American Institute of Mathematical Sciences

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2015, 5(3): 251-266. doi: 10.3934/naco.2015.5.251

A gradient algorithm for optimal control problems with model-reality differences

Sie Long Kek 1, , Mohd Ismail Abd Aziz 2, and  Kok Lay Teo 3,

1. 

Department of Mathematics, Universiti Tun Hussein Onn Malaysia, 86400 Parit Raja, Malaysia
2. 

Department of Mathematics, Universiti Teknologi Malaysia, 81310 UTM, Skudai, Malaysia
3. 

Department of Mathematics and Statistics, Curtin University, Perth, W.A. 6845

Received  May 2014 Revised  March 2015 Published  August 2015

In this paper, we propose a computational approach to solve a model-based optimal control problem. Our aim is to obtain the optimal solution of the nonlinear optimal control problem. Since the structures of both problems are different, only solving the model-based optimal control problem will not give the optimal solution of the nonlinear optimal control problem. In our approach, the adjusted parameters are added into the model used so as the differences between the real plant and the model can be measured. On this basis, an expanded optimal control problem is introduced, where system optimization and parameter estimation are integrated interactively. The Hamiltonian function, which adjoins the cost function, the state equation and the additional constraints, is defined. By applying the calculus of variation, a set of the necessary optimality conditions, which defines modified model-based optimal control problem, parameter estimation problem and computation of modifiers, is then derived. To obtain the optimal solution, the modified model-based optimal control problem is converted in a nonlinear programming problem through the canonical formulation, where the gradient formulation can be made. During the iterative procedure, the control sequences are generated as the admissible control law of the model used, together with the corresponding state sequences. Consequently, the optimal solution is updated repeatedly by the adjusted parameters. At the end of iteration, the converged solution approaches to the correct optimal solution of the original optimal control problem in spite of model-reality differences. For illustration, two examples are studied and the results show the efficiency of the approach proposed.
Keywords: gradient algorithm, adjusted parameters, iterative solution., Nonlinear optimal control, model-reality differences.
Mathematics Subject Classification: Primary: 93C05, 93C10; Secondary: 93B4.
Citation: Sie Long Kek, Mohd Ismail Abd Aziz, Kok Lay Teo. A gradient algorithm for optimal control problems with model-reality differences. Numerical Algebra, Control & Optimization, 2015, 5 (3) : 251-266. doi: 10.3934/naco.2015.5.251
References:
[1]

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

[2]

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

[3]

S. L. Kek, Nonlinear programming approach for optimal control problems,, Proceeding of the 2nd International Conference on Global Optimization and Its Applications, (2013), 20.   Google Scholar

[4]

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

[5]

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

[6]

Q. Lin, R. Loxton and K. L. Teo, The control parameterization method for nonlinear optimal control: a survey,, Journal of Industrial and Management Optimization, 10 (2014), 275.  doi: 10.3934/jimo.2014.10.275.  Google Scholar

[7]

R. Loxton, K. L. Teo and V. Rehbock, Computational method for a class of switched system optimal control problems,, IEEE Transactions on Automatic Control, 54 (2009), 2455.  doi: 10.1109/TAC.2009.2029310.  Google Scholar

[8]

R. Loxton, K. L. Teo, V. Rehbock and K. F. C. Yiu, Optimal control problems with a continuous inequality constraint on the state and the control,, Automatica, 45 (2009), 2250.  doi: 10.1016/j.automatica.2009.05.029.  Google Scholar

[9]

L. F. Lupián and J. R. Rabadán-Martin, LQR control methods for trajectory execution in omnidirectional mobile robots,, Recent Advances in Mobile Robotics, (2011), 385.   Google Scholar

[10]

L. H. Nguyen, S. Park, A. Turnip and K. S. Hong, Application of LQR control theory to the design of modified skyhook control gains for semi-active suspension systems,, Proceeding of ICROS-SICE International Joint Conference, (2009), 4698.   Google Scholar

[11]

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

[12]

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

[13]

R. C. H. del Rosario and R. C. Smith, LQR control of shell vibrations via piezocreramic actuators,, NASA Contractor Report 201673, (2016), 97.   Google Scholar

[14]

J. Saak and P. Benner, Application of LQR techniques to the adaptive control of quasilinear parabolic PDEs,, Proceedings in Applied Mathematics and Mechanics, (2007).   Google Scholar

[15]

K. L. Teo, C. J. Goh and K. H. Wong, A Unified Computational Approach to Optimal Control Problem,, Longman Scientific and Technical, (1991).   Google Scholar

[16]

L. X. Wang, A Course in Fuzzy Systems and Control,, Upper Saddle River, (1997).   Google Scholar

[17]

C. Z. Wu, K. L. Teo and V. Rehbock, Optimal control of piecewise affine systems with piecewise affine state feedback,, Journal of Industrial and Management Optimization, 5 (2009), 737.  doi: 10.3934/jimo.2009.5.737.  Google Scholar

[18]

B. Yang and B. Xiong, Application of LQR techniques to the anti-sway controller of overhead crane,, Advanced Material Research, 139-141 (2010), 139.   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,, 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, Nonlinear programming approach for optimal control problems,, Proceeding of the 2nd International Conference on Global Optimization and Its Applications, (2013), 20.   Google Scholar

[4]

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

[5]

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

[6]

Q. Lin, R. Loxton and K. L. Teo, The control parameterization method for nonlinear optimal control: a survey,, Journal of Industrial and Management Optimization, 10 (2014), 275.  doi: 10.3934/jimo.2014.10.275.  Google Scholar

[7]

R. Loxton, K. L. Teo and V. Rehbock, Computational method for a class of switched system optimal control problems,, IEEE Transactions on Automatic Control, 54 (2009), 2455.  doi: 10.1109/TAC.2009.2029310.  Google Scholar

[8]

R. Loxton, K. L. Teo, V. Rehbock and K. F. C. Yiu, Optimal control problems with a continuous inequality constraint on the state and the control,, Automatica, 45 (2009), 2250.  doi: 10.1016/j.automatica.2009.05.029.  Google Scholar

[9]

L. F. Lupián and J. R. Rabadán-Martin, LQR control methods for trajectory execution in omnidirectional mobile robots,, Recent Advances in Mobile Robotics, (2011), 385.   Google Scholar

[10]

L. H. Nguyen, S. Park, A. Turnip and K. S. Hong, Application of LQR control theory to the design of modified skyhook control gains for semi-active suspension systems,, Proceeding of ICROS-SICE International Joint Conference, (2009), 4698.   Google Scholar

[11]

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

[12]

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

[13]

R. C. H. del Rosario and R. C. Smith, LQR control of shell vibrations via piezocreramic actuators,, NASA Contractor Report 201673, (2016), 97.   Google Scholar

[14]

J. Saak and P. Benner, Application of LQR techniques to the adaptive control of quasilinear parabolic PDEs,, Proceedings in Applied Mathematics and Mechanics, (2007).   Google Scholar

[15]

K. L. Teo, C. J. Goh and K. H. Wong, A Unified Computational Approach to Optimal Control Problem,, Longman Scientific and Technical, (1991).   Google Scholar

[16]

L. X. Wang, A Course in Fuzzy Systems and Control,, Upper Saddle River, (1997).   Google Scholar

[17]

C. Z. Wu, K. L. Teo and V. Rehbock, Optimal control of piecewise affine systems with piecewise affine state feedback,, Journal of Industrial and Management Optimization, 5 (2009), 737.  doi: 10.3934/jimo.2009.5.737.  Google Scholar

[18]

B. Yang and B. Xiong, Application of LQR techniques to the anti-sway controller of overhead crane,, Advanced Material Research, 139-141 (2010), 139.   Google Scholar

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