# Effect of Pole Zero Cancellation on Nyquist Plot/Stability Criterion

• maverick280857
In summary: Nyquist decomposition) and then state space analysis. There are several good books on the topic. I would recommend "Nyquist and Beyond" by Stanley Lieb and "Nyquist and the Art of Feedback Control" by Jan Willems. However, Lieb's book is more mathematical and Willems' book is more conceptual.
maverick280857
Hi

Suppose a closed feedback system has the transfer function

$$\frac{C(s)}{R(s)} = \frac{G(s)}{1 + G(s)H(s)}$$

In order to employ Nyquist's method to judge the stability of the system, I consider the loop gain

$$L(s) = G(s)H(s)$$

and map the s-plane contour into the $L(s)$ plane. But what if there is a pole of G(s) common with a zero of H(s) (or a zero of G(s) common with a pole of H(s))?

Since the Nyquist plot depends only on the loop gain and not on G and H separately, the effect of this canceled pole-zero pair will not be evident in the L(s)-plane contour. What is the physical significance of this? Does it have a direct relation to the notion of observability in control theory?

In other words, what is the effect of a pole-zero cancellation in the feedforward and feedback transfer functions, on the system vis a vis its stability and other time/frequency domain characteristics?

EDIT: Strictly, I can cancel the pole-zero pair only when I am not at that point. Therefore, the reduced expression (after cancellation) will hold only for those neighborhoods of the s-plane that do not enclose the pole/zero pair. Suppose the pole-zero pair is at s = 0, i.e. G(s) has a pole of order 1 at s = 0 and H(s) has a zero of order 1 at s = 0. Then, L(s) will have no pole or zero at s = 0. But am I supposed to consider a semicircular contour at s = 0 while constructing the Nyquist s-plane contour?

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Indeed, that is a big problem in control theory but! first

If you talk about a rational function which is any transfer function for example, even though in the literature they say "well you shouldn't cancel them", it is just nonsense. If a transfer function is given such that there are common factors in the numerator and the denominator, they are just canceled out, period.

In practice, this is a problem because it destroys either controllability (if common factor is a zero in G and pole on H) or observability (vice versa). Think of a basic case a differentiator is connected to an integrator. This is theoretically allowed but practically dangerous because there might be a signal blowing up in between that you cannot observe... etc

Therefore, Nyquist plot only cares about the transfer function and rational functions with common factors forms eqivalence classes hence you have to cancel them to make sense out of it. Wouldn't you if you had 4/8 instead of 1/2.

If you want to keep that information, you have to use the behavioral approach of the great Jan Willems. He keeps the information as (...)Y(s) = (....)U(s) and never forms a transfer function.

Thanks trambolin, quite interesting. Books haven't really talked about this in detail. I did find a reference to it in Ogata, who says that Root Locus analysis considers the reduced transfer function, but if its a pole of G and a zero of H that are common, the pole of G does in fact appear as a closed loop pole too, so its not lost as such. A similar argument would hold for a zero of G.

However, the Nyquist analysis description in standard textbooks at least, does not cover such a case (or perhaps I overlooked it).

tramolin is correct. the problem is that, say the system is not completely observable, although you cannot see the evil of instability (assuming the pole that was canceled is unstable) at the output, inside the box there are states that are blowing up and, because of some arithmetic combination of them, the result you see at the output looks fine (until those internal states go non-linear, then you'll know you're in trouble).

the way to check for this is with a state-variable representation and then look at the s-plane roots of the determinant of the sI-A matrix.

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rbj said:
the way to check for this is with a state-variable representation and then look at the s-plane roots of the determinant of the sI-A matrix.

Yes, I did this with a numerical problem and it confirmed the fact that one of the eigenvalues doesn't show up in the output expression but does show up explicitly in one of the states. The state space model used a controllable canonical form, however, and I didn't try the observable canonical form...maybe I should try that as well.

Is there some place I could read about this particular aspect of control theory (pole zero cancellation analyzed from a state space and frequency domain perspective, and contrasted) in more detail..and perhaps more rigor?

You can first search for "minimal realization" and then move on to Kalman decomposition (not filter).

Another point to remember is that in practice pole-zero cancellation is basically impossible.

The reason is that there will always be some parameter uncertainty in your system.

The danger I'm addressing here is the pole-zero cancellation of a RHP pole or a RHP zero. If you don't cancel the RHP pole/zero exactly you end up with an unstable system on your hands, and for the reasons above it is very likely that you won't cancel them exactly.

## 1. What is pole zero cancellation and how does it affect the Nyquist plot?

Pole zero cancellation is a phenomenon that occurs when a pole and zero in a transfer function are located at the same complex frequency. This results in the elimination of that particular frequency from the transfer function, causing a change in the shape and behavior of the Nyquist plot. The cancellation of poles and zeros can either improve or deteriorate the stability of a system, depending on the location and number of poles and zeros.

## 2. How does pole zero cancellation affect the stability criterion in a Nyquist plot?

The stability criterion in a Nyquist plot is based on the number of encirclements of the critical point (-1,0) in the complex plane. When pole zero cancellation occurs, the number of encirclements may change, potentially causing the system to become unstable. Therefore, pole zero cancellation has a direct impact on the stability of a system as determined by the Nyquist stability criterion.

## 3. Can pole zero cancellation improve the stability of a system?

Yes, pole zero cancellation can improve the stability of a system in some cases. If the poles and zeros that are cancelled were causing instability in the system, their elimination can result in a more stable system. However, it is important to note that pole zero cancellation can also introduce new instabilities, so it is not always a guaranteed solution for improving stability.

## 4. Are there any drawbacks to pole zero cancellation in a Nyquist plot?

Yes, there are potential drawbacks to pole zero cancellation in a Nyquist plot. As mentioned before, it can introduce new instabilities if not properly accounted for. Additionally, pole zero cancellation can also change the gain margin and phase margin of a system, which can affect its robustness and performance.

## 5. How can I determine the effect of pole zero cancellation on a Nyquist plot?

The effect of pole zero cancellation on a Nyquist plot can be determined by analyzing the transfer function before and after the cancellation. Additionally, performing sensitivity analysis and simulation studies can help understand the impact of pole zero cancellation on the stability and performance of a system. It is important to carefully consider all factors and perform thorough analysis before making any changes to a system's poles and zeros.

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