Tight Remainder-Form Decomposition Functions With Applications to Constrained Reachability and Guaranteed State Estimation

In this article, we propose a tractable family of remainder-form mixed-monotone decomposition functions that are useful for overapproximating the image set of nonlinear mappings in reachability and estimation problems. Our approach applies to a new class of nonsmooth, discontinuous nonlinear systems that we call either-sided locally Lipschitz semicontinuous systems, which we show to be a strict superset of locally Lipschitz continuous systems, thus expanding the set of systems that are formally known to be mixed-monotone. In addition, we derive lower and upper bounds for the overapproximation error and show that the lower bound is achieved with our proposed approach, i.e., our approach constructs the tightest, tractable remainder-form mixed-monotone decomposition function. Moreover, we introduce a set inversion algorithm that along with the proposed decomposition functions can be used for constrained reachability analysis and guaranteed state estimation for continuous- and discrete-time systems with bounded noise.