SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Consider the setting of the example in Section 3.2. Let  $b_{c}$  denote the NCBF defined in the example, which fails our defined safety conditions. For comparison, we trained an NCBF  $b_{\theta}$  and verified it using our proposed approach. We then constructed a nominal controller  $\mu_{nom}$  as a Linear Quadratic Regulator (LQR) controller that drives the system from initial point  $(0,0.1)$  to the origin. We compared the trajectories arising from the optimization-based controller defined by Eq. (10) using the  $b_{\theta}$  and  $b_{c}$ . For the unsafe NCBF  $b_{c}$ , the optimization-based controller is unable to satisfy the safety constraints at the boundary point  $(0,1)$ , resulting in a safety violation. On the other hand, while the NCBF  $b_{\theta}$  contained multiple non-differentiable points, it is possible to choose  $u$  to ensure safety at these points. For example, the point  $(-0.19,2.91)$  is a non-differentiable point on the boundary  $b_{\theta} = 0$ . There are four activation sets intersecting at this point, with corresponding values of  $\frac{\partial b_{c}}{\partial x} g(x)$  given by  $\{-0.0455, -0.053, -0.025, -0.033\}$ . Since any control input  $u$  with negative sign and sufficiently large magnitude will satisfy  $\frac{\partial b_{c}}{\partial x} (f(x) + g(x)u) \geq 0$  for all of these values, this non-differentiable point does not compromise safety of the system, and the trajectory of the system constrained by  $b_{\theta}$  remains in the safe region for all time.
Figure 4: Comparison of optimization-based controller using trained NCBF  $b_{\theta}$  and unsafe NCBF  $b_{c}$ .