Some Results on Parameter Identification of Nonlinear Systems

A priori global identifiability deals with the (theoretical) uniqueness of solutions of model parameters from noise free input–output data. This issue is particularly critical when dealing with physiological systems where a different numerical estimate can distinguish a pathological state from a normal state. Recently, differential algebra tools have been applied to study identifiability of dynamic systems described by polynomial or rational equations. These methods are based on elimination theory for algebraic differential systems, the main tool being the computation of the so-called characteristic set of a differential ideal associated to the polynomials defining the dynamic system. This characteristic set can be found by symbolic computation and provides the so-called exhaustive summary of the model.The reparametrization of the input–output relation of the system by the exhaustive summary plays a major role not only in a priori identifiability but also in parameter estimation of nonlinear models. It is in fact a linear reparametrization and can be used to derive explicit one-shot least-squares estimates of the parameters. This allows to avoid the usual bottleneck of nonlinear parameter optimization which has to be performed by iterative optimization routines which are often unreliable, e.g. they give no guarantee of converging to a true minimum, and hence require time-consuming random search in the parameter space. Our algorithm has been tested in one- and two-dimensional Michaelis–Menten model where the choice of initial values of the parameters is critical since nonlinear least-squares minimization problem shows many local minima. Our method structurally has only one minimum and does not require initial values for the unknown parameters. However, when noise in the data is present, the optimization problem is highly ill-conditioned and the solution of the problem in the general case is still under study.