A Discrete-Time Algorithm for the Resolution of the Nonlinear Riccati Matrix Differential Equation for the Optimal Control

Tahar Latreche

doi:doi:10.11648/j.ajce.20140202.11

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A Discrete-Time Algorithm for the Resolution of the Nonlinear Riccati Matrix Differential Equation for the Optimal Control

Tahar Latreche

Published in American Journal of Civil Engineering (Volume 2, Issue 2)

Received: 7 November 2013 Accepted: Published: 10 March 2014

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Abstract

The Riccati Matrix Differential Equation (RMDE) is an interesting equation in different fields of science and engineering practice. In fact, that the arithmetic solution for this matrix differential equation in the general case of varying-time matrices is very difficult to find. The literature offers the solution of this differential equation in the case of dependant constant matrices (i.e. invariant-time matrices). The present approach is an approximate discrete-time method for the resolution of the matrix differential equation of Riccati in the general case of varying-time (dependant of time) matrices; the method in fact, is a discrétisation of the exact matrix solution, that evaluates for any so small step of time, and which is function of the solution of the preceding step of time and the constitute equation matrices. The proposed algorithm is verified, for a controlled structure under Modified El-Centro earthquake by a comparison with the same uncontrolled structure, which constitutes by a two Degrees Of Freedom (2DOF) system. The results of this comparison offer good differences between the controlled and the uncontrolled systems.

Published in	American Journal of Civil Engineering (Volume 2, Issue 2)
DOI	10.11648/j.ajce.20140202.11
Page(s)	12-17
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

Riccati Matrix Differential Equation, Discrete-Time Algorithm, Varying-Time Matrices, Optimal Control, Nonlinear Quadratic Regulator

References

[1]	Anderson, B. D. O. et Moore, J. B., (1971). Linear optimal control. First Edition, Prentice-Hall, Inc.
[2]	Anderson, B. D. O. et Moore, J. B., (1979). Optimal Filtering. First Edition, Prentice-Hall, Inc.
[3]	Anderson, B. D. O. et Moore, J. B., (1989). Optimal control, linear quadratic methods. First Edition, Prentice-Hall, Inc.
[4]	Arfiadi, Y., (2000). Optimal passive and active control mechanisms for seismically excited buildings. PhD Thesis, University of Wollongong
[5]	Astolfi, A. et Marconi, L., (2008). Analysis and design of nonlinear control systems. First Edition, Springer Publishers
[6]	Elliott, D. L., (2009). Bilinear control systems. First Edition, Springer Publishers
[7]	Grewal, M. S. et Andrews, A. P., (2008). Kalman Filtering: Theory and practice. Third Edition, John Wiley and Sons, Inc.
[8]	Grune, L. et Pannek, J., (2011). Nonlinear model predictive control. First Edition, Springer Publishers
[9]	Isidori, A., (1999). Nonlinear control systems 2. First Edition, Springer Publishers
[10]	Marazzi, F., (2002). Semi-active control for civil structures: Implementation aspects. PhD Thesis, University of Pavia
[11]	Preumont, A., (2002). Vibration control of active structures: An Introduction. Second Edition, Kluwer Academic Publishers
[12]	Vér, I. L. et Beranek, L. L., (2006). Noise and vibration control engineering. First Edition, John Wiley and Sons, Inc.
[13]	William, S. L., (2011). Control system: Fundamentals. Second Edition, Taylor and Francis Group, LLC
[14]	William, S. L., (2011). Control system: Applications. Second Edition, Taylor and Francis Group, LLC
[15]	William, S. L., (2011). Control system: Advanced methods. Second Edition, Taylor and Francis Group, LLC

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APA Style

Tahar Latreche. (2014). A Discrete-Time Algorithm for the Resolution of the Nonlinear Riccati Matrix Differential Equation for the Optimal Control. American Journal of Civil Engineering, 2(2), 12-17. https://doi.org/10.11648/j.ajce.20140202.11

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ACS Style

Tahar Latreche. A Discrete-Time Algorithm for the Resolution of the Nonlinear Riccati Matrix Differential Equation for the Optimal Control. Am. J. Civ. Eng. 2014, 2(2), 12-17. doi: 10.11648/j.ajce.20140202.11

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AMA Style

Tahar Latreche. A Discrete-Time Algorithm for the Resolution of the Nonlinear Riccati Matrix Differential Equation for the Optimal Control. Am J Civ Eng. 2014;2(2):12-17. doi: 10.11648/j.ajce.20140202.11

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@article{10.11648/j.ajce.20140202.11,
  author = {Tahar Latreche},
  title = {A Discrete-Time Algorithm for the Resolution of the Nonlinear Riccati Matrix Differential Equation for the Optimal Control},
  journal = {American Journal of Civil Engineering},
  volume = {2},
  number = {2},
  pages = {12-17},
  doi = {10.11648/j.ajce.20140202.11},
  url = {https://doi.org/10.11648/j.ajce.20140202.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajce.20140202.11},
  abstract = {The Riccati Matrix Differential Equation (RMDE) is an interesting equation in different fields of science and engineering practice.  In fact, that the arithmetic solution for this matrix differential equation in the general case of varying-time matrices is very difficult to find. The literature offers the solution of this differential equation in the case of dependant constant matrices (i.e. invariant-time matrices). The present approach is an approximate discrete-time method for the resolution of the matrix differential equation of Riccati in the general case of varying-time (dependant of time) matrices; the method in fact, is a discrétisation of the exact matrix solution, that evaluates for any so small step of time, and which is function of the solution of the preceding step of time and the constitute equation matrices. The proposed algorithm is verified, for a controlled structure under Modified El-Centro earthquake by a comparison with the same uncontrolled structure, which constitutes by a two Degrees Of Freedom (2DOF) system. The results of this comparison offer good differences between the controlled and the uncontrolled systems.},
 year = {2014}
}

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TY  - JOUR
T1  - A Discrete-Time Algorithm for the Resolution of the Nonlinear Riccati Matrix Differential Equation for the Optimal Control
AU  - Tahar Latreche
Y1  - 2014/03/10
PY  - 2014
N1  - https://doi.org/10.11648/j.ajce.20140202.11
DO  - 10.11648/j.ajce.20140202.11
T2  - American Journal of Civil Engineering
JF  - American Journal of Civil Engineering
JO  - American Journal of Civil Engineering
SP  - 12
EP  - 17
PB  - Science Publishing Group
SN  - 2330-8737
UR  - https://doi.org/10.11648/j.ajce.20140202.11
AB  - The Riccati Matrix Differential Equation (RMDE) is an interesting equation in different fields of science and engineering practice.  In fact, that the arithmetic solution for this matrix differential equation in the general case of varying-time matrices is very difficult to find. The literature offers the solution of this differential equation in the case of dependant constant matrices (i.e. invariant-time matrices). The present approach is an approximate discrete-time method for the resolution of the matrix differential equation of Riccati in the general case of varying-time (dependant of time) matrices; the method in fact, is a discrétisation of the exact matrix solution, that evaluates for any so small step of time, and which is function of the solution of the preceding step of time and the constitute equation matrices. The proposed algorithm is verified, for a controlled structure under Modified El-Centro earthquake by a comparison with the same uncontrolled structure, which constitutes by a two Degrees Of Freedom (2DOF) system. The results of this comparison offer good differences between the controlled and the uncontrolled systems.
VL  - 2
IS  - 2
ER  -

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Author Information

Tahar Latreche

Doctorate student at Constantine University, Algeria, B.P. 129 Salem Lalmi, 40003 Khenchela, Algeria

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Plain Text BibTeX RIS

APA Style

Tahar Latreche. (2014). A Discrete-Time Algorithm for the Resolution of the Nonlinear Riccati Matrix Differential Equation for the Optimal Control. American Journal of Civil Engineering, 2(2), 12-17. https://doi.org/10.11648/j.ajce.20140202.11

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ACS Style

Tahar Latreche. A Discrete-Time Algorithm for the Resolution of the Nonlinear Riccati Matrix Differential Equation for the Optimal Control. Am. J. Civ. Eng. 2014, 2(2), 12-17. doi: 10.11648/j.ajce.20140202.11

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AMA Style

Tahar Latreche. A Discrete-Time Algorithm for the Resolution of the Nonlinear Riccati Matrix Differential Equation for the Optimal Control. Am J Civ Eng. 2014;2(2):12-17. doi: 10.11648/j.ajce.20140202.11

Copy | Download

@article{10.11648/j.ajce.20140202.11,
  author = {Tahar Latreche},
  title = {A Discrete-Time Algorithm for the Resolution of the Nonlinear Riccati Matrix Differential Equation for the Optimal Control},
  journal = {American Journal of Civil Engineering},
  volume = {2},
  number = {2},
  pages = {12-17},
  doi = {10.11648/j.ajce.20140202.11},
  url = {https://doi.org/10.11648/j.ajce.20140202.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajce.20140202.11},
  abstract = {The Riccati Matrix Differential Equation (RMDE) is an interesting equation in different fields of science and engineering practice.  In fact, that the arithmetic solution for this matrix differential equation in the general case of varying-time matrices is very difficult to find. The literature offers the solution of this differential equation in the case of dependant constant matrices (i.e. invariant-time matrices). The present approach is an approximate discrete-time method for the resolution of the matrix differential equation of Riccati in the general case of varying-time (dependant of time) matrices; the method in fact, is a discrétisation of the exact matrix solution, that evaluates for any so small step of time, and which is function of the solution of the preceding step of time and the constitute equation matrices. The proposed algorithm is verified, for a controlled structure under Modified El-Centro earthquake by a comparison with the same uncontrolled structure, which constitutes by a two Degrees Of Freedom (2DOF) system. The results of this comparison offer good differences between the controlled and the uncontrolled systems.},
 year = {2014}
}

Copy | Download

TY  - JOUR
T1  - A Discrete-Time Algorithm for the Resolution of the Nonlinear Riccati Matrix Differential Equation for the Optimal Control
AU  - Tahar Latreche
Y1  - 2014/03/10
PY  - 2014
N1  - https://doi.org/10.11648/j.ajce.20140202.11
DO  - 10.11648/j.ajce.20140202.11
T2  - American Journal of Civil Engineering
JF  - American Journal of Civil Engineering
JO  - American Journal of Civil Engineering
SP  - 12
EP  - 17
PB  - Science Publishing Group
SN  - 2330-8737
UR  - https://doi.org/10.11648/j.ajce.20140202.11
AB  - The Riccati Matrix Differential Equation (RMDE) is an interesting equation in different fields of science and engineering practice.  In fact, that the arithmetic solution for this matrix differential equation in the general case of varying-time matrices is very difficult to find. The literature offers the solution of this differential equation in the case of dependant constant matrices (i.e. invariant-time matrices). The present approach is an approximate discrete-time method for the resolution of the matrix differential equation of Riccati in the general case of varying-time (dependant of time) matrices; the method in fact, is a discrétisation of the exact matrix solution, that evaluates for any so small step of time, and which is function of the solution of the preceding step of time and the constitute equation matrices. The proposed algorithm is verified, for a controlled structure under Modified El-Centro earthquake by a comparison with the same uncontrolled structure, which constitutes by a two Degrees Of Freedom (2DOF) system. The results of this comparison offer good differences between the controlled and the uncontrolled systems.
VL  - 2
IS  - 2
ER  -

Copy | Download