For any initial conditions, the problem control of chaotic oscillations in FOSG system is equivalent to that of the states of the FOSG system remaining on the surface st for all t >0.
Considering that adaptive sliding mode control input law in (15) is used to control the FOSG system in (10) with parameter estimate in (32), then the controller (31) is globally asymptotically stable.
The FOSG system (10) with the robust adaptive sliding mode controller (31) and parameter estimate (32) with initial conditions as in Section 2 and initial condition for the parameter estimate [i.sub.0](0) = 0.1 are numerically analyzed using Matlab.
where [u.sub.i] is the fractional order PID action control, [delta], [beta] are the fractional order differential and integral operators [42-44], [e.sub.i] is the error signal given by [e.sub.i] = 0 - [x.sub.i], and [K.sub.P], [K.sub.I], [K.sub.D] are the proportional, integral, and the derivative gains to suppress the nonlinearity in the FOSG system.
Table 1 shows the FOPID gain values after running the GA solver from the optimization tool with the options cited above, we get the best solutions tracked over generations for the complete chaos suppression of the FOSG system via fractional order PID controllers using gains values genetically optimized; Figure 12 shows the time history of the FOSG states ([x.sub.1], [x.sub.2], [x.sub.3], [x.sub.4]) and Figure 13 shows the time history of parameter estimate.
FPGA Implementation of the FOSG System and GAFOPID Controllers
The fractional order of the FOSG system and GAFOPID controllers are kept as q = 0.992.
Power utilization for various fractional orders is presented and it is shown that the system uses maximum power when the FOSG system exhibits its largest Lyapunov exponent.
Caption: FIGURE 12: Time history of GAFOPID controlled FOSG system states.
Caption: FIGURE 14: Xilinx schematics of the FOSG system implemented in Kintex 7 (device = 7k160t; package = fbg484 S).
Caption: FIGURE 15: 3D phase portraits of the FPGA implemented FOSG system.
Caption: FIGURE 16: (a) Power utilized by FOSG system.