Tuning of PIλDμ controllers based on sensitivity constraint

• Stabilizing regions in integral-derivative plane of PIλDμ controller.
• Optimization of fractional orders to expect bigger stabilizing regions.
• Sensitivity function constraint in the stabilizing region.

A graphical tuning method for fractional-order PID (PIλDμ) controllers is proposed based on the sensitivity function constraint of the closed-loop, which provides the information on robustness to plant uncertainties. The stabilizing regions in integral-derivative plane of the controller are first identified using a graphical stability criterion applicable to fractional-delay systems. Then, via Leibniz Sector Formula, the stabilizing region is optimized with respect to the two fractional orders of the controller to expect bigger stabilizing regions. Finally, the sensitivity function constraint of the closed-loop is mapped into stabilizing region by means of the explicit algebraic equations which can be solved efficiently. Numerical examples of a second-order integrating delay process are followed in each design procedure to show the effectiveness of the method.