# Regularity/Controllability/Observability of an NDS with Descriptor Form Subsystems and Generalized LFTs

## TL;DR

This paper provides necessary and sufficient conditions for regularity, controllability, and observability of large-scale networked dynamic systems with descriptor subsystems, using matrix rank conditions based on the Kronecker canonical form.

## Contribution

It introduces a novel framework using generalized linear fractional transformations and matrix rank conditions for analyzing NDS properties independently on subsystems.

## Key findings

- Matrix rank conditions for regularity, controllability, and observability.
- Conditions depend affinely on subsystem parameters and connections.
- Explicit requirements for subsystem parameters to ensure controllability and observability.

## Abstract

This paper investigates regularity, controllability and observability for a networked dynamic system (NDS) with its subsystems being described in a descriptor form and system matrices of each subsystem being represented by a generalized linear fractional transformation (GLFT) of its parameters. Except a well-posedness condition, no any other constraints are put on either parameters or connections of a subsystem. Based on the Kronecker canonical form (KCF) of a matrix pencil, some matrix rank based necessary and sufficient conditions are established respectively for the regularity and complete controllability/observability of the NDS, in which the associated matrix depends affinely on both subsystem parameters and subsystem connections. These conditions keep the property that all the involved numerical computations are performed on each subsystem independently, which is attractive in the analysis and synthesis of a large scale NDS. Moreover, some explicit and easily checkable requirements are derived for subsystem dynamics/parameters with which a completely controllable/observable NDS can be constructed more easily.