Tracking control stabilization of systems manipulated by constrained parabolic nonlinear actuator

The model-based control approach proves highly efficient when an accurate system model is available. However, a common oversight in conventional model-based control methods is the neglect of hard constraints governing the actuation system during the design phase. To ensure the effectiveness of the control algorithm in practical systems, it becomes crucial to incorporate the dynamic model of the actuator, accounting for operational limitations like hard constraints on the actuation system's command input and natural frequency. This complexity intensifies when dealing with nonlinear relationships between the actuator's input and output. While some research has addressed controller design for systems with linear actuators, the challenge of systems where the actuator follows second-order parabolic nonlinear dynamic equations with hard constraints on command input and natural frequency remains unexplored. In this article, the equations for the modified open-loop system are derived, incorporating the actuator's dynamics, focusing on a second-order parabolic nonlinear model with hard constraints. The design introduces a Lyapunov-based control input, utilizing a stable reaching law with time-varying coefficients. By aligning the discrete-time tracking error equation with the reaching law, the control input is determined. Filtering desired values according to actuator constraints guides the system in a stable manner, achieved through an optimal searching algorithm for finding suitable coefficients. Despite actuator limitations, the controller incorporates position and velocity modes, dynamically switching based on minimizing a predictive cost function containing the position error in the next step. This optimal control algorithm guarantees Lyapunov stability of the closed-loop system without resorting to computationally expensive model predictive control methods, ensuring accurate trajectory following. Simulation results controlling a nonlinear Newtonian mechanical system with a second-order parabolic nonlinear actuator, subject to hard constraints on command input and natural frequency, demonstrate the algorithm's adept tracking performance. The controlled system, when applying the presented algorithm for desired square pulse and harmonic trajectories, exhibits minimal over/undershoot, no offset error, and demonstrates appropriate settling time and rise time.