Parameter Subset Selection in Differential Equation Models with Dead Time A. Duong Vo*. Aly Elraghy** Kim B. McAuley*** Department of Chemical Engineering, Queen’s University, Kingston, K7L2N6 Canada *(e-mail: anhduong.vo@queensu.ca). **(e-mail: aly.elraghy@queensu.ca). *** (Tel: 613-533-6000 ext 77562; e-mail: kim.mcauley@queensu.ca). Abstract: A methodology is proposed for parameter ranking and parameter subset selection for nonlinear ordinary differential equation (ODE) models with time delay, in which delay is treated as an unknown model parameter. The methodology builds on earlier algorithms for ranking model parameters in systems without time delay (Yao et al., 2003; Thompson et al., 2009) and for finding the optimum number of parameters for estimation (Wu et al., 2011; McLean and McAuley, 2012a). A polymerization reactor system for producing bio-source polyether is used to illustrate the effectiveness of the proposed method in comparison with prior results obtained by Cui et al. (2015) who neglected the time delay. Keywords: mathematical models, time-delay estimation, identifiability, parameter estimation, differential equations, nonlinear equations, algorithms 1. INTRODUCTION Fundamental models are used for scale-up, control, and optimization of chemical processes. Obtaining accurate model predictions requires estimation of model parameters and often making decisions about which parameters should be estimated from available data and which should be fixed at reasonable values or removed via model simplification (Walter and Pronzato, 1997; Chu et al., 2009; McLean and McAuley, 2012a; Kravaris et al., 2013). Several algorithms have been developed to determine which parameters can and/or should be estimated. The most popular methods rely on forward-selection to rank the parameters from most estimable to least estimable (Yao et al., 2003; Lund and Foss, 2008; Thompson et al., 2009). A mean-squared-error selection criterion is then used to find an appropriate number of parameters to estimate to obtain reliable predictions (Wu et al., 2011; McLean et al., 2012b; Eghtesadi and McAuley, 2016). For example, Table 1 shows an orthogonalization- based parameter-ranking algorithm and Table 2 shows an algorithm for selecting parameters to obtain low mean- squared prediction errors. Algorithms in Tables 1 and 2 and other related subset selection methods were developed for parameter selection in dynamic models without time delay (e.g., Chu et al., 2009). Sometimes, however, modelers need to account for significant delay. In situations, where delays are associated with measurements or with piping that is not part of a recycle stream, deadtime can be handled during parameter estimation either by shifting the experimental data backward in time or shifting the predictions forward. If the delay arises in an internal recycle stream, deadtime must be considered within the model (e.g., using delay differential equations). The objective of this article is to show how algorithms in Tables 1 and 2 can be extended to rank parameters and select appropriate subsets for estimation when deadtime appears as an unknown model parameter. We use a polymerization system with unknown time delay and unknown mass hold-up due to an overhead condenser to illustrate the proposed methodology. 2. PROPOSED METHODOLOGY Consider an ordinary differential equation (ODE) model with time delay of the form: ) , , ( ) ( m θ u x f x  t  (1a) 0 x x  ) ( 0 t (1b) ε θ u x g y m d   ) , , , ( ) ( t ,θ t d (1c) where x  is a vector of state variables, t is time, f is a vector of non-linear functions, u is a vector of input variables, θm is the vector of unknown model parameters that appear in the ODEs, x0 is a vector of initial conditions for the state variables, y is a vector of measured output variables (some of which are affected by an unknown time delay θd), gd is a vector of model predictions that accounts for this time delay in the affected responses and ε is a vector of zero-mean random variables. For simplicity, the time delay θd does not influence any of the state variables within the ODEs in (1a), but the proposed methodology could be readily used for more complex systems with time delay in the state variables rather than simple additive delay. In (1c) the delay influences some but not all of the outputs. For example, a prediction of the i th delayed response is ) , , , ( ) , , , , ( d d θ t t θ   m i m di θ u x g θ u x g where gi is the Preprints, 10th IFAC International Symposium on Advanced Control of Chemical Processes Shenyang, Liaoning, China, July 25-27, 2018 Copyright © 2018 IFAC 655