An improved nonlinear proportional-integral-differential controller combined with fractional operator and symbolic adaptation algorithm

Parameter adjustment is usually applied for designing the proportional-integral-differential (PID) controllers. However, the ability to improve control performance by adjusting parameters is limited. Hence, with the goal to achieve ideal closed-loop response, this paper takes advantage of a structural optimization method for modifying the controller model. A symbolic adaptation algorithm for fractional order PID (FOPID) controller is employed to obtain precise nonlinear controller model. Firstly, a modeling comparison for nonlinear duffing system is carried out to highlight the efficiency of the symbolic adaptation algorithm. The case study indicates the proposed algorithm can establish compact dynamic models by amending the shortcomings of symbolic regression. Secondly, the proposed controller is restructured with the linear FOPID controller and its nonlinearity is increased by adjusting controllers' components in symbolic form. The proposed controllers are simulated in an unstable second-order system, a time-delay system and a nonlinear VanderPol system. Compared with the IOPID and the FOPID controller, the symbolic adaptation algorithm improves the structural flexibility of these linear controllers. Meanwhile, the system response can better approximate the desired response and the structural integrity of the nonlinear controller model is guaranteed simultaneously. Finally, the nonlinear FOPD controllers for trajectory tracking experiments are carried out on a rotary inverted pendulum control system.