Fractional Complex Dynamical Systems for Trajectory Tracking using Fractional Neural Network via Fractional Order PID Control Law

FPGA Realization and Lyapunov–Krasovskii Analysis for a Master-Slave Synchronization Scheme Involving Chaotic Systems and Time-Delay Neural Networks

“…, 4) are positive real parameters. System (27) has two equilibrium points E 1 (0, 0, 0) and E 2 (a 1 /a 4 , 0, 0). e parameters and initial conditions of ( 27) are chosen as…”

“…e corresponding eigenvalues of equilibria E 1 , E 2 and their stability are described in Table 1. e Lyapunov exponents for system (27) are LE 1 � 0.0818, LE 2 � 0, LE 3 � − 0.5227 which according to [34] make the system chaotic. e algorithm employed for determining Lyapunov exponents was proposed in [35].…”

“…Its Kaplan-Yorke dimension is D KY � 2.1564, and the phase plane of the chaotic system is displayed in Figure 1. In addition, the bifurcation diagrams of system (27) in terms of the parameters a 1 and a 3 are plotted in Figures 2(a By considering the following delayed neural network defined as in equation ( 1), we get…”

“…In Table 3, some synchronization schemes in a drive-response configuration are displayed 3, reference [40] considers the implementation of a synchronization scheme on an Arduino chip while the rest of the references only consider numerical results. In this paper, we implement the chaotic system (27), the delayed recurrent neural network (28), and the control laws ( 29) and ( 30) in an FPGA. Table 4 presents a brief review, including some FPGAbased implementations of chaotic systems over the last ten years.…”

“…e Genesio-Tesi and Liu systems are considered. Although the proposed scheme could be useful to follow other nonlinear systems, we decide to restrict this work using only these two chaotic systems because their tracking can be more challenging than other tracking problems such as the reproduction of the gait cycle in humanoids [25] or the trajectory tracking in different mechanical systems [26] and complex systems [27]. In all the former cases, the movement may be softer and periodic.…”