Parameter Estimation of Discrete and Continuous-Time Physical Models: A Similarity Transformation Approach

The fitting of physical dynamical models to stimulus-response data such as the chemical concentration measured after a gas has been released to the environment, or the plasma concentration measured after an intravenous or oral input of a drug, are important problems in the area of system identification. Using models of different structures, one can obtain relevant statistical information on the parameters of the model from an array of software packages available in the literature. A meaningful interpretation of these parameters requires that in the presence of error-free data and an error-free model structure, a unique solution for the model parameters is guaranteed. This problem is known as a priori identifiability. Once the model is deemed identifiable, the parameters are then obtained, usually via a nonlinear least squares technique. In addition to identifiability, there is the problem of convergence of the parameters to the true values. It is a known fact that nonlinear parameter estimation algorithms do not always converge to the true parameter set. This is due to the fact that estimating the parameters of a nonlinear model can at times be an ill-conditioned problem. In this paper we use the same state space analysis techniques used to determine identifiability, to estimate the model parameters in a linear fashion. We approach the problem from a system identification point of view and then take advantage of the similarity transformation between the physical model and the identified model. We formulate the similarity relations and then transform them into a null space problem whose solution leads to the physical parameters. The novelty of our approach is in the use of a state space system identification algorithm to identify a black-box system, followed by a physical parameter extraction step using robust numerical tools such as the singular value decomposition.