Observability and identifiability of nonlinear systems with applications in biology

Doctoral thesis, 2007

This thesis concerns the properties of observability and identifiability of nonlinear systems. It consists of two parts, the first dealing with systems of ordinary differential equations and the second with delay-differential equations with discrete time delays. The first part presents a review of two different approaches to study the observability of nonlinear ODE-systems found in literature. The differential-geometric and algebraic approaches both lead to the so-called rank test where the observability of a control system is determined by calculating the dimension of the space spanned by gradients of the time-derivatives of its output functions. We show that for analytic systems affine in the input variables, the number of time-derivatives of the output that have to be considered in the rank test is limited by the number of state variables. Parameter identifiability is a special case of the observability problem. A case study is presented in which the parameter identifiability of a previously published kinetic model for the metabolism of S. cerevisiae (baker's yeast) has been analysed. The results show that some of the model parameters cannot be identified from any set of experimental data. The general features of kinetic models of metabolism are examined and shown to allow a simplified identifiability analysis, where all sources of structural unidentifiability are to be found in single reaction rate expressions. We show how the assumption of an algebraic relation between concentrations in metabolic models can cause parameters to be unidentifiable. The second part concerning delay systems begins by an introduction to the algebraic framework of modules over noncommutative rings. We then present both previously published and new results on the problem of observability. New results are shown on the problems of state elimination and characterisation of the identifiability of time-lag parameters. Their identifiability is determined by the form of the system's input-output representation. Linear-algebraic criteria are formulated to decide the identifiability of the delay parameters which eliminate the need for explicit computation of the input-output equations. The criteria are applied in the analysis of biological models from the literature.