# Optimal Feedback Control for Linear Systems with Input Delays Revisited

## Author

Listed:
• Yusheng Zhou

(Nanjing University of Aeronautics and Astronautics)

• Zaihua Wang

(Nanjing University of Aeronautics and Astronautics)

## 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.

## Suggested Citation

• Yusheng Zhou & Zaihua Wang, 2014. "Optimal Feedback Control for Linear Systems with Input Delays Revisited," Journal of Optimization Theory and Applications, Springer, vol. 163(3), pages 989-1017, December.
• Handle: RePEc:spr:joptap:v:163:y:2014:i:3:d:10.1007_s10957-014-0532-8
DOI: 10.1007/s10957-014-0532-8
as

File Function: Abstract

File URL: https://libkey.io/10.1007/s10957-014-0532-8?utm_source=ideas
LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
---><---

As the access to this document is restricted, you may want to search for a different version of it.

## References listed on IDEAS

as
1. N. Haddadi & Y. Ordokhani & M. Razzaghi, 2012. "Optimal Control of Delay Systems by Using a Hybrid Functions Approximation," Journal of Optimization Theory and Applications, Springer, vol. 153(2), pages 338-356, May.
2. Lindset, Snorre & Lund, Arne-Christian & Matsen, Egil, 2009. "Optimal information acquisition for a linear quadratic control problem," European Journal of Operational Research, Elsevier, vol. 199(2), pages 435-441, December.
Full references (including those not matched with items on IDEAS)

## Citations

Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
as

Cited by:

1. Yusheng Zhou & Zaihua Wang, 2016. "Motion Controller Design of Wheeled Inverted Pendulum with an Input Delay Via Optimal Control Theory," Journal of Optimization Theory and Applications, Springer, vol. 168(2), pages 625-645, February.

## Most related items

These are the items that most often cite the same works as this one and are cited by the same works as this one.
1. Mohamed Karim Bouafoura & Naceur Benhadj Braiek, 2019. "Hybrid Functions Direct Approach and State Feedback Optimal Solutions for a Class of Nonlinear Polynomial Time Delay Systems," Complexity, Hindawi, vol. 2019, pages 1-14, April.
2. Keshavarz, E. & Ordokhani, Y. & Razzaghi, M., 2019. "The Bernoulli wavelets operational matrix of integration and its applications for the solution of linear and nonlinear problems in calculus of variations," Applied Mathematics and Computation, Elsevier, vol. 351(C), pages 83-98.
3. Postavaru, Octavian & Toma, Antonela, 2022. "A numerical approach based on fractional-order hybrid functions of block-pulse and Bernoulli polynomials for numerical solutions of fractional optimal control problems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 194(C), pages 269-284.
4. Weber, Thomas A. & Nguyen, Viet Anh, 2018. "A linear-quadratic Gaussian approach to dynamic information acquisition," European Journal of Operational Research, Elsevier, vol. 270(1), pages 260-281.
5. Zogheib, Bashar & Tohidi, Emran, 2016. "A new matrix method for solving two-dimensional time-dependent diffusion equations with Dirichlet boundary conditions," Applied Mathematics and Computation, Elsevier, vol. 291(C), pages 1-13.

### Keywords

Input delay; Optimal control; Delayed state feedback; Dynamic programming;
All these keywords.

## Corrections

All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:joptap:v:163:y:2014:i:3:d:10.1007_s10957-014-0532-8. See general information about how to correct material in RePEc.

If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

Please note that corrections may take a couple of weeks to filter through the various RePEc services.

IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.