right-half s-plane.
Stability Condition for Closed-Loop System
For a closed-loop control system to be stable, all its poles must lie in the left-half s-plane. This means that the number of closed-loop poles in the right-half s-plane, \(Z\), must be equal to zero (\(Z = 0\)).
Substituting \(Z = 0\) into the Nyquist criterion equation:
\(N = P - 0\)
\(N = P\)
This implies that for a closed-loop system to be stable, the Nyquist plot of the open-loop transfer function \(G(s)H(s)\) must encircle the critical point \((-1 + j0)\) a number of times equal to \(P\) (the number of open-loop right-half s-plane poles) in the clockwise direction.
Analysis of the Provided Option
Let's analyze the given correct option: "Encircles the s-plane point (\(\minus\)1 + j0) in the counterclockwise direction as many times as the number of right-half s-plane poles."
If we denote \(N_{ccw}\) as the number of counterclockwise encirclements, the option states:
\(N_{ccw} = P\)
We know that a counterclockwise encirclement is the negative of a clockwise encirclement. Therefore, \(N_{ccw} = -N\).
Substituting this into the statement from the option:
\(-N = P\)
From our stability condition, we know that \(N = P\) (where \(N\) is clockwise encirclements). Substituting this into the above equation:
\(-(P) = P\)
\(-P = P\)
This equation \(-P = P\) is only true if \(P = 0\).
Therefore, the statement "Encircles the s-plane point (\(\minus\)1 + j0) in the counterclockwise direction as many times as the number of right-half s-plane poles" is accurate for a stable closed-loop system specifically when the open-loop system itself is stable (i.e., has no poles in the right-half s-plane, so \(P=0\)). In this particular case, for the closed-loop system to be stable, the Nyquist plot must not encircle the \((-1 + j0)\) point at all (i.e., \(N_{ccw} = 0\)), which is consistent with \(P=0\).
In summary, for a general case where the open-loop system might have right-half s-plane poles (\(P > 0\)), the Nyquist plot must encircle the \((-1 + j0)\) point \(P\) times in the clockwise direction for the closed-loop system to be stable. However, the provided option points to a specific condition that holds true for open-loop stable systems.
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
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:
The critical point (-1, j0) is mapped to ________ on the Nichols chart.
