Spectrum computation and optimization for controllability Gramian of networked Laplacian systems with limited control placement

This paper investigates the problem of placing a given number of controls to optimize energy efficiency for a family of linear dynamical systems, whose structure is induced by the Laplacian of a square-grid network. To quantify the performance of control combinations, several metrics have been proposed based on the spectrum of the controllability Gramian. But commonly used algorithms to compute the spectrum are usually time-consuming. In this paper, we first classify five anchor symmetries of the network systems. Then motivated by various advantages of symmetric control combinations, we provide a method to compute the eigenvalues and eigenvectors of their controllability Gramians more efficiently. Specifically, we show that they can be expressed by those of two lower-dimensional matrices. Furthermore, our method can be applied for non-symmetric cases to provide upper and lower bounds for the spectrum of the controllability Gramians. Finally, by employing the sum of eigenvalues, i.e., the trace of controllability Gramian, as the objective function, we provide a closed-form algorithm to the spectrum optimization problem with a given number of controls subject to system controllability.