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The fractional-order proportional-integral (FOPI) controller tuning rules based on the fractional calculus for the parallel cascade control systems are systematically proposed in this paper. The modified parallel cascade control structure (PCCS) with the Smith predictor is addressed for stable, unstable, and integrating process models with time delays. Normally, the PCCS consists of three controllers, including a stabilized controller, for a class of unstable and integrating models, a disturbance rejection controller in the secondary loop, and a primary servomechanism controller. Accordingly, the ideal controller is obtained by using the internal model control (IMC) approach for the inner loop. The proportional-derivative (PD) controller is suggested for the stabilized controller and is designed based on a stability criterion. Based on the fractional calculus, the analytical tuning rules of the FOPI controller for the outer loop can be established in the frequency domain. The simulation study is considered for three mentioned cases of process models and the results demonstrate the flexibility and effectiveness of the proposed method for the PCCS in comparison with the other methods. The robustness of the proposed method is also justified by perturbed process models with ±20% of process parameters including gain, time constant, and delay time.

Analytical Design of Fractional-Order PI Controller for Parallel Cascade Control Systems