# Poles/Zeros of a Control System

1. May 24, 2015

### KiltedEngineer

ME student here. I have managed to get through two control theory courses at my university, which I have managed to get A+s in despite having no idea why I am doing all of it. I understand the math (i.e. why RHP poles are not good for the system stability) but do not understand what any of it means and how it can be applied to the real world. I was wondering if anyone here can clear up some of my confusion in laymen terms.

I have read on some website about how electric motors have two poles, subway cars have 9, aircraft may have 23 and spacecraft may have upwards of 100. What exactly do the poles represent, and what induces them into a system? Why do these systems have poles and what causes poles to be present?

Any help is very much appreciated.

2. May 24, 2015

### Hesch

Zeros and poles are not physical quantities, but mathematical quantities. You can calculate models and transfer function of a dc-motor by means of Laplace transforms or z_transforms, or you can measure them. For a typical dc-motor you could find the transfer function:

By Laplace transform: H(s) = 3.2 / ( s2+10.1s+1). Setting the denominator = 0 you can find the roots: s = -0.1 ∨ s = -10. These roots are the poles of the motor. Setting the numerator = 0, you can in the same way find the zeroes ( there are none here ).

By z-transform: H(z) = ( 3z + 3.6 ) / (z2 - 1.05z + 0.095) you can find the poles: z = 0.1 ∨ z = 0.95 and the zero: z = -1.2.

So mathematical quantities that depends on which transformation is used.

Now you want to control this motor, making a control loop wherein you add filter blocks, for example an integrator ( G(s) = 1/s or G(z) = z/(z-1) ). Thereby you add an extra pole to the system. In the same way you can add extra zeroes to the system. The purpose of doing so is to change the qualities of the controlled system. You may analyze the system by means of root-locus or other tools, determining where to add poles and zeroes. Example: If you make a speed controller for a dc-motor and you want its speed to be independent on the load, you must add an integrator in the control loop, but maybe now the loop will be unstable, so you must add a zero to stabilize it. You can now vary the location of the zero and/or you can vary the amplification in the loop, thereby plotting a root-locus and determine the limits wherein the controlled system is stable. Changing the amplification, A, the root-locus will start at the poles ( A = 0 ) and will be attracted by the zeroes as A is increased.

Last edited: May 24, 2015
3. May 24, 2015

### jim hardy

That's why educators should include practical problems in their textbooks.

That math predicts the behavior of physical systems.
Historically it's relatively new. German textbooks were among the War Prizes brought home at end of WW2. Hence the phrase "Rocket Science".

You might search on terms servomechanisms and motor control for practical applications.