2015, 5(3): 275-288. doi: 10.3934/naco.2015.5.275

Output regulation for discrete-time nonlinear stochastic optimal control problems with model-reality differences

1. 

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

2. 

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

Received  May 2014 Revised  March 2015 Published  August 2015

In this paper, we propose an output regulation approach, which is based on principle of model-reality differences, to obtain the optimal output measurement of a discrete-time nonlinear stochastic optimal control problem. In our approach, a model-based optimal control problem with adding the adjustable parameters is considered. We aim to regulate the optimal output trajectory of the model used as closely as possible to the output measurement of the original optimal control problem. In doing so, an expanded optimal control problem is introduced, where system optimization and parameter estimation are integrated. During the computation procedure, the differences between the real plant and the model used are measured repeatedly. In such a way, the optimal solution of the model is updated. At the end of iteration, the converged solution approaches closely to the true optimal solution of the original optimal control problem in spite of model-reality differences. It is important to notice that the resulting algorithm could give the output residual that is superior to those obtained from Kalman filtering theory. The accuracy of the output regulation is therefore highly recommended. For illustration, a continuous stirred-tank reactor problem is studied. The results obtained show the efficiency of the approach proposed.
Citation: 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
References:
[1]

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

[2]

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

[3]

S. L. Kek and A. A. Mohd Ismail, Optimal control of discrete-time linear stochastic dynamic system with model-reality differences,, in \emph{Proceeding of International Conference on Machine Learning and Computing (ICML 2009)}, (2009), 10. Google Scholar

[4]

S. L. Kek, K. L. Teo and A. A. Mohd Ismail, An integrated optimal control algorithm for discrete-time nonlinear stochastic system,, \emph{International Journal of Control}, 83 (2010), 2536. doi: 10.1080/00207179.2010.531766. Google Scholar

[5]

S. L. Kek, K. L. Teo and A. A. Mohd Ismail, Filtering solution of nonlinear stochastic optimal control problem in discrete-time with model-reality differences,, \emph{Numerical Algebra, 2 (2012), 207. doi: 10.3934/naco.2012.2.207. Google Scholar

[6]

S. L. Kek, A. A. Mohd Ismail, K. L. Teo and A. Rohanin, An iterative algorithm based on model-reality differences for discrete-time nonlinear stochastic optimal control problems,, \emph{Numerical Algebra, 3 (2013), 109. doi: 10.3934/naco.2013.3.109. Google Scholar

[7]

D. E. Kirk, Optimal Control Theory: An Introduction,, Mineola, (2004). Google Scholar

[8]

F. L. Lewis and V. L. Syrmos, Optimal Control,, 2nd ed, (1995). Google Scholar

[9]

A. A. Mohd Ismail and S. L. Kek, Optimal control of nonlinear discrete-time stochastic system with model-reality differences,, in \emph{2009 IEEE International Conference on Control and Automation}, (2009), 9. Google Scholar

[10]

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 \emph{Proceeding of the 2010 IRAST Internation Congress on Computer Applications and Computational Science (CACS 2010)}, (2010), 4. Google Scholar

[11]

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

[12]

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

show all references

References:
[1]

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

[2]

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

[3]

S. L. Kek and A. A. Mohd Ismail, Optimal control of discrete-time linear stochastic dynamic system with model-reality differences,, in \emph{Proceeding of International Conference on Machine Learning and Computing (ICML 2009)}, (2009), 10. Google Scholar

[4]

S. L. Kek, K. L. Teo and A. A. Mohd Ismail, An integrated optimal control algorithm for discrete-time nonlinear stochastic system,, \emph{International Journal of Control}, 83 (2010), 2536. doi: 10.1080/00207179.2010.531766. Google Scholar

[5]

S. L. Kek, K. L. Teo and A. A. Mohd Ismail, Filtering solution of nonlinear stochastic optimal control problem in discrete-time with model-reality differences,, \emph{Numerical Algebra, 2 (2012), 207. doi: 10.3934/naco.2012.2.207. Google Scholar

[6]

S. L. Kek, A. A. Mohd Ismail, K. L. Teo and A. Rohanin, An iterative algorithm based on model-reality differences for discrete-time nonlinear stochastic optimal control problems,, \emph{Numerical Algebra, 3 (2013), 109. doi: 10.3934/naco.2013.3.109. Google Scholar

[7]

D. E. Kirk, Optimal Control Theory: An Introduction,, Mineola, (2004). Google Scholar

[8]

F. L. Lewis and V. L. Syrmos, Optimal Control,, 2nd ed, (1995). Google Scholar

[9]

A. A. Mohd Ismail and S. L. Kek, Optimal control of nonlinear discrete-time stochastic system with model-reality differences,, in \emph{2009 IEEE International Conference on Control and Automation}, (2009), 9. Google Scholar

[10]

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 \emph{Proceeding of the 2010 IRAST Internation Congress on Computer Applications and Computational Science (CACS 2010)}, (2010), 4. Google Scholar

[11]

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

[12]

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

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