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

This paper proposes a tractable family of remainder-form mixed-monotone decomposition functions that are useful for over-approximating 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 (ELLS) systems, which we show to be a strict superset of locally Lipschitz continuous (LLC) 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 over-approximation 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.