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Switching Time and Parameter Optimization in Nonlinear Switched Systems with Multiple Time-Delays

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Abstract

In this paper, we consider a dynamic optimization problem involving a general switched system that evolves by switching between several subsystems of nonlinear delay-differential equations. The optimization variables in this system consist of: (1) the times at which the subsystem switches occur; and (2) a set of system parameters that influence the subsystem dynamics. We first establish the existence of the partial derivatives of the system state with respect to both the switching times and the system parameters. Then, on the basis of this result, we show that the gradient of the cost function can be computed by solving the state system forward in time followed by a costate system backward in time. This gradient computation procedure can be combined with any gradient-based optimization method to determine the optimal switching times and parameters. We propose an effective optimization algorithm based on this idea. Finally, we consider three numerical examples, one involving the 1,3-propanediol fed-batch production process, to illustrate the effectiveness and applicability of the proposed algorithm.

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Acknowledgments

The first author is supported by the Natural Science Foundation for the Youth of China (Grant Nos. 11201267 and 11001153), the Tian Yuan Special Funds of the National Natural Science Foundation of China (Grant No. 11126077), and the Shandong Province Natural Science Foundation of China (Grant Nos. ZR2010AQ016 and ZR2011AL003). The second author is supported by the Natural Science Foundation of China (Grant No. 11350110208) and the Australian Research Council (Discovery Grant DP110100083). The third author is supported by the Australian Research Council (Discovery Grant DP110100083).

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Correspondence to Kok Lay Teo.

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Communicated by Alexander S. Strekalovsky.

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Liu, C., Loxton, R. & Teo, K.L. Switching Time and Parameter Optimization in Nonlinear Switched Systems with Multiple Time-Delays. J Optim Theory Appl 163, 957–988 (2014). https://doi.org/10.1007/s10957-014-0533-7

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  • DOI: https://doi.org/10.1007/s10957-014-0533-7

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