Abstract
In this chapter, we show how the problem of controller synthesis can be posed as a form of convex optimization in an operator-theoretic framework. Furthermore, we show how to: (a) Parameterize the integral and multiplier operator-valued decision variables using finite-dimensional vectors; (b) Verify and enforce positivity and negativity of multiplier and integral operators using positive matrices; (c) Invert positive integral and multiplier operators through the use of a new formula based on algebraic manipulation. Finally, we show how these 3 parts can be combined into a computational procedure for finding a stabilizing state-feedback controller for systems defined by differential-difference equations—a class which includes differential systems with discrete delays. Finally, a numerical example is used to illustrate the form of the resulting stabilizing controller.
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Acknowledgements
This work was supported by National Natural Science Foundation of PR China under Grant 61503189, the Natural Science Foundation of Jiangsu Province under Grant BK20150926. This work was also supported by NSF Grants 1538374, 1301660, 1301851.
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Miao, G., Peet, M.M., Gu, K. (2019). Inversion of Separable Kernel Operator and Its Application in Control Synthesis. In: Valmorbida, G., Seuret, A., Boussaada, I., Sipahi, R. (eds) Delays and Interconnections: Methodology, Algorithms and Applications. Advances in Delays and Dynamics, vol 10. Springer, Cham. https://doi.org/10.1007/978-3-030-11554-8_17
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