Controllability maximization of large-scale systems using projected gradient method

Kazuhiro Sato,Akiko Takeda

doi:10.1109/lcsys.2020.2993983

Abstract

In this work, 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 non-trivial results. 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.

Highlights

C ONTROLLABILITY, which refers to the possibility to change the present network state to a desired state is a fundamental concept in large-scale networked dynamical systems [1]–[9]
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 that determines the structure of the matrix A
We proved that a sequence generated by our method has global convergence with locally linear convergence rate

Summary

INTRODUCTION

C ONTROLLABILITY, which refers to the possibility to change the present network state to a desired state is a fundamental concept in large-scale networked dynamical systems [1]–[9]. BV (t) was considered as the matrix B, where B ∈ Rn×m and V (t) ∈ {0, 1}m×m denote a fixed constant matrix and time-varying diagonal matrix, respectively, and an optimization problem to determine the diagonal entries of V (t) was examined. The methods to determine the M candidate column vectors of B in [15]– [17] and the matrix B in [20], [21] for a large-scale networked dynamical system remain unclear To overcome this limitation, in this work, we consider the controllability maximization problems from a different perspective than those considered in [15]–[17], [20], [21].

PROBLEM SETTINGS

PROJECTED GRADIENT METHODS FOR PROBLEMS 1 AND 2

NUMERICAL EXPERIMENTS

CONCLUSION