Adaptive state observers for a class of nonlinear systems: The non-triangular and non-state-affine case

We are interested in this paper in the problem of designing a globally convergent adaptive observer for nonlinear systems that contain nonlinear terms in the unmeasurable part of the state. Therefore, they do not fit into the category of state-affine systems for which several adaptive observers have been reported. Furthermore, the system is not in a normal triangular form, hence it is not amenable for high-gain observer designs. To solve the problem we introduce two new technical tools, first, the construction of a mixed linear regression equation, dependent on the unmeasurable state and the unknown parameters. Second, the proof that the error dynamics can be immersed in a lower dimension target dynamics with the desired convergence characteristics. Although it is assumed that the derivative of the unmeasurable part of the state is known, it is shown in the paper that this is the case for several practical examples. Simulation results with a van der Pol-like system are used to illustrate the performance of the proposed adaptive observer.