Nyquist stability criterion

The Nyquist stability ciiterion is a method for determining the stability of systenas in the frequency domain. It is almost always applied to closedloop systems. A working, but not completely general, statement of the Nyquist stabiU ity criterion is  

If a polar plot of the total openloop tranter function of the system B( , wraps around the ( — 1,0) point in the Gf/B plane as frequency co goes from zero to infinity, the system is closedloop unstable. 

The two polar plots sketched in Fig. 13.1a show that system A is closedloop unstable whereas system B is closedloop stable. 

The Nyquist stability criterion is, on the surface, quite remarkable. We are able to deduce something about the stability of the closedloop system by making a frequency response plot of the openloop system And the encirclement of the mystical, magical (— 1, 0) point somehow tells us that the system is closedloop unstable. This all looks like blue smoke and mirrors However, as we will prove below, it all goes back to finding out if there are any roots of the closedloop characteristic equation in the RHP. 

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The Nyquist stability criterion can be stated as A closed-loop control system is stable if, and only if, a contour in the G s)H s) plane describes a number of counterclockwise encirclements of the (—l,jO) point, the number of encirclements being equal to the number of poles of G s)H s) with positive real parts . 

In practice, only the frequencies lu = 0 to+oo are of interest and since in the frequency domain. v = jtu, a simplified Nyquist stability criterion, as shown in Figure 6.18 is A closed-loop system is stable if, and only if, the locus of the G(iLu)H(iuj) function does not enclose the (—l,j0) point as lu is varied from zero to infinity. Enclosing the (—1, jO) point may be interpreted as passing to the left of the point . The G(iLu)H(iLu) locus is referred to as the Nyquist Diagram. 

Robust stability can be investigated in the frequency domain, using the Nyquist stability criterion, defined in section 6.4.2. 

From the Nyquist stability criterion, let N k, G(iuj)) be the net number of clockwise encirclements of a point (k, 0) of the Nyquist contour. Assume that all plants in the family tt, expressed in equation (9.132) have the same number ( ) of right-hand plane (RHP) poles. 

The method is suited to the complex-variable theory associated with the Nyquist stability criterion [1]. 

A preview We can derive the ultimate gain and ultimate period (or frequency) with stability analyses. In Chapter 7, we use the substitution s = jco in the closed-loop characteristic equation. In Chapter 8, we make use of what is called the Nyquist stability criterion and Bode plots. 

This equation, of course, contains information regarding stability, and as it is written, implies that one may match properties on the LHS with the point (-1,0) on the complex plane. The form in (7-2a) also imphes that in the process of analyzing the closed-loop stability property, the calculation procedures (or computer programs) only require the open-loop transfer functions. For complex problems, this fact eliminates unnecessary algebra. We just state the Nyquist stability criterion here.1... 

Nyquist stability criterion Given the closed-loop equation 1 + Gol (joi) = 0, if the function G0l(J ) has P open-loop poles and if the polar plot of GOL(](o) encircles the (-1,0) point... 

Do not panic Without the explanation in our Web Support, this statement makes little sense. On the other hand, we do not really need this full definition because we know that just one unstable closed-loop pole is bad enough. Thus the implementation of the Nyquist stability criterion is much simpler than the theory. 

Figure 8.3. Illustration of the stable versus unstable possibilities under the Nyquist stability criterion.

Once we understand the origin of Nyquist stability criterion, putting it to use is easy. Suppose we have a closed-loop system with characteristic equation 1 + GCGP = 0. With the point (-1,0) as a reference and the Gc(jco)Gp(jco) curve on a... 

The Nyquist stability criterion can be applied to Bode plots. In fact, the calculation using the Bode plot is much easier. To obtain the gain margin, we find the value of GCGP which corresponds to a phase lag of-180°. To find the phase margin, we look up the phase lag corresponding to when GCGP is 1. 

A. COMPLEX VARIABLE THEOREM. The Nyquist stability criterion is derived... 

RHP. This is exactly the result we get from the Nyquist stability criterion... 

In Chap. 12 we presented three different kinds of graphs that were used to represent the frequency response of a system Nyquist, Bode, and Nichols plots. The Nyquist stability criterion was developed in the previous section for Nyquist or polar plots. The critical point for closedloop stability was shown to be the 1,0) point on the Nyquist plot. 

Use the Nyquist stability criterion to find the values of for which the system is closedloop stable. 

An openloop unstable, second-order process has one positive pole at + 1/ti and one negative pole at — I/tj. If a proportional controller is used and if ti < show by using a root locus plot and then by using the Nyquist stability criterion that the system is always unstable. 

The Nyquist stability criterion that we developed in Chap. 13 can be directly applied to multivariable processes. As you should recall, the procedure is based on a complex variable theorem which says that the dilTerence between the number of zeros and poles that a function has inside a dosed contour can be found by plotting the function and looking at the number of times it endrdes the origin. 

Thus the Nyquist stability criterion for a multivariable openloop-stable process is ... 

The usual way to use the Nyquist stability criterion in SISO systems is to not plot 1 + and look at encirclements of the origin. Instead we simply... 

Sampled-data control systems can be designed in the frequency domain by using the same techniques that we employed for continuous systems. The Nyquist stability criterion is applied to the appropriate closedloop characteristic equation to find the number of zeros outside the unit circle. 

Thus the Nyquist stability criterion can be applied directly to sampled-data... 

The Nyquist stability criterion may be employed in cases where the Bode criterion is not applicable (e.g. where the phase shift has a value of -180° for more than one value of frequency). It is usually stated in the form071 ... 

Example 18.5 Stability Characteristics of a Third-Order System Using the Nyquist Stability Criterion... 

Using the Nyquist stability criterion, show that feedback systems with first-and second-order open-loop responses are always stable. 

State the Nyquist stability criterion and give some examples of stable and unstable feedback control systems different from those presented in this chapter. Explain the concept of encirclement of the point (-1,0) by the Nyquist plot, which is so central for the Nyquist criterion. 

IV.73 Using the Nyquist stability criterion, find which of the closed-loop systems of Problem IV. 54 are stable and which are not. 

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