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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

 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.
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. 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. Google Scholar [3] Z. Artstein, Linear systems with delayed controls: A reduction,, IEEE Transactions on Automatic Control, 27 (1982), 869. 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, 31 (2001), 836. Google Scholar [5] Z.-F. Dai, Two modified HS type conjugate gradient methods for unconstrained optimization problems,, Nonlinear Analysis, 74 (2011), 927. 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. 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. Google Scholar [8] M. Ikeda, Decentralized control of large scale systems,, in Three Decades of Mathematical System Theory, (1989), 219. 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. 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. 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. 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). 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1016/0005-1098(92)90024-A. Google Scholar [25] D. D. Šiljak, Decentralized Control of Complex Systems,, Mathematics in Science and Engineering, (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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Google Scholar [3] Z. Artstein, Linear systems with delayed controls: A reduction,, IEEE Transactions on Automatic Control, 27 (1982), 869. 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, 31 (2001), 836. Google Scholar [5] Z.-F. Dai, Two modified HS type conjugate gradient methods for unconstrained optimization problems,, Nonlinear Analysis, 74 (2011), 927. 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. 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. Google Scholar [8] M. Ikeda, Decentralized control of large scale systems,, in Three Decades of Mathematical System Theory, (1989), 219. 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. 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. 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. 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). 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1016/0005-1098(92)90024-A. Google Scholar [25] D. D. Šiljak, Decentralized Control of Complex Systems,, Mathematics in Science and Engineering, (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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1080/10556780701223293. Google Scholar
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