Synthesis of Input-to-State Attractivity Controllers for Transition Systems With Disturbances

In this article, we introduce the notion of input-to-state attractivity (ISA) controllers for a class of finite transition systems with disturbances. The performances of an ISA controller are characterized by a gain function that quantifies the deviation of closed-loop trajectories from the target set as a function of the amplitude of past disturbances on a bounded time window. We prove the existence of controllers that are gain-optimal (GO) in the sense that their gain function is minimal (with respect to a given order on the set of gain functions) over all possible ISA controllers. Then, we consider the problem of synthesizing ISA controllers. We present an approach based on successive refinements of controllers: starting from a controller synthesized against worst-case disturbances, the controller is iteratively refined in order to improve the closed-loop behavior under lower disturbances. We prove that our method makes it possible to synthesize an ISA controller that is shown to be a GO-ISA controller (for the colexicographic order) when a condition, which can be easily checked a posteriori, is satisfied. Finally, an application of adaptive cruise control demonstrates the effectiveness of our approach.