This article presents a new method for computing sharp bounds on the solutions of nonlinear dynamic systems subject to uncertain initial conditions, parameters, and time-varying inputs. Such bounds are widely used in algorithms for uncertainty propagation, robust state estimation, system verification, global dynamic optimization, and more. Recently, it has been shown that bounds computed via differential inequalities can often be made much less conservative by exploiting state constraints that are known to hold for all trajectories of interest (e.g., path constraints that describe feasible trajectories in the context of dynamic optimization, or constraints that explicitly describe invariant sets containing all system trajectories). However, effective bounding algorithms of this type are currently only available for problems with linear constraints. Moreover, the theoretical results underlying these algorithms do not apply to constraints that depend on time-varying inputs and rely on assumptions that prove to be very restrictive for nonlinear constraints. This article contributes a new differential inequalities theorem that permits the use of a very general class of nonlinear state constraints. Moreover, a new algorithm is presented for efficiently exploiting nonlinear constraints to achieve tighter bounds. The proposed approach is shown to produce very sharp bounds for two challenging case studies.

本文提出了一种新的方法，可以在不确定的初始条件，参数和时变输入的情况下，对非线性动力系统的解进行计算。这样的界限广泛用于不确定性传播，鲁棒状态估计，系统验证，全局动态优化等算法中。最近，研究表明，通过利用已知的对所有感兴趣轨迹都适用的状态约束（例如，在动态优化的情况下描述可行轨迹的路径约束），通常可以使通过微分不等式计算的边界的保守性降低得多。明确描述包含所有系统轨迹的不变集的约束）。但是，这种有效的边界算法目前仅可用于线性约束问题。此外，这些算法的理论结果不适用于依赖于随时间变化的输入的约束，并且不适用于对非线性约束非常严格的假设。本文提出了一个新的微分不等式定理，该定理允许使用非常通用的一类非线性状态约束。此外，提出了一种新算法，可有效利用非线性约束条件以实现更严格的边界。结果表明，所提出的方法为两个具有挑战性的案例研究提供了非常清晰的界限。本文提出了一个新的微分不等式定理，该定理允许使用非常通用的一类非线性状态约束。此外，提出了一种新算法，可有效利用非线性约束条件以实现更严格的边界。结果表明，所提出的方法为两个具有挑战性的案例研究提供了非常清晰的界限。本文提出了一个新的微分不等式定理，该定理允许使用非常通用的一类非线性状态约束。此外，提出了一种新算法，可有效利用非线性约束条件以实现更严格的边界。结果表明，所提出的方法为两个具有挑战性的案例研究提供了非常清晰的界限。