Implicit Fixed-Time Convergence ISS Safe Control Barrier Functions for Signal Temporal Logic Tasks

Fixed-time convergence control barrier functions (FxTC-CBFs) are proved to be powerful to deal with a typical signal temporal logic (STL) specification, "reachability", which drives the state of a dynamical system to a given set in a specific time interval and enforces the system trajectories to evolve in the safe set at all times. However, existing literature rarely discussed the effect of external disturbances on "reachability" specifications, and the corresponding robust FxTC-CBFs are not well studied. To address these problems, in this paper, the FxTC input-to-state safety CBFs (FxTC-ISSf-CBFs) is proposed. As a result, the original reachability specification updates to a new reachability specification. Meantime, instead of the original target set of the reachability specification, the FxTC-ISSf-CBFs guarantees forward invariance of a safety set of a larger size. Moreover, motivated by the fact that computing the derivatives of a CBF with explicit expression is often complicated and time-consuming, the implicit FxTC-ISSf-CBFs (IFxTC-ISSf-CBFs) is introduced based on an adaption of the implicit Lyapunov condition. Finally, we formulate a quadratic program (QP) to compute the control input that satisfies the conditions of the IFxTC-ISSf-CBFs.