Consider a negative unity feedback system with forward path transfer function \(G\left( s \right) = \frac{K}{{\left( {s + a} \right)\left( {s - b} \right)\left( {s + c} \right)}}\), where K, a, b, c are positive real numbers. For a Nyquist path enclosing the entire imaginary axis and right half of the s-plane is the clockwise direction, the Nyquist plot of (1 + G(s)), encircles the origin (1 + G(s)) –plane once in the clockwise direction and never passes through this origin for a certain value of K. then, the number of poles of \(\frac{{G\left( s \right)}}{{1 + G\left( s \right)}}\) lying in the open right half of the s-plane is ______.
Concept:
D(s) = 1 + G(s)H(s)
D(s) gives the roots of characteristic equation i.e. closed-loop poles.
Nyquist stability criteria state that the number of unstable closed-loop poles is equal to the number of unstable open-loop poles plus the number of encirclements of the origin of the Nyquist plot of the complex function D(s).
It can be slightly simplified if instead of plotting the function D(s) = 1 + G(s)H(s), we plot only the function G(s)H(s) around the point and count encirclement of the Nyquist plot of around the point (-1, j0).
From the principal of argument theorem, the number of encirclements about (-1, j0) is
N = P - Z
Where
Where P = Number of open-loop poles on the right half of s plane
Z = Number of closed-loop poles on the right half of s plane
Calculation:
Given the open-loop transfer function, \(G\left( s \right) = \frac{K}{{\left( {s + a} \right)\left( {s - b} \right)\left( {s + c} \right)}}\)
Clockwise encirclements = 1 i.e. N = -1
Open loop poles on right half of s plane, P = 1
The closed loop poles lying in the open right half of the s-plane,
Z = P – N = 1 – (-1) = 2
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:
