Pareto Optimal Design Of Decoupled Sliding Mode Control Based On A New Multi-Objective Particle Swarm Optimization Algorithm

Document Type : Research Article


1 School of Mechanical and Aerospace Engineering, the George Washington University, Washington DC, USA.

2 Department of Mechanical Engineering, Sirjan University of Technology, Sirjan, Iran.


One of the most important applications of multi-objective optimization is adjusting parameters ofpractical engineering problems in order to produce a more desirable outcome. In this paper, the decoupled sliding mode control technique (DSMC) is employed to stabilize an inverted pendulum which is a classic example of inherently unstable systems. Furthermore, a new Multi-Objective Particle Swarm Optimization (MOPSO) algorithm is implemented for optimizing the DSMC parameters in order to decrease the normalized angle error of the pole and normalized distance error of the cart, simultaneously. The results of simulation are presented which consist of results with and without disturbances. The proposed Pareto front for the DSMC problem demonstrates that the Ingenious-MOPSO operates much better than other multi-objective evolutionary algorithms.


[1] J. Kennedy, R.C. Eberhart, “Particle swarm optimization,” in Proceedings of the IEEE
International Conference on Neural Networks, vol.IV, Perth, Australia, pp. 1942-1948, 1995.
[2] P.J. Angeline, “Using selection to improve particle swarm optimization,” in Proceedings of the IEEE
Congress on Evolutionary Computation,Anchorage, AK, pp. 84-89, May 1998.
[3] R. C. Eberhart, Y. Shi, “Comparison between genetic algorithms and particle swarm optimization,” in Proceedings of the IEEE Congress on Evolutionary Computation,Anchorage, AK, pp. 611-616, May 1998.
[4] H. Yoshida, K. Kawata, Y. Fukuyama, “A particle swarm optimization for reactive power and voltage
control considering voltage security assessment,” IEEE Transactions on Power Systems, vol. 15, pp.
1232-1239, 2000.
[5] X. Hu, R. C. Eberhart, “Multi-objective optimization using dynamic neighborhood particle swarm optimization,” In Proceedings of the IEEE World Congress on Computational Intelligence (CEC’02), pp. 1677-1681, 2002.
[6] J. E. Fieldsend, S. Singh, “A multi-objective algorithm based upon particle swarm optimization
and efficient data structure and turbulence,” In Workshop on Computational Intelligence, pp. 34-
44, 2002.
[7] S. Mostaghim, J. Teich, “Strategies for finding good local guides in multi- objective particle
swarm optimization (MOPSO),” In Proceedings of the IEEE Swarm Intelligence Symposium, pp. 26-
33, 2003.
[8] K. E. Parsopoulos, D. K. Tasoulis, M. N. Vrahatis, “Multi-objective optimization using parallel vector
evaluated particle swarm optimization,” Proceedings of the IASTED International Conference on Artificial Intelligence and Applications, vol. 2, pp. 823-828, 2004.
[9] P. K. Tripathi, S. Bandyopadhyay, S. K. Pal,“Multi-objective particle swarm optimization with
time variant inertia and acceleration coefficients,”Information Sciences, vol. 177, pp. 5033-5049,
[10] S. J. Tsai, T. Y. Sun, C. C. Liu, S. T. Hsieh, W. C. Wu, S. Y Chiu, “An improved multi-objective
particle swarm optimizer for multi-objective problems,” Expert Systems with Applications, vol.
37, pp. 5872-5886, 2010.
[11] Y. Wang, Y. Yang, “Particle swarm optimization with preference order ranking for multi-objective
optimization,” Information Sciences, vol. 179, pp.1944-1959, 2009.
[12] S. Mostaghim, J. Teich, “The role of ε-dominance in multi objective particle swarm optimization
methods,” Congress on Evolutionary Computation,vol. 3, pp. 1764-1771, 2003.
[13] A. G. Hernandez-Diaz, L. V. Santana-Quintero, C.A. Coello Coello, J. Molina, R. Caballero,
“Improving the efficiency of ε-dominance based grids, Information Sciences,” vol. 181, no. 15, pp.
3101-3129, 2011.
[14] K. Atashkari, N. Nariman-Zadeh, M. Golcu, A. Khalkhali, A. Jamali, “Modelling and multi-objective optimization of a variable valve-timing spark-ignition engine using polynomial neural networks and evolutionary algorithms,” Energy Conversion and Management, vol. 48, pp. 1029-1041, 2007.
[15] M. J. Mahmoodabadi, A. Bagheri, S. Arabani Mostaghim, M. Bisheban, “Simulation of stability using Java application for Pareto design of controllers based on a new multi-objective particle swarm optimization,” Mathematical and Computer Modelling, vol. 54, no. 5-6, pp. 1584-1607, 2011.
[16] M. J. Mahmoodabadi, M. Taherkhorsandi, A. Bagheri, “Optimal robust sliding mode tracking control of a biped robot based on ingenious multi-objective PSO,” Neurocomputing, vol. 124, pp. 194–209.
[17] P. Fleming, R. Purshouse, “Evolutionary algorithms in control systems engineering: a survey,” Control Engineering Practice, vol. 10, no. 11, pp. 1223-1241, 2002.
[18] C. Fonseca, P. Fleming, “Multi-objective optimal controller design with genetic algorithms,” International Conference on Control, vol. 1, March, pp. 745-749, 1994.
[19] G. Sanchez, M. Villasana, M. Strefezza, “Multi-objective pole placement with evolutionary algorithms,” Lecture Notes in Computer Science, vol. 4403, pp. 417, 2007.
[20] W. Qiao, G. Venayagamoorthy, R. Harley, “Design of optimal PI controllers for doubly fed induction generators driven by wind turbines using particle swarm optimization,” International Joint Conference on Neural Networks, pp. 1982-1987, 2006.
[21] Z. L. Gaing, “A particle swarm optimization approach for optimum design of PID controller in AVR system,” IEEE Transaction on Energy Conversion, vol. 19, no. 2, pp. 384-391, 2004.
[22] K. C. Ng, Y. Li, D. Munay-Smith, K.C. Shaman, “Genetic algorithm applied to fuzzy sliding mode controller design,” Genetic Algorithms in Engineering Systems, Innovations and Applications 12-14, Conference Publication No. 414, IEE, pp. 220-225, 1995.
[23] C. C Wong, S.Y. Chang, “Parameter selection in the sliding mode control design using genetic algorithms,” Tamkang Journal of Science and Engineering vol. 1 no. 2, pp. 115-122, 1998.
[24] C. C. Kung, T. H. Chen, L. H. Kung, “Modified adaptive fuzzy sliding mode controller for uncertain nonlinear systems,” IEICE Transaction on Fundamentals of Electronics, Communications
and Computer Sciences, vol. 88, no. 5, pp. 1328-1334, 2005.
[25] N. Yagiz, Y. Hacioglu, “Robust control of a spatial robot using fuzzy sliding modes,” Mathematical and Computer Modeling vol. 49, no. 1-2, pp. 114-127, 2009.
[26] C. Zhi-mei, M. Wen-jun, Z. Jing-gang, Z. Jian-chao, “Scheme of sliding mode control based on modified particle swarm optimization,” Systems Engineering-Theory & Practice, vol. 29, no. 5, pp. 137-141, 2009.
[27] E. Alfaro-Cid, E. W. McGookina, D. J. Murray-Smith, T. I. Fossen, “Genetic algorithms optimization of decoupled Sliding Mode controllers: simulated and real results,” Control Engineering Practice, vol. 13, pp. 739-748, 2005.
[28] M. J. Mahmoodabadi, A. Bagheri, N. Nariman-zadeh, A. Jamali, R. Abedzadeh Maafi, “Pareto design of decoupled sliding-mode controllers for nonlinear systems based on a multiobjective genetic algorithm,” Journal of Applied Mathematics pp. 22, 2012.
[29] M. J. Mahmoodabadi, S. Arabani Mostaghim, A. Bagheri, N. Nariman-zadeh, “Pareto optimal design of the decoupled sliding mode controller for an inverted pendulum system and its stability simulation via Java programming,” Mathematical and Computer Modelling, vol. 57, pp. 1070–1082, 2013.
[30] A. P. Engelbrecht, Fundamentals of Computational Swarm Intelligence, John Wiley & Sons, 2005.
[31] R. C. Eberhart, J. Kennedy, “A new optimizer using particle swarm theory,” Proceedings of the Sixth International Symposium on Micro Machine and Human Science, Nagoya, Japan, pp. 39-43, 1995.
[32] A. Ratnaweera, S. K. Halgamuge, “Self-organizing hierarchical particle swarm optimizer with time-varying acceleration coefficient,” IEEE Transactions on Evolutionary Computation, vol. 8, no. 3, pp. 240-255, 2004.
[33] C. A. Coello Coello, D. A. Van Veldhuizen, G. B. Lamont, In: Evolutionary Algorithms for Solving Multi-Objective Problems, Kluwer Academic, Dordrecht, 2002.
[34] A. Jamali, A. Hajiloo, N. Nariman-zadeh, “Reliability-based robust Pareto design of linear state feedback controllers using a multi-objective uniform- diversity genetic algorithm,” (MUGA), Expert Systems with Applications, vol. 37, no. 1, pp. 401–413, 2010.
[35] W. S. Lin, C. S. Chen, “Robust adaptive sliding mode control using fuzzy modeling for a class of uncertain MIMO nonlinear systems,” ControlTheory and Applications, IEE, vol. 149, no. 3, pp. 193-201, 2002.
[36] J. Jing, Q.H. Wuan, “Intelligent sliding mode control algorithm for position tracking servo system,” International Journal of Information Technology, vol. 12, no. 7, pp. 57-62, 2006.
[37] Utkin, Sliding Modes and Their Application in Variable Structure Systems, Central Books Ltd, 1978
[38] M. Dotoli, P. Lino, B. Turchiano, “A decoupled fuzzy sliding mode approach to swing-Up and stabilize an inverted pendulum,” The CSD03, The 2nd IFAC Conference on Control Systems Design, Bratislava, Slovak Republic, pp. 113-120, September 2003.
[39] R. Toscana, “A simple robust PI/PID controller design via numerical optimization approach,” Journal of Process Control, vol. 15, pp. 81–88, 2005.
[40] W.A. Wolovich, Automatic Control Systems, Harcourt Brace College Publication Orlando, Saunders College Publishing, USA, 1994.