Fractional-order PID feedback synthesis controller including some external influences on insulin and glucose monitoring

The article aims to develop a fractional-order proportional integral derivative (PID) controller to monitor insulin and glucose levels in humans under the influences of stress, excitement, and trauma. A novel fractional-order diabetes mellitus model is proposed, incorporating a nonsingular, nonlocal kernel (Mittag-Leffler function) to account for the effect of epinephrine on suppressing insulin secretion and the dynamics of beta-cell mass. As beta-cell mass increases in the presence of adrenaline, the system remains highly responsive to rising blood glucose and falling insulin levels, driven by the hormone’s suppressive effects. The key advantage of this model is its ability to incorporate these physiological stressors and use fractional-order derivatives to describe the nonlocal dynamics within the system. The innovations of this work include a fractional-order diabetes mellitus model that captures the biological memory and hereditary effects of glucose regulation under stress, and a fractional-order PID controller that offers greater stability and robustness compared to conventional controllers, particularly in managing adrenaline-induced hyperglycemia. The model’s positivity, boundedness, and equilibrium solutions are rigorously analyzed to ensure feasibility. Additionally, a new theorem is proven using fixed-point theory, confirming the existence and uniqueness of the fractional-order model. Ulam–Hyers stability analysis further demonstrates the model’s robustness and well-posedness, while qualitative properties are explored. Numerical simulations to explore which is done by solutions with a two-step Lagrange polynomial for generalized Mittag Leffler kernel showed that prolonged and severe hyperglycemia was caused by regular release of adrenaline into the blood at different fractional order values and fractal dimensions by changing initial values for normal and diabetes patients. PID and controller results are analyzed to increase the stability of the system to monitor and assess of glucose–insulin system with beta cell mass to control the hyperglycemia. Lastly, the results are obtained and visually shown using graphical representations, which provide empirical evidence in support of our theoretical findings. At the end comparison of numerical simulations is constructed to show the efficiency, convergence, and accuracy of proposed techniques at different fractional values with power law and exponential kernels. Numerical simulations, mathematical modeling, and analysis work together to shed light on the dynamics of diabetes mellitus and make important advances in the knowledge and treatment of this common disease.