Fractional Order Control of Unstable and Time-Delayed Linear Time-Invariant (LTI) Plants

The present work describes the formulation of the fractional order (FO) proportional–integral–derivative \(PI^{\lambda } D^{\mu }\) controllers entailing FO integrator and FO differentiator, a procedure which allows defining the numerical terms of the FO controller. These are inferences to be the general form of PIDs, the output of which is a linear combination of the input, a fractional derivative of the input, and a fractional integral of the input. They have more tuning freedom and a wider region of parameters that may stabilize the plant under control surpassing the performance of the integer-order (IO) counterparts, specifically in servo control applications. A consequential issue in control engineering is presented by plants and processes that are unstable. Therefore, in this work unstable as well as time-delayed plants have been considered which are frequently encountered in the field of control.