Observer-Based Realization of Control Systems

TL;DR

This paper introduces a new model reduction method for large-scale systems using function-type control systems and semi-tensor products, enabling observer-based realizations and extensions to nonlinear systems.

Contribution

It presents a novel framework utilizing semi-tensor products and bridge matrices for constructing observer-based realizations of control systems, including nonlinear extensions.

Findings

Established necessary and sufficient conditions for exact observer realizations.

Developed extended observer-based realizations when exact ones do not exist.

Connected the minimal feedback extended realization to Kalman's minimal realization.

Abstract

A novel model reduction framework for large-scale complex systems is proposed by introducing function-type dynamic control systems via the dimension-keeping semi-tensor product (DK-STP) of matrices. Utilizing bridge matrices, the DK-STP facilitates the construction of an approximate observer-based realization (OR) of a linear control system in the form of a function-type control system, where the functions serve as observers. A necessary and sufficient condition is established for the OR-system to admit exact observer dynamics. When an exact OR-system does not exist, an extended OR-system is developed by incorporating the original system's observers into its state. Furthermore, a minimal feedback extended OR-system is constructed, and its relationship to Kalman's minimal realization is analyzed. Finally, the proposed approach is extended to nonlinear control-affine systems.