Cascade observer design for a class of nonlinear uncertain systems: Application to bioreactor

The present work proposes a state observer with a cascade structure for a class of nonlinear systems in the presence of uncertainties in the state equations and an arbitrarily long delay in the output. The first system in the cascade allows to estimate the delayed state while each of the remaining systems is a predictor. Each predictor estimates the state of the preceding one with a prediction horizon equal to a fraction of the time delay in such a way that the state of the last predictor is an estimate of the system actual state. The design of the observer is achieved by assuming a set of conditions under which the ultimate boundedness of the estimation error is established. It is in particular shown that in the absence of uncertainties, the observation error converges exponentially to zero. In the presence of uncertainties, the asymptotic observation error remains in a ball which radius depends on the delay magnitude and can be decreased by appropriately choosing the cascade length and the observer design parameters. The performance of the proposed observer and its main properties are highlighted through a typical bioreactor model.