Gradient Descent Approach to Optimal Mode Scheduling in Hybrid Dynamical Systems |
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Authors: | H Axelsson Y Wardi M Egerstedt E I Verriest |
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Institution: | (1) Kencast, Stamford, CT, USA;(2) School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA, USA |
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Abstract: | This paper concerns the problem of optimally scheduling the sequence of dynamic response functions in nonlinear switched-mode
hybrid dynamical systems. The control parameter has a discrete component and a continuous component, namely the sequence of
modes and the duration of each mode, while the performance criterion consists of a cost functional on the state trajectory.
The problem is naturally cast in the framework of optimal control. This framework has established techniques sufficient to
address the continuous part of the parameter, but lacks adequate tools to consider the discrete element. To get around this
difficulty, the paper proposes a bilevel hierarchical algorithm. At the lower level, the algorithm considers a fixed mode
sequence and minimizes the cost functional with respect to the mode durations; at the upper level, it updates the mode sequence
by using a gradient technique that is tailored to the special structure of the discrete variable (mode sequencing). The resulting
algorithm is not defined on a single parameter space, but rather on a sequence of Euclidean spaces of increasing dimensions,
an unusual setting for which there is no established notion of convergence. The paper suggests first a suitable definition
of convergence based on the concepts of optimality functions; then, it proves that the proposed algorithm converges in that
sense.
Research supported in part by the National Science Foundation under Grant #0509064. |
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Keywords: | Switched-mode systems Gradient descent Optimality functions Optimal control |
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