Hybrid Set-theoretic Approaches for Constrained Control and Estimation with Applications to Autonomous Sailing Boats

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Set-theoretic or set-membership methods generally refer to all the set-membership techniques theoretically based on some properties of subsets of the state space. On the one hand, sets are a pertinent language and formulation to specify uncertainties, constraints, estimation errors, design specifications and system performances for control systems. On the other hand, sets also play an active role in the solutions of the problems due to their specific mathematical formats and the associated set-theoretic methods. For example, branch-and-bound of an admissible domain can lead to the global minimum or maximum of a non-convex optimization problem and a Lyapunov function for guaranteeing the stability of a system may be synthesized from positively invariant sub-level sets.Various types of sets such as intervals, ellipsoids, zonotopes and polytopes have been extensively studied in the literature and each individual type of set has its own advantages and disadvantages in the context of problem formulation and solving. Making full use of the specific advantages provided by each set-membership tool, this research project aims to develop hybrid set-theoretic approaches that can integrate all set-membership tools as well as their specific advantages. For instance, the idea of bisecting an interval has been introduced to bisect a zonotope and a polytope can be represented as the intersection of zonotopes for enabling exact polytopic set computation via zonotopic set computation. The developed hybrid set-theoretic methods and algorithms are to used for solving constrained control and estimation issues encountered in various kinds of control systems with improved accuracy and/or efficiency such as the search of a robust control invariant set via the bisection of zonotopes or even polytopes and guaranteed state estimation via exact polytopic set computation for nonlinear discrete-time systems. These developed hybrid set-theoretic methods and algorithms are also to be used for reachability analysis and fault detection of piecewise affine and hybrid systems. Finally, these theoretical progresses on constrained control and estimation via the developed hybrid set-theoretic approaches are to be applied to solve path planning, obstacle avoidance, control and estimation issues of autonomous sailing boats under complex marine environments where set-based solutions are more pertinent than point-based solutions.