Controllability and Observability of Heterogeneous Networked Systems with Non-Uniform Node Dimensions and Distinct Inner-Coupling Matrices

In this paper, controllability and observability of a heterogeneous networked system with Linear Time Invariant (LTI) nodal systems having Multiple-Inputs and Multiple-Outputs (MIMO) aligned in a weighted and directed network topology are studied. Apart from the heterogenity in nodal dynamics, the inner-coupling matrices that quantify the interactions among nodes are also different. In contrast to the existing literature, the system under consideration has distinct node dimensions, which adds to the generality. Necessary and sufficient conditions for controllability and observability as well as certain necessary conditions for controllability of a class of networked systems are established. These conditions show the dependence of network controllability and observability on various node and network-specific factors. As a practical application, a three-sector economy is modelled as a heterogeneous networked system with distinct node dimensions and its controllability is analysed. Computational time in floating point operations (flops) of the proposed methods are estimated, which indicates their efficiency on comparison with the classical conditions. This is illustrated by computational comparison of the existing and proposed schemes, applied to a randomly generated networked system. Also, robustness of the proposed schemes are analysed with the example of randomly generated networked systems. All the results are supported with illustrative numerical examples.