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July  2014, 10(3): 883-903. doi: 10.3934/jimo.2014.10.883

A nonlinear conjugate gradient method for a special class of matrix optimization problems

El-Sayed M.E. Mostafa 1,

1. 

Department of Mathematics and Computer Science, Faculty of Science, Alexandria University, Moharam Bey 21511, Alexandria, Egypt

Received  August 2012 Revised  July 2013 Published  November 2013

In this article, a nonlinear conjugate gradient method is studied and analyzed for finding the local solutions of two matrix optimization problems resulting from the decentralized static output feedback problem for continuous and discrete-time systems. The global convergence of the proposed method is established. Several numerical examples of decentralized static output feedback are presented that demonstrate the applicability of the considered approach.
Keywords: Nonlinear conjugate gradient methods, decentralized output feedback control., nonlinear programming.
Mathematics Subject Classification: Primary: 90C30, 90C52, 93B52, 49N10; Secondary: 93D1.
Citation: El-Sayed M.E. Mostafa. A nonlinear conjugate gradient method for a special class of matrix optimization problems. Journal of Industrial and Management Optimization, 2014, 10 (3) : 883-903. doi: 10.3934/jimo.2014.10.883
References:
[1]

A. G. Aghdam, E. J. Davison and R. Becerril-Arreola, Structural modification of systems using discretization and generalized sampled-data hold functions, Automatica, 42 (2006), 1935-1941. doi: 10.1016/j.automatica.2006.06.005.

[2]

M. Aldeen and J. F. Marsh, Decentralized observer-based control scheme for interconnected dynamical systems with unknown inputs, IEEE Proc. Control Theory Appl., 146 (1999), 349-357.

[3]

Z. Artstein, Linear systems with delayed controls: A reduction, IEEE Transactions on Automatic Control, 27 (1982), 869-879. doi: 10.1109/TAC.1982.1103023.

[4]

Y.-Y. Cao and J. Lam, A computational method for simultaneous LQ optimal control design via piecewise constant output feedback, IEEE Transaction on Systems, MAN, and Cybernetics-Part B: Cybernetics, 31 (2001), 836-842.

[5]

Z.-F. Dai, Two modified HS type conjugate gradient methods for unconstrained optimization problems, Nonlinear Analysis, 74 (2011), 927-936. doi: 10.1016/j.na.2010.09.046.

[6]

Z. Gong, Decentralized robust control of uncertain interconnected systems with prescribed degree of exponential convergence, IEEE Transaction on Automatic Control, 40 (1995), 704-707. doi: 10.1109/9.376105.

[7]

W. W. Hager and H. Zhang, A survey of nonlinear conjugate gradient methods, Pacific Journal of Optimization, 2 (2006), 35-58.

[8]

M. Ikeda, Decentralized control of large scale systems, in Three Decades of Mathematical System Theory, Lecture Notes in Control and Inform. Sci., 135, Springer, Berlin, 1989, 219-242. doi: 10.1007/BFb0008464.

[9]

M. S. Mahmoud, M. F. Hassan and S. J. Saleh, Decentralized structures for a stream water quality control problems, Optimal Control Applications & Methods, 6 (1985), 167-168. doi: 10.1002/oca.4660060209.

[10]

D. Jiang and J. B. Moore, A gradient flow approach to decentralized output feedback optimal control, Systems & Control Letters, 27 (1996), 223-231. doi: 10.1016/0167-6911(96)80519-6.

[11]

K. H. Lee, J. H. Lee and W. H. Kwon, Sufficient LMI conditions for $H_\infty$ output feedback stabilization of linear discrete-time systems, IEEE Transactions on Automatic Control, 51 (2006), 675-680. doi: 10.1109/TAC.2006.872766.

[12]

F. Leibfritz, COMPlib: COnstraint Matrix-Optimization Problem library-A Collection of Test Examples for Nonlinear Semi-Definite Programs, Control System Design and Related Problems, Technical Report, 2004. Available from: http://www.complib.de.

[13]

T. Liu, Z.-P. Jiang and D. J. Hill, Decentralized output-feedback control of large-scale nonlinear systems with sensor noise, Automatica J. IFAC, 48 (2012), 2560-2568. doi: 10.1016/j.automatica.2012.06.054.

[14]

W. Q. Liu and V. Sreeram, New algorithm for computing LQ suboptimal output feedback gains of decentralized control systems, Journal of Optimization Theory and Applications, 93 (1997), 597-607. doi: 10.1023/A:1022647230641.

[15]

P. M. Mäkilä and H. T. Toivonen, Computational methods for parametric LQ problems-a survey, IEEE Transactions on Automatic Control, 32 (1987), 658-671. doi: 10.1109/TAC.1987.1104686.

[16]

E. M. E. Mostafa, A trust region method for solving the decentralized static output feedback design problem, Journal of Applied Mathematics & Computing, 18 (2005), 1-23. doi: 10.1007/BF02936553.

[17]

E. M. E. Mostafa, Computational design of optimal discrete-time output feedback controllers, Journal of the Operations Research Society of Japan, 51 (2008), 15-28.

[18]

E. M. E. Mostafa, On the computation of optimal static output feedback controllers for discrete-time systems, Numerical Functional Analysis and Optimization, 33 (2012), 591-610. doi: 10.1080/01630563.2012.661381.

[19]

E. M. E. Mostafa, A conjugate gradient method for discrete-time output feedback control design, Journal of Computational Mathematics, 30 (2012), 279-297. doi: 10.4208/jcm.1109-m3364.

[20]

E. M. E. Mostafa, Nonlinear conjugate gradient method for continuous time output feedback design, Journal of Applied Mathematics and Computing, 40 (2012), 529-549. doi: 10.1007/s12190-012-0574-8.

[21]

W. Nakamura, Y. Narushima and H. Yabe, Nonlinear conjugate gradient methods with sufficient descent properties for unconstrained optimization, Journal of Industrial and Management Optimization, 9 (2013), 595-619. doi: 10.3934/jimo.2013.9.595.

[22]

P. R. Pagilla and Y. Zhu, A decentralized output feedback controller for a class of large-scale interconnected nonlinear systems, ASME J. Dynam. Syst. Meas. Control, 127 (2004), 167-172. doi: 10.1115/1.1870047.

[23]

T. Rautert and E. W. Sachs, Computational design of optimal output feedback controllers, SIAM Journal on Optimization, 7 (1997), 837-852. doi: 10.1137/S1052623495290441.

[24]

M. Saif and Y. Guan, Decentralized state estimation in large-scale interconnected dynamical systems, Automatica J. IFAC, 28 (1992), 215-219. doi: 10.1016/0005-1098(92)90024-A.

[25]

D. D. Šiljak, Decentralized Control of Complex Systems, Mathematics in Science and Engineering, 184, Academic Press, Inc., Boston, MA, 1991.

[26]

D.D. Šiljak and D. M. Stipanović, Robust stabilization of nonlinear systems: The LMI approach, Math. Problems Eng., 6 (2000), 461-493. doi: 10.1155/S1024123X00001435.

[27]

V. L. Syrmos, C. T. Abdallah, P. Dorato and K. Grigoriadis, Static output feedback-a survey, Automatica J. IFAC, 33 (1997), 125-137. doi: 10.1016/S0005-1098(96)00141-0.

[28]

S. Tong, Y. Li and T. Wang, Adaptive fuzzy decentralized output feedback control for stochastic nonlinear large-scale systems using DSC technique, International Journal of Robust and Nonlinear Control, 23 (2013), 381-399. doi: 10.1002/rnc.1834.

[29]

R. J. Veilette, J. V. Medanić and W. R. Perkins, Design of reliable control systems, IEEE Transaction on Automatic Control, 37 (1992), 290-304. doi: 10.1109/9.119629.

[30]

Z. Wei G. Li and L. Qi, New nonlinear conjugate gradient formulas for large-scale unconstrained optimization problems, Applied Mathematics and Computation, 179 (2006), 407-430. doi: 10.1016/j.amc.2005.11.150.

[31]

G. Yu, L. Guan and Z. Wei, Globally convergent Polak-Ribière-Polyak conjugate gradient methods under a modified Wolfe line search, Applied Mathematics and Computation, 215 (2009), 3082-3090. doi: 10.1016/j.amc.2009.09.063.

[32]

G. Zhai, M. Ikeda and Y. Fujisaki, Decentralized Hinf controller design: A matrix inequality approach using a homotopy method, Automatica J. IFAC, 37 (2001), 565-572. doi: 10.1016/S0005-1098(00)00190-4.

[33]

L. Zhang, W. Zhou and D. Li, Some descent three-term conjugate gradient methods and their global convergence, Optimization Methods and Software, 22 (2007), 697-711. doi: 10.1080/10556780701223293.

show all references

References:
[1]

A. G. Aghdam, E. J. Davison and R. Becerril-Arreola, Structural modification of systems using discretization and generalized sampled-data hold functions, Automatica, 42 (2006), 1935-1941. doi: 10.1016/j.automatica.2006.06.005.

[2]

M. Aldeen and J. F. Marsh, Decentralized observer-based control scheme for interconnected dynamical systems with unknown inputs, IEEE Proc. Control Theory Appl., 146 (1999), 349-357.

[3]

Z. Artstein, Linear systems with delayed controls: A reduction, IEEE Transactions on Automatic Control, 27 (1982), 869-879. doi: 10.1109/TAC.1982.1103023.

[4]

Y.-Y. Cao and J. Lam, A computational method for simultaneous LQ optimal control design via piecewise constant output feedback, IEEE Transaction on Systems, MAN, and Cybernetics-Part B: Cybernetics, 31 (2001), 836-842.

[5]

Z.-F. Dai, Two modified HS type conjugate gradient methods for unconstrained optimization problems, Nonlinear Analysis, 74 (2011), 927-936. doi: 10.1016/j.na.2010.09.046.

[6]

Z. Gong, Decentralized robust control of uncertain interconnected systems with prescribed degree of exponential convergence, IEEE Transaction on Automatic Control, 40 (1995), 704-707. doi: 10.1109/9.376105.

[7]

W. W. Hager and H. Zhang, A survey of nonlinear conjugate gradient methods, Pacific Journal of Optimization, 2 (2006), 35-58.

[8]

M. Ikeda, Decentralized control of large scale systems, in Three Decades of Mathematical System Theory, Lecture Notes in Control and Inform. Sci., 135, Springer, Berlin, 1989, 219-242. doi: 10.1007/BFb0008464.

[9]

M. S. Mahmoud, M. F. Hassan and S. J. Saleh, Decentralized structures for a stream water quality control problems, Optimal Control Applications & Methods, 6 (1985), 167-168. doi: 10.1002/oca.4660060209.

[10]

D. Jiang and J. B. Moore, A gradient flow approach to decentralized output feedback optimal control, Systems & Control Letters, 27 (1996), 223-231. doi: 10.1016/0167-6911(96)80519-6.

[11]

K. H. Lee, J. H. Lee and W. H. Kwon, Sufficient LMI conditions for $H_\infty$ output feedback stabilization of linear discrete-time systems, IEEE Transactions on Automatic Control, 51 (2006), 675-680. doi: 10.1109/TAC.2006.872766.

[12]

F. Leibfritz, COMPlib: COnstraint Matrix-Optimization Problem library-A Collection of Test Examples for Nonlinear Semi-Definite Programs, Control System Design and Related Problems, Technical Report, 2004. Available from: http://www.complib.de.

[13]

T. Liu, Z.-P. Jiang and D. J. Hill, Decentralized output-feedback control of large-scale nonlinear systems with sensor noise, Automatica J. IFAC, 48 (2012), 2560-2568. doi: 10.1016/j.automatica.2012.06.054.

[14]

W. Q. Liu and V. Sreeram, New algorithm for computing LQ suboptimal output feedback gains of decentralized control systems, Journal of Optimization Theory and Applications, 93 (1997), 597-607. doi: 10.1023/A:1022647230641.

[15]

P. M. Mäkilä and H. T. Toivonen, Computational methods for parametric LQ problems-a survey, IEEE Transactions on Automatic Control, 32 (1987), 658-671. doi: 10.1109/TAC.1987.1104686.

[16]

E. M. E. Mostafa, A trust region method for solving the decentralized static output feedback design problem, Journal of Applied Mathematics & Computing, 18 (2005), 1-23. doi: 10.1007/BF02936553.

[17]

E. M. E. Mostafa, Computational design of optimal discrete-time output feedback controllers, Journal of the Operations Research Society of Japan, 51 (2008), 15-28.

[18]

E. M. E. Mostafa, On the computation of optimal static output feedback controllers for discrete-time systems, Numerical Functional Analysis and Optimization, 33 (2012), 591-610. doi: 10.1080/01630563.2012.661381.

[19]

E. M. E. Mostafa, A conjugate gradient method for discrete-time output feedback control design, Journal of Computational Mathematics, 30 (2012), 279-297. doi: 10.4208/jcm.1109-m3364.

[20]

E. M. E. Mostafa, Nonlinear conjugate gradient method for continuous time output feedback design, Journal of Applied Mathematics and Computing, 40 (2012), 529-549. doi: 10.1007/s12190-012-0574-8.

[21]

W. Nakamura, Y. Narushima and H. Yabe, Nonlinear conjugate gradient methods with sufficient descent properties for unconstrained optimization, Journal of Industrial and Management Optimization, 9 (2013), 595-619. doi: 10.3934/jimo.2013.9.595.

[22]

P. R. Pagilla and Y. Zhu, A decentralized output feedback controller for a class of large-scale interconnected nonlinear systems, ASME J. Dynam. Syst. Meas. Control, 127 (2004), 167-172. doi: 10.1115/1.1870047.

[23]

T. Rautert and E. W. Sachs, Computational design of optimal output feedback controllers, SIAM Journal on Optimization, 7 (1997), 837-852. doi: 10.1137/S1052623495290441.

[24]

M. Saif and Y. Guan, Decentralized state estimation in large-scale interconnected dynamical systems, Automatica J. IFAC, 28 (1992), 215-219. doi: 10.1016/0005-1098(92)90024-A.

[25]

D. D. Šiljak, Decentralized Control of Complex Systems, Mathematics in Science and Engineering, 184, Academic Press, Inc., Boston, MA, 1991.

[26]

D.D. Šiljak and D. M. Stipanović, Robust stabilization of nonlinear systems: The LMI approach, Math. Problems Eng., 6 (2000), 461-493. doi: 10.1155/S1024123X00001435.

[27]

V. L. Syrmos, C. T. Abdallah, P. Dorato and K. Grigoriadis, Static output feedback-a survey, Automatica J. IFAC, 33 (1997), 125-137. doi: 10.1016/S0005-1098(96)00141-0.

[28]

S. Tong, Y. Li and T. Wang, Adaptive fuzzy decentralized output feedback control for stochastic nonlinear large-scale systems using DSC technique, International Journal of Robust and Nonlinear Control, 23 (2013), 381-399. doi: 10.1002/rnc.1834.

[29]

R. J. Veilette, J. V. Medanić and W. R. Perkins, Design of reliable control systems, IEEE Transaction on Automatic Control, 37 (1992), 290-304. doi: 10.1109/9.119629.

[30]

Z. Wei G. Li and L. Qi, New nonlinear conjugate gradient formulas for large-scale unconstrained optimization problems, Applied Mathematics and Computation, 179 (2006), 407-430. doi: 10.1016/j.amc.2005.11.150.

[31]

G. Yu, L. Guan and Z. Wei, Globally convergent Polak-Ribière-Polyak conjugate gradient methods under a modified Wolfe line search, Applied Mathematics and Computation, 215 (2009), 3082-3090. doi: 10.1016/j.amc.2009.09.063.

[32]

G. Zhai, M. Ikeda and Y. Fujisaki, Decentralized Hinf controller design: A matrix inequality approach using a homotopy method, Automatica J. IFAC, 37 (2001), 565-572. doi: 10.1016/S0005-1098(00)00190-4.

[33]

L. Zhang, W. Zhou and D. Li, Some descent three-term conjugate gradient methods and their global convergence, Optimization Methods and Software, 22 (2007), 697-711. doi: 10.1080/10556780701223293.

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