# Integrating PSO with modified hybrid GA for solving nonlinear optimal control problems

## DOI:

https://doi.org/10.14419/ijamr.v3i4.3538## Published:

2014-10-21## Abstract

Here, a two-phase algorithm based on integrating particle swarm optimization (PSO) with modified hybrid genetic algorithm (MHGA) is proposed for solving the associated nonlinear programming problem of a nonlinear optimal control problem. In the first phase, PSO starts with a completely random initial swarm of particles, where each of them contains two random matrices in time nodes. After phase 1, to achieve more accurate solutions, the number of time nodes is increased. The values of the associated new control inputs are estimated by linear or spline interpolations using the curves computed in the phase 1. In addition, to maintain the diversity in the population, some additional individuals are added randomly. Next, in the second phase, MHGA, starts by the new population constructed by the above procedure and tries to improve the obtained solutions at the end of phase 1. MHGA combines a GA with a successive quadratic programming, SQP, as a local search. Finally, we implement the proposed algorithm on some well-known nonlinear optimal control problems. The numerical results show that the proposed algorithm can find almost better solution than other proposed algorithms.Â

**Keywords: **Optimal control problem, Hybrid genetic algorithm, ?particle swarm optimization, Spline interpolation.

## References

Zaer S. Abo-Hammour, Ali Ghaleb Asasfeh, Adnan M. Al-Smadi, and Othman M. K. Alsmadi, A novel continuous genetic algorithm for the solution of optimal control problems, Optimal Control Applications and Methods, 32 (2011), no. 4, 414432.

M.M. Ali, C. Storey, and A.Trn, Application of stochastic global optimization algorithms to practical problems, Journal of Optimization Theory and Applications, 95 (1997), no. 3, 545563 (English).

M. Senthil Arumugam, G. Ramana Murthy, and C. K. Loo, On the optimal control of the steel annealing processes as a two stage hybrid systems via PSO algorithms, International Journal Bio-Inspired Computing, 1 (2009), no. 3, 198209.

M. Senthil Arumugam and M. V. C. Rao, On the improved performances of the particle swarm optimization algorithms with adaptive parameters, cross-over operators and root mean square (RMS) variants for computing optimal control of a class of hybrid systems, Application Soft Computing, 8 (2008), no. 1, 324336.

Saman Babaie-Kafaki, Reza Ghanbari, and Nezam Mahdavi-Amiri, Two effective hybrid metaheuristic algorithms for minimization of multimodal functions, International Journal Computing Mathematics, 88 (2011), no. 11, 24152428.

Saman Babaie-Kafaki, Reza Ghanbari, and Nezam Mahdavi-Amiri, An efficient and practically robust hybrid metaheuristic algorithm for solving fuzzy bus terminal location problems, Asia-Pacific Journal of Operational Research, 29 (2012), no. 2, 125.

J.J.F. Bonnans, J.C. Gilbert, C. Lemarchal, and C.A. Sagastizbal, Numerical optimization: Theoretical and practical aspects, Springer London, Limited, 2006.

I.L. Lopez Cruz, L.G. Van Willigenburg, and G. Van Straten, Efficient differential evolution algorithms for multimodal optimal control problems, Applied Soft Computing, 3 (2003), no. 2, 97 122.

A.P. Engelbrecht, Computational intelligence: An introduction, Wiley, 2007.

Brian C. Fabien, Numerical solution of constrained optimal control problems with parameters, Applied Mathematics and Computation, 80 (1996), no. 1, 43 62.

Brian C. Fabien, Some tools for the direct solution of optimal control problems, Advances Engineering Software, 29 (1998), no. 1, 4561.

F Ghomanjani, M.H Farahi, and M Gachpazan, Bzier control points method to solve constrained quadratic optimal control of time varying linear systems , Computational and Applied Mathematics, 31 (2012), 433 456 (en).

Arnob Ghosh, Swagatam Das, Aritra Chowdhury, and Ritwik Giri, An ecologically inspired direct search method for solving optimal control problems with Bzier parameterization, Engineering Applications of Artificial Intelligence, 24 (2011), no. 7, 1195 1203.

C.J. Goh and K.L. Teo, Control parametrization: A unified approach to optimal control problems with general con- straints, Automatica 24 (1988), no. 1, 3 18.

Fernando Herrera and Jie Zhang, Optimal control of batch processes using particle swam optimisation with stacked neural network models, Computers and Chemical Engineering 33 (2009), no. 10, 1593 1601.

J. Kennedy and R. Eberhart, Particle swarm optimization, Neural Networks, 1995. Proceedings., IEEE International Conference on, vol. 4, 1995, pp. 19421948.

Donald E. Kirk, Optimal control theory: An introduction, Dover Publications, 2004.

A. Vincent Antony Kumar and P. Balasubramaniam, Optimal control for linear system using genetic programming, Optimal Control Applications and Methods, 30 (2009), no. 1, 4760.

Moo Ho Lee, Chonghun Han, and Kun Soo Chang, Dynamic optimization of a continuous polymer reactor using a modified differential evolution algorithm, Industrial and Engineering Chemistry Research, 38 (1999), no. 12, 48254831.

Wichaya Mekarapiruk and Rein Luus, Optimal control of inequality state constrained systems, Industrial and Engineering Chemistry Research, 36 (1997), no. 5, 16861694.

Hamidreza Modares and Mohammad-Bagher Naghibi-Sistani, Solving nonlinear optimal control problems using a hybrid IPSO - SQP algorithm, Engineering Applications of Artificial Intelligence, 24 (2011), no. 3, 476 484.

Debasis Sarkar and Jayant M. Modak, Optimization of fed-batch bioreactors using genetic algorithm: multiple control variables., Computers and Chemical Engineering, 28 (2009), no. 5, 789798.

X. H. Shi, L. M. Wan, H. P. Lee, X. W. Yang, L. M. Wang, and Y. C. Liang, An improved genetic algorithm with variable population-size and a PSO-GA based hybrid evolutionary algorithm, Machine Learning and Cybernetics, 2003 International Conference on, vol. 3, 2003, pp. 17351740.

Y. C. Sim, S. B. Leng, and V. Subramaniam, A combined genetic algorithms-shooting method approach to solving optimal control problems, International Journal of Systems Science, 31 (2000), no. 1, 8389.

Fan SUN, Wenli DU, Rongbin QI, Feng QIAN, and Weimin ZHONG, A hybrid improved genetic algorithm and its application in dynamic optimization problems of chemical processes, Chinese Journal of Chemical Engineering 21 (2013), no. 2, 144 154.

K.L. Teo, C.J. Goh, and K.H. Wong, A unified computational approach to optimal control problems, Pitman monographs and surveys in pure and applied mathematics, Longman Scientific and Technical, 1991.

Jelmer Marinus van Ast, Robert Babu? ska, and Bart De Schutter, Novel ant colony optimization approach to optimal control, International Journal of Intelligent Computing and Cybernetics, 2 (2009), no. 3, 414434.

Feng-Sheng Wang and Ji-Pyng Chiou, Optimal control and optimal time location problems of differential-algebraic systems by differential evolution, Industrial and Engineering Chemistry Research, 36 (1997), no. 12, 53485357.

Wen Zhang and He-ping Ma, Chebyshev-legendre method for discretizing optimal control problems, Journal of Shanghai University (English Edition), 13 (2009), no. 2, 113118.