Minimal Controllability Problems

Abstract

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 by a simple greedy heuristic which sequentially picks variables to maximize the rank increase of the controllability matrix. Experiments on Erdos-Renyi random graphs demonstrate this heuristic almost always succeeds at findings 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 Bii = 1, and else we set Bii = 0. Consequently, both of these variations are easily seen to be equivalent to the problem of finding the sparsest diagonal matrix.

Controllability metrics, limitations and algorithms for complex networks

“…It can be shown that, while Trace(W −1 K,T ) measures the average control energy over random target states, Det(W K,T ) is proportional to the volume of the ellipsoid containing the states that can be reached with a unit-energy control input. The selection of the control nodes for the optimization of these metrics is usually a computationally hard combinatorial problem [13], for which heuristics without performance guarantees and non-scalable optimization procedures have been proposed [21], [24], [25].…”

“…We depart from [5], [6], [8], [11], and analogously from [12], [13], [14], by adopting a quantitative measure of network controllability, namely the worst-case control energy, by characterizing tradeoffs between the difficulty of the control task and the number of control nodes and, finally, by proposing an open-loop control strategy suitable for complex networks.…”

Optimality of Fast-Matching Algorithms for Random Networks With Applications to Structural Controllability

“…ky k (t) > } for some arbitrary > 0, the first two conditions are satisfied by (23), (24). By the definition of asymptotic empirical degree distributions lim…”

Positive controllability of large-scale networks

“…A recent trend in the literature is to investigate the controllability of large-scale complex networks such as those appearing in a broad spectrum of scientific disciplines, ranging from Biology to Social Sciences, from Technology to Engineering. When no a-priori information is available, then an interesting problem investigated for instance in [2], [3] is to determine what is the minimal number of driver nodes that allows to guarantee controllability.…”

“…When instead a set of weights is given, then the construction of a minimal set of driver nodes can rely upon other well-established controllability tests. For instance in [6] a method based on the PBH test is introduced, while in [3] a thorough estimate of the minimal number of driver nodes is computed. Other characterizations of such "input selection" problem appear in [7], [8], [9].…”