Numerical nonlinear observers using pseudo‐Newton‐type solvers

In constructing a globally convergent numerical nonlinear observer of Newton‐type for a continuous‐time nonlinear system, a globally convergent nonlinear equation solver with a guaranteed rate of convergence is necessary. In particular, the solver should be Jacobian free, because an analytic form of the state transition map of the nonlinear system is generally unavailable. In this paper, two Jacobian‐free nonlinear equation solvers of pseudo‐Newton type that fulfill these requirements are proposed. One of them is based on the finite difference approximation of the Jacobian with variable step size together with the line search. The other uses a similar idea, but the estimate of the Jacobian is mostly updated through a BFGS‐type law. Then, by using these solvers, globally stable numerical nonlinear observers are constructed. Numerical results are included to illustrate the effectiveness of the proposed methods.