# In PID control, what do the poles and zeros represent?

Whenever I read a text about control (e.g. PID control) it often mentions 'poles' and 'zeros'. What do they mean by that? What physical state does a pole or a zero describe?

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Ah I remember we learned those stuff in control, but I have forgotten them. Something about where some function gets to 0 or infinity (zeros and poles) and that there were some curves starting from zeros to poles in the s-space (was that after laplas transform?) or something like that. I remember the diagrams looked beautiful, but I don't remember anything else! –  Shahbaz Nov 19 '12 at 15:13

The function $T(\mathbf{x})$ that describes how ones input to a system maps to the output of the system is referred to as a transfer function.

For linear systems the transfer function can be written as $N(\mathbf{x})/D(\mathbf{x})$ where $N$ and $D$ are polynomials, i.e. $$T(\mathbf{x}) = {N(\mathbf{x})\over D(\mathbf{x})}$$

The zeros of the system are the values of $x$ that satisfy the statement $N(\mathbf{x}) = 0$. In other words they are the roots of the polynomial $N(\mathbf{x})$. As $N(\mathbf{x})$. approaches a zero, the numerator of the transfer function (and therefore the transfer function itself) approaches the value 0.

Similarly the poles of the system are the values of $x$ that satisfy the statement $D(\mathbf{x}) = 0$. In other words they are the roots of the polynomial $D(\mathbf{x})$. When $D(\mathbf{x})$ approaches a pole, the denominator of the transfer function approaches zero, and the value of the transfer function approaches infinity.

The poles and zeros allow us to understand how a system will react to various inputs. The zeros are interesting for their ability to block frequencies while the poles provide us information about the stability of the system. Generally we plot the poles and zeros in the complex plane and we say a system is bounded-input bounded-output (BIBO) stable if the poles are located in the left half of the complex plane (LHP - Left Half Plane).

Finally, when we design a controller we are in effect manipulating it's poles and zeros so as to achieve specific design parameters.

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Thanks, but I don't feel any the wiser. Can you explain what zeros and poles mean in a control context? –  Rocketmagnet Nov 19 '12 at 18:17
I have added a bit more per your request. I hope that helps. –  DaemonMaker Nov 19 '12 at 19:46
I think the problem here @Rocketmagnet is that this is a pretty broad topic. I would probably put it in the category of If you can imagine an entire book that answers your question, you’re asking too much. –  Mark Booth Nov 20 '12 at 0:27

These polynomial transfer functions occur, when you perform a Laplace transform on some linear differential equation which either actually describes your robot or is the result of linearizing the robot's dynamics at some desired state. Think of it like a "Taylor expansion" around that state.

The Laplace transform is the generalization of the Fourier transform to functions which are not periodic. In electrical engineering, the Laplace transform is interpreted as the representation of the system in the frequency domain, i.e. it describes, how the system transmits any frequencies from the input signal. Zeros then describe frequencies that don't get transmitted. And as already mentioned by DaemonMaker, poles are important when considering the system's stability: The transfer function of the system goes to infinity near the poles.

What they mean in a control context:

Poles: They tell you, if a system (that can also be a new system, in which you have inserted a feedback loop with a control law) is stable or not. Usually you want a system to be stable. So, you want all the poles of the system to be in the left half plane (i.e. the real parts of the poles must be smaller than zero). The poles are the eigenvalues of your system matrix. How far they are on the left half-plane tells you how fast the system converges to it's resting state. The further they are away from the imaginary axis, the faster the system converges.

Zeros: They can be convenient if you have a pole on the right half plane or still on the left half plane, but too close to the imaginary axis: By clever modification of your system, you can shift the zeros onto your unwanted poles to annihilate them.

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Can you add some images to illustrate this? –  Ian Nov 26 '12 at 20:34
Sorry for my long absense. Has to do with a lot of studying work I currently have to do. If still desired, I can add one as soon as I have time for it. –  Daniel Eberts Jan 26 '13 at 16:40