Niels Haverbeke

Efficient Numerical Methods for Nonlinear MPC and Moving Horizon Estimation

This overview paper reviews numerical methods for solution of optimal control problems in real-time, as they arise in nonlinear model predictive control (NMPC) as well as in moving horizon estimation (MHE). In the first part, we review numerical optimal control solution methods, focussing exclusively on a discrete time setting. We discuss several algorithmic ”building blocks” that can be combined to a multitude of algorithms. We start by discussing the sequential and simultaneous approaches, the first leading to smaller, the second to more structured optimization problems. The two big families of Newton type optimization methods, Sequential Quadratic Programming (SQP) and Interior Point (IP) methods, are presented, and we discuss how to exploit the optimal control structure in the solution of the linear-quadratic subproblems, where the two major alternatives are “condensing” and band structure exploiting approaches. The second part of the paper discusses how the algorithms can be adapted to the real-time challenge of NMPC and MHE. We recall an important sensitivity result from parametric optimization, and show that a tangential solution predictor for online data can easily be generated in Newton type algorithms. We point out one important difference between SQP and IP methods: while both methods are able to generate the tangential predictor for fixed active sets, the SQP predictor even works across active set changes. We then classify many proposed real-time optimization approaches from the literature into the developed categories.

Glycemia Prediction in Critically Ill Patients Using an Adaptive Modeling Approach

Background: Strict blood glucose control by applying nurse-driven protocols is common nowadays in intensive care units (ICUs). Implementation of a predictive control system can potentially reduce the workload for medical staff but requires a model for accurately predicting the glycemia signal within a certain time horizon. Methods: GlucoDay (A. Menarini Diagnostics, Italy) data coming from 19 critically ill patients (from a surgical ICU) are used to estimate the initial ICU “minimal” model (based on data of the first 24 hours) and to reestimate the model as new measurements are obtained. The reestimation is performed every hour or every 4 hours. For both approaches the optimal size of the data set for each reestimation is determined. Results: The prediction error that is obtained when applying the 1-hour reestimation strategy is significantly smaller than when the model is reestimated only every 4 hours (p < 0.001). The optimal size of the data set to be considered in each reestimation process of the ICU minimal model is found to be 4 hours. The obtained average “mean percentage error” is 7.6% (SD 3.1%) and 14.6% (SD 7.0%) when the model is reestimated every hour and 4 hours, respectively. Conclusions: Implementation of the ICU minimal model in the appropriate reestimation process results in clinically acceptable prediction errors. Therefore, the model is able to predict glycemia trends of patients admitted to the surgical ICU and can potentially be used in a predictive control system.

Nonlinear model predictive control with moving horizon state and disturbance estimation - Application to the normalization of blood glucose in the critically ill

Abstract In this paper we present a nonlinear model predictive control (NMPC) strategy that can be used to tackle nonlinear control problems with changing model parameters, unknown disturbance factors and specifications on the rates of change of the inputs. The closed-loop performance of the proposed NMPC strategy is demonstrated by applying it to the problem of blood glucose normalization in critically ill patients. A nonlinear patient model, that is particularly developed for describing the glucose and the insulin dynamics of these patients, is used for online state and disturbance estimation and control under a realistic disturbance realization. The results are satisfactory both in terms of control behavior (set point tracking and the suppression of unknown disturbance factors) and clinical acceptability.

A structure exploiting interior-point method for moving horizon estimation

In this article a primal barrier interior-point method for moving horizon estimation (MHE) is presented. It exploits the structure of the KKT systems yielding an algorithm with linear complexity in the horizon length as opposed to cubically as in unstructured solvers. Ideas of square root covariance Kalman filtering are proposed in order to update covariance matrices occurring in the factorization of the KKT matrix efficiently and in a numerically stable way. The algorithm is able to compute - without any additional costs - the covariance of the last estimate within the horizon, which reflects the accuracy of the estimate.

Prediction Performance Comparison between three Intensive Care Unit Glucose Models

Abstract In this paper the prediction performance of two models that were particularly developed for predicting the blood glucose signal in critically ill patients, and a third (rather ‘naive’) model were compared. The imposed real-life conditions were challenging as the prediction processes started at time step 1 (comparable to the admission of a patient at the intensive care unit) and the prediction horizon was set at 4 hours (although accurate prediction of the blood glucose signal in the initial phase after admission is difficult due to lack of patient-specific data). The results of one of the models was satisfactory in terms of forecasting ability and showed its potential to be validated for use in a predictive control system in real-life. Copyright

Adaptive modeling for control of glycemia in critically ill patients

Abstract In this paper we propose an optimized adaptive ‘minimal’ modeling approach for predicting glycemia of critically ill patients and a corresponding Model based Predictive Control (MPC) setting for controlling glycemia. Re-estimations of the model, based on a real-life dataset from 19 critically ill patients, are performed every hour or every four hours by only considering recently passed data. The contributions of this study are the determination of the best dataset size for the re-estimations and the proposed MPC design. The results are satisfactory both in terms of forecasting ability and in qualitative controller performance.