Resumen Este trabajo presenta un análisis de desempeño de dos nuevos algoritmos de control desarrollados como parte del proyecto conocido como “Páncreas Artificial”, siendo este el tratamiento más promisorio para la diabetes mellitus tipo 1. Se desarrolló una versión del controlador proporcional, integral y derivativo el cual cumple restricciones de positividad en los estados y entradas, además de una nueva versión impulsiva del control predictivo basado en modelo. El desempeño de las estrategias es evaluado en 50 pacientes virtuales extraídos de la literatura y del Simulador UVa/Padova aprobado por la Food and Drug Administration de EEUU. La robustez se evalúa realizando variaciones paramétricas en los parámetros importantes del modelo. Las estrategias se validan en el simulador considerando diferencia planta-modelo. Los resultados muestran que el control predictivo presenta mejor desempeño, pero puede presentar infactibilidad si aumenta la diferencia planta-modelo. En contraste, el controlador proporcional, integral y derivativo presenta una respuesta más rápida, manteniendo un desempeño aceptable bajo perturbaciones de comida y seguimiento de referencia.

Abstract This paper examines the performance of two new closed-loop control strategies developed as part of the Artificial Pancreas project, this being the most promising treatment for type 1 diabetes mellitus. The first strategy uses a new version of the well-known proportional, integral and derivative control, developed to respect state and input positivity constraints. The second is a new formulation of model-based predictive control with an impulsive input. The strategies’ performance is evaluated with 50 virtual patients taken from the literature and the UVa/Padova metabolic simulator, approved by the US Food and Drug Administration. Also, a robustness analysis is added to evaluate the strategies under the parametric variations of the most important physiological parameters. The results show that both strategies have a good performance with low to moderate plant-model mismatch.

An iterative method based on Picard's approach to ODEs' initial-value problems is proposed to solve first-order quasilinear PDEs with matrix-valued unknowns, in particular, the recently discovered variational PDEs for the missing boundary values in Hamilton equations of optimal control. As illustrations the iterative numerical solutions are checked against the analytical solutions to some examples arising from optimal control problems for nonlinear systems and regular Lagrangians in finite dimension, and against the numerical solution obtained through standard mathematical software. An application to the (n + 1)-dimensional variational PDEs associated with the n-dimensional finite-horizon time-variant linear-quadratic problem is discussed, due to the key role the LQR plays in two-degrees-of freedom control strategies for nonlinear systems with generalized costs. Mathematical subject classification: Primary: 35F30; Secondary: 93C10.

The recently discovered variational PDEs (partial differential equations) for finding missing boundary conditions in Hamilton equations of optimal control are applied to the extended-space transformation of time-variant linear-quadratic regulator (LQR) problems. These problems become autonomous but with nonlinear dynamics and costs. The numerical solutions to the PDEs are checked against the analytical solutions to the original LQR problem. This is the first validation of the PDEs in the literature for a nonlinear context. It is also found that the initial value of the Riccati matrix can be obtained from the spatial derivative of the Hamiltonian flow, which satisfies the variational equation. This last result has practical implications when implementing two-degrees-of freedom control strategies for nonlinear systems with generalized costs.