Stability via the Nyquist Diagram – The Nyquist Criteria

The Nyquist criterion can tell us if the system is stable or unstable by determining how many closed-loop poles are in the right half-plane of the closed-loop system of Figure 1:

Figure 1

Consider the contour A defined in s-plane of Figure 2:

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 of the mapping; that is, Z:

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

This mapping is called the Nyquist diagram, or Nyquist plot, of G(s)H(s).

To understand the Nyquist criteria for stability, we must first establish four important concepts that will be used during its application:

(1) the relationship between the poles of 1+ G(s)H(s) and the poles of G(s)H(s); (2) the relationship between the zeros of 1+ G(s)H(s) and the poles of the closed-loop transfer function (3) the concept of mapping points; and (4) the concept of mapping contours.

We could demonstrate that the poles of 1+ G(s)H(s) are the same as the
poles of G(s)H(s), the open-loop system, and (2) the zeros of  1+ G(s)H(s) are the
same as the poles of closed-loop transfer function of the system.

Example:

Step 1. Find the open-loop transfer function G_(s)H_(s) of the system.

Consider the closed-loop control system as follows:

The system characteristic equation is as follows:

The factor form of this characteristic equation is:

To determine the previous factor form:

Where the open-loop transfer function G_(s)H_(s) of the system is:

Step 2. Use Command Window of Matlab to draw the Nyquist Diagram, applying the following commands:

>> s=tf(‘s’);

>> G=10/(s^3+2*s^2+5*s);

>> nyquist(G);

We can see at the previous Diagram that for:

To reach stability, Z must be equal to zero:

Recalling that the poles of 1+ G(s)H(s), are the same as the poles of G(s)H(s), the open-loop system, we can determine P, the number of open-loop poles enclosed by the contour A from:

A detour around the poles on the contour is required:

In the Nyquist Diagram obtained for the system of Task 2, the point -1+j0 is highlighted in red:

We can see that N=0, so:

However, the Nyquist diagram intersects the real axis at -1+j0. Hence, according to the Nyquist Criteria, the system is marginally stable.

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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