Mixed Monotonicity for Efficient Reachability with Applications to Robust Safe Autonomy
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Abate, Matthew Christian Licciardi
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Abstract
Reachability analysis of control systems plays a crucial role in system verification and controller synthesis. However, many reachability techniques fall short, being only applicable to certain classes of systems or too computationally burdensome for real-time applications. The subject of this thesis is the mixed monotonicity property of dynamical systems which is known to be a general property and which provides a computationally efficient technique for over-approximating reachable sets using hyperrectangles. Specifically, the mixed monotonicity of a dynamical system is tied to the existence of a related decomposition function that separates the system's vector field into cooperative and competitive state interactions. Reachable sets for the mixed monotone system can then be computed simply using a decomposition function and foundational results from monotone dynamical systems theory.
In this thesis, we establish that all continuous-time dynamical systems bearing a locally Lipschitz continuous vector field are mixed monotone and we provide a construction for the unique tight decomposition function of a dynamical system that attains the tightest possible over-approximations of reachable sets. We then provide a suite of new analysis tools for mixed monotone systems that can be applied to attain, for example, over- and under-approximations of both forward- and backward-time reachable sets, and also robustly forward invariant sets. As a final point, we study conservatism in mixed monotone reachable set approximations, and we provide new tools for reducing conservatism using, for example, the decomposition function of a separate dynamical system, formed via a transformation of the initial system's vector field. We conclude with a case study of a five-dimensional spacecraft system and a hardware demonstration of in-the-loop reachability analysis and enforced system safety. Numerous illustrative numerical examples are also provided.
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2022-08-02
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Dissertation