  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.
