Differential Stability and Design of Reduced Order Observers for Nonlinear Systems

In this paper, differential stability is introduced for nonlinear systems, and this concept is further exploited in reduced order observer design for nonlinear systems with nonlinearities of unmeasured state variables, a more general class of nonlinear systems than the systems with linear observer errors. It has been shown that if the dynamics of unmeasured state variables under a certain state transformation is differentially stable, a reduced order observer can be designed to produce an asymptotic estimate of unmeasured state variables. A systematic design method is then introduced for a class of multi-output nonlinear systems when the nonlinear terms of unmeasured state variables enter the system through a coupling matrix. For such a system, it is found that a reduced observer can be designed if the linear part with the coupling matrix as the input matrix has no unstable invariant zeros. A further exploitation is presented for a class of single-output nonlinear systems with nonlinearity of unmeasured state variables. In this case, the coupling vector is allowed to be a vector field which depends on the system output.