Discrete-Time Fractional Order PIλDµ Controller Design

Abstract

This thesis presents a comprehensive framework for the design, discretization, optimization, and tuning of Discrete-Time Fractional Order PID (DT FOPID) controllers. Fractional-order controllers offer superior flexibility and performance over classical PID controllers, particularly in systems with memory effects and non-integer-order dynamics. However, their implementation in digital systems remains limited due to challenges in accurate discretization and the complexity of multi-parameter tuning.
The research introduces a structured and practical methodology for tuning DT FOPID controllers based on system behavior. During this process, a novel second-order direct discretization approach—referred to as the HK method—was developed to address short-comings in existing discretization techniques. The HK method demonstrated strong stability characteristics, accurate frequency-domain behavior, and consistent performance across varying sampling periods and fractional orders.
The proposed methodology was validated using seven benchmark systems representing a wide range of dynamic behaviors, including instability, underdamped and overdamped responses, and non-minimum phase characteristics. Tuning was performed primarily using Particle Swarm Optimization (PSO), with additional support from Genetic Algorithm (GA) and Gravitational Search Optimization (GSO) when appropriate. Step response metrics—including rise time, settling time, overshoot, steady-state error, and pole location—were used to evaluate control performance.
The results confirm that well-tuned and properly discretized DT FOPID controllers can significantly improve transient and steady-state system behavior. However, the findings also indicate that in certain well-behaved systems, classical PID controllers may offer comparable performance, highlighting the need for careful justification of FOPID use.
Finally, a detailed sensitivity analysis was conducted to extract general tuning trends and formulate practical design guidelines. The insights gained provide a foundation for future work toward analytical tuning models and AI-assisted controller design. Overall, this thesis offers a unified and reproducible approach to DT FOPID controller implementation, bridging the gap between theoretical development and digital application.

Date of Award	Jul 2025
Original language	American English
Supervisor	Reyad El Khazali (Supervisor)