Backward Reachability Analysis for Nonlinear Dynamical Systems via Pseudospectral Method
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ISSN:1598-6446 eISSN:2005-4092 http://www.springer.com/12555
Backward Reachability Analysis for Nonlinear Dynamical Systems via Pseudospectral Method Myoung Hoon Lee and Jun Moon* Abstract: In this paper, we propose a new approach to solving the backward reachability problem for nonlinear dynamical systems. Previously, this class of problems has been studied within frameworks of optimal control and zero-sum differential games, where a backward reachable set can be expressed as the zero sublevel set of the value function that can be characterized by solving the Hamilton-Jacobi-Bellman (HJB) partial differential equation (PDE). In many cases, however, a high computational cost is incurred in numerically solving such HJB PDEs due to the curse of dimensionality. We use the pseudospectral method to convert the associated optimal control problem into nonlinear programs (NLPs). We then show that the zero sublevel set obtained by the optimal cost of the NLP is the corresponding backward reachable set. Note that our approach does not require solving complex HJB PDEs. Therefore, it can reduce computation time and handle high-dimensional dynamical systems, compared with the numerical software package developed by I. Mitchell, which has been used widely in the literature to obtain backward reachable sets by solving HJB equations. We provide several examples to validate the effectiveness of the proposed approach. Keywords: Backward reachable set, nonlinear programming, optimal control, pseudospectral method.
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INTRODUCTION
In recent years, researchers have increasingly studied (forward and backward) reachability analysis for nonlinear dynamical systems under various operational constraints. The problem entails the identification of a set of initial and terminal states with which the system can accomplish its control objective, such as collision avoidance, target tracking, and reachability. For example, it may be impossible to transfer arbitrary initial states to arbitrary terminal states if there exist some constraints of state and control variables in dynamical systems, which implies a failure of reachability. Therefore, reachability analysis of a nonlinear dynamical system under various constraints takes a large portion among several control problems [1–3]. The reachability analysis has been applied to various control applications of autonomous systems, such as optimal target tracking, optimal collision avoidance, pursuit-evasion differential games, and reach-avoid problems of unmanned vehicles [1–12]. Generally, for a dynamical system, the forward reachable set is described by the set of all terminal states for
a given set of initial states, in which the system can be transferred under the state and control constraints. On the other hand, the backward reachable set is described by the set of all initial states for a given set of terminal states, in which the system can be transferred under the state and control constraints [4]. To characterize the reachable set, it is necessary to solve optimal control problems (OCPs)
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