# Practical Implications of Transfer Function Zeros

K
Thread Starter

#### Klaus

- Zeros on the left hand side of the jw axis indicates that the system is minimum phase.

- Zeros on the right hand side of the jw axis indicates that the system is non-minimum phase.

- Zeros on the jw axis indicates that the system is 'marginally' minimum phase. The magnitude and frequency response of such a system indicates a phase shift of 2*pi rad between the input and output while the magnitude of the input/output response is 0db for a major part of the frequency range.

My Questions are as follows:

1. What is a transfer function without zeros at all referred to?

2. If the relative degree of a linear system is the difference between its poles and zeros, a transfer function without zeros has its relative degree = number of poles.
What does this mean in terms of the practical response of the system?

3. Is it preferable to have a system with left hand zeros or no zeros in terms of its transient response? Looking at the bode plots, no zeros seem to support a wide frequency response. However, at a certain frequency, the input is dramatically attenuated and the phase shift is undefined!

4. The zeros of a transfer function dictates wether its zero dynamics is stable. Zeros of a transfer function coincide with the roots of its zero dynamics. Does this mean that a system without zeros has no zero dynamics and therefore is advantageous when designing a controller?

Thanks,
Klaus

J

#### Justin Shriver

> - Zeros on the left hand side of the jw axis indicates that the system is minimum phase.
>
> - Zeros on the right hand side of the jw axis indicates that the system is non-minimum phase.
>
> - Zeros on the jw axis indicates that the system is 'marginally' minimum phase. <

The magnitude and frequency response of such a system indicates a phase shift of 2*pi rad between the input and output while the magnitude of the input/output response is 0db for a major part of the frequency range.

> My Questions are as follows:
>
> 1. What is a transfer function without zeros at all referred to? <

Good, question let me know if you get a good name.

> 2. If the relative degree of a linear system is the difference between its poles and zeros, a transfer function without zeros has its relative degree = number of poles.
> What does this mean in terms of the practical response of the system? <

This is a little more subtle, the pat answer is it is how many times you have to differentiate the output before an input shows up. In other words its a measure of how many integrators lie between your input and output. Is this a good or bad thing? That depends, many results require a transfer function to be strictly proper relative degree>=1. However, the sytem y=u (no poles) is pretty easy to control.

> 3. Is it preferable to have a system with left hand zeros or no zeros in terms of its transient response? Looking at the bode plots, no zeros seem to support a wide frequency response. However, at a certain frequency, the input is dramatically attenuated and the phase shift is undefined! <

No/Yes

Ok, to be a little less flip. If you are asking about the zeros of your plant, then the question is not really relevant your stuck with what the plant gives you. Do you want to introduce zeros into the controller transfer function. That is quite likely for example a standard lead/lag controller introduces zeros as lead (damping) elements.

> 4. The zeros of a transfer function dictates wether its zero dynamics is stable. Zeros of a transfer function coincide with the roots of its zero dynamics. Does this mean that a system without zeros has no zero dynamics and therefore is advantageous when designing a controller? <

Zero dynamics is used in many contexts, for example when one feedback linearizes a plant you often have unobservable dynamics (zero dynamics) which if unstable are bad (ok very bad). In terms of hard and easy it all depends, non-minimum phase zeros are in general bad.

Back to the linear case
Given poles at -1 -2 -3 and the choice of no zeros
or zeros at -5 and -6 the later gives faster performance (at a control input cost) for the same control gains. This is not suprising as we could think of this plant as being our original plant plus a lead controller.

I think the best way to view it is in terms of lead lag controllers for the linear case.

K

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