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

TL;DR

This paper introduces a controllability-based iterative method for solving linear algebraic equations by linking LAE solvability to control system controllability, enabling finite-step solutions and least squares solutions through a novel tracking property.

Contribution

It presents a new controllability-based design framework for iterative methods solving LAEs, including finite-step deadbeat solutions and least squares solutions, connecting control theory with linear algebra.

Findings

LAE solvability is equivalent to control system controllability.

The proposed ILC method can find all solutions, including least squares, for unsolvable LAEs.

Finite iteration convergence is achieved via deadbeat control design.

Abstract

In the interaction between control and mathematics, mathematical tools are fundamental for all the control methods, but it is unclear how control impacts mathematics. This is the first part of our paper that attempts to give an answer with focus on solving linear algebraic equations (LAEs) from the perspective of systems and control, where it mainly introduces the controllability-based design results. By proposing an iterative method that integrates a learning control mechanism, a class of tracking problems for iterative learning control (ILC) is explored for the problem solving of LAEs. A trackability property of ILC is newly developed, by which analysis and synthesis results are established to disclose the equivalence between the solvability of LAEs and the controllability of discrete control systems. Hence, LAEs can be solved by equivalently achieving the perfect tracking tasks of resulting ILC systems via the classic state feedback-based design and analysis methods. It is shown that the solutions for any solvable LAE can all be calculated with different selections of the initial input. Moreover, the presented ILC method is applicable to determining all the least squares solutions of any unsolvable LAE. In particular, a deadbeat design is incorporated to ILC such that the solving of LAEs can be completed within finite iteration steps. The trackability property is also generalized to conventional two-dimensional ILC systems, which creates feedback-based methods, instead of the common used contraction mapping-based methods, for the design and convergence analysis of ILC.