Navigating the Control Landscape: A Mathematical Survey and Operational Decision Matrix for 34 Control Methodologies

Source abstract

Control engineering has undergone a transformative evolution, progressing from rudimentary relay-based regulators to highly sophisticated adaptive, robust, and data-driven intelligent architectures. This paper provides an intensive and structured survey of thirty-four distinct control techniques, categorized into classical compensators (P, PI, PD, PID, Lead-Lag), structural extensions (Feedforward, Cascade, Smith Predictor, IMC), modern state-space methods (LQR, LQG, Kalman Filter, MPC), robust designs ($H_\infty$, $\mu$-Synthesis, QFT), adaptive strategies (MRAC, Self-Tuning, Gain Scheduling), nonlinear approaches (Sliding Mode, Backstepping, Feedback Linearization, Lyapunov), and intelligent soft-computing systems (Fuzzy, Neural Networks, Reinforcement Learning, ANFIS). A defining feature of this review is the rigorous mapping of every control methodology to its original foundational reference, providing a comprehensive historical and theoretical lineage for each system. For each category, we derive the fundamental mathematical models governing controller behavior. The paper concludes with a comprehensive classification table and an operational decision matrix, evaluating techniques based on implementation domain, practical difficulty, and model-dependency to serve as an evidence-based, expert-informed decision framework for selecting appropriate control paradigms based on plant complexities and environmental constraints.