Optimization-based PID tuning for nonlinear systems via multiple-shooting and collocation on finite elements

[摘要] We present a direct, optimization-based framework for tuning proportional–integral–derivative (PID) controllers on general nonlinear systems. The method treats the PID gains as decision variables in a finite-horizon optimal control problem and enforces the full closed-loop dynamics using multiple-shooting combined with collocation on finite elements. The derivative action is incorporated algebraically, eliminating auxiliary derivative states and avoiding local linearization or explicit decoupling. The resulting sparse nonlinear program supplies exact sensitivities with respect to the gains, enabling fast convergence with large-scale interior-point or SQP solvers. Case studies on an unstable first-order plant, an inverted pendulum on a moving cart, and a nonlinear quadruple (four-tank) system demonstrate accurate tracking and robustness to cross-couplings while using only PID structure. The approach is simple to implement, scalable, and compatible with standard MATLAB toolchains.

[发布日期] 2026-07-01

[关键词] proportional integral derivative controller (PID controller);nonlinear mechanical systems;multiple shooting method;controller parameter optimization;nonlinear programming (NLP)