From Control to Mathematics-Part II: Observability-Based Design for Iterative Methods in Solving Linear Equations

TL;DR

This paper introduces an observer-based iterative method inspired by control theory to solve linear algebraic equations efficiently, achieving exponential or finite-step convergence and bridging the gap between classical iterative learning control and feedback control.

Contribution

It presents a novel observer-based approach for solving LAEs, connecting control design with mathematical problem-solving, and enabling finite-iteration solutions and improved convergence properties.

Findings

All solutions can be found exponentially fast or monotonically.

Finite iteration convergence is achieved using deadbeat control design.

The method bridges classical ILC and feedback control approaches.

Abstract

The control approaches generally resort to the tools from the mathematics, but whether and how the mathematics can benefit from the control approaches is unclear. This paper aims to bring the "control design" idea into the mathematics by providing an observer-based iterative method that focuses on solving linear algebraic equations (LAEs). An inherent relationship is revealed between the problem-solving of LAEs and the design of observer-based control systems, with which the iterative method for solving LAEs is exploited based on the design of the basic state observers. It is shown that all (least squares) solutions for any (un)solvable LAEs can be determined exponentially fast or monotonically with different selections of initial conditions. By integrating the design idea of the deadbeat control, the solving of LAEs can be achieved within only finite iterations. In particular, our proposed iterative method can be leveraged to develop a new observer-based design algorithm to realize the perfect tracking objective of conventional two-dimensional iterative learning control (ILC) systems, where the gap between classical ILC design and popular feedback-based control design is narrowed.