Observer Design and Sensor Location in Distributed Parameter Bioreactors 2

Abstract

Abstract A nonlinear distributed parameter bioreactor is studied in this paper. The system model comprises two state variables which describe the biomass of the microorganisms and the concentration of the growth limiting substrate distributed along the reactor. An orthogonal collocation method is applied to convert the partial differential equation (PDE) model into a nonlinear ordinary differential equation (ODE) system. The resulting ODE system is linearized around the steady state of the operational profile of the original PDE model. Observability of the PDE system is studied based on the linearized ODE model of the system. A criterion of relative observability, the concept also discussed in the paper, is analyzed as a function of the number and position of the substrate sensors located along the reactor tube. The ratio of the smallest and largest singular values of the linearized observability matrix modified by taking into account the numerical accuracy of the personal computer is used in the observability calculations. Relative observability surfaces as a function of sensor positions are calculated in the case of one, two, three, and four sensors, and for several different locations and numbers of the collocation points at which the ODE model describes the behaviour of the PDE system. The actual observer design is based on placing the observer poles of the model linearized around the steady state. The final observer is nonlinear including the measurement error gain of the linearized design. This technique results in a locally stable observer. Some typical state observation results are also presented having optimally and also poorly located sensors.