State observers for nonlinear systems with slowly varying inputs

In recent years many authors pointed out that for nonlinear systems the property of state observability for zero input (drift-observability) does not imply observability for any input (uniform observability). This paper considers the problem of state observation for systems that do not satisfy the restrictive condition of uniform observability. In a previous work the authors have shown that drift-observability ensures existence of exponential observers if the input function is sufficiently small. In this work a condition weaker than uniform observability (denoted almost-uniform observability) is found for existence of exponential observers in those cases in which the input is bounded and its derivative is sufficiently small (slowly varying input). The observer for small inputs and the observer for small input derivative are both presented in this paper. Numerical results show the effectiveness of the proposed observers.