| Principal Differential Analysis (PDA) is an alternative parameter estimation technique for differential equation models in which basis functions (e.g., B-splines) are fitted to dynamic data. Derivatives of the resulting empirical expressions are used to avoid solving differential equations numerically when estimating parameters. Benefits and shortcomings of PDA were examined using a simple continuous stirred-tank reactor (CSTR) model with 2 kinetic model parameters to estimate, kref and E/ R. Although PDA required considerably less computational effort than traditional nonlinear regression, parameter estimates from PDA were less precise. Sparse and noisy data resulted in poor spline fits and misleading derivative information, leading to poor parameter estimates. These problems were addressed by a new iterative PDA algorithm (iPDA) in which the spline fits are improved using model-based penalties. Parameter estimates from iPDA using a single-state CSTR model, as well as a 2-state model with 1 input, were unbiased and more precise than those from standard PDA. Application to a more complex 2-state, 5-input version of the model (which had 4 parameters in total) yielded kinetic parameter estimates that were unbiased and of comparable precision to the single-input model. Estimates of heat transfer constants a and b had a slight bias that was smaller than the NLS estimates, but were not as precise as traditional NLS. Computation time for iPDA was increased because more parameters were being estimated, however it was still less than traditional NLS. Issues that need to be resolved before iPDA can be used for more complex models are discussed and recommendations for future areas of work are given. |
