2012, 2(1): 207-222. doi: 10.3934/naco.2012.2.207

Filtering solution of nonlinear stochastic optimal control problem in discrete-time with model-reality differences

1. 

Department of Mathematics, Universiti Tun Hussein Onn Malaysia, 86400 Parit Raja, Malaysia

2. 

Department of Mathematics and Statistics, Curtin University, G.P.O. Box U1987, Perth, WA 6845

3. 

Department of Mathematics, Universiti Teknologi Malaysia, 81310 UTM, Skudai, Malaysia

Received  March 2011 Revised  July 2011 Published  March 2012

In this paper, we propose an efficient algorithm for solving a nonlinear stochastic optimal control problem in discrete-time, where the true filtered solution of the original optimal control problem is obtained through solving a linear model-based optimal control problem with adjustable parameters iteratively. The adjustments of these parameters are based on the differences between the real plant and the linear model that are measured. The main feature of the algorithm proposed is the integration of system optimization and parameter estimation in an interactive way so that the correct filtered solution of the original optimal control problem is obtained when the convergence is achieved. For illustration, a nonlinear continuous stirred reactor tank problem is studied. The simulation results obtained demonstrate the efficiency of the algorithm proposed.
Citation: Sie Long Kek, Kok Lay Teo, Mohd Ismail Abd Aziz. Filtering solution of nonlinear stochastic optimal control problem in discrete-time with model-reality differences. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 207-222. doi: 10.3934/naco.2012.2.207
References:
[1]

V. M. Becerra, "Development and Applications of Novel Optimal Control Algorithms,", Ph.D. thesis, (1994).   Google Scholar

[2]

V. M. Becerra and P. D. Roberts, Dynamic integrated system optimization and parameter estimation for discrete time optimal control of nonlinear systems,, Int. J. Control, 63 (1996), 257.  doi: 10.1080/00207179608921843.  Google Scholar

[3]

V. M. Becerra and P. D. Roberts, Application of a novel optimal control algorithm to a benchmark fed-batch fermentation process,, Trans. Inst. Measurement Control, 20 (1998), 11.  doi: 10.1177/014233129802000103.  Google Scholar

[4]

A. E. Bryson and Y. C. Ho, "Applied Optimal Control,", Hemisphere Publishing Company, (1975).   Google Scholar

[5]

A. E. Bryson, "Applied Linear Optimal Control, Examples and Algorithms,", Cambridge University Press, (2002).   Google Scholar

[6]

Y. Y. Haimes and D. A. Wismer, A computational approach to the combined problem of optimization and parameter estimation,, Automatica, 8 (1972), 337.  doi: 10.1016/0005-1098(72)90052-0.  Google Scholar

[7]

M. H. Hu, Q. Gao and H. H. Shao, Optimal control of a class of non-linear discrete-continuous hybrid systems,, in, (2006), 835.   Google Scholar

[8]

M. H. Hu, Y. S. Wang and H. H Shao, Costate prediction based optimal control for non-linear hybrid systems,, ISA Transactions, 47 (2008), 113.  doi: 10.1016/j.isatra.2007.06.001.  Google Scholar

[9]

S. L. Kek and A. A. Mohd Ismail, Optimal control of discrete-time linear stochastic dynamic system with model-reality differences,, in, (2009), 10.   Google Scholar

[10]

J. S. Kong and B. W. Wan, The study of integrated optimal control approach for complex system under network environment,, Computing Technology and Automation, 22 (2003), 23.   Google Scholar

[11]

F. L. Lewis, "Optimal Control,", John Wiley and Sons, (1986).   Google Scholar

[12]

F. L. Lewis, "Applied Optimal Control and Estimation: Digital Design and Implementation,", Prentice Hall, (1992).   Google Scholar

[13]

J. M. Li, B. W. Wan and Z. L. Huang, Optimal control of nonlinear discrete systems with model-reality differences,, Control Theory and Applications, 16 (1999), 32.   Google Scholar

[14]

A. A. Mohd Ismail and S. L. Kek, Optimal control of nonlinear discrete-time stochastic system with model-reality differences,, in, (2009), 9.   Google Scholar

[15]

A. A. Mohd Ismail, A. Rohanin, S. L. Kek and K. L. Teo, Computational integrated optimal control and estimation with model information for discrete-time nonlinear stochastic dynamic system,, in, (2010), 4.   Google Scholar

[16]

W. H. Ray, "Advanced Process Control,", McGraw-Hill, (1989).   Google Scholar

[17]

P. D. Roberts, An algorithm for steady-state system optimization and parameter estimation,, Int. J. Systems Science, 10 (1979), 719.  doi: 10.1080/00207727908941614.  Google Scholar

[18]

P. D. Roberts and T. W. C. Williams, On an algorithm for combined system optimization and parameter estimation,, Automatica, 17 (1981), 199.  doi: 10.1016/0005-1098(81)90095-9.  Google Scholar

[19]

P. D. Roberts, Optimal control of nonlinear systems with model-reality differences,, Proceedings of the 31st IEEE Conference on Decision and Control, 1 (1992), 257.   Google Scholar

[20]

P. D. Roberts and V. M. Becerra, Optimal control of a class of discrete-continuous non-linear systems decomposition and hierarchical structure,, Automatica, 37 (2001), 1757.  doi: 10.1016/S0005-1098(01)00141-8.  Google Scholar

[21]

Y. Zhang and S. Y. Li, DISOPE distributed model predictive control of cascade systems with network communication,, Journal of Control Theory and Applications, 2 (2005), 131.  doi: 10.1007/s11768-005-0005-6.  Google Scholar

show all references

References:
[1]

V. M. Becerra, "Development and Applications of Novel Optimal Control Algorithms,", Ph.D. thesis, (1994).   Google Scholar

[2]

V. M. Becerra and P. D. Roberts, Dynamic integrated system optimization and parameter estimation for discrete time optimal control of nonlinear systems,, Int. J. Control, 63 (1996), 257.  doi: 10.1080/00207179608921843.  Google Scholar

[3]

V. M. Becerra and P. D. Roberts, Application of a novel optimal control algorithm to a benchmark fed-batch fermentation process,, Trans. Inst. Measurement Control, 20 (1998), 11.  doi: 10.1177/014233129802000103.  Google Scholar

[4]

A. E. Bryson and Y. C. Ho, "Applied Optimal Control,", Hemisphere Publishing Company, (1975).   Google Scholar

[5]

A. E. Bryson, "Applied Linear Optimal Control, Examples and Algorithms,", Cambridge University Press, (2002).   Google Scholar

[6]

Y. Y. Haimes and D. A. Wismer, A computational approach to the combined problem of optimization and parameter estimation,, Automatica, 8 (1972), 337.  doi: 10.1016/0005-1098(72)90052-0.  Google Scholar

[7]

M. H. Hu, Q. Gao and H. H. Shao, Optimal control of a class of non-linear discrete-continuous hybrid systems,, in, (2006), 835.   Google Scholar

[8]

M. H. Hu, Y. S. Wang and H. H Shao, Costate prediction based optimal control for non-linear hybrid systems,, ISA Transactions, 47 (2008), 113.  doi: 10.1016/j.isatra.2007.06.001.  Google Scholar

[9]

S. L. Kek and A. A. Mohd Ismail, Optimal control of discrete-time linear stochastic dynamic system with model-reality differences,, in, (2009), 10.   Google Scholar

[10]

J. S. Kong and B. W. Wan, The study of integrated optimal control approach for complex system under network environment,, Computing Technology and Automation, 22 (2003), 23.   Google Scholar

[11]

F. L. Lewis, "Optimal Control,", John Wiley and Sons, (1986).   Google Scholar

[12]

F. L. Lewis, "Applied Optimal Control and Estimation: Digital Design and Implementation,", Prentice Hall, (1992).   Google Scholar

[13]

J. M. Li, B. W. Wan and Z. L. Huang, Optimal control of nonlinear discrete systems with model-reality differences,, Control Theory and Applications, 16 (1999), 32.   Google Scholar

[14]

A. A. Mohd Ismail and S. L. Kek, Optimal control of nonlinear discrete-time stochastic system with model-reality differences,, in, (2009), 9.   Google Scholar

[15]

A. A. Mohd Ismail, A. Rohanin, S. L. Kek and K. L. Teo, Computational integrated optimal control and estimation with model information for discrete-time nonlinear stochastic dynamic system,, in, (2010), 4.   Google Scholar

[16]

W. H. Ray, "Advanced Process Control,", McGraw-Hill, (1989).   Google Scholar

[17]

P. D. Roberts, An algorithm for steady-state system optimization and parameter estimation,, Int. J. Systems Science, 10 (1979), 719.  doi: 10.1080/00207727908941614.  Google Scholar

[18]

P. D. Roberts and T. W. C. Williams, On an algorithm for combined system optimization and parameter estimation,, Automatica, 17 (1981), 199.  doi: 10.1016/0005-1098(81)90095-9.  Google Scholar

[19]

P. D. Roberts, Optimal control of nonlinear systems with model-reality differences,, Proceedings of the 31st IEEE Conference on Decision and Control, 1 (1992), 257.   Google Scholar

[20]

P. D. Roberts and V. M. Becerra, Optimal control of a class of discrete-continuous non-linear systems decomposition and hierarchical structure,, Automatica, 37 (2001), 1757.  doi: 10.1016/S0005-1098(01)00141-8.  Google Scholar

[21]

Y. Zhang and S. Y. Li, DISOPE distributed model predictive control of cascade systems with network communication,, Journal of Control Theory and Applications, 2 (2005), 131.  doi: 10.1007/s11768-005-0005-6.  Google Scholar

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