# Stability of two systems in series (Controls Engineering)

1. Mar 28, 2013

### nebbione

Hi everyone, one my textbook there is written (if two system blocks are asintotically stable and are in series, their product will be asintotically stable), but I've heard that sometimes the transfer function of the controller could be not asintotically stable in some cases (see p.i.p. condition), so my question is, since the transfer function of the controller and the transfer function ofmy system are in series, my question is, their product in series will not be asintotycally stable hence my total system won't be stable ??
How does it works for a instable transfer function of my controller ?? And in which cases should i use it ?

2. Mar 28, 2013

### milesyoung

Systems in series are equivalent to a single system with zeros and poles that are the union of the zeros and poles of the individual systems, respectively. Unless you have zeros to cancel any unstable poles, which you won't in practice, the equivalent system will be unstable.

If you see an example with a controller that has poles in the right half-plane then it's probably in some kind of feedback configuration.

Last edited: Mar 28, 2013
3. Mar 28, 2013

### nebbione

so the series of an unstable and a stable system can be a stable system ? right ? Becuase i may have an unstable controller with zeros that cancel some unstable pole of the transfer function of the system right ?

4. Mar 28, 2013

### milesyoung

You mean a stable controller? Otherwise you'd have to deal with those poles aswell.

It's perfectly valid to cancel a pole with a zero, mathematically speaking. In practice, though, you won't be able to exactly cancel a pole (you won't know its exact value and if you did you'd have no hope of producing a controller with a zero to match) and anything but an exact cancellation would yield an unstable system.