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Furthermore, how can you see from the pole/zero plot if the system has feed forward/feedback elements?

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  • $\begingroup$ You've got three "close" votes because your question appears to lack detail or clarity. I suggest that if you change "How can" to "can" in the two places where it occurs, it'll be a pretty good question. To really make it good, it's good practice to keep the title short, and to put the whole question in the title. So "Using a transfer function to determine if a system has feedback" might be a good title, and then just move what is now the title into the body of the question. $\endgroup$
    – TimWescott
    Commented Dec 6, 2021 at 22:09

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It depends, if you have a graphical representation of your system, you can see that the system has feedback or feedforward elements.

Or you can look at the difference equation or transfer function. For example a discrete integrator $y[n] = x[n] - y[n-1]$ has feedback since the previous output is fed back to the current output. On the other hand, a differentiator $y[n] = x[n] - x[n-1]$ has no feedback as the output does not depend on previous outputs.

Edit : The transfer function does not tell everything. It is possible that pole/zero cancellation will hide away feedback elements.

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Because the definition of "feedback" is contextual, you can't.

If you're talking about physical systems, any transfer function that's not of the form $\frac{A}{s^n}$ has feedback, because the only way to make a non-zero pole is with feedback.

So in theory, non-zero poles means there's feedback. By this same theory, any zeros that aren't at infinity (i.e., a polynomial of any degree in the numerator) at least means that there are parallel paths -- but there's no way of knowing if these paths are due to feedforward, or simply parallel paths.

However, if you mean that you want to hypothesize a block diagram that has blocks with "full" transfer functions (polynomials of non-zero degree) in the numerator and the denominator, then no, you can't tell from the resulting transfer function whether there's feedback or not.

As an extreme example, if I were designing an industrial machine, I might want to buy a motor for it. These days, it's more common than not to buy a motor that comes with its own built-in controller, to which you provide power, and commands for torque, speed, or certain motions.

If you were designing that motor controller, then your block diagram would have feedback, and possibly feed-forward. But when I buy the motor, at best I'm just going to get the motor's transfer function*. So for me that motor is just a transfer function, and I would just have to know that because I'm buying a motor-controller unit, that feedback is happening inside, somewhere.

* More likely, I'm going to get a guarantee that it's got certain maximum settling times, or minimum frequencies at which the amplitude drop and phase shift become significant.

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All realizable causal IIR (Infinite Impulse Response) systems essentially have feedback. In discrete time systems this is given by poles that are not at the origin. (If the poles are inside the unit circle, we further know the system is stable).

Given a generic transfer function as follows with zeros in the numerator ($z_1, z_2 \ldots$) and poles in the denominator ($p_1, p_2 \ldots$):

$$H(z) = \frac{(z-z_1)(z-z_2) \ldots}{(z-p_1)(z-p_2) \ldots}$$

If the denominator has all poles equal to zeros (such that then denominator is simply $z^M$ for some power $M$, then the system does not have feedback, or at least is functionally identical to such a system that has no feedback (in case internally there is pole zero cancellation and a sub-component does have feedback which we would not be able to discern).

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