Is there a formalized control-theory-like approach for uncontrollable systems?

Question

I'm basically trying to control a system to achieve a given set of outputs, but I don't actually have enough inputs to control all the outputs. Is there any formalized theory on how to achieve the set of outputs closest to the desired outputs? All of the literature I've read works on the assumption that the system is completely controllable. Is there perhaps a good way to transform the system in such a manner that controlling the controllable subset of the transformed systems states results in appropriate control over the original states?

Answer 1

You don't need one input to control each output. All you need is controllability. Controllability allows you to do pole assignment and, should you so choose, optimal control.

In the case of systems which are not controllable, you can use the Kalman decomposition to separate the controllable modes and the uncontrollable ones. If the latter are stable, you have a stabilizable system.

Look up the concepts mentioned above in a good linear control systems textbook and you shall find the answers to your questions, at least for the case of linear time-invariant systems given in state-space form.