On Sliding Mode Observers for Systems with Unknown Inputs

This paper considers the problem of designing an observer for a linear system subject to unknown inputs. This problem has been extensively studied in the literature with respect to both linear and nonlinear (sliding mode) observers. Necessary and sufficient conditions to enable a linear unknown input observer to be designed have been established for many years. One way to express these conditions is that the transfer function matrix between the unknown input and the measured output must be minimum phase and relative degree one. Identical conditions must be met in order to design a 'classical' sliding mode observer for the same problem. This paper shows how the relative degree condition can be weakened if a classical sliding mode observer is combined with sliding mode exact differentiators to essentially generate additional independent output signals from the available measurements. A simple pedagogical example demonstrates the efficacy of the approach