Control of networked systems in the graph-frequency domain
Juan Andrés Bazerque, Pablo Monzón · 2017
This paper studies the stability and control of networked systems from the perspective of the new field of signal processing over graphs. Specifically, we reformulate the linear quadratic optimal controller as a graph filter by demonstrating that it becomes separable when applying the graph Fourier transform. Accordingly, the graph-frequency components of the input signal are processed independently by a set of parallel controllers which are given in closed form by solving their Riccati equations. Additionally, we prove that the joint graph-and-temporal-frequency transfer function of the controlled system satisfies the condition for stability, that is, that the complement of its region of convergence fits inside a cylinder of unit radius. The results are universal in the sense that they hold true for any graph-frequency and do not depend on the specific eigenvalues of the network shift operator. Numerical tests in an directed circulant graph show that the unstable poles corresponding to graph-frequencies larger than one are shrunk towards the origin.