Feedback of Control on Mathematics: Bettering Iterative Methods by Observer System Design

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 article 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 the different selections of initial conditions. Moreover, the general solution subspace and particular (least squares) solutions of LAEs are related closely with the unobservable subspace and observable states of their associated observer systems, respectively. Through incorporating the design idea of the deadbeat control, the solving of LAEs can be realized within only finite iterations. In particular, our proposed iterative method can be leveraged to develop a new observer-based design algorithm for traditional two-dimensional iterative learning control to realize the perfect tracking tasks.