Date: Tuesday July 19th 2022

Time: 1:30 - 3:30 pm ET

Location: TSRB 523

Matthew Abate

Robotics PhD Candidate

School of Mechanical Engineering

Georgia Institute of Technology

Committee

Dr. Samuel Coogan (Advisor) - Department of Eletrical and Computer Engineering, Georgia Tech

Dr. Eric Feron (Advisor) - Division of Computer, Electrical and Mathematical Sciences and Engineering, KAUST

Dr. Matthieu Bloch - Department of Eletrical and Computer Engineering, Georgia Tech
Dr. Panagiotis Tsiotras - Department of Aerospace Engineering, Georgia Tech
Dr. Yorai Wardi - Department of Eletrical and Computer Engineering, Georgia Tech

Summary 

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 seven-dimensional spacecraft system and a hardware demonstration of in-the-loop reachability analysis and enforced system safety.  Numerous illustrative numerical examples are also provided.