(669b) Adaptive Data-Based Model Predictive Control of Batch Systems

Batch (and fed-batch) processes have found widespread applications in a variety of sectors in the chemical industry, specifically those with an emphasis on quality control. The primary control objective in batch processes is to reach a target product quality, which may not be an equilibrium point, by batch termination. However, key characteristics of batch processes, such as the absence of measurements related to the final end-use quality coupled with strong nonlinear and time-varying dynamics over a wide range of operating conditions, complicate the control problem and limit the control performance achievable by implementing controllers designed for continuous processes (characterized by control at a steady-state and small variations in operating conditions). The industry response to these issues has been to operate batch processes with an entirely open-loop control policy (with incorporation of process knowledge and/or history) or indirectly pursue the primary control objective through process variable trajectory tracking approaches.

In trajectory tracking methods, trajectories for a set of measurable process variables related to the end-use quality are generated off-line or re-calculated at specific time points during the batch by solving a dynamic optimization problem. These trajectories are subsequently tracked using local model-based controllers with the most common control strategy being model predictive control (MPC). MPC performance is contingent on the accuracy of the underlying model’s predictions. When available, a deterministic state-space model with well identified parameters can be used; however, there are situations, particularly in industry, when a good first-principles model is unavailable. This has motivated the use of data-based or empirical models, particularly those identified via latent variable tools (e.g. see [1,2]) for formulating computationally tractable MPC designs. For a batch system, the available ‘training’ data for building an empirical model is essentially limited to historical databases (possibly appended with one or two identification experiments) of previous batches. Further complicating the model identification task is the fact that batch process dynamics are highly nonlinear and time-varying, rendering conventional system identification approaches, where a single input-output linear model is identified around one operating condition, ill-suited for identifying an accurate model. One general strategy to describe nonlinear behavior while retaining the simplicity of linear models is to partition/cluster the training data into a number of different regions, identify local linear models for each region, and combine them with appropriate weights to describe the global nonlinear behavior. This idea was formalized in our previous work [3,4] that unified the concepts of auto-regressive exogenous (ARX) modeling, latent variable regression techniques, fuzzy c-means clustering, and multiple local linear models in an integrated framework capable of capturing the nonlinearities and multivariate nature of batch data. The key delineating aspects of the work in [3, 4] from existing multi-model approaches (e.g. see [2], [5], and [6]) were the clustering algorithm used to partition the training data, the use of latent variable tools to estimate the model parameters, and the derivation of a generalized continuous weighting function that is entirely data dependent and does not require precise process knowledge. In this work, we generalize our previously developed modeling framework to account for time-varying dynamics by incorporating online learning ability into the model, making it adaptive. First, the standard recursive least squares (RLS) algorithm with a forgetting factor (to account for time-varying dynamics) is applied to update the model parameters. However, applying the standard RLS algorithm leads to a simultaneous or global update of all the models, which may be unnecessary depending on the operating conditions of the process. We address this issue by developing a probabilistic RLS (PRLS) estimator (also with a forgetting factor) for each model that takes the probability of the model being representative of the current plant dynamics into account in the update. The main advantage of adopting this local update approach is adaptation tuning flexibility. Specifically, the forgetting factor, which controls the speed and smoothness of adaptation, for each local PRLS estimator can be specified independently. More importantly, the model adaptations can be made more aggressive while maintaining better parameter precision (i.e., low variances) compared to the the standard RLS algorithm. The benefits from incorporating both RLS algorithms are demonstrated via simulations of a nylon-6,6 batch polymerization reactor. Closed-loop simulation results illustrate how using an adaptive model in a trajectory tracking predictive controller can substantially improve tracking performance (over the non-adaptive model based design). The adaptation of the model is also shown to be crucial for achieving acceptable control performance when encountering large disturbances in the initial conditions.