**4. Numerical Examples**

Some of the previously presented autotuning methods are used to determine the parameters of various types of fractional-order controllers for a series of processes that exhibit time delays, integrative effects, overdamped and poorly damped responses, higher orders. For simplicity, only the most recent and widely used autotuning methods are considered. All resulting fractional-order controllers are implemented using the same method and the same approximation parameters [50]. For the numerical examples considered in this manuscript, all fractional-order controllers are implemented with the proportional, fractional integration and differentiation actions on the error signal. The peaks in some

FO-PID controller output signals are not the result of the tuning; they are simply the result of using derivative action on a setpoint step. These can be removed by implementing the FO-PI action on the error signal and FO-D action on the output signal.
