Tuning Fractional-Order <italic/><inline-formula> <tex-math notation="LaTeX">${PI}^{\lambda } {D} ^{\mu }$ </tex-math></inline-formula> Controllers for a Solid-Core Magnetic Bearing System

Abstract

This brief presents the design of a fractional-order (FO) PI <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">λ</sup> D <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">μ</sup> controller and its application to stabilizing an FO solid-core magnetic bearing (MB) system. A practical numerical search method of designing the FO PI <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">λ</sup> D <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">μ</sup> controller is proposed. By presearching the orders λ and μ in their calculated ranges, the remaining three undetermined controller parameters can be first analytically solved based on three given basic specifications, and thus, a series of controllers are obtained. Then, a relative optimal FO PI <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">λ</sup> D <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">μ</sup> controller is achieved by evaluating exact performance indexes. In addition, an integer order (IO) PID controller based on the same specifications is derived and the corresponding feasible control zone consisting of specifications is calculated. The performance comparisons between the IO PID and the FO PI <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">λ</sup> D <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">μ</sup> controllers are implemented by simulation and experiments. The results show that this tuning method is effective, and the FO PI <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">λ</sup> D <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">μ</sup> controller can achieve more favorable dynamic performance than the IO PID controller for this FO MB system.