Hi @Milad
A Google search reveals that the third-party FOMCON toolbox on FileExchange includes a 9-parameter function named oustapid(), designed for obtaining the Oustaloup approximation of the fractional-order PID controller:
Gc = oustapid(Kp, Ki, lam, Kd, mu, wb, wh, N, type, red)
While this toolbox is valuable, it takes input from an expert fractional human designer to accurately define the 9 parameters through advanced mathematical transient and stability analyses. This process ensures that the controller achieves the desired performance metrics such as percent overshoot, settling time, and disturbance rejection.
For those lacking technical expertise, the complexity of the nine powerful parameters may be daunting, leading them to employ metaheuristic optimization algorithms or reinforcement learning. These methods statistically determine optimal values for the parameters within specified search regions, making them more accessible for untrained application designers.
In summary, one can either apply mathematical calculations to derive control parameter values or delegate this task to intelligent machines. Although the latter approach is potent, it still demands human intervention to carefully craft a custom objective function (not necessarily quadratic) and configure hyperparameters to govern the optimization or learning process.
Or, consider the third option, which I refer to as the pseudo-design approach: Fine-tune the integer-order PID controller to achieve satisfactory performance, and then transfer the tuned values {Kp, Ki, Kd} to the fractional-order control design. Initially, construct a fractional-order 
controller that replicates the functionality of the integer-order PID controller. Subsequently, gradually adjust the fractional-order parameters, lambda (λ) and mu (μ), by employing intelligent machines to search within a small region around the nominal values.
controller that replicates the functionality of the integer-order PID controller. Subsequently, gradually adjust the fractional-order parameters, lambda (λ) and mu (μ), by employing intelligent machines to search within a small region around the nominal values.
