Controllability maximization of large-scale systems using projected gradient method

Abstract

In this letter, we formulate two controllability maximization problems for large-scale networked dynamical systems such as brain networks: The first problem is a sparsity constraint optimization problem with a box constraint. The second problem is a modified problem of the first problem, in which the state transition matrix is Metzler. In other words, the second problem is a realization problem for a positive system. We develop a projected gradient method for solving the problems, and prove global convergence to a stationary point with locally linear convergence rate. The projections onto the constraints of the first and second problems are given explicitly. Numerical experiments using the proposed method provide some results that are difficult to deduce theoretically. In particular, the controllability characteristic is observed to change with increase in the parameter specifying sparsity, and the change rate appears to be dependent on the network structure.

Controllability scores for selecting control nodes of large-scale network systems

“…• h(W ) = − log det W , then Problem (17) coincides with Problem (9). • h(W ) = tr(W −1 ), then Problem (17) coincides with Problem (13).…”

“…Moreover, the authors in [19], [20] formulated the selection problems as discrete optimization problems which choose a column vector of B in (1) to optimize the metrics from a given candidate set. Furthermore, [16], [17], [20] have studied the problems using continuous optimization approaches.…”

“…Although it has been known that the approximate solutions are good in some numerical experiments, the solutions may be actually far away from the optimal solutions. • The objective function of the continuous optimization problems formulated in [16], [17] was the trace of the controllability Gramian. Although it is an approximation of the trace of the inverse controllability Gramian which denotes the the average of the minimum energy as explained in Section II-B, the maximization does not ensure controllability [14].…”

Minimal controllability problem on linear structural descriptor systems with forbidden nodes

“…• quantitative problems; the maximization problems of controllability metrics [8]- [12]. • qualitative problems; selecting input problems that render the system controllable [7], [13]- [16].…”

“…Fig.11:The bipartite graph representation of descriptor system (1) with (5) and(12). The bold edges represent s-arcs.…”