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?
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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.
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$\begingroup$ I know you don't need an equivalent number of inputs and outputs but I know my controllability matrix is not of full rank. Thanks for the comment on Kalman decomposition, which looks like it might be useful in my case. Any recommendations on books? (I don't have any formal training in the control theory as you might have guessed, but my linear algebra is decent, so I'm working off Wikipedia and other stuff I can find online.) $\endgroup$ Commented Feb 15, 2014 at 18:41
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$\begingroup$ The 3 books below are excellent. The 1st is more basic; the 2nd very clear, excellent textbook; the last more mathematical, covers more topics in less detail. There are other fine books, I prefer these 3. Linear systems theory, JP Hespanha. Linear System Theory, 2nd edition, WJ Rugh. Mathematical Control Theory, 2nd edition, ED Sontag. $\endgroup$– PaitCommented Feb 17, 2014 at 13:23