Control Systems Nyquist Plots - Control Systems

What are Nyquist Plots?ϕ=tan−1ωτ−tan−1βωτ

Nyquist plots are the sign of polar plots to find the stability of the closed loop control systems by changing ω from −∞ to ∞. This means that Nyquist plots are used to draw the complete frequency response of the open loop transfer function.

Nyquist Stability Criterion

The Nyquist stability criterion shows that the principle of argument. It means that if any P poles and Z zeros are showed by the ‘s’ plane closed path, then the related G(s)H(s) plane must encircle the origin P−Ztimes. So that we can mention 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’s apply the principle of argument to the entire right half of the ‘s’ plane once you select the closed path. This selected path is called the Nyquist contour.
Here the closed loop control system is known to be stable when the poles of the closed loop transfer function are included in the left half of the ‘s’ plane. The poles of the closed loop transfer function are included in the roots of the characteristic equation. When the order of the characteristic equation increases, then it will not be easy to find the roots. So, find 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.
When the open loop control system is stable then there is no open loop pole in the the right half of the ‘s’ plane.
i.e.,P=0⇒N=−Z
If the closed loop control system is stable then there is no closed loop pole available in the right half of the ‘s’ plane.
i.e.,Z=0⇒N=P
Nyquist stability criterion indicates the total number of encirclements about the critical point (1+j0) which must be equal to the poles of characteristic equation, which is equal to the poles of the open loop transfer function included in the right half of the ‘s’ plane. The major shift in origin to (1+j0) provides the characteristic equation plane.

Rules for Drawing Nyquist Plots

Here are some 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 ωω from zero to infinity. If pole or zero present at s = 0, then varying ωω from 0+ to infinity for drawing polar plot.
• Draw the mirror image of above polar plot for values of ωωranging 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.
Once you draw the Nyquist plot, then you can find then 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

We can identify the Nyquist plots, to know 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 shows the negative real axis (phase angle is 1800) which is known as the phase cross over frequency. It is represented by ωpc.

Gain Cross over Frequency

The frequency at which the Nyquist plot is including the magnitude of one is known as the gain cross over frequency. It is represented by ωgc.
The stability of the control system will be depend upon the relation between phase cross over frequency and gain cross over frequency is mentioned as below.
• If the phase cross over frequency wpc is greater than the gain cross over frequency wgc, then the control system is stable.
• If the phase cross over frequency wpc is equal to the gain cross over frequency wgc, then the control system is marginally stable.
• If phase cross over frequency wpc is less than gain cross over frequency wgc, then the control system is unstable.

Gain Margin

When the gain margin GMGM is equal to the reciprocal of the magnitude of the Nyquist plot then at the phase cross over frequency.
Where, Mpc is denoted as the magnitude in normal scale at the phase cross over frequency.

Phase Margin

The phase margin PMPM is similar to the sum of 1800 and the phase angle at the gain cross over frequency.
Where, is known as 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.