(748a) Estimating Parameters and Model Uncertainty in Fundamental Dynamic Models Using Historical Data

Nonlinear models based on fundamental mass and energy balances describe dynamic behavior of chemical processes in process monitoring, control and optimization applications. Predictions from these models are always (at least slightly) wrong due to inaccurate parameter values and model imperfections associated with unmodelled disturbances, inadequate process understanding, simplifying assumptions, and uncertain inputs. We have developed new methods to use batches of old dynamic process data to simultaneously estimate model parameters and the “badness” of the associated nonlinear dynamic model.

To account for model imperfections, stochastic terms are added to the right-hand sides of the dynamic model equations, resulting in stochastic differential equation (SDE) models of the form:

xdot(t) = f(x(t),u(t),θ)+η(t)

x(t₀) = x₀

y(t_j) = g(x(t_j),u(t_j),θ)+ε(t_j)

where x is the vector of state variables, t is time, f is a vector of nonlinear functions (derived based on scientific knowledge and assumptions), u is the vector of input variables, θ is the parameter vector and η(t) is a vector of stochastic independent continuous white-noise processes corresponding to a power spectral density matrix Q whose diagonal elements are disturbance intensities. These disturbance intensities indicate the level of imperfection of each corresponding ODEs (i.e., the differential equations above with η(t) removed). Large disturbance intensities correspond to poor predictive power of the model and large mismatch due to stochastic disturbances or other model imperfections.

Our proposed SDE parameter estimation methods provide estimates of the model parameters θ (and their associated confidence intervals) along with estimates of the disturbance intensities in Q and the measurement noise variances, if they are unknown. These techniques readily accommodate situations where: i) some or all of the initial conditions in are unknown and must also be estimated from the data; ii) different types of measurements are made at different sampling frequencies or some measurements are made at irregular intervals; iii) the modeler is willing to specify prior knowledge about some or all of the model parameters and/or noise variances (e.g., initial guesses and corresponding initial uncertainty levels). Time-varying parameters and nonstationary disturbances can also be accommodated by augmenting the state vector with additional unmeasured parameter or disturbance states.

An Approximate Bayesian Expectation Maximization (ABEM) methodology and a Laplace Approximation Bayesian (LAB) methodology are proposed for estimating parameters and uncertainties associated with SDE models and data. These methodologies are more powerful than our previous maximum-likelihood methodologies for SDEs because they enable modelers to account for prior information about unknown parameters and initial conditions. The ABEM methodology is suitable for situations where the modeler can assume that measurement noise variances are well known, whereas LAB includes measurement noise variances among the parameters that require estimation. Both techniques estimate the magnitudes of stochastic terms included in the differential equations to account for model mismatch and unknown process disturbances. The proposed ABEM and LAB methodologies are illustrated using a two-state nonlinear continuous stirred tank reactor (CSTR) case study, with simulated data sets generated using a variety of scenarios. The ABEM and LAB objective functions derived for the case study have additional terms and result in improved estimates of model parameters and noise parameters compared to our previous maximum-likelihood objective functions, especially in situations where data available for parameter estimation are sparse. Benefits of including prior information about unknown parameters and noise variances are illustrated and advice is provided regarding which objective function should be used for parameter estimation in different situations. Because the proposed ABEM and LAB methodologies rely on B-spline basis functions rather than Markov Chain Monte Carlo techniques, they are straightforward to implement using available optimizers and modeling software and require only modest computational effort.

We hope that the proposed ABEM and LAB objective functions for SDE parameter estimation will be attractive to practitioners who develop dynamic models for online monitoring and control and for dynamic optimization. We anticipate that they will be able to readily adopt the proposed objective function structure in their own modeling and parameter estimation problems, thereby obtaining useful uncertainty information that can aid model improvement and/or implementation.