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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 & 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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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