Widely convergent method for finding solutions of simultaneous nonlinear equations

The authors propose a new approach for solving the nonlinear problem, based on a model-reference adaptive control (MRAC). The algorithm is composed of three parts: a plant containing unknown parameters, a reference model for compactly specifying the desired output and an adaptation mechanism for updating the adjustable parameters. The structure of the adaptation mechanism is explicitly designed so that the asymptotical stability is guaranteed according to Lyapunov theorem. The method reformulates the algebraic nonlinear equations into a set of ordinary differential equations, whose equilibrium point corresponds to the solution of the original nonlinear problems. The stability and region of attraction coincide with entire state space no matter whether the parameters reside in solvable region or unsolvable region.

► A new approach for solving the nonlinear problem is proposed.
► An artificial dynamic system is built based on model-reference adaptive control.
► The asymptotical stability is guaranteed by Lyapunov theorem.
► The method overcomes numerical difficulties arising from the ill-conditioned cases.