Optimal impulsive disturbance rejection in linear systems with application to diabetes treatment

This study introduces a novel mathematical framework for optimizing the rejection of impulsive disturbances by adjusting the parameters of compensating impulses. This is motivated by the need of optimizing the glycemia response in diabetic patients under insulin bolus treatment. The methodology aims to ensure general applicability by formulating the optimization problem using a continuous-time linear state-space model.

The optimization problem is based on an integral quadratic criterion that quantifies the system's response to impulsive inputs. This criterion is minimized by adjusting the magnitude and relative timing of the compensating impulse.

Using eigenvalue decomposition of the system matrices yielded an explicit form of the cost function. The gradient and Hessian of the cost function are derived, and the optimal magnitude of the compensating impulse is determined analytically. Newton's method is applied to find the optimal relative time delay as the solution to a transcendental equation. Conditions for the existence of an optimal solution and the convergence of the numerical method are established.

These simulations showed that optimizing the amount and timing of insulin administration with respect to carbohydrate intake can effectively suppress glycemia fluctuations, thus significantly improving the overall quality of insulin therapy.