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

Previous Article
A quasi-Newton trust region method based on a new fractional model
NACO Home
This Issue
Next Article
Matrix group monotonicity using a dominance notion

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

Abstract
Related Papers

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.
[2]	A. E. Bryson and Y. C. Ho, Applied Optimal Control,, Hemisphere Publishing Company, (1975).
[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.
[4]	D. E. Kirk, Optimal Control Theory: An Introduction,, Mineola, (2004).
[5]	F. L. Lewis and V. L. Syrmos, Optimal Control,, 2nd ed, (1995).
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[13]	R. C. H. del Rosario and R. C. Smith, LQR control of shell vibrations via piezocreramic actuators,, NASA Contractor Report 201673, (2016), 97.
[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).
[15]	K. L. Teo, C. J. Goh and K. H. Wong, A Unified Computational Approach to Optimal Control Problem,, Longman Scientific and Technical, (1991).
[16]	L. X. Wang, A Course in Fuzzy Systems and Control,, Upper Saddle River, (1997).
[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.
[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.

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.
[2]	A. E. Bryson and Y. C. Ho, Applied Optimal Control,, Hemisphere Publishing Company, (1975).
[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.
[4]	D. E. Kirk, Optimal Control Theory: An Introduction,, Mineola, (2004).
[5]	F. L. Lewis and V. L. Syrmos, Optimal Control,, 2nd ed, (1995).
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[13]	R. C. H. del Rosario and R. C. Smith, LQR control of shell vibrations via piezocreramic actuators,, NASA Contractor Report 201673, (2016), 97.
[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).
[15]	K. L. Teo, C. J. Goh and K. H. Wong, A Unified Computational Approach to Optimal Control Problem,, Longman Scientific and Technical, (1991).
[16]	L. X. Wang, A Course in Fuzzy Systems and Control,, Upper Saddle River, (1997).
[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.
[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.

[1]	Sie Long Kek, Mohd Ismail Abd Aziz, Kok Lay Teo, Rohanin Ahmad. An iterative algorithm based on model-reality differences for discrete-time nonlinear stochastic optimal control problems. Numerical Algebra, Control & Optimization, 2013, 3 (1) : 109-125. doi: 10.3934/naco.2013.3.109
[2]	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
[3]	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
[4]	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
[5]	Rein Luus. Optimal control of oscillatory systems by iterative dynamic programming. Journal of Industrial & Management Optimization, 2008, 4 (1) : 1-15. doi: 10.3934/jimo.2008.4.1
[6]	Ellina Grigorieva, Evgenii Khailov. Optimal control of a nonlinear model of economic growth. Conference Publications, 2007, 2007 (Special) : 456-466. doi: 10.3934/proc.2007.2007.456
[7]	Yumei Liao, Wei Wei, Xianbing Luo. Existence of solution of a microwave heating model and associated optimal frequency control problems. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-14. doi: 10.3934/jimo.2019045
[8]	Zhiqing Meng, Qiying Hu, Chuangyin Dang. A penalty function algorithm with objective parameters for nonlinear mathematical programming. Journal of Industrial & Management Optimization, 2009, 5 (3) : 585-601. doi: 10.3934/jimo.2009.5.585
[9]	Sören Bartels, Marijo Milicevic. Iterative finite element solution of a constrained total variation regularized model problem. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1207-1232. doi: 10.3934/dcdss.2017066
[10]	Guangzhou Chen, Guijian Liu, Jiaquan Wang, Ruzhong Li. Identification of water quality model parameters using artificial bee colony algorithm. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 157-165. doi: 10.3934/naco.2012.2.157
[11]	Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. II. Convergence of the method of finite differences. Inverse Problems & Imaging, 2016, 10 (4) : 869-898. doi: 10.3934/ipi.2016025
[12]	Hamid Reza Marzban, Hamid Reza Tabrizidooz. Solution of nonlinear delay optimal control problems using a composite pseudospectral collocation method. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1379-1389. doi: 10.3934/cpaa.2010.9.1379
[13]	Rong Liu, Feng-Qin Zhang, Yuming Chen. Optimal contraception control for a nonlinear population model with size structure and a separable mortality. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3603-3618. doi: 10.3934/dcdsb.2016112
[14]	V.N. Malozemov, A.V. Omelchenko. On a discrete optimal control problem with an explicit solution. Journal of Industrial & Management Optimization, 2006, 2 (1) : 55-62. doi: 10.3934/jimo.2006.2.55
[15]	Yanqin Bai, Yudan Wei, Qian Li. An optimal trade-off model for portfolio selection with sensitivity of parameters. Journal of Industrial & Management Optimization, 2017, 13 (2) : 947-965. doi: 10.3934/jimo.2016055
[16]	Qun Lin, Ryan Loxton, Kok Lay Teo. The control parameterization method for nonlinear optimal control: A survey. Journal of Industrial & Management Optimization, 2014, 10 (1) : 275-309. doi: 10.3934/jimo.2014.10.275
[17]	Yi Zhang, Xiao-Li Ma. Research on image digital watermarking optimization algorithm under virtual reality technology. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1427-1440. doi: 10.3934/dcdss.2019098
[18]	Filipe Rodrigues, Cristiana J. Silva, Delfim F. M. Torres, Helmut Maurer. Optimal control of a delayed HIV model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 443-458. doi: 10.3934/dcdsb.2018030
[19]	Cristiana J. Silva, Helmut Maurer, Delfim F. M. Torres. Optimal control of a Tuberculosis model with state and control delays. Mathematical Biosciences & Engineering, 2017, 14 (1) : 321-337. doi: 10.3934/mbe.2017021
[20]	Judy Day, Jonathan Rubin, Gilles Clermont. Using nonlinear model predictive control to find optimal therapeutic strategies to modulate inflammation. Mathematical Biosciences & Engineering, 2010, 7 (4) : 739-763. doi: 10.3934/mbe.2010.7.739

Impact Factor:

American Institute of Mathematical Sciences