$α$-stability of Differentially Flat Systems with Application to Newton-Raphson Tracking Control for Vehicle Dynamics

Abstract: This paper studies the $α$-stability property of differentially flat nonlinear dynamical systems. The results build off the recently introduced notion of $α$-stability, which is particularly amenable to characterize the ability of a system to track dynamic output reference signals. We consider systems controlled using the Newton-Raphson tracking controller, which results in closed-form control policies, therefore it is computationally efficient, and it has been shown to be effective to control a large variety of mobile robots, including autonomous vehicles. The main results of the paper consist in sufficient conditions for the $α$-stability of differentially flat systems and for the equivalence between the proposed control algorithm and the Newton-Raphson tracking controller applied directly to the nonlinear dynamics. We demonstrate the behavior of the proposed controller applied to the kinematic unicycle and dynamic bicycle models.