Control System Analysis, The Nyquist Criteria

Sketching the Nyquist Diagram

The Nyquist diagram is also known as “The Polar Trace” of a transfer function G(jω), is a graph of the magnitude of G(jω) against the phase angle of G(jω) in polar coordinates, according to ω variables of zero to infinity. Therefore, The Nyquist diagram is the geometric place of vectors:

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conforming ω modified from zero to infinity. Note that, in polar graphs, the phase angles are positive (negative) if they are measured counterclockwise (clockwise) from the positive real axis.

The following Figure shows an example of a Nyquist Diagram:

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All points of the polar trace of G(jω) represent the end point of a vector at a given value of ω. The projections of G(jω) on the real and imaginary axes are their real and imaginary components. The magnitude and phase angle of G(jω) must be calculated directly for each frequency ω in order to construct polar traces.

Conceptually, the Nyquist diagram is plotted by substituting the points
of the contour that encloses the right half-plane into the function G(s)H(s).  This process is called mapping. Next Figure shows the process of mapping:

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Consider the closed-loop control system of Figure 1:

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Figure 1

Thus, in the Nyquist diagram, the contour that encloses the right half-plane, shown in Figure 2, can be mapped through the function G(s)H(s), derived from Figure 1,  by substituting points along the contour into G(s)H(s):

null

Figure 2

If a contour, A, that encircles the entire right half-plane of the root-locus of the system determined by the characteristic equation 1+ G(s)H(s), is mapped through G(s)H(s), then the number of closed-loop poles, Z, in the right half-plane equals the number of open-loop poles, P, that are in the right half-plane minus the number of counterclockwise revolutions, N, around -1+j0 of the mapping; that is, Z:

null

Thus, Z must be equal to zero to reach stability.

This mapping is called the Nyquist diagram, or Nyquist plot, of G(s)H(s). For more information an examples, see: The Nyquist Criteria

 

Example

Consider the control system whose block diagram and diagram are shown in the following Figure 3:

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Figure 3

Conceptually, the Nyquist diagram is plotted by substituting the points of the contour shown in Figure 4(a) into G(s)H(s):

null

Each Pole and Zero term of G(s) shown in Figure 3(b) is a vector in Figure 4(a) and 4(b). The resultant vector, , found at any point along the contour is in general the product of the Zero vectors divided by the product of the Pole vectors (see Figure 4(c)). Thus, the magnitude of the resultant is the product of the Zero lengths divided by the product of the Pole lengths, and the angle of the resultant is the sum of the Zero angles minus the sum of the Pole angles.

null

Figure 4

The mapping from point A to point C can also be explained analytically. From
A to C the collection of points along the contour is imaginary. Hence, from A to C,
G(s)H(s)=G(s)*1=G(s)=G(jω), or from Figure 3(b):

nullAt zero frequency:

null

Thus, the Nyquist diagram starts at 50/3 at an angle of 0°. As ω increases the real part remains positive, and the imaginary part remains negative.

At null the real part becomes negative. At null, the Nyquist diagram crosses the negative real axis since the imaginary term goes to zero. The real value at the axis crossing, point Q in Figure 4(c), is -0.874. Continuing toward , the real part is negative, and the imaginary part is positive. At infinite frequency:

nullor approximately zero at 90°.

Around the infinite semicircle from point C to point D shown in Figure 4(b), the vectors rotate clockwise, each by 180°. Hence, the resultant undergoes a counterclockwise rotation of 3×180, starting at point C’ and ending at point D’ of Figure 4(c).

Nyquist diagram with Matlab

Consider the following open loop transfer function:

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To create the Nyquist Diagram of the system, use the following commands in the command window of Matlab:

>> s=tf(‘s’)

>> G=1/(s^2+0.8*s+1)

>> nyquist(G)

This line of commands yields:

null

We can obtain information of points of interest in the Nyquist Diagram by cliking once over the contour. This yields:

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Sources:

  1. Modern_Control_Engineering, Ogata 4t
  2. Control Systems Engineering, Nise
  3. Sistemas de Control Automatico, Kuo

Literature review by:

Prof. Larry Francis Obando – Technical Specialist – Educational Content Writer

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