Adaptive Control Barrier Functions with Vanishing Conservativeness Under Persistency of Excitation

Abstract

This article presents a closed-form adaptive controlbarrier-function (CBF) approach for satisfying state constraints in systems with parametric uncertainty. This approach uses a sampled-data recursive-least-squares algorithm to estimate the unknown model parameters and construct a nonincreasing upper bound on the norm of the estimation error. Together, this estimate and upper bound are used to construct a CBF-based constraint that has nonincreasing conservativeness. Furthermore, if a persistency of excitation condition is satisfied, then the CBFbased constraint has vanishing conservativeness in the sense that the CBF-based constraint converges to the ideal constraint corresponding to the case where the uncertainty is known. In addition, the approach incorporates a monotonically improving estimate of the unknown model parameters thus, this estimate can be effectively incorporated into a desired control law. We demonstrate constraint satisfaction and performance using 2 two numerical examples, namely, a nonlinear pendulum and a nonholonomic robot.

Figures (11)

• Figure 2: $\xi$ given by \ref{['ex:xi']}.
• Figure 3: $\gamma$, $\dot{\gamma}$ and $u$ for Cases 1, 2 and 3. Note that $\gamma_{\rm{d}}$, $\dot{\gamma}_{\rm{d}}$ and $u_{\rm{d}}$ are shown with dashed lines.
• Figure 4: $\psi_{0}$, $\psi_{1}$ and $\psi$ for Cases 1, 2 and 3.
• Figure 5: $\theta$ for Cases 1, 2 and 3. Note that $\theta_{*}$ is shown using dashed line.
• Figure 6: $\nu$ for Cases 1, 2 and 3. Note that $||\tilde{\theta}_{k}||$ is shown with dashed line.

Theorems & Definitions (5)

• proof : Proof
• proof : Proof
• proof : Proof
• proof : Proof
• proof