Application of Conjugate Gradient Approach for Nonlinear Optimal Control Problem with Model-Reality Differences
Abstract:
In this paper, an efficient computational algorithm is proposed to solve the nonlinear optimal control problem. In our approach, the linear quadratic optimal control model, which is adding the adjusted parameters into the model used, is employed. The aim of applying this model is to take into account the differences between the real plant and the model used during the calculation procedure. In doing so, an expanded optimal control problem is introduced such that system optimization and parameter estimation are mutually interactive. Accordingly, the optimality conditions are derived after the Hamiltonian function is defined. Specifically, the modified model-based optimal control problem is resulted. Here, the conjugate gradient approach is used to solve the modified model-based optimal control problem, where the optimal solution of the model used is calculated repeatedly, in turn, to update the adjusted parameters on each iteration step. When the convergence is achieved, the iterative solution approaches to the correct solution of the original optimal control problem, in spite of model-reality differences. For illustration, an economic growth problem is solved by using the algorithm proposed. The results obtained demonstrate the efficiency of the algorithm proposed. In conclusion, the applicability of the algorithm proposed is highly recommended.
Cite this paper: Kek, S. , Leong, W. , Sim, S. and Teo, K. (2018) Application of Conjugate Gradient Approach for Nonlinear Optimal Control Problem with Model-Reality Differences. Applied Mathematics, 9, 940-953. doi: 10.4236/am.2018.98064.
References

[1]   Kek, S.L., Teo, K.L. and Mohd Ismail, A.A. (2010) An Integrated Optimal Control Algorithm for Discrete-Time Nonlinear Stochastic System. International Journal of Control, 83, 2536-2545.
https://doi.org/10.1080/00207179.2010.531766

[2]   Kek, S.L., Teo, K.L. and Mohd Ismail, A.A. (2012) Filtering Solution of Nonlinear Stochastic Optimal Control Problem in Discrete-Time with Model-Reality Differences. Numerical Algebra, Control and Optimization, 2, 207-222.
https://doi.org/10.3934/naco.2012.2.207

[3]   Kek, S.L., Mohd Ismail, A.A., Teo, K.L. and Rohanin, A. (2013) An Iterative Algorithm Based on Model-Reality Differences for Discrete-Time Nonlinear Stochastic Optimal Control Problems. Numerical Algebra, Control and Optimization, 3, 109-125.
https://doi.org/10.3934/naco.2013.3.109

[4]   Kek, S.L., Teo, K.L. and Mohd Ismail, A.A. (2015) Efficient Output Solution for Nonlinear Stochastic Optimal Control Problem with Model-Reality Differences. Mathematical Problems in Engineering, 2015, Article ID: 659506.
https://doi.org/10.1155/2015/659506

[5]   Kek, S.L., Mohd Ismail, A.A. and Teo, K.L. (2015) A Gradient Algorithm for Optimal Control Problems with Model-Reality Differences. Numerical Algebra, Control and Optimization, 5, 252-266.

[6]   Kek, S.L. and Mohd Ismail, A.A. (2015) Output Regulation for Discrete-Time Nonlinear Stochastic Optimal Control Problems with Model-Reality Differences. Numerical Algebra, Control and Optimization, 5, 275-288.
https://doi.org/10.3934/naco.2015.5.275

[7]   Kek, S.L., Li, J. and Teo, K.L. (2017) Least Squares Solution for Discrete Time Nonlinear Stochastic Optimal Control Problem with Model-Reality Differences. Applied Mathematics, 8, 1-14.
https://doi.org/10.4236/am.2017.81001

[8]   Kek, S.L., Li, J., Leong, W.J. and Mohd Ismail, A.A. (2017) A Gauss-Newton Approach for Nonlinear Optimal Control Problem with Model-Reality Differences. Open Journal of Optimization, 6, 85-100.
https://doi.org/10.4236/ojop.2017.63007

[9]   Kek, S.L., Sim, S.Y., Leong, W.J. and Teo, K.L. (2018) Discrete-Time Nonlinear Stochastic Optimal Control Problem Based on Stochastic Approximation Approach. Advances in Pure Mathematics, 8, 232-244.
https://doi.org/10.4236/apm.2018.83012

[10]   Becerra, V.M. and Roberts, P.D. (1996) Dynamic Integrated System Optimization and Parameter Estimation for Discrete Time Optimal Control of Nonlinear Systems. International Journal of Control, 63, 257-281.
https://doi.org/10.1080/00207179608921843

[11]   Roberts, P.D. and Becerra, V.M. (2001) Optimal Control of a Class of Discrete-Continuous Nonlinear Systems Decomposition and Hierarchical Structure. Automatica, 37, 1757-1769.
https://doi.org/10.1016/S0005-1098(01)00141-8

[12]   Roberts, P.D. (1979) An Algorithm for Steady-State System Optimization and Parameter Estimation. International Journal of Systems Science, 10, 719-734.
https://doi.org/10.1080/00207727908941614

[13]   Roberts, P.D. and Williams, T.W.C. (1981) On an Algorithm for Combined System Optimization and Parameter Estimation. Automatica, 17, 199-209.
https://doi.org/10.1016/0005-1098(81)90095-9

[14]   Chong, E.K.P. and Zak, S.H. (2013) An Introduction to Optimization. 4th Edition, John Wiley & Sons, Inc., Hoboken, New Jersey.

[15]   Mostafa, E.M.E. (2014) A Nonlinear Conjugate Gradient Method for A Special Class of Matrix Optimization Problems. Journal of Industrial and Management Optimization, 10, 883-903.
https://doi.org/10.3934/jimo.2014.10.883

[16]   Lasdon, L.S. and Mitter, S.K. (1967) The Conjugate Gradient Method for Optimal Control Problems. IEEE Transactions on Automatic Control, 12, 132-138.
https://doi.org/10.1109/TAC.1967.1098538

[17]   Nwaeze, E. (2011) An Extended Conjugate Gradient Method for Optimizing Continuous-Time Optimal Control Problems. Canadian Journal on Computing in Mathematics, Natural Sciences, Engineering & Medicine, 2, 39-44.

[18]   Raji, R.A. and Oke, M.O. (2015) Higher-Order Conjugate Gradient Method for Solving Continuous Optimal Control Problems. IOSR Journal of Mathematics, 11, 88-90.

[19]   Brock, W.A. and Mirman, L. (1972) Optimal Economic Growth and Uncertainty: The Discounted Case. Journal of Economy Theory, 4, 479-513.
https://doi.org/10.1016/0022-0531(72)90135-4

[20]   Bryson, A.E. and Ho, Y.C. (1975) Applied Optimal Control. Hemisphere, Washington DC.

[21]   Lewis, F.L., Vrabie, V. and Symos, V.L. (2012) Optimal Control. 3rd Edition, John Wiley & Sons, Inc., New York.
https://doi.org/10.1002/9781118122631

[22]   Kirk, D.E. (2004) Optimal Control Theory: An Introduction. Dover Publications, New York.

[23]   Grüne, L., Semmler, W. and Stieler, M. (2015) Using Nonlinear Model Predictive Control for Dynamic Decision Problems in Economics. Journal of Economic Dynamics and Control, 60, 112-133.
https://doi.org/10.1016/j.jedc.2015.08.010

[24]   Santos, M.S. and Vigo-Aguiar, J. (1998) Analysis of a Numerical Dynamic Programming Algorithm Applied to Economic Models. Econometrica, 66, 409-426.
https://doi.org/10.2307/2998564

Top