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Optimal Control of a Constrained Bilinear Dynamic System

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Abstract

In this paper, an optimal feedback, for a free vibrating semi-active controlled plant, is derived. The problem is represented as a constrained optimal control problem of a single input, free vibrating bilinear system, and a quadratic performance index. It is solved by using Krotov’s method and to this end, a novel sequence of Krotov functions that suits the addressed problem, is derived. The solution is arranged as an algorithm, which requires solving the states equation and a differential Lyapunov equation in each iteration. An outline of the proof for the algorithm convergence is provided. Emphasis is given on semi-active control design for stable free vibrating plants with a single control input. It is shown that a control force, derived by the proposed technique, obeys the physical constraint related with semi-active actuator force without the need of any arbitrary signal clipping. The control efficiency is demonstrated with a numerical example.

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Notes

  1. Recall that \({\mathbf {M}}>0\), \({\mathbf {K}}>0\) iff \({\mathbf {z}}^T{\mathbf {M}}{\mathbf {z}}>0\), \({\mathbf {z}}^T{\mathbf {K}}{\mathbf {z}}>0\) for all \({\mathbf {z}}\in \mathbb {R}^{n_z}\), \({\mathbf {z}}\ne {\mathbf {0}}\) and \({\mathbf {C}}_d\ge 0\) iff \({\mathbf {z}}^T{\mathbf {C}}_d{\mathbf {z}}\ge 0\) for all \({\mathbf {z}}\in \mathbb {R}^{n_z}\).

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Acknowledgements

Ido Halperin is grateful for the support of The Irving and Cherna Moskowitz Foundation for his scholarship.

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Authors and Affiliations

  1. Ariel University, Ariel, Israel

    Ido Halperin, Grigory Agranovich & Yuri Ribakov

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  1. Ido Halperin
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  2. Grigory Agranovich
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  3. Yuri Ribakov
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Correspondence to Ido Halperin.

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Halperin, I., Agranovich, G. & Ribakov, Y. Optimal Control of a Constrained Bilinear Dynamic System. J Optim Theory Appl 174, 803–817 (2017). https://doi.org/10.1007/s10957-017-1095-2

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