Taylor-Series-Based Hybrid Adaptive Sliding Mode Control with Online Meal Disturbance Estimation for Artificial Pancreas Systems

Technical Summary: Taylor-Series-Based Hybrid Adaptive Sliding Mode Control for Artificial Pancreas Systems

Problem Statement
The regulation of blood glucose in Type 1 Diabetes Mellitus (T1DM) via Artificial Pancreas (AP) systems faces significant challenges due to physiological uncertainties and unpredictable external disturbances, specifically meal intake. While Sliding Mode Control (SMC) offers inherent robustness and finite-time convergence, conventional SMC formulations suffer from two critical limitations in clinical settings:

1. Fixed Gains: Most SMC-based controllers utilize fixed gains that cannot adapt to patient-specific dynamics or time-varying insulin sensitivity.
2. Disturbance Bounds: Standard Higher-Order Sliding Mode Control (HOSMC) and adaptive approaches often require prior knowledge of disturbance bounds (e.g., maximum meal size) to ensure stability. In practice, meal carbohydrate content is unknown and unpredictable, making conservative worst-case tuning necessary, which often leads to excessive insulin delivery or "chattering" (high-frequency oscillations).

Existing strategies like Model Predictive Control (MPC) rely heavily on model accuracy and computational resources, while PID controllers lack adaptability to time-varying conditions. Intelligent control methods (fuzzy/neural) often lack formal stability guarantees. Consequently, there is a need for a control framework that ensures robustness, rapid postprandial recovery, and smooth insulin delivery without requiring prior knowledge of meal disturbance magnitudes.

Methodology
The paper proposes a unified Adaptive Sliding Mode Control (ASMC) framework based on an extended Bergman Minimal Model (BMM). The methodology integrates three core components:

1. System Modeling: The glucose-insulin dynamics are modeled using the BMM, extended with an explicit meal disturbance term $D(t)$. The disturbance is modeled as an exponentially decaying function with an unknown amplitude $A$ (representing meal carbohydrate content) and a known decay rate $\beta$.
2. Taylor-Series Parameterization: To handle the unknown disturbance amplitude $A$, the authors apply Taylor's theorem to the disturbance function. This allows the nonlinear disturbance dynamics to be decomposed into a linear regressor form. Specifically, the lumped nonlinear term in the system dynamics is separated into nominal system dynamics and a disturbance-related component parameterized by $A$. This yields a known regressor vector $\Psi$ dependent on measurable states ($G, X, I$) and time, enabling the estimation of $A$ online.
3. Adaptive Control Law:
   • A second-order sliding surface $S$ is defined based on the glucose tracking error and its derivatives.
   • The control input $u$ is split into an equivalent control (based on the estimated dynamics) and a discontinuous switching term.
   • Crucially, the unknown amplitude $A$ is replaced by an online estimate $\hat{A}$.
   • A Lyapunov-based adaptive law is derived to update $\hat{A}$ in real-time. The update law is designed such that the time derivative of the Lyapunov function is non-positive, ensuring the boundedness of all closed-loop signals.
   • Theoretical analysis proves finite-time convergence of the sliding surface to zero without requiring prior knowledge of the upper bound of the disturbance.

Key Contributions
The paper identifies three primary contributions:

1. Development of an ASMC with Online Estimation: The formulation of an adaptive controller that estimates the unknown meal disturbance amplitude online using a Taylor-series parameterized regressor, eliminating the need for prior disturbance bounds or meal announcements.
2. Rigorous Stability Analysis: A formal Lyapunov-based stability analysis that establishes finite-time convergence for the closed-loop system, proving that the adaptive law ensures boundedness of all signals (glucose, insulin, and estimation error).
3. Comprehensive Comparative Evaluation: A simulation-based validation against a conventional SMC using physiologically validated patient parameters, demonstrating superior performance in terms of peak reduction, settling time, and chattering elimination.

Simulation Results
The proposed ASMC was evaluated in MATLAB/Simulink over a 24-hour horizon with three meal disturbances (breakfast, lunch, dinner) across three distinct patient profiles with varying metabolic parameters. Key quantitative findings include:

• Glucose Regulation: The ASMC reduced postprandial glucose peaks by 70.8% compared to conventional SMC (e.g., lunch peak reduced from 380.65 mg/dL to 108.42 mg/dL). The conventional SMC exceeded the ADA-recommended 180 mg/dL threshold, while ASMC maintained levels within safe limits.
• Convergence Speed: The ASMC achieved a 28.3% faster settling time (1.70 hours vs. 2.37 hours) across meal events.
• Tracking Accuracy: The ASMC demonstrated significantly lower error metrics: 87.20% lower ITAE, 14.71% lower IAE, and 5.22% lower ISE and MSE compared to SMC.
• Control Smoothness: The ASMC eliminated chattering, evidenced by an 85.95% reduction in Maximum Amplitude Change (MAC) and a 78.40% reduction in Input Signal Variation (ISV). This resulted in a smoother insulin infusion profile, avoiding the high-frequency oscillations seen in conventional SMC.
• Safety: In all simulated scenarios, including patients with high insulin resistance, the ASMC maintained glucose levels strictly above the 70 mg/dL hypoglycemia threshold, whereas the conventional SMC exhibited unsafe hyperglycemic excursions.

Significance and Claims
The authors claim that the proposed framework offers a robust, computationally efficient, and clinically reliable solution for artificial pancreas systems. The significance of the work lies in its ability to address the critical challenge of unknown meal disturbances without relying on conservative worst-case tuning or prior knowledge of disturbance bounds.

By integrating Taylor-series estimation with adaptive sliding mode control, the system achieves:

• Clinical Safety: Prevention of both severe hyperglycemia and hypoglycemia.
• Patient Comfort: Elimination of chattering leads to smoother insulin delivery, reducing the risk of pump fatigue and tissue irritation.
• Personalization: The controller adapts to individual patient dynamics (insulin sensitivity and glucose clearance) through the online estimation mechanism.

The paper concludes that while conventional SMC is theoretically robust, it is clinically impractical for AP systems due to its inability to handle unknown disturbances without excessive gains. The proposed ASMC bridges this gap, offering a viable alternative that ensures finite-time convergence and smooth control action, making it suitable for embedded implementation in insulin pumps. Future work is noted to focus on clinical validation and integration with noisy sensor data.