State Observers for Nonlinear Systems with Smooth/Bounded input

It is known that for nonlinear systems the drift-observability property (i.e. observability for zero input) is not sufficient to guarantee the existence of an asymptotic observer for any input. Many authors studied conditions on systems structure that ensure uniform observability (i.e. observability for any input). Conditions are available that define restrict classes of uniformly observable systems. This work considers the problem of state observation with exponential error rate for smooth nonlinear systems that do not meet conditions of uniform observability: conditions are given on the input, instead of on the system structure. It is shown that drift-observability, together with a smoothness/ boundedness condition on the input, is sufficient to ensure the existence of an exponential observer. Three types of observers are presented, that can be constructed under drift-observability assumption. One works well for systems with maximal relative degree or in the case of input sufficiently small. A second type of observer succeeds for systems with any relative degree in the case of input sufficiently smooth. The input derivatives up to a certain order are required for its implementation. Both observers ensure exponential convergence of exponential error to zero. A third observer suitable in the case of smooth input does not require input derivatives, and ensures exponential decay of the observation error below a prescribed level. Computer simulations close the paper.