Observer design for a class of disturbed nonlinear systems with time-varying delayed outputs using mixed time-continuous and sampled measurements

A state observer is proposed for a class of uniformly observable systems involving bounded disturbances. The system outputs are delayed with a known (time-varying) delay and their measurements are continuously available over some time horizons and sampled over other ones. The rational behind the proposed observer design consists in the consideration of an appropriate corrective term provided by a linear Delay Differential Equation (DDE), which accounts for the form under which the output measurements are available as well as for the presence of delay, through a well defined piece-wise continuous function depending on the sampling instants and on the output (time-varying) delay. Of particular interest, the proposed observer has the same dimension as the system and is of the high gain variety in the case where the magnitude of the output (time-varying) delay is relatively small. The arbitrarily long delays are handled thanks to a suitable observer structure composed of cascaded subsystems each one of which has the same dimension as the system. More specifically, the cascade observer is first designed in the constant long delay case before being extended to the time-varying long delay case using a suitable procedure. The convergence analysis is performed thanks to a comprehensive Lyapunov approach under a well-defined condition on the maximum value of the sampling partition diameter. The effectiveness of the proposed observer is emphasized through illustrative simulation results.