Stabilization Theory for Active Multi Port Networks

TL;DR

This paper develops a stabilization theory for active multi-port networks, enabling stable interconnections without relying on complex feedback signal flow graphs, thus facilitating the design of stable interconnected active networks.

Contribution

It introduces a direct port connection-based stabilization theory for active multi-port networks, avoiding the need for signal flow graph formulations and enabling all stabilizing port compensations.

Findings

Provides a stable port interconnection framework.

Results in an affine parametrized network function.

Enables design of all stabilizing port compensations.

Abstract

This paper proposes a theory for designing stable interconnection of linear active multi-port networks at the ports. Such interconnections can lead to unstable networks even if the original networks are stable with respect to bounded port excitations. Hence such a theory is necessary for realising interconnections of active multiport networks. Stabilization theory of linear feedback systems using stable coprime factorizations of transfer functions has been well known. This theory witnessed glorious developments in recent past culminating into the $H_{\infty}$ approach to design of feedback systems. However these important developments have seldom been utilized for network interconnections due to the difficulty of realizing feedback signal flow graph for multi-port networks with inputs and outputs as port sources and responses. This paper resolves this problem by developing the stabilization theory directly in terms of port connection description without formulation in terms of signal flow graph of the implicit feedback connection. The stable port interconnection results into an affine parametrized network function in which the free parameter is itself a stable network function and describes all stabilizing port compensations of a given network.