An Output Stabilization Problem of Distributed Linear Systems Approaches and Simulations

Affiliation(s)

Department of Mathematics and Computer Science, Faculty of Science, University of Moulay Isma?l, Meknès, Morocco.

Department of Mathematics and Computer Science, Faculty of Science, University of Moulay Isma?l, Meknès, Morocco.

ABSTRACT

The goal of this paper is to study an output stabilization problem: the gradient stabilization for linear distributed systems. Firstly, we give definitions and properties of the gradient stability. Then we characterize controls which stabilize the gradient of the state. We also give the stabilizing control which minimizes a performance given cost. The obtained results are illustrated by simulations in the case of one-dimensional distributed systems.

The goal of this paper is to study an output stabilization problem: the gradient stabilization for linear distributed systems. Firstly, we give definitions and properties of the gradient stability. Then we characterize controls which stabilize the gradient of the state. We also give the stabilizing control which minimizes a performance given cost. The obtained results are illustrated by simulations in the case of one-dimensional distributed systems.

Cite this paper

E. Zerrik and Y. Benslimane, "An Output Stabilization Problem of Distributed Linear Systems Approaches and Simulations,"*Intelligent Control and Automation*, Vol. 3 No. 2, 2012, pp. 159-167. doi: 10.4236/ica.2012.32018.

E. Zerrik and Y. Benslimane, "An Output Stabilization Problem of Distributed Linear Systems Approaches and Simulations,"

References

[1] W. M. Wonham, “Linear Multivariable Control: A Geometric Approach,” Springer-Verlag, Berlin, 1974.

[2] A. V. Balakrishnan, “Strong Stabilizabity and the Steady State Riccati Equation,” Applied Mathematics and Optimization, Vol. 7, No. 1, 1981, pp. 335-345. doi:10.1007/BF01442125

[3] R. F. Curtain and H. J. Zwart, “An Introduction to Infinite Dimensional Linear Systems Theory,” SpringerVerlag, Berlin, 1995.

[4] A. J. Pritchard and J. Zabczyk, “Stability and Stabilizability of Infinite Dimensional Systems,” SIAM Review, Vol. 23, No.1, 1981, pp. 25-51. doi:10.1137/1023003

[5] T. Kato, “Perturbation Theory for Linear Operators,” Springer-Verlag, Berlin, 1980.

[6] R. Triggiani, “On the Stabilizability Problem in Banach Space,” Journal of Mathematical Analysis and Applications, Vol. 52, No. 3, 1979, pp. 383-403. doi:10.1016/0022-247X(75)90067-0

[7] R. F. Curtain, and A. J. Pritchard, “Infinite Dimensional Linear Systems Theory,” Springer-Verlag, Berlin, 1978.

[8] H. T. Banks and K. Kunisch, “The Linear Regulator Problem for Parabolic Systems,” SIAM Journal on Control and Optimization, No. 22, Vol. 5, 1984, pp. 684-696. doi:10.1137/0322043

[9] N. J, Higham, “The Scaling and Squaring Method for the Matrix Exponential,” SIAM Journal on Matrix Analysis and Applications, Vol. 26, No. 4, 2005, pp. 1179-1193. doi:10.1137/04061101X

[10] A. J. Laub, “A Schur Method for Solving Algebraic Riccati Equations,” IEEE Transactions on Automatic Control, Vol. 24, No. 6, 1979, pp. 913-921. doi:10.1109/TAC.1979.1102178

[11] W. F Arnold and A. J. Laub, “Generalized Eigenproblem Algorithms and Soft-Ware for Algebraic Riccati Equations,” Proceedings of the IEEE, Vol. 72, No. 12, 1984, pp. 1746-1754. doi:10.1109/PROC.1984.13083

[1] W. M. Wonham, “Linear Multivariable Control: A Geometric Approach,” Springer-Verlag, Berlin, 1974.

[2] A. V. Balakrishnan, “Strong Stabilizabity and the Steady State Riccati Equation,” Applied Mathematics and Optimization, Vol. 7, No. 1, 1981, pp. 335-345. doi:10.1007/BF01442125

[3] R. F. Curtain and H. J. Zwart, “An Introduction to Infinite Dimensional Linear Systems Theory,” SpringerVerlag, Berlin, 1995.

[4] A. J. Pritchard and J. Zabczyk, “Stability and Stabilizability of Infinite Dimensional Systems,” SIAM Review, Vol. 23, No.1, 1981, pp. 25-51. doi:10.1137/1023003

[5] T. Kato, “Perturbation Theory for Linear Operators,” Springer-Verlag, Berlin, 1980.

[6] R. Triggiani, “On the Stabilizability Problem in Banach Space,” Journal of Mathematical Analysis and Applications, Vol. 52, No. 3, 1979, pp. 383-403. doi:10.1016/0022-247X(75)90067-0

[7] R. F. Curtain, and A. J. Pritchard, “Infinite Dimensional Linear Systems Theory,” Springer-Verlag, Berlin, 1978.

[8] H. T. Banks and K. Kunisch, “The Linear Regulator Problem for Parabolic Systems,” SIAM Journal on Control and Optimization, No. 22, Vol. 5, 1984, pp. 684-696. doi:10.1137/0322043

[9] N. J, Higham, “The Scaling and Squaring Method for the Matrix Exponential,” SIAM Journal on Matrix Analysis and Applications, Vol. 26, No. 4, 2005, pp. 1179-1193. doi:10.1137/04061101X

[10] A. J. Laub, “A Schur Method for Solving Algebraic Riccati Equations,” IEEE Transactions on Automatic Control, Vol. 24, No. 6, 1979, pp. 913-921. doi:10.1109/TAC.1979.1102178

[11] W. F Arnold and A. J. Laub, “Generalized Eigenproblem Algorithms and Soft-Ware for Algebraic Riccati Equations,” Proceedings of the IEEE, Vol. 72, No. 12, 1984, pp. 1746-1754. doi:10.1109/PROC.1984.13083