Nonlinear models describing chemical and physical phenomena are used in a variety of industrial 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 and simplifying assumptions. 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:
$$x ̇(t)=f(x(t),u(t),θ)+η(t), \qquad x(t_0 )=x_0 $$
$$ y(t_j )=g(x(t_j ),u(t_j ),θ)+ε(t_j)$$
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 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 ODE (i.e., the model equations with η(t) removed). Large disturbance intensities correspond to poor predictive power of the deterministic portion of the model.
The proposed SDE parameter estimation methods (an Approximate Bayesian Expectation Maximization (ABEM) methodology and a Laplace Approximation Bayesian (LAB) methodology) provide estimates of model parameters θ along with estimates of disturbance intensities in Q and any unknown measurement noise variances. These techniques accommodate situations where: i) some or all of the initial conditions in x_0 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. Time-varying parameters and more complex nonstationary stochastic disturbances can be accommodated by augmenting the state vector with additional parameter and disturbance states. The proposed methods are illustrated using a chemical reactor case study. 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.