I teach the undergraduate course "Modern Control Theory".
Recently, I find this simple question puzzling:
Suppose we've got state equation dx/dt = Ax+Bu+f(t), where x is the state vector and f(t) is an arbitrary perturbation.
Why is the state x completely controllable if and only if the pair of matrices (A, B) are controllable, despite f(t)? How can this be proved?
It must be very simple to answer but I cannot find any reference by hand.
From the definition, it does not consider any disturbance/perturbation when talking about whether an LTI system is controllable. The problem is not well-posed when you say controllability of a system with disturbances.
See https://en.wikipedia.org/wiki/Controllability
Hi Ning Cai,
Your question is indeed interesting. I answer something from the top of my mind, it might be of some help. Consider f(t) a second input. It makes sense to ask if the state of the system can be driven anywhere in space by f(t). The answer should come from the same theory... so suppose f(t) is a ector of disturbances... then the "B" matrix associated to f(t) is the identity matrix (missing in your text). Then, of course the pair (A, I) is trivially controllable and what really matters is the other pair: (A, B).
Dr. Polat understands my question very well.
Yes it is a pertuabation cacelling problem. Here I'm only concerning the existence of cancellation, by u, but not the construction of it.
Whether or not there exists u to cancel the effect of f(t) is still puzzling to me.
Now I can only give a conservative condition, any f(t) can be counteracted by u(t) at any time if and only if the dimension of u is no less than x, meanwhile B's row rank is full.
Just to add something to the great answer provided by Ilham, note that controllability is defined for some T>0. It is one of these concepts that have little to do with real life, but we love due to its nice mathematical properties, and it allows us to define an essential concept such as minimal representation when we are jumping between time-domain and frequency-domain.
If you think in moving your state in a very small period of time, your designed input will be something very similar to a delta function. Evidently, any function f without deltas will have negligible with respect to my deltas. For the design of my deltas and derivative of deltas as input, the only real condition is the one given by Ilham, so f is irrelevant by moving your state fast enough such that \int_{0}^{T}exp(At)f(t)dt->0. If you are still not happy, then given the reachable state x(T), you always can design an input that move you to x(T)-\int_{0}^{T}exp(At)f(t)dt and the linearity of the system will do the rest.
I tried to write this argument of the deltas and its derivatives in my lecture notes when T=0^+, i.e. T=\lim_{h->0^+}h. In case it is helpful see page 64 here:
http://personalpages.manchester.ac.uk/staff/joaquin.carrascogomez/SS_notes.pdf
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