(204f) Stochastic Model Predictive Control with Adaptive Probabilistic Disturbance Models: Application to Automated Insulin Delivery in Diabetes

Model predictive control (MPC) has demonstrated the ability to regulate complex systems in numerous applications ranging from industrial chemical processes to pharmaceutical manufacturing and automated drug delivery. MPC can handle complex system dynamics, multivariable architectures, and constraints. The MPC algorithm, a receding horizon control problem, is formulated as an open-loop optimization problem that minimizes, at each sampling instance, an objective function over a prediction horizon to determine the optimal trajectory of input variables given the current state of the system. The first instance of the computed manipulated inputs is then implemented, and the optimization problem is reinitialized and repeated at subsequent sampling instance to solve for new trajectories of input variables, resulting in an implicit feedback control law able to accommodate certain system uncertainties and unmeasured disturbances [1], [2].

The deterministic formulation of conventional MPC limits its ability to deliberately and methodically handle uncertainties and unmeasured disturbances. Robust MPC formulations, using min-max optimization problems, are developed to solve for the optimal input variables with respect to worst-case scenarios for uncertainty realization. The worst-case scenarios might be overly conservative or may render the optimization problem infeasible. Extending robust MPC, tube-based MPC is developed to contain the predicted state trajectories within a set and guarantee that constraints are satisfied [3], [4]. However, in many applications, the uncertainty may be probabilistic in nature, and the probabilities for the uncertainty realization may need updating based on recent data.

Stochastic MPC (SMPC) incorporates probabilistic uncertainty descriptions to solve for the optimal control actions with an expected objective function based on a distribution of uncertainty and soft constraints. Therefore, SMPC requires the distributional information of the uncertainty, and discrepancies between estimated and actual distributions of the uncertainties may deteriorate control performance. Furthermore, the distribution of the uncertainty may vary over time, and updating the distribution based on samples of the disturbance realization is required [5]. The distribution of the disturbance may also be estimated through historical data and expert knowledge.

Automated drug delivery is an application that has benefited from advanced control techniques, such as MPC. An example is automated insulin delivery in people with type 1 diabetes, where unannounced meals constitute unmeasured disturbances [6]. Historical data and expert knowledge of behavior can be used to train explainable machine learning models that characterize the time-varying distribution of the disturbance. Estimated realizations of the disturbance and associated probabilities can be used to construct an SMPC formulation that is tractable and ensures satisfaction of the probabilistic constraints.

In this work, we develop an adaptive SMPC formulation for the regulation of blood glucose in people with type 1 diabetes by modulating insulin dosing. We use real-world labeled data from 50 subjects with type 1 diabetes collected over 2 years to train a Bayesian network that estimates the probabilities of meals over the future horizon, which are then used to generate likely meal scenarios over the future horizon. The adaptive SMPC formulation then determines the optimal insulin given the likely future scenarios for meals. The Bayesian network is interpretable and can be updated as new data becomes available. Simulation studies involving the multivariable glucose-insulin-physiological variable simulator (mGIPsim) demonstrate the performance of the adaptive SMPC using virtual patients with type 1 diabetes under unannounced meals and physical activities [7]. The adaptive SMPC improves glucose control and increases the time subjects spend in the desired target glucose range. Subjects using the adaptive SMPC spend on average 88.3% of the time in the desired glucose range, compared to an adaptive MPC without stochastic future meal predictions that results in only 83.3% of time spent in the desired target range. The results demonstrate that an adaptive SMPC incorporating adaptive probabilistic models to estimate potential scenarios of future disturbances, such as unannounced meals, can improve the control of blood glucose in people with type 1 diabetes.

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[7] M. Rashid et al., “Simulation software for assessment of nonlinear and adaptive multivariable control algorithms: Glucose–insulin dynamics in Type 1 diabetes,” Comput Chem Eng, vol. 130, p. 106565, 2019, doi: 10.1016/j.compchemeng.2019.106565.