Hybrid Model and Data-Driven Optimal PID Control With Enhanced Disturbance Rejection

This paper proposes an optimal proportional-integral-derivative (PID) control method with improved disturbance rejection for nonlinear uncertain systems. The controller achieves optimal closed-loop dynamics by integrating constant and time-varying control gains. Initially, the system is transformed into a linear model with a nonlinear uncertainty term that lumps unmodeled dynamics and disturbances. Constant PID gains are subsequently derived using an optimal tuning rule based on the frequency and gain of the linear model at $-180^{o}$ and $-120^{o}$ phase lag points for systems with and without ultimate point, respectively, by employing a fractional-order integrator and Tchebyshev polynomials to simplify computation. Then an extended state observer (ESO)-assisted recursive optimization approach is introduced to adaptively adjust the time-varying PID gains in real time based on the uncertainty estimation from the ESO and operational data, improving adaptation under nonlinear uncertainties. This hybrid tuning enhances the robustness of the closed-loop system to uncertain dynamics and mitigates the instability caused by poor initial parameters, outperforming traditional model-driven and data-driven methods. The stability of the proposed approach is proven, and its effectiveness is validated through simulations and an industrial case study on the pressure control of the hematite regrinding process. Note to Practitioners—This paper addresses practical challenges in controlling nonlinear uncertain systems and proposes an optimal proportional-integral-derivative (PID) control strategy by integrating the advantages of model- and data-driven methods. The hybrid tuning strategy demonstrates superior performance compared to traditional model-based tuning methods, which struggle with adapting to nonlinear uncertain dynamics, and data-driven methods, which are prone to instability due to inappropriate initial values. Particularly, the designed extended state observer (ESO)-assisted recursive optimization method can improve the disturbance rejection performance of the closed-loop system through the continual learning of PID parameters, which enhances the robustness of the closed-loop system. This paper provides practical guidance for applying the proposed method to pressure control in hematite regrinding processes. By following a similar framework, the approach can be extended to most systems in the process industry with a PID controller. First- or second-order linear models are recommended in practice when considering the complexity of implementation. For systems of greater complexity, additional auxiliary strategies may be required, such as decoupling techniques for multivariable systems and Smith predictors for systems with large time delays.