A Discrete-Time Quasi-Theoretical Solution of the Modified Riccati Matrix Algebraic Equation

Tahar Latreche

doi:doi:10.11648/j.acis.20140205.13

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A Discrete-Time Quasi-Theoretical Solution of the Modified Riccati Matrix Algebraic Equation

Tahar Latreche

Published in Automation, Control and Intelligent Systems (Volume 2, Issue 5)

Received: 31 October 2014 Accepted: 7 November 2014 Published: 20 November 2014

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Abstract

In this paper, based on MacLaurin’s series and the Riccati equation, an algebraic quadratic equation will be developed and hence, its two roots, which represent the minimizing and maximizing optimal control matrices, would be deducted easier. Otherwise, a step-by-step algorithm to compute the control matrix for every step of time according to the preceding responses and a new signal pick will be explained. The proposed method presents a new discrete-time solution for the problem of optimal control in the linear or nonlinear cases of systems subjected to arbitrary signals. As an example, a system (structure) of three degrees of freedom, subjected to a strong earthquake is analyzed. The displacements versus time and the stiffness forces versus displacements of the system, for the two uncontrolled and controlled cases are graphically shown. Therefore, the curves of variations of the elements of the optimal control matrix versus discrete-time are also presented and clearly show the effect of the nonlinearity, of the system, which is the cause of the great responses in the uncontrolled case, and that it is optimally treated by the proposed solution. The results obtained clarify a great reduction of the controlled system results, in comparison with the uncontrolled system ones. The percentage of the differences between the controlled and uncontrolled results (displacements or stiffness forces) could even surpass 90 %, which demonstrates that the adopted solution is good even than that of the original ones of the differential or the algebraic Riccati equation.

Published in	Automation, Control and Intelligent Systems (Volume 2, Issue 5)
DOI	10.11648/j.acis.20140205.13
Page(s)	87-92
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Optimal Control, Modified Riccati Equation, Quasi-Theoretical Solution, Discrete-Time Algorithm, Nonlinear Systems

References

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[6]	W. F. Arnold, A. J. Laub, “Generalized eigenproblem algorithms and software for algebraic Riccati equations”, Proceedings of IEEE, vol. 72, no. 12, 1984.
[7]	A. Astolfi, L. Marconi, Analysis and design of nonlinear control systems. Springer Publishers, 2008.
[8]	P. Benner, J-R. Li and P. Thilo, “Numerical solution of large-scale Lyapunov equations, Riccati equations, and linear-quadratic optimal control problems”, Numerical Linear Algebra with Applications, no. 15, pp. 755 – 777, 2008.
[9]	D. L. Elliott, Bilinear control systems. Springer Publishers, 2009.
[10]	P. H. Geering, Optimal control with engineering applications. Springer Publishers, 2007.
[11]	M. S. Grewal, A. P. Andrews, Kalman Filtering: Theory and practice. John Wiley, 2008.
[12]	L. Grune, J. Pannek, Nonlinear model predictive control. Springer Publishers, 2011.
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[14]	P. L. Kogut, G. R. Leugering, Optimal control problems for practical differential equations on reticulated domains. Springer Publishers, 2011.
[15]	T. Latreche, “A discrete-time algorithm for the resolution of the Nonlinear Riccati Matrix Differential Equation for the optimal control”, American J. of Civil Engineering, no. 2, pp. 12-17, 2014.
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[23]	S. L. William, Control system: Advanced methods. Taylor and Francis, 2011.

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APA Style

Tahar Latreche. (2014). A Discrete-Time Quasi-Theoretical Solution of the Modified Riccati Matrix Algebraic Equation. Automation, Control and Intelligent Systems, 2(5), 87-92. https://doi.org/10.11648/j.acis.20140205.13

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ACS Style

Tahar Latreche. A Discrete-Time Quasi-Theoretical Solution of the Modified Riccati Matrix Algebraic Equation. Autom. Control Intell. Syst. 2014, 2(5), 87-92. doi: 10.11648/j.acis.20140205.13

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AMA Style

Tahar Latreche. A Discrete-Time Quasi-Theoretical Solution of the Modified Riccati Matrix Algebraic Equation. Autom Control Intell Syst. 2014;2(5):87-92. doi: 10.11648/j.acis.20140205.13

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@article{10.11648/j.acis.20140205.13,
  author = {Tahar Latreche},
  title = {A Discrete-Time Quasi-Theoretical Solution of the Modified Riccati Matrix Algebraic Equation},
  journal = {Automation, Control and Intelligent Systems},
  volume = {2},
  number = {5},
  pages = {87-92},
  doi = {10.11648/j.acis.20140205.13},
  url = {https://doi.org/10.11648/j.acis.20140205.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acis.20140205.13},
  abstract = {In this paper, based on MacLaurin’s series and the Riccati equation, an algebraic quadratic equation will be developed and hence, its two roots, which represent the minimizing and maximizing optimal control matrices, would be deducted easier. Otherwise, a step-by-step algorithm to compute the control matrix for every step of time according to the preceding responses and a new signal pick will be explained. The proposed method presents a new discrete-time solution for the problem of optimal control in the linear or nonlinear cases of systems subjected to arbitrary signals. As an example, a system (structure) of three degrees of freedom, subjected to a strong earthquake is analyzed. The displacements versus time and the stiffness forces versus displacements of the system, for the two uncontrolled and controlled cases are graphically shown. Therefore, the curves of variations of the elements of the optimal control matrix versus discrete-time are also presented and clearly show the effect of the nonlinearity, of the system, which is the cause of the great responses in the uncontrolled case, and that it is optimally treated by the proposed solution. The results obtained clarify a great reduction of the controlled system results, in comparison with the uncontrolled system ones. The percentage of the differences between the controlled and uncontrolled results (displacements or stiffness forces) could even surpass 90 %, which demonstrates that the adopted solution is good even than that of the original ones of the differential or the algebraic Riccati equation.},
 year = {2014}
}

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TY  - JOUR
T1  - A Discrete-Time Quasi-Theoretical Solution of the Modified Riccati Matrix Algebraic Equation
AU  - Tahar Latreche
Y1  - 2014/11/20
PY  - 2014
N1  - https://doi.org/10.11648/j.acis.20140205.13
DO  - 10.11648/j.acis.20140205.13
T2  - Automation, Control and Intelligent Systems
JF  - Automation, Control and Intelligent Systems
JO  - Automation, Control and Intelligent Systems
SP  - 87
EP  - 92
PB  - Science Publishing Group
SN  - 2328-5591
UR  - https://doi.org/10.11648/j.acis.20140205.13
AB  - In this paper, based on MacLaurin’s series and the Riccati equation, an algebraic quadratic equation will be developed and hence, its two roots, which represent the minimizing and maximizing optimal control matrices, would be deducted easier. Otherwise, a step-by-step algorithm to compute the control matrix for every step of time according to the preceding responses and a new signal pick will be explained. The proposed method presents a new discrete-time solution for the problem of optimal control in the linear or nonlinear cases of systems subjected to arbitrary signals. As an example, a system (structure) of three degrees of freedom, subjected to a strong earthquake is analyzed. The displacements versus time and the stiffness forces versus displacements of the system, for the two uncontrolled and controlled cases are graphically shown. Therefore, the curves of variations of the elements of the optimal control matrix versus discrete-time are also presented and clearly show the effect of the nonlinearity, of the system, which is the cause of the great responses in the uncontrolled case, and that it is optimally treated by the proposed solution. The results obtained clarify a great reduction of the controlled system results, in comparison with the uncontrolled system ones. The percentage of the differences between the controlled and uncontrolled results (displacements or stiffness forces) could even surpass 90 %, which demonstrates that the adopted solution is good even than that of the original ones of the differential or the algebraic Riccati equation.
VL  - 2
IS  - 5
ER  -

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Author Information

Tahar Latreche

Magistère in Civil Engineering, B.P. 129 Salem Lalmi, 40003 Khenchela, Algeria

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Plain Text BibTeX RIS

APA Style

Tahar Latreche. (2014). A Discrete-Time Quasi-Theoretical Solution of the Modified Riccati Matrix Algebraic Equation. Automation, Control and Intelligent Systems, 2(5), 87-92. https://doi.org/10.11648/j.acis.20140205.13

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ACS Style

Tahar Latreche. A Discrete-Time Quasi-Theoretical Solution of the Modified Riccati Matrix Algebraic Equation. Autom. Control Intell. Syst. 2014, 2(5), 87-92. doi: 10.11648/j.acis.20140205.13

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AMA Style

Tahar Latreche. A Discrete-Time Quasi-Theoretical Solution of the Modified Riccati Matrix Algebraic Equation. Autom Control Intell Syst. 2014;2(5):87-92. doi: 10.11648/j.acis.20140205.13

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@article{10.11648/j.acis.20140205.13,
  author = {Tahar Latreche},
  title = {A Discrete-Time Quasi-Theoretical Solution of the Modified Riccati Matrix Algebraic Equation},
  journal = {Automation, Control and Intelligent Systems},
  volume = {2},
  number = {5},
  pages = {87-92},
  doi = {10.11648/j.acis.20140205.13},
  url = {https://doi.org/10.11648/j.acis.20140205.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acis.20140205.13},
  abstract = {In this paper, based on MacLaurin’s series and the Riccati equation, an algebraic quadratic equation will be developed and hence, its two roots, which represent the minimizing and maximizing optimal control matrices, would be deducted easier. Otherwise, a step-by-step algorithm to compute the control matrix for every step of time according to the preceding responses and a new signal pick will be explained. The proposed method presents a new discrete-time solution for the problem of optimal control in the linear or nonlinear cases of systems subjected to arbitrary signals. As an example, a system (structure) of three degrees of freedom, subjected to a strong earthquake is analyzed. The displacements versus time and the stiffness forces versus displacements of the system, for the two uncontrolled and controlled cases are graphically shown. Therefore, the curves of variations of the elements of the optimal control matrix versus discrete-time are also presented and clearly show the effect of the nonlinearity, of the system, which is the cause of the great responses in the uncontrolled case, and that it is optimally treated by the proposed solution. The results obtained clarify a great reduction of the controlled system results, in comparison with the uncontrolled system ones. The percentage of the differences between the controlled and uncontrolled results (displacements or stiffness forces) could even surpass 90 %, which demonstrates that the adopted solution is good even than that of the original ones of the differential or the algebraic Riccati equation.},
 year = {2014}
}

Copy | Download

TY  - JOUR
T1  - A Discrete-Time Quasi-Theoretical Solution of the Modified Riccati Matrix Algebraic Equation
AU  - Tahar Latreche
Y1  - 2014/11/20
PY  - 2014
N1  - https://doi.org/10.11648/j.acis.20140205.13
DO  - 10.11648/j.acis.20140205.13
T2  - Automation, Control and Intelligent Systems
JF  - Automation, Control and Intelligent Systems
JO  - Automation, Control and Intelligent Systems
SP  - 87
EP  - 92
PB  - Science Publishing Group
SN  - 2328-5591
UR  - https://doi.org/10.11648/j.acis.20140205.13
AB  - In this paper, based on MacLaurin’s series and the Riccati equation, an algebraic quadratic equation will be developed and hence, its two roots, which represent the minimizing and maximizing optimal control matrices, would be deducted easier. Otherwise, a step-by-step algorithm to compute the control matrix for every step of time according to the preceding responses and a new signal pick will be explained. The proposed method presents a new discrete-time solution for the problem of optimal control in the linear or nonlinear cases of systems subjected to arbitrary signals. As an example, a system (structure) of three degrees of freedom, subjected to a strong earthquake is analyzed. The displacements versus time and the stiffness forces versus displacements of the system, for the two uncontrolled and controlled cases are graphically shown. Therefore, the curves of variations of the elements of the optimal control matrix versus discrete-time are also presented and clearly show the effect of the nonlinearity, of the system, which is the cause of the great responses in the uncontrolled case, and that it is optimally treated by the proposed solution. The results obtained clarify a great reduction of the controlled system results, in comparison with the uncontrolled system ones. The percentage of the differences between the controlled and uncontrolled results (displacements or stiffness forces) could even surpass 90 %, which demonstrates that the adopted solution is good even than that of the original ones of the differential or the algebraic Riccati equation.
VL  - 2
IS  - 5
ER  -

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