Control Systems - Nyquist Plots
Nyquist plots are the continuation of polar plots for finding the stability of the closed loop control systems by varying ω from −∞ to ∞. That means, Nyquist plots are used to draw the complete frequency response of the open loop transfer function.
Nyquist Stability Criterion
The Nyquist stability criterion works on the principle of argument. It states that if there are P poles and Z zeros are enclosed by the ‘s’ plane closed path, then the corresponding $G(s)H(s)$ plane must encircle the origin $P − Z$ times. So, we can write the number of encirclements N as,
$$N=P-Z$$
If the enclosed ‘s’ plane closed path contains only poles, then the direction of the encirclement in the $G(s)H(s)$ plane will be opposite to the direction of the enclosed closed path in the ‘s’ plane.
If the enclosed ‘s’ plane closed path contains only zeros, then the direction of the encirclement in the $G(s)H(s)$ plane will be in the same direction as that of the enclosed closed path in the ‘s’ plane.
Let us now apply the principle of argument to the entire right half of the ‘s’ plane by selecting it as a closed path. This selected path is called the Nyquist contour.
We know that the closed loop control system is stable if all the poles of the closed loop transfer function are in the left half of the ‘s’ plane. So, the poles of the closed loop transfer function are nothing but the roots of the characteristic equation. As the order of the characteristic equation increases, it is difficult to find the roots. So, let us correlate these roots of the characteristic equation as follows.
The Poles of the characteristic equation are same as that of the poles of the open loop transfer function.
The zeros of the characteristic equation are same as that of the poles of the closed loop transfer function.
We know that the open loop control system is stable if there is no open loop pole in the the right half of the ‘s’ plane.
i.e.,$P=0 \Rightarrow N=-Z$
We know that the closed loop control system is stable if there is no closed loop pole in the right half of the ‘s’ plane.
i.e.,$Z=0 \Rightarrow N=P$
Nyquist stability criterion states the number of encirclements about the critical point (1+j0) must be equal to the poles of characteristic equation, which is nothing but the poles of the open loop transfer function in the right half of the ‘s’ plane. The shift in origin to (1+j0) gives the characteristic equation plane.
Rules for Drawing Nyquist Plots
Follow these rules for plotting the Nyquist plots.
Locate the poles and zeros of open loop transfer function $G(s)H(s)$ in ‘s’ plane.
Draw the polar plot by varying $\omega$ from zero to infinity. If pole or zero present at s = 0, then varying $\omega$ from 0+ to infinity for drawing polar plot.
Draw the mirror image of above polar plot for values of $\omega$ ranging from −∞ to zero (0− if any pole or zero present at s=0).
The number of infinite radius half circles will be equal to the number of poles or zeros at origin. The infinite radius half circle will start at the point where the mirror image of the polar plot ends. And this infinite radius half circle will end at the point where the polar plot starts.
After drawing the Nyquist plot, we can find the stability of the closed loop control system using the Nyquist stability criterion. If the critical point (-1+j0) lies outside the encirclement, then the closed loop control system is absolutely stable.
Stability Analysis using Nyquist Plots
From the Nyquist plots, we can identify whether the control system is stable, marginally stable or unstable based on the values of these parameters.
- Gain cross over frequency and phase cross over frequency
- Gain margin and phase margin
Phase Cross over Frequency
The frequency at which the Nyquist plot intersects the negative real axis (phase angle is 1800) is known as the phase cross over frequency. It is denoted by $\omega_{pc}$.
Gain Cross over Frequency
The frequency at which the Nyquist plot is having the magnitude of one is known as the gain cross over frequency. It is denoted by $\omega_{gc}$.
The stability of the control system based on the relation between phase cross over frequency and gain cross over frequency is listed below.
If the phase cross over frequency $\omega_{pc}$ is greater than the gain cross over frequency $\omega_{gc}$, then the control system is stable.
If the phase cross over frequency $\omega_{pc}$ is equal to the gain cross over frequency $\omega_{gc}$, then the control system is marginally stable.
If phase cross over frequency $\omega_{pc}$ is less than gain cross over frequency $\omega_{gc}$, then the control system is unstable.
Gain Margin
The gain margin $GM$ is equal to the reciprocal of the magnitude of the Nyquist plot at the phase cross over frequency.
$$GM=\frac{1}{M_{pc}}$$
Where, $M_{pc}$ is the magnitude in normal scale at the phase cross over frequency.
Phase Margin
The phase margin $PM$ is equal to the sum of 1800 and the phase angle at the gain cross over frequency.
$$PM=180^0+\phi_{gc}$$
Where, $\phi_{gc}$ is the phase angle at the gain cross over frequency.
The stability of the control system based on the relation between the gain margin and the phase margin is listed below.
If the gain margin $GM$ is greater than one and the phase margin $PM$ is positive, then the control system is stable.
If the gain margin $GM$ is equal to one and the phase margin $PM$ is zero degrees, then the control system is marginally stable.
If the gain margin $GM$ is less than one and / or the phase margin $PM$ is negative, then the control system is unstable.
