Control System Analysis, The Nyquist Criteria

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:

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Figure 1

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

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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:

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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.

In construction…

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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