Sure robust global versus robust global almost sure convergence: Implementing a dynamic random coefficient selection in a quaternion-hysteresis-based distributed hybrid algorithm

By employing Lyapunov-like conditions along with the stochastic invariance principle, we develop a quaternion-hysteresis-based distributed stochastic hybrid feedback algorithm aimed at achieving robust global attitude synchronization, addressing both sure and almost sure convergence. The distributed hybrid algorithm integrates a hybrid control variable that experiences continuous changes and instantaneous resets according to a unit quaternion-based reset rule and the associated hysteresis-based conditions. In this hysteresis-based hybrid framework, by treating the reset coefficients as random variables, these coefficients introduce the unique source of randomness; these coefficients are dynamically and randomly selected by the reset rule, leading to varying hysteresis half-widths and, consequently, different nominal levels of robustness; the orientation of the spring force, which reversely pulls the rigid body along the rotation axis to prevent the unwinding phenomenon, is determined by the deterministic sign of the hybrid control variable, whereas the hysteresis half-widths can delay this pulling. By dynamically and randomly adjusting the stochastic hysteresis half-widths, we create a flexible tradeoff between mitigating the amount of unwinding and enhancing the nominal level of robustness to measurement noise. By imposing different constraints on the support domain of random reset coefficients, the hybrid algorithm achieves both sure robust global and robust global almost sure attitude synchronization.