Interdependence quantification for compositional control synthesis with an application in vehicle safety systems

Composing controllers designed individually for interacting subsystems, while preserving the guarantees that each controller provides for each subsystem, is a challenging task. Motivated by this challenge, we consider in this paper the problem of synthesizing safety controllers for linear parameter-varying subsystems, where the system matrices of each subsystem depend (possibly nonlinearly) on the states of the other subsystems. In particular, we propose a method for synthesis of controlled invariant sets and associated controllers, that is robust against affine parametric uncertainties in the system matrices. Then we show for certain classes of parameter dependencies how to quantify the uncertainty imposed on the other subsystems by convexifying, with an affine map, the effects of these parameters. An analysis of this quantification is provided. In the second part of the paper, we focus on an application of this method to vehicle safety systems. We demonstrate how controllers for lane-keeping and adaptive cruise control can be synthesized in a compositional way using the proposed method. Our simulations illustrate how these controllers keep their individual safety guarantees when implemented simultaneously, as the theory suggests.