Fractional-Order 𝑃𝐼 Control of First Order Plants with Guaranteed Time Specifications

Table 2

Implementation of the controllers.

Controller	Original transfer function	Implemented structure	Operator to implement
𝐼^𝛼	𝐾𝛼𝑛𝑠^𝛼	𝐾𝛼𝑛𝑠𝑠^1−𝛼	𝐷^1−𝛼
𝑃𝐼^𝛼	𝐾𝛽𝑛𝑠^𝛼+𝐾𝛼𝑛𝑠^𝛼	𝐾𝛽𝑛+𝐾𝛼𝑛𝑠𝑠^𝛼	𝐷^1−𝛼
𝐼𝐼^𝛽	𝐾𝛽𝑛𝑠^1−𝛽+𝐾𝛼𝑛𝑠	𝐾𝛽𝑛𝑠^1−𝛽+𝐾𝛼𝑛𝑠	𝐷^1−𝛽
𝐼^𝛼𝐷^1−𝛼	𝐾𝛽𝑛𝑠+𝐾𝛼𝑛𝑠^𝛼	𝐾𝛽𝑛+𝐾𝛼𝑛𝑠𝑠^1−𝛼	𝐷^1−𝛼