PID Stabilization of Nonlinear Systems: Integrating Stabilizing Sets, Lyapunov Analysis, and Gain Scheduling

This research investigates the use of PID controllers to stabilize nonlinear systems by combining stabilizing set analysis with Lyapunov-based verification techniques. The proposed method begins with linearizing the nonlinear system around equilibrium points, computing the PID stabilizing set, and verifying closed-loop stability via Lyapunov functions. For systems with multiple operating points, a gain-scheduled PID controller is constructed, and a common Lyapunov function is introduced to assess the possibility of global stability. Choose an appropriate weighting function to establish a composite Lyapunov function, which ensures the stability among all operating points. Case studies on the Van der Pol oscillator and the pendulum equation demonstrate the approach, showing consistent regulation, improved transient performance, and robustness to moderate disturbances. MATLAB simulations illustrate that appropriate PID gain scheduling, combined with Lyapunov verification, can extend local linear designs to wider nonlinear scenarios. This framework combines traditional PID design with modern nonlinear stability analysis and offers a systematic strategy to extend local linear controller performance to broader nonlinear conditions.