Controllability of a Class of Nonlinear Networked Systems

In this paper, controllability of a heterogeneous networked system, where each node is under a nonlinear perturbation is studied. The nodal systems are Multi-Input Multi-Output (MIMO) and Linear Time Invariant (LTI) with distinct nodal dynamics, distinct dimensions and distinct inner-coupling matrices. Each node in the networked system is subjected to a nonlinear perturbation of the Hölder class. A sufficient controllability condition for networked systems where nonlinearity occurs as nodal perturbations is derived. Note that this result works for a wider class of systems, as the class of Hölder continuous functions include sub-linear and Lipschitz functions. Further, as a consequence of this result, a sufficient controllability condition for systems where interactions among nodes are nonlinear is also proved. The results are established using the Boyd-Wong fixed point theorem, which is a generalization of the Banach's contraction principle. A computational algorithm for steering control is also established based on the fixed point iteration. Numerical examples are provided to illustrate the results.