Abstract
Optimal design of controllers without considering uncertainty in the plant dynamics can induce feedback instabilities and lead to obtaining infeasible controllers in practice. This paper presents a multi-objective evolutionary algorithm integrated with Monte Carlo simulations (MCS) to perform the optimal stochastic design of robust controllers for uncertain time-delay systems. Each potential optimal solution represents a controller in the form of a transfer function with the optimal numerator and denominator polynomials. The proposed methodology uses genetic programming to evolve robust controllers. Using GP enables the algorithm to optimize the structure of the controller and tune the parameters in a holistic approach. The proposed methodology employs MCS to apply robust optimization and uses a new adaptive operator to balance exploration and exploitation in the search space. The performance of controllers is assessed in the closed-loop system with respect to three objective functions as (1) minimization of mean integral time absolute error (ITAE), (2) minimization of the standard deviation of ITAE and (3) minimization of maximum control effort. The new methodology is applied to the first-order and second-order systems with dead time. We evaluate the performance of obtained robust controllers with respect to the upper and lower bounds of step responses and control variables. We also perform a post-processing analysis considering load disturbance and external noise; we illustrate the robustness of the designed controllers by cumulative distribution functions of objective functions for different uncertainty levels. We show how the proposed methodology outperforms the state-of-the-art methods in the literature.
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References
Assimi H, Jamali A, Nariman-zadeh N (2017) Sizing and topology optimization of truss structures using genetic programming. Swarm Evolution Comput 37:90–103
Awad NH, Ali MZ, Mallipeddi R, Suganthan PN (2019) An efficient differential evolution algorithm for stochastic OPF based active–reactive power dispatch problem considering renewable generators. Appl Soft Comput 76:445–458
Balandina GI (2017) Control system synthesis by means of cartesian genetic programming. Proc Computer Sci 103:176–182
Bi S, Deng Z, Chen Z (2013) Stochastic validation of structural FE-models based on hierarchical cluster analysis and advanced Monte Carlo simulation. Finite Elem Anal Des 67:22–33
Biswas PP, Suganthan PN, Mallipeddi R, Amaratunga GAJ (2019) Optimal reactive power dispatch with uncertainties in load demand and renewable energy sources adopting scenario-based approach. Appl Soft Comput 75:616–632
Biwer A, Griffith S, Cooney C (2005) Uncertainty analysis of penicillin V production using Monte Carlo simulation. Biotechnol Bioeng 90(2):167–179
Blasco X, Herrero JM, Sanchis J, Martínez M (2008) A new graphical visualization of n-dimensional Pareto front for decision-making in multiobjective optimization. Inf Sci 178(20):3908–3924
Bodla KK, Murthy JY, Garimella SV (2013) Optimization under uncertainty applied to heat sink design. ASME J Heat Transf, vol 135
Chiou S-W (2018) A data-driven bi-level program for knowledge-based signal control system under uncertainty. Knowl-Based Syst 160:210–227
Darbra RM, Eljarrat E, Barceló D (2008) How to measure uncertainties in environmental risk assessment. TrAC Trends Anal Chem 27(4):377–385
Diwekar UM, Kalagnanam JR (1997) Efficient sampling technique for optimization under uncertainty. AIChE J 43(2):440–447
Eldred M (2009) Recent advances in non-intrusive polynomial chaos and stochastic collocation methods for uncertainty analysis and design
Field RV, Voulgaris PG, Bergman LA (1996) Methods to compute probabilistic measures of robustness for structural systems. J Vibration Control 2(4):447–463
Fukunaga A, Hiruma H, Komiya K, Iba H (2012) Evolving controllers for high-level applications on a service robot: a case study with exhibition visitor flow control. Genet Program Evolvable Mach 13:239–263
Gholaminezhad I, Jamali A (2016) A multi-objective differential evolution approach based on ε-elimination uniform-diversity for mechanisms design. Struct Multidisciplinary Optim 52(5):861–877
Gholaminezhad I, Jamali A, Assimi H (2014) Automated synthesis of optimal controller using multi-objective genetic programming for two-mass-spring system. In: presented at the 2nd RSI/ISM international conference on robotics and mechatronics, ICRoM 2014
Gomes FM, Pereira FM, Silva AF, Silva MB (2019) Multiple response optimization: analysis of genetic programming for symbolic regression and assessment of desirability functions. Knowl-Based Syst 179:21–33
Hajiloo A, Nariman-Zadeh N, Jamali A, Bagheri A, Alasti A (2008) Pareto optimum design of robust PI controllers for systems with parametric uncertainty. Int Rev Mech Eng 1(6):628–640
Hu N, Zhong J, Zhou JT, Zhou S, Cai W, Monterola C (2018) Guide them through: an automatic crowd control framework using multi-objective genetic programming. Appl Soft Comput 66:90–103
Jamali A (2009) Pareto Robust design of controllers with probabilistic uncertainties using multi objective evolutionary algorithms. Ph.D. Thesis, University of Guilan
Jamali A, Nariman-Zadeh N, Atashkari K (2008) Multi-objective uniform diversity genetic algorithm (MUGA). In: Kosinski W (ed) In advanced in evolutionary algorithms. IN-TECH, Vienna
Jamali A, Hajiloo A, Nariman-zadeh N (2010) Reliability-based robust Pareto design of linear state feedback controllers using a multi-objective uniform-diversity genetic algorithm (MUGA). Expert Syst Appl 37(1):401–413
Jamali A, Ghamati M, Ahmadi B, Nariman-zadeh N (2013a) Probability of failure for uncertain control systems using neural networks and multi-objective uniform-diversity genetic algorithms (MUGA). Eng Appl Artif Intell 26(2):714–723
Jamali A, Salehpour M, Nariman-zadeh N (2013b) Robust Pareto active suspension design for vehicle vibration model with probabilistic uncertain parameters. Multi-body Syst Dyn 30:265–285
Jamali A, Khaleghi E, Gholaminezhad I, Nariman-zadeh N, Gholaminia B, Jamal-Omidi A (2014a) Multi-objective genetic programming approach for robust modeling of complex manufacturing processes having probabilistic uncertainty in experimental data. J Intell Manuf 28(1):149–163
Jamali A, Khaleghi E, Gholaminezhad I, Nariman-zadeh N (2014b) Modelling and prediction of complex non-linear processes by using Pareto multi-objective genetic programming. Int J Syst Sci 47(7):1675–1688
Kadlic B, Sekaj I, Pernecký D (2014) Design of continuous-time controllers using cartesian genetic programming. IFAC Proc Vol 47(3):6982–6987
Kalat AA (2019) A robust direct adaptive fuzzy control for a class of uncertain nonlinear MIMO systems. Soft Comput 23(19):9747–9759
Kalos MH, Whitlock PA (1998) Monte Carlo methods. Wiley, New York
Kang Z (2005) Robust design optimization of structures under uncertainties. Institut fur Statik und Dynamik der Luft- und Raumfahrkonstruktionen, Universit¨at Stuttgart
Koza JR, Keane MA, Streeter MJ, Mydlowec W, Yu J, Lanaz G (2003) Genetic Programming IV: routine human-competitive machine intelligence. Kluwer Academic Publishers, Berlin
Krishnan K, Karpagam G (2013) Comparison of PID controller tuning techniques for a FOPDT system. Int J Current Eng Technol 4:2667–2670
Kumaresan N, Ratnavelu K (2014) Optimal control for stochastic linear quadratic singular neuro Takagi-Sugeno fuzzy system with singular cost using genetic programming. Appl Soft Comput 24:1136–1144
Li HS, Ma C (2013) Hybrid dimension-reduction method for robust design optimization. AIAA J 51:138–144
Mallipeddi R, Suganthan PN, Pan QK, Tasgetiren MF (2011) Differential evolution algorithm with ensemble of parameters and mutation strategies. Appl Soft Comput 11(2):1679–1696
Mohammadzadeh A, Taghavifar H (2020) A robust fuzzy control approach for path-following control of autonomous vehicles. Soft Comput 24(5):3223–3235
Nariman-Zadeh N, Jamali A, Hajiloo A (2007) Frequency-based reliability Pareto optimum design of proportional-integral-derivative controllers for systems with probabilistic uncertainty. Proc Inst Mech Eng Part I J Syst Control Eng 221:1061–1075
Nariman-Zadeh N, Salehpour M, Jamali A, Haghgoo E (2010) Pareto optimization of a five-degree of freedom vehicle vibration model using a multi-objective uniform-diversity genetic algorithm (MUGA). Eng Appl Artif Intell 23(4):543–551
Nejlaoui M, Houidi A, Affi Z, Romdhane L (2013) Multiobjective robust design optimization of rail vehicle moving in short radius curved tracks based on the safety and comfort criteria. Simul Model Pract Theory 30:21–34
Pettersson MP, Iaccarino G, Nordstrom J (2015) Polynomial chaos methods for hyperbolic partial differential equations. In: Mathematical engineering, Springer
Pettersson MP, Iaccarino G, Nordstrom J (2015b) Polynomial chaos methods for hyperbolic partial differential equations. Springer, Berlin
Sekaj I, Perkacz J (2007) Genetic programming—based controller design. In: 2007 IEEE congress on evolutionary computation, pp 1339–1343
Smith BA, Kenny SP, Crespo LG (2005) Probabilistic parameter uncertainty analysis of single input single output control systems. NASA
Toscano R (2005) A simple robust PI/PID controller design via numerical optimization approach. J Process Control 15(1):81–88
Uyeh DD et al (2018) Interactive livestock feed ration optimization using evolutionary algorithms. Comput Electron Agric 155:1–11
Wang Q, Stengel RF (2002) Robust control of nonlinear systems with parametric uncertainty. Automatica 38(9):1591–1599
Witteveen J, Iaccarino G (2012) Simplex stochastic collocation with random sampling and extrapolation for nonhypercube probability spaces. SIAM J Sci Comput 34(2):A814–A838
Zhao Q, Chen X, Ma ZD, Lin Y (2015) Robust topology optimization based on stochastic collocation methods under loading uncertainties. Math Probl Eng
Zio E, Bazzo R (2011) Level diagrams analysis of Pareto Front for multiobjective system redundancy allocation. Reliab Eng Syst Saf 96(5):569–580
Acknowledgement
This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology under the Grant NRF2015R1C1A1A01055669.
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Rammohan Mallipeddi has received research grants from the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology under the Grant NRF2015R1C1A1A01055669. Iman Gholaminezhad declares that he has no conflict of interest. Mohammad S. Saeedi declares that he has no conflict of interest. Hirad Assimi declares that he has no conflict of interest. Ali Jamali declares that he has no conflict of interest.
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Mallipeddi, R., Gholaminezhad, I., Saeedi, M.S. et al. Robust controller design for systems with probabilistic uncertain parameters using multi-objective genetic programming. Soft Comput 25, 233–249 (2021). https://doi.org/10.1007/s00500-020-05133-x
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DOI: https://doi.org/10.1007/s00500-020-05133-x