State observers and state-feedback controllers for a class of non-linear systems

A design methodology for observers and controllers for a class of single-input, single-output non-linear systems is developed. A non-linear observer form is defined, and a corresponding observer using non-linear observer gains is specified. The resulting error dynamics, which are non-linear, are asymptotically stable for a proper choice of those gains and bounded inputs and outputs. A non-linear controller form is defined and a corresponding controller, which results in linear closed-loop dynamics with arbitrary eigenvalue placement, is devised. Transformations from the class of interest to the defined forms are derived. In the observer case, the transformation to the observer form and the non-linear entries in the state matrix of that form can be determined separately, leading to a system of linear partial-differential equations—a major simplification. In the controller case, a simple, well-known system of linear partial-differential equations is obtained. The efficacy of the developed methodology is shown, by employing a non-linear, third-order model of a vehicle's longitudinal dynamics in the design process. Both here, and in other simulations, the latter results in a considerable improvement over a linearization/classical-design approach—especially in situations where the nonlinear effects are pronounced. Moreover, it has definite advantages over parameter scheduling, especially for systems where the non-linearities are functions of many state variables, resulting in a large number of operating points.