State observer design for non-linear uncertain time-varying delay systems with unmatched uncertainties

State feedback is a quite useful method in system control design; however, some of the system states cannot be directly measured. Thus, acquiring the desired properties of state feedback control is a barrier. Fortunately, the state observer could be an elegant solution to such a problem. In this paper, based on sliding mode control (SMC) theory, a new sliding mode observer (SMO) for a class of non-linear uncertain time-varying delay systems with unmatched uncertainties is proposed. As is well known, during the sliding mode, the system behaviour has an invariance property, which is independent of matched uncertainty, but it is not invariant to the unmatched uncertainty and this usually restricts the applications of the results. Therefore, we make an attempt to extend the famous Luenberger observer to generate a new SMO scheme. With the proposed SMO, the estimation errors asymptotically tend to zero if the matching condition is satisfied, and the effects of the unmatched parts on the estimation errors can be uniformly ultimately bounded if the matching condition is not satisfied. A numerical example with a series of simulation studies is given to demonstrate the effectiveness of the proposed approach.