Eigenvalue problems arising in the control of a distributed-parameter bioreactor

A distributed-parameter model of a continuous-flow fixed-bed reactor is studied. The main emphasis lies on a structural property of the partial differential equation (PDE) system model. This property, which in lumped parameter systems is called the nonminimum phase property, has certain implications in the controller design. The controller design for the PDE model is constructed via semidiscretisation. The only space variable of the PDE model is discretised by using Galerkin's finite element method (FEM). For some parameter values of the PDE model the linearising control of the semidiscretised model results in unstable behaviour in the sense that the zero dynamics of the model is unstable. This same unstable behaviour was also earlier observed in using the orthogonal collocation for the semidiscretisation. Connections between the location of the zeros of the original PDE model linearised around its steady-state solution and the stability/instability properties of the linearising control of the semidiscrete model are discussed in relation to the same issues in lumped-parameter differential system models.