American Institute of Mathematical Sciences

Advanced
  • Previous Article
    A hybrid approach for index tracking with practical constraints
  • JIMO Home
  • This Issue
  • Next Article
    Second order optimality conditions and reformulations for nonconvex quadratically constrained quadratic programming problems
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

[1]

Wataru Nakamura, Yasushi Narushima, Hiroshi Yabe. Nonlinear conjugate gradient methods with sufficient descent properties for unconstrained optimization. Journal of Industrial & Management Optimization, 2013, 9 (3) : 595-619. doi: 10.3934/jimo.2013.9.595
[2]

Ta T.H. Trang, Vu N. Phat, Adly Samir. Finite-time stabilization and $H_\infty$ control of nonlinear delay systems via output feedback. Journal of Industrial & Management Optimization, 2016, 12 (1) : 303-315. doi: 10.3934/jimo.2016.12.303
[3]

Predrag S. Stanimirović, Branislav Ivanov, Haifeng Ma, Dijana Mosić. A survey of gradient methods for solving nonlinear optimization. Electronic Research Archive, 2020, 28 (4) : 1573-1624. doi: 10.3934/era.2020115
[4]

Rohit Gupta, Farhad Jafari, Robert J. Kipka, Boris S. Mordukhovich. Linear openness and feedback stabilization of nonlinear control systems. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1103-1119. doi: 10.3934/dcdss.2018063
[5]

Shishun Li, Zhengda Huang. Guaranteed descent conjugate gradient methods with modified secant condition. Journal of Industrial & Management Optimization, 2008, 4 (4) : 739-755. doi: 10.3934/jimo.2008.4.739
[6]

Sie Long Kek, Mohd Ismail Abd Aziz. Output regulation for discrete-time nonlinear stochastic optimal control problems with model-reality differences. Numerical Algebra, Control & Optimization, 2015, 5 (3) : 275-288. doi: 10.3934/naco.2015.5.275
[7]

Magdi S. Mahmoud. Output feedback overlapping control design of interconnected systems with input saturation. Numerical Algebra, Control & Optimization, 2016, 6 (2) : 127-151. doi: 10.3934/naco.2016004
[8]

Jian Chen, Tao Zhang, Ziye Zhang, Chong Lin, Bing Chen. Stability and output feedback control for singular Markovian jump delayed systems. Mathematical Control & Related Fields, 2018, 8 (2) : 475-490. doi: 10.3934/mcrf.2018019
[9]

Zhenyu Lu, Junhao Hu, Xuerong Mao. Stabilisation by delay feedback control for highly nonlinear hybrid stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4099-4116. doi: 10.3934/dcdsb.2019052
[10]

V. Rehbock, K.L. Teo, L.S. Jennings. Suboptimal feedback control for a class of nonlinear systems using spline interpolation. Discrete & Continuous Dynamical Systems, 1995, 1 (2) : 223-236. doi: 10.3934/dcds.1995.1.223
[11]

H. T. Banks, R.C. Smith. Feedback control of noise in a 2-D nonlinear structural acoustics model. Discrete & Continuous Dynamical Systems, 1995, 1 (1) : 119-149. doi: 10.3934/dcds.1995.1.119
[12]

Yigui Ou, Haichan Lin. A class of accelerated conjugate-gradient-like methods based on a modified secant equation. Journal of Industrial & Management Optimization, 2020, 16 (3) : 1503-1518. doi: 10.3934/jimo.2019013
[13]

Zhong Wan, Chaoming Hu, Zhanlu Yang. A spectral PRP conjugate gradient methods for nonconvex optimization problem based on modified line search. Discrete & Continuous Dynamical Systems - B, 2011, 16 (4) : 1157-1169. doi: 10.3934/dcdsb.2011.16.1157
[14]

Gaohang Yu, Lutai Guan, Guoyin Li. Global convergence of modified Polak-Ribière-Polyak conjugate gradient methods with sufficient descent property. Journal of Industrial & Management Optimization, 2008, 4 (3) : 565-579. doi: 10.3934/jimo.2008.4.565
[15]

Brian D. O. Anderson, Shaoshuai Mou, A. Stephen Morse, Uwe Helmke. Decentralized gradient algorithm for solution of a linear equation. Numerical Algebra, Control & Optimization, 2016, 6 (3) : 319-328. doi: 10.3934/naco.2016014
[16]

N. U. Ahmed. Existence of optimal output feedback control law for a class of uncertain infinite dimensional stochastic systems: A direct approach. Evolution Equations & Control Theory, 2012, 1 (2) : 235-250. doi: 10.3934/eect.2012.1.235
[17]

Hao Sun, Shihua Li, Xuming Wang. Output feedback based sliding mode control for fuel quantity actuator system using a reduced-order GPIO. Discrete & Continuous Dynamical Systems - S, 2021, 14 (4) : 1447-1464. doi: 10.3934/dcdss.2020375
[18]

Shui-Hung Hou, Qing-Xu Yan. Nonlinear locally distributed feedback stabilization. Journal of Industrial & Management Optimization, 2008, 4 (1) : 67-79. doi: 10.3934/jimo.2008.4.67
[19]

Gonzalo Robledo. Feedback stabilization for a chemostat with delayed output. Mathematical Biosciences & Engineering, 2009, 6 (3) : 629-647. doi: 10.3934/mbe.2009.6.629
[20]

Abderrahim Azouani, Edriss S. Titi. Feedback control of nonlinear dissipative systems by finite determining parameters - A reaction-diffusion paradigm. Evolution Equations & Control Theory, 2014, 3 (4) : 579-594. doi: 10.3934/eect.2014.3.579

2020 Impact Factor: 1.801

Tools

Metrics

  • PDF downloads (54)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences