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Optimal Feedback Control for Linear Systems with Input Delays Revisited

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Abstract

The design problem of optimal feedback control for linear systems with input delays is very important in many engineering applications. Usually, the linear systems with input delays are firstly converted into linear systems without delays, and then all the design procedures are based on the delay-free linear systems. In this way, the feedback controllers are not designed in terms of the original states. This paper presents some new closed-form formula in terms of the original states for the delayed optimal feedback control of linear systems with input delays. We firstly reveal the essential role of the input delay in the optimal control design of the linear system with a single input delay: the input delay postpones the action of the optimal control only. Based on this fact, we calculate the delayed optimal control and find that the optimal state can be represented by a simple closed-form formula, so that the delayed optimal feedback control can be obtained in a simple way. We show that the delayed feedback gain matrix can be “smaller” than that for the controlled system with zero input delay, which implies that the input delay can be considered as a positive factor. In addition, we give a general formula for the delayed optimal feedback control of time-variant linear systems with multiple input delays. To show the effectiveness and advantages of the main results, we present five illustrative examples with detailed numerical simulation and comparison.

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Acknowledgments

The authors thank the financial support of NSF of China under Grants 11032009 and 11372354, Funding of Jiangsu Innovation Program for Graduate Education and the Priority Academic Program Development of Jiangsu Higher Education Institutions.

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Authors and Affiliations

  1. State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nangjing, 210016, China

    Yusheng Zhou & Zaihua Wang

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  1. Yusheng Zhou
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  2. Zaihua Wang
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Correspondence to Zaihua Wang.

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Communicated by Felix L. Chernousko.

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Zhou, Y., Wang, Z. Optimal Feedback Control for Linear Systems with Input Delays Revisited. J Optim Theory Appl 163, 989–1017 (2014). https://doi.org/10.1007/s10957-014-0532-8

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  • DOI: https://doi.org/10.1007/s10957-014-0532-8

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