Given a linear system, we consider the problem of finding a small set of variables to affect with an input so that the resulting system is controllable. We show that this problem is NP-hard; indeed, we show that even approximating the minimum number of variables that need to be affected within a multiplicative factor of c log n is NP-hard for some positive c. On the positive side, we show it is possible to find sets of variables matching this inapproximability barrier in polynomial time. This can be done with a simple greedy heuristic which sequentially picks variables to maximize the rank increase of the controllability matrix. Experiments on Erdos-Renyi random graphs that demonstrate this heuristic almost always succeed at findingng the minimum number of variables. 1 One may also consider variations in which we search for a matrix B ∈ R n×n renderingẋ = Ax + B u controllable while seeking to minimize either the number of nonzero entries of B , or the number of rows of B with a nonzero entry (representing the number of components of the system affected). However, for any such matrix B , we can easily construct a diagonal matrix B rendering the system controllable without increasing the number of nonzero entries, or the number of rows with a nonzero entry: indeed, if the i'th row of B contains a nonzero entry, we simply set B ii = 1, and else we set B ii = 0. Consequently, both of these variations are easily seen to be equivalent to the problem of finding the sparsest diagonal matrix. 2325-5870
Reference REF demonstrates that the problem of finding a small set of variables to affect with an input which guarantees controllability of the resulting system is NP-hard and provides a greedy heuristic that has good empirical performance.
Minimal Controllability Problems
IEEE Transactions on Control of Network Systems
IEEE Transactions on Control of Network Systems
Mathematics Computer Science