The Nyquist stability criterion

A frequency domain stability criterion developed by Nyquist (1932) is based upon Cauchy s theorem. If the function F(s) is in fact the characteristic equation of a closed-loop control system, then 

Note that the roots of the eharaeteristie equation are the elosed-loop poles, whieh are the zeros of F s). 

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 . 

An important difference between analysis of stability in the. v-plane and stability in the frequency domain is that, in the former, system models in the form of transfer functions need to be known. In the latter, however, either models or a set of input-output measured open-loop frequency response data from an unknown system may be employed. 

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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... 

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.

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. 

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. 

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. 

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

On the surface, the Nyquist stability criterion is 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. 

Remember also that for gains greater than the ultimate gain, the root locus plot showed two roots of the closedloop characteristic equation in the RHP. This is exactly the result we get from the Nyquist stability criterion (N = 2 = Z). ... 

The Nyquist stability criterion can be used for openloop-unstable processes, but we have to use the complete, rigorous version with P (the number of poles of the closedloop characteristic equation in the RHP) no longer equal to zero. 

We found in Chapter 9, using Laplace-domain root locus plots, that we could make this system closedloop stable by using a proportional controller with a gain Kc greater than MKp. Let us see if the Nyquist stability criterion leads us to the same conclusion. It certainly should if it is any good, because a table in Chinese must be a table in Russian ... 

If the. system is openloop stable, there will be no poles of HGm z) (z) outside the unit circle, and P = 0. Thus, the Nyquist stability criterion can be applied directly sampled-data systems.. ... 

Sometimes we would like to resolve system stability without modeling impedances or determining the zeros and poles of the impedance. This can be done using the Nyquist stability criterion [587, 588, 596] developed from the theory of complex functions and Cauchy s integral theorem, which can be stated as follows an electrochemical system is stable if and only if the number of clockwise encirclements ( N) of the origin of the Z (—Z") plane, going from low to high frequencies, equals the... 

If approximation by the transfer function in the form of Eq. (13.17) is possible, then poles of the transfer function can be analyzed because the stable transfer function cannot have positive poles. When such an analysis is not possible, one can use the Nyquist stability criterion. Only validated data can be used for analysis. 

The Bode stability criterion is very useful for a wide variety of process control problems. However, for any Gql ) that does not satisfy the required conditions, the Nyquist stability criterion of Section J.3 can be applied. 

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