Design and Evaluation of a Robust Observer Using Dead-Zone Lyapunov Functions—Application to Reaction Rate Estimation in Bioprocesses

Abstract

This paper addresses the design and evaluation of a robust observer for second order bioprocesses considering unknown bounded disturbance terms and uncertainty in the dynamics of the unknown and known states. The observer design and the stability analysis are based on dead-zone Lyapunov functions, and a detailed procedure is provided. The transient response bounds and the convergence region of the unknown observer error are determined in terms of the disturbance bounds, considering persistent but bounded disturbances in the dynamics of both the known and unknown observer errors. This is a significant contribution to closely related observer design studies, in which the transient response bounds are determined, but persistent and bounded disturbances are not considered in the dynamics of the known observer error. Other important contributions are: (i) the procedure for defining the observer parameters is significantly simpler than common observer designs, since a solution to the Ricatti equation, solution to LMI constraints, or the accomplishment of eigenvalue inequality conditions are not required; (ii) discontinuous signals are not used in the observer; and (iii) the effect of the gain sign associated with the unknown state in the dynamics of the known state is explicitly and clearly considered in the observer design and in the convergence study. In addition, the guidelines for selecting the observer parameters are provided. Numerical simulation confirms the stability analysis results: the observer errors converge within a short time, with a low estimation error, if observer-parameters are properly defined.