# Affine Dependence of Network Controllability/Observability on Its Subsystem Parameters and Connections

## TL;DR

This paper presents an affine parameter-dependent condition for network controllability and observability, allowing independent subsystem analysis and avoiding NP-hard structural conditions, thus advancing the design of controllable and observable networked systems.

## Contribution

It introduces a necessary and sufficient affine condition for network controllability/observability based on subsystem parameters, simplifying analysis and design.

## Key findings

- Affine conditions depend on subsystem parameters independently.
- Subsystem input/output rank conditions influence controllability/observability.
- Normal rank conditions are removed, simplifying system analysis.

## Abstract

This paper investigates observability/controllability of a networked dynamic system (NDS) in which system matrices of its subsystems are expressed through linear fractional transformations (LFT). Some relations have been obtained between this NDS and descriptor systems about their observability/controllability. A necessary and sufficient condition is established with the associated matrices depending affinely on subsystem parameters/connections. An attractive property of this condition is that all the required calculations are performed independently on each individual subsystem. Except well-posedness, not any other conditions are asked for subsystem parameters/connections. This is in sharp contrast to recent results on structural observability/controllability which is proven to be NP hard. Some characteristics are established for a subsystem which are helpful in constructing an observable/controllable NDS. It has been made clear that subsystems with an input matrix of full column rank are helpful in constructing an observable NDS, while subsystems with an output matrix of full row rank are helpful in constructing a controllable NDS. These results are extended to an NDS with descriptor form subsystems. As a byproduct, the full normal rank condition of previous works on network observability/controllability has been completely removed. On the other hand, satisfaction of this condition is shown to be appreciative in building an observable/controllability NDS.