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Format:
Online
Author:
Perez Avellaneda, Ivan
Dept./Program:
Electrical and Biomedical Engineering
Year:
2023
Degree:
Ph. D.
Abstract:
A reachable set is the set of all possible states produced by applying a set of inputs, initial states, and parameters. The fundamental problem of reachability is checking if a set of states is reached provided a set of inputs, initial states, and parameters, typically, in a finite time. In the engineering field, reachability analysis is used to test the guarantees of the operation's safety of a system. In the present work, the reachability analysis of nonlinear control affine systems is studied by means of the Chen-Fliess series. Different perspectives for addressing the reachability problem, such as interval arithmetic, mixed-monotonicity, and optimization, are used in this dissertation. The first two provide, in general, an overestimation of the reachable set that is not guaranteed to be the smallest. To improve these methods and obtainthe minimum bounding box of the reachable set, the derivative-based optimization of Chen-Fliess series is developed. To achieve this, the closed form of the Gâteaux and Fréchet derivatives of Chen-Fliess series and several other tools from analysis are obtained. To provide a representation of these tools practically and systematically, an abstract algebraic derivative acting on words of a monoid is defined. Three nonconvex optimization algorithms are implemented for Chen-Fliess series. The problem of computing an inner approximation of the reachable set via Chen-Fliess series is also solved by means of convex analysis tools. Furthermore, a method for the computation of the backward reachable set of an output set is also provided. In this case, different from forward reachability analysis, the feasibility problem represents a challenge and requires using the Positivestellensatz. Examples and simulations are provided for every method presented. The application of control barrier functions via Chen-Fliess series is outlined. Finally, the future work and conclusions are stated in the last chapter.