On New Laplacian Matrix with a User-Assigned Nullspace in Distributed Control of Multiagent Systems

Most distributed control results utilize the benchmark consensus algorithm, which is built on the well-known Laplacian matrix whose nullspace spans the vector of ones. Since this algorithm is the key building block for many distributed control architectures, extensions of this algorithms are also predicated on this Laplacian matrix. To this end, we explore how one can generalize the Laplacian nullspace, which can span any vector with positive elements, to pave the way for composing complex cooperative behaviors in multiagent systems. Specifically, a new Laplacian matrix is introduced for undirected and connected graphs that generalizes the well-known, standard Laplacian matrix, where it is based on a desired, user-assigned nullspace. We first give the mathematical definition of this Laplacian matrix and show that it inherits some fundamental properties of the standard Laplacian matrix. We then present distributed control architectures for convergence to the desired nullspace and for convergence to a specific vector within that nullspace. Finally, an application of the proposed Laplacian matrix to formation tracking and scaling problem is given.