On the observer matching condition and unknown input observer design based on the system left-invertibility concept:

This paper considers the problems of unknown input observer (UIO) designs when the so-called observer matching condition (OMC) is not satisfied. Firstly, the system left-invertibility (SLI) concept is investigated in detail and some new criteria are given. Secondly, based on the SLI concept, an augmented output vector formed partly by the original measurable output and partly by the auxiliary output is obtained. By this constructed output vector, the new system is allowed to be transformed into an equivalent linear dynamic system which contains no unknown input, and thus a simple Luenberger observer can be constructed to simultaneously estimate the original states and unknown inputs. Compared with the existing results, the present UIO design can be taken under the single strong detectability condition and the OMC is not needed any more. Finally, the simulations of a single-link flexible joint robotic model and a linearized vertical take-off and landing helicopter model are given to show the effectiveness of our method and its superiority over existing ones.