On the need for switched-gain observers for non-monotonic nonlinear systems

This paper focuses on the challenges in observer design for nonlinear systems which are non-monotonic. A class of nonlinear systems is considered in which the process dynamics and output equations are both composed of nonlinear vector functions of scalar combinations of the states. The nonlinear functions are assumed to be differentiable with bounded derivatives. An observer design algorithm that requires solving just a single linear matrix inequality for exponentially convergent state estimation is developed. The developed algorithm works effectively when the involved nonlinear functions are monotonic. However, it fails when all or even some of the system functions are non-monotonic. Both numerical computation and analytical results show that the observer design LMI has no feasible solutions when either all output functions or all process dynamics functions are non-monotonic. Further, other constant gain LMI-based observer design methods from literature also fail when the involved nonlinear functions are all non-monotonic, no matter how small the Lipschitz constant or the Jacobian bounds of the nonlinearities. This limitation has not previously been recognized in observer design literature. To overcome this limitation, a hybrid observer that switches between multiple constant observer gains is developed that can provide global asymptotic stability for systems with non-monotonic nonlinear functions. Hybrid observers with switched gains enable existing observer design methods to be utilized for non-monotonic nonlinear functions with finite local extrema. The application of the developed hybrid observer to two motion estimation applications, one a vehicle position tracking problem on roads and another a piston position estimation problem for an industrial actuator, are demonstrated.