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Special Issues Volume 2, Issue 5, October 2014, Page: 87-92
A Discrete-Time Quasi-Theoretical Solution of the Modified Riccati Matrix Algebraic Equation
Tahar Latreche, Magistère in Civil Engineering, B.P. 129 Salem Lalmi, 40003 Khenchela, Algeria
Received: Oct. 31, 2014;       Accepted: Nov. 7, 2014;       Published: Nov. 20, 2014
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.
Keywords
Optimal Control, Modified Riccati Equation, Quasi-Theoretical Solution, Discrete-Time Algorithm, Nonlinear Systems
Tahar Latreche, A Discrete-Time Quasi-Theoretical Solution of the Modified Riccati Matrix Algebraic Equation, Automation, Control and Intelligent Systems. Vol. 2, No. 5, 2014, pp. 87-92. doi: 10.11648/j.acis.20140205.13
Reference

H. M. Amman, H. Neudecker, “Numerical solutions of algebraic Riccati equation”, J. of Economic Dynamics and Control, no. 21, pp. 363 - 369, 1997.

B. D. O. Anderson, J. B. Moore, Linear optimal control. Prentice-Hall, 1971.

B. D. O. Anderson, J. B. Moore, Optimal Filtering. Prentice-Hall, 1979.

B. D. O. Anderson, J. B. Moore, Optimal control, linear quadratic methods. Prentice-Hall, 1989.

Y. Arfiadi, Optimal passive and active control mechanisms for seismically excited buildings. PhD Thesis, University of Wollongong, 2000.

W. F. Arnold, A. J. Laub, “Generalized eigenproblem algorithms and software for algebraic Riccati equations”, Proceedings of IEEE, vol. 72, no. 12, 1984.

A. Astolfi, L. Marconi, Analysis and design of nonlinear control systems. Springer Publishers, 2008.

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.

D. L. Elliott, Bilinear control systems. Springer Publishers, 2009.

P. H. Geering, Optimal control with engineering applications. Springer Publishers, 2007.

M. S. Grewal, A. P. Andrews, Kalman Filtering: Theory and practice. John Wiley, 2008.

L. Grune, J. Pannek, Nonlinear model predictive control. Springer Publishers, 2011.

A. Isidori, Nonlinear control systems 2. Springer Publishers, 1999.

P. L. Kogut, G. R. Leugering, Optimal control problems for practical differential equations on reticulated domains. Springer Publishers, 2011.

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.

T. Latreche, “A numerical algorithm for the resolution of scalar and matrix algebraic equations using Runge-Kutta method”, Applied and Computational Mathematics, no. 3, pp. 68-74, 2014.

A. Locatelli, Optimal control: an introduction. Birkhäuser Virlag, 2004.

T. K. Nguyen, “Numerical solution of discrete-time algebraic Riccati equation”, Website: http://www.ictp.trieste.it/~pub-off

A. Preumont, Vibration control of active structures: An Introduction. Kluwer Academic Publishers, 2002.

I. L. Vér, L. L. Beranek, Noise and vibration control engineering. John Wiley, 2006.

S. L. William, Control system: Fundamentals. Taylor and Francis, 2011.

S. L. William, Control system: Applications. Taylor and Francis, 2011.

S. L. William, Control system: Advanced methods. Taylor and Francis, 2011. 