Double Inverted Pendulum

The double inverted pendulum system [6] (see Figure 1) is a standard benchmark in the field of control. The system consists of a cart and a two-part pole attached to the cart. The system has six states which are cart position on a track, cart velocity, and the angles and the angular velocities of the two attached pendulums. The single control signal is the lateral force applied to the cart. The dynamical equations for the double inverted pendulum system can be found e.g. in [6], in this experiment a discrete system with a time step of $\Delta t=0.05$    s was simulated using the MATLAB ordinary differential equation solver ode23.

To make sure that the learning scheme did not favour the proposed algorithm, standard backpropagation algorithm was used to learn an MLP network to model the system dynamics using a relatively small sample of 2000 input-output pairs. To make this problem more challenging, only the velocity and position of the cart and the angle of the upper pendulum were available as observations, and the rest of the state had to be inferred from these. Experiments were run on ten different data sets with 50 samples each using 5 different initialisations. The final results can be seen in Figure 1.

IEKS suffered from quite serious convergence problems with this data set. These problems were especially bad during the early iterations, but several runs failed to converge to a meaningful result even after the iteration limit was reached. IUKS performed somewhat better, but suffered from some stability problems too. The proposed method was much more robust and did not suffer from stability issues and also performed better on average than the two Kalman filter based algorithms. It should be noted, however, that in some experiments both IEKS and IUKS converged in only a few iterations, resulting in a superior performance compared to the proposed method. Therefore the problem with IEKS and IUKS may at least partially be related to poor choice of initialisations.

Figure 1: Inference with the double inverted pendulum system. On the left the schematic of the system, on the right root mean square error plotted against computation time.

[cc][cc]$\phi_1$ [cc][cc]$\phi_2$ [cc][cc] $l_1=0.5$   m [cc][cc] $l_2=0.6$   m [cc][cc] $M=1.0$   kg [cc][cc] $m_1=0.1$   kg [cc][cc] $m_2=0.12$   kg [cc][cc]$F$ [cc][cc]$s$