FPGA implementation of nonlinear equations with delay

Abstract

The study of dynamic systems has been the subject of growing interest in recent decades. Its purpose is to reveal the phenomena or behaviors of the systems, and even those hidden, for given initial conditions and parameter values. Their implementations allow them to be exploited in various fields of application. The present work deals with the study and FPGA implementation of nonlinear time-delay equations. These are the Johnson and Moon equation and the Mackey-Glass equation. The stability analysis of these two models allowed to plot the stability chart in the parameter space of each system, thus delimiting the areas of stability and instability. Numerical simulation for different values of the control parameter revealed a damped, periodic and even chaotic oscillation for the two systems. An architecture based on the fourth-order Runge-Kutta method (RK4) taking into account the delay, in which the calculations are performed according to the 32-bit IEEE-754–1985 floating-point numbers standard, has been proposed for the hardware description of the systems. The corresponding VHDL code in each case was synthesized on the Xilinx Virtex-6 family XCV6VLX130T-3FF484 FPGA chip and the maximum operating frequencies obtained are of the order of 431.727 MHz and 432.280 MHz respectively for the Johnson and Moon and Mackey-Glass equation. The results obtained from the Modelsim simulation of the FPGA implementation of the models are identical to those of the numerical simulation. This contribution highlighted a new approach for the FPGA implementation of time-delay systems and can be exploited for other time-delay systems.

Keywords

• Nonlinear equations
• Delay
• FPGA implementation
• Stability analysis
• Chaos