Contribuições à síntese de controladores fracionários: simulação numérica, identificação, sintonia de parâmetros, análise de estabilidade e implementação embarcada

Abstract

In this thesis, innovative proposals are developed and implemented for different stages that constitute the efficient design of fractional-order controllers, namely: numerical simulation, system identification, analytical stability analysis, parameter tuning, and embedded implementation. For the stability analysis stage, a completely novel approach is developed using the Lambert-Tsallis function to determine the poles of fractional-order control systems whose characteristic equation assumes the generic form of a trinomial given by $a_n s^n + a_m s^m + a_0 = 0$, $\forall , n, m \in~\mathbb{Q}^*$ and $a_n a_m a_0 \neq 0$. As a consequence, an analytical equation is also developed to define the bounds of the controller parameters that ensure the stability of such systems. To illustrate the use of the developed tools, both hypothetical plants from the literature and real-world plants are considered. This analytical capability is extended to parameter tuning, so that the search space of an optimization algorithm is constrained to the hypervolume corresponding to the stability region of the control system under consideration. To validate the approach in a real application context, a discrete fractional-order controller is developed and embedded in a neonatal incubator for temperature control. The plant model was obtained using a recursive identification algorithm for fractional-order systems proposed in this thesis, based on the LMS (least mean square) algorithm. The resulting fractional-order model achieved a VAF (variance accounted for) of 97.84\%. During the controller tuning process, the computational simulations are carried out using discrete equations developed in this thesis, resulting in a processing time approximately 250 times shorter than the elapsed time required by toolboxes available in the literature. Finally, an implementation of this controller is proposed for the 8-bit microcontroller of the incubator, using an MLP (multilayer perceptron) neural network representation to mitigate the loss of optimality in parameter tuning observed in traditional discretization methods for such controllers. The proposed approach achieved an RMSE (root mean square error) of 0.0715, compared to 0.5032 obtained using the Merrikh-Bayat technique. In the embedded system, with only 4 neurons in the hidden layer, the neural fractional-order controller required just 6.511 ms for execution, occupying 1556 bytes of program memory and 241 bytes of RAM. The response of the control system exhibited an overshoot of 1.66\% and a settling time of 8183 s.