Stability of two systems in series (Controls Engineering)

[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 ?

 

[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.

 

[nebbione]

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

 

[milesyoung]

... because i may have an unstable controller with zeros that cancel some unstable pole of the transfer function of the system right ?

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.

 

[alphysicist]

The statement in your textbook is correct, as long as both system blocks are asymptotically stable, their product will also be asymptotically stable. However, as you mentioned, there are cases where the transfer function of the controller may not be asymptotically stable, which can affect the stability of the overall system.

In these cases, it is important to carefully design and analyze the controller to ensure that it does not introduce instability into the system. This can be done through methods such as root locus analysis, frequency response analysis, and pole placement techniques.

In general, it is best to use an asymptotically stable controller in order to maintain stability of the overall system. However, there may be situations where an unstable controller may be necessary, such as in systems with high gain or non-minimum phase systems.

Overall, the key is to carefully analyze and design the controller to ensure stability of the overall system. It is also important to continuously monitor and tune the controller to maintain stability as the system may change over time.