(522d) Adaptive Stochastic Predictive Control with Recursive Subspace Identification and Learning

Authors: 
Hajizadeh, I., Illinois Institute of Technology
Askari, M. R., Illinois Institute of Technology
Kumar, R., University of Wisconsin-Madison
Zavala, V. M., University of Wisconsin-Madison
Cinar, A., Illinois Institute of Technology
In this work, we present a computational framework that integrates disturbance forecasting, uncertainty quantifications, learning and recursive subspace identification with model predictive control (MPC) to develop an adaptive stochastic MPC. MPC is widely employed in many fields due to its inherent ability to effectively handle complex systems with constraints and many input and output variables. MPC algorithms use dynamic models of the system in the optimization problem to predict the future evolution of the outputs over a finite-time horizon to determine the optimal manipulated variables with respect to a specified performance index. Furthermore, MPC can explicitly consider the system constraints and multivariable interactions in the optimization problem, and the MPC formulations are not inexorably restricted by the type of model, objective function, or constraints [1-4]. However, the accuracy of the model, formulation of the objective function and system constraints and realization of unknown stochastic disturbances for use in output predictions affect the MPC performance.

In a standard MPC implementation, a deterministic representation of uncertain disturbances (i.e., a forecast) is used to compute control actions. In other words, the forecast acts as a summarizing statistic of the entire disturbance uncertainty space. The forecast is typically the most likely realization of the disturbance (usually the mean). Due to the inability to handle disturbances that cannot be adequately represented by mean (most likely) forecasts, stochastic MPC (sMPC) formulations have been developed [5-7]. In sMPC, one often uses historical data to create uncertainty characterizations of the model disturbances and use such characterizations to generate the most probable scenarios for the evolution of the outputs over the prediction horizon. Consequently, the sMPC provides a more systematic framework to account for diverse disturbances, satisfy constraints, and maximize control performance [6-7].

In this work, a novel approach based on partial least squares (PLS) models is used for forecasting and uncertainty quantification from historical operational data. PLS is a multivariate regression method, especially convenient for a large number of highly correlated data sets [8]. PLS models summarize the original data matrix (input variables X) to extract the most predictive information for the response variable (Y) and maximize the covariance between X and Y. It can analyze data with many noisy, collinear, and even incomplete variables in both X and Y. At each sampling time, a past window of recent observed (actual) disturbance history and derived features from measured outputs and control actions are considered as inputs of the PLS model (X) to forecast the most probable realizations of model disturbances (the outputs of PLS model Y). In order to adapt to the changes in the system, it is necessary to avoid the influence of old data as new data become available. So, recursive PLS (rPLS) algorithms are considered for the online uncertainty modeling and scenario generation. When new data are available, the PLS regression is performed using the updated data matrices. In a time-varying process, the mean levels of the variables may be changing with time. Therefore, in this work, for the recursive application, updated mean and variance is calculated for the most recent data.

The historical data can also be utilized to identify behaviors and patterns of the underlying system. Incorporating online learning of probable times of unknown disturbances from the amassed historical data using an rPLS would improve the control performance by proactively mitigating the effects of impending disturbances. In this work, the controller set-point, the weights in the MPC objective functions, and the system constraints are appropriately modified in advance for the anticipated periods of the disturbance effects. To detect rapid deviations from the desired trajectories caused by significant disturbances in real-time, a feature extraction method based on outputs measurements is also designed. The key parameters of the MPC optimization problem are modified using the information that the feature extraction method provides about the rate and shape of variations in the outputs measurements to improve the effectiveness of the controller when the presence of major disturbances is detected.

To guarantee that the underlying model can provide accurate output predictions for use in model-based predictive control algorithms, we extended the optimized version of the recursive predictor-based subspace identification method [9] to obtain a stable time-varying state space model of the process. This is done by incorporation of constraints on the fidelity and accuracy of the identified models, the correctness of the sign of the input-to-output gains, and the integration of heuristics to ensure the stability of the recursively identified models [10].

The performance of the methods developed are illustrated by regulating the blood glucose concentrations (BGC) in people with type 1 diabetes (T1D) by means of controlled insulin delivery with artificial pancreas (AP) systems. Biological processes are complex dynamical systems with significant uncertainties and exogenous disturbances that render their modeling and control a challenging problem. The human body, a time-varying system with nonlinear behavior, is affected by several stochastic and unknown disturbances. Using a multivariable simulation platform for T1D, the adaptive stochastic predictive control with recursive subspace identification and learning is applied to the problem of regulating BGC. The case studies illustrate a significant improvement in the AP system performance, and the potential in developing a fully automated AP that can function without any manual information and accommodate major disturbances to the BGC.

References

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