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

Table 1

Tuning equation resume.

Controller	Transfer function	Controller parameters	Tuning equations
𝑃𝐼	𝑅0𝑛(𝑠)=𝐾𝛽𝑛+𝐾𝛼𝑛𝑠	𝐾𝛽𝑛 and 𝐾𝛼𝑛	𝐾𝛽𝑛=𝑧𝑟𝐾𝛼𝑛=−𝜔𝑐𝑛𝑧𝑖
𝐼^𝛼	𝑅1𝑛𝐾(𝑠)=^𝛼𝑛𝑠^𝛼	𝐾𝛼𝑛 and 𝛼	𝐾𝛼𝑛=sign(𝑧𝑟)𝜔^𝛼𝑐𝑛(𝑧^2𝑟+𝑧^2𝑖)^1/22𝛼=𝜋tan^−1−𝑧𝑖𝑧𝑟
𝑃𝐼^𝛼	𝑅2𝑛𝐾(𝑠)=^𝛽𝑛𝑠^𝛼+𝐾𝛼𝑛𝑠^𝛼	𝐾𝛽𝑛, 𝐾𝛼𝑛 and 𝛼	𝐾𝛽𝑛(𝛼)=𝑧𝑟+𝑧𝑖)𝛼)𝐾tan((𝜋/2𝛼𝑛𝜔(𝛼)=−^𝛼𝑐𝑛)𝛼)𝑧sin((𝜋/2^𝑖
𝐼𝐼^𝛽	𝑅3𝑛𝐾(𝑠)=^𝛽𝑛𝑠^1−𝛽+𝐾𝛼𝑛𝑠	𝐾𝛽𝑛, 𝐾𝛼𝑛 and 𝛽	𝐾𝛽𝑛𝜔(𝛽)=^𝛽𝑐𝑛()𝛽𝑧cos𝜋/2^𝑟𝐾𝛼𝑛(𝛽)=−𝜔𝑐𝑛𝑧𝑖+𝑧𝑟𝜋tan2𝛽
𝐼^𝛼𝐷^1−𝛼	𝑅4𝑛𝐾(𝑠)=^𝛽𝑛𝑠+𝐾𝛼𝑛𝑠^𝛼	𝐾𝛽𝑛, 𝐾𝛼𝑛 and 𝛼	𝐾𝛽𝑛1(𝛼)=𝜔^1−𝛼𝑐𝑛𝑧𝑟𝜋sin2𝛼+𝑧𝑖𝜋cos2𝛼𝐾𝛼𝑛(𝛼)=𝜔^𝛼𝑐𝑛𝑧𝑟𝜋cos2𝛼−𝑧𝑖𝜋sin2𝛼