Damped Pendulum

Next, genetic programming tackled the damped pendulum system, given by Eqs. 4 and 5:

$$\dot x_1 = x_2$$ (4)

$$\dot x_2 = -\sin x_1-x_2$$ (5)

The ranges for $x_1$ and $x_2$ are $x_1\in[-\pi/2,\pi/2]$ and $x_2\in[-1,1]$. The nonterminal set was $\{+,-,\times,\div,\cdot ^2,\sqrt{\cdot },\sin,\cos,\mathrm{arctan}\}$.

For this system, the tests produced direct hits quite quickly (not more than eight generations) in almost all cases. Figs. 8 and Fig. 9 show the results of two test runs.

Fig. 8 shows one of the oddest functions produced: a function composed of a lot of arctangents. This the interesting thing about genetic programming: it can produce functions that don't look at all as we would expect them. But, assuming this function did not latently stray across the $V=0$ plane, it proves the stability nonetheless.

Fig. 9 shows a false hit. This function was returned as 100% Lyapunov, although there are regions of positive $\dot V$ near the $x_2$-axis, which weren't tested and slipped through. The fitness approximation needs improvement.

Figure 8: Result of a genetic programming Lyapunov search for a damped pendulum system.

$$V(x_1,x_2) = \left( \mathrm{arctan}(x_1 \mathrm{arctan} x_... ...{\sqrt{1 + (\mathrm{arctan}(x_2 + \sin x_1)) ^ 2}} \right) / 8$$

$$\dot x_1 = x_2$$

$$\dot x_2 = -\sin x_1-x_2$$

Figure 9: False result of a genetic programming Lyapunov search for a damped pendulum system.

$$V(x_1,x_2) = \mathrm{arctan}(\mathrm{arctan}(x_1^2+(x_1+x_2)^2))$$

$$\dot x_1 = x_2$$

$$\dot x_2 = -\sin x_1-x_2$$