Advertisement

Arabian Journal for Science and Engineering

, Volume 44, Issue 3, pp 2213–2241 | Cite as

Accelerated Opposition-Based Antlion Optimizer with Application to Order Reduction of Linear Time-Invariant Systems

  • Shail Kumar DinkarEmail author
  • Kusum Deep
Research Article - Electrical Engineering

Abstract

This paper proposes a novel variant of antlion optimizer (ALO), namely accelerated opposition-based antlion optimizer (OB-ac-ALO). This modified version is conceptualized with opposition-based learning (OBL) model and integrated with acceleration coefficient (ac). The OBL model approximates the original as well as opposite candidate solutions simultaneously during evolution process. The implementation of OBL technique is collaborated with an exploitation acceleration coefficient which is useful in local searching and tends to find global optimum efficiently. A position update equation is formulated using these strategies. To validate the proposed technique, a broad set of 21 benchmark test suit of extensive variety of features is chosen. To analyse the performance of proposed algorithm, various analysis metrics such as search history, trajectories and average distance between search agents before and after improving the algorithm are performed. A nonparametric Wilcoxon ranksum test is applied to show its statistical significance. It is applied to solve a real-world application for approximating the higher-order linear time-invariant system to its corresponding lower-order invariant system. Three single-input single-output problems have been considered in terms of integral square error.

Keywords

Antlion optimizer Opposition-based learning Acceleration coefficient Order reduction Integral square error 

List of symbols

\(H_\mathrm{ant} =\left( {H_{A,1} ,H_{A,2} ,\ldots H_{A,n} ,\ldots ,H_{A,N} } \right) ^{\mathrm{T}}\)

Initial population of ant

\(H_{A,n} =\left( {H_{A,n}^1 ,\ldots H_{A,n}^d ,\ldots ,H_{A,n}^D } \right) \)

nth ant

\(H_{A,n}^d \)

dth variable of the nth ant

\(\textit{MH}_\mathrm{ant} =\left( {\textit{MH}_{A,1} ,\textit{MH}_{A,2} \ldots \textit{MH}_{A,n} ,\ldots \textit{MH}_{A,N} } \right) ^{\mathrm{T}}\)

Fitness matrix of ant

\(\textit{MH}_{A,n} = f\left( {H_{A,n}^1 ,\ldots ,H_{A,n}^d ,\ldots ,H_{A,n}^D } \right) \)

Fitness value of nth ant

\(H_\mathrm{antlion} =\left( {H_{\mathrm{AL},1} ,H_{\mathrm{AL},2} ,\ldots ,H_{\mathrm{AL},n} ,\ldots ,H_{\mathrm{AL},N} } \right) ^{\mathrm{T}}\)

Antlion population

\(H_{\mathrm{AL},n}=\left( {H_{\mathrm{AL},n}^1 ,\ldots H_{\mathrm{AL},n}^d ,\ldots H_{\mathrm{AL},n}^D } \right) \)

nth antlion

\(H_{\mathrm{AL},n}^d \)

dth variable of the nth antlion

\(\textit{MH}_\mathrm{antlion} =\left( {\textit{MH}_{\mathrm{AL},1} ,\ldots ,\textit{MH}_{\mathrm{AL},n} ,\ldots ,\textit{MH}_{\mathrm{AL},N} } \right) \)

Fitness matrix antlion

\(e\hbox {max}=1\), \(e\hbox {min}-0.00001\)

Fitness value of nth antlion

\({it}_\mathrm{curr}, {it}_\mathrm{max} \)

Current and maximum iteration

\(\hbox {LB}, \hbox {UB}\)

Lower and upper bound

\(H_\mathrm{sel} \)

Selected antlion

\(H_\mathrm{elite}\)

Elite (best) antlion

\(\hbox {Rw}_\mathrm{A} \)

Random walk around \(H_{\mathrm{sel}} \)

\(\hbox {Rw}_\mathrm{E} \)

Random walk around \(H_{\mathrm{elite}} \)

\(e_\mathrm{ac} \)

Acceleration coefficient

\(e\hbox {max}=1\), \(e\hbox {min}-0.00001\)

Max and min value of constant

G(s)

Original LTI system

s

Complex variable

n

Order of original system

\(a_p \) and \(b_p \)

Const. coefficient of s for original LTI

\(\widetilde{{G(s)}}\)

Reduced LTI system

r

Order of reduced system

\(\widetilde{a_p }\) and \({\widetilde{b_p }}\)

Const. coefficient of s for reduced LTI

\(y\left( t \right) \)

Step response of original system

\(\widetilde{y\left( t \right) }\)

Step response of reduced system

Abbreviation

ALO

Antlion optimizer

OB-ac-ALO

Accelerated opposition-based ALO

OB-L-ALO

Opposition-based Laplacian ALO

GA

Genetic algorithm

DE

Differential evolution

PSO

Particle swarm optimization

ABC

Artificial bee colony

GSA

Gravitation search algorithm

HSA

Harmony search algorithm

BBO

Biogeography-based optimization

ACO

Ant colony optimization

CS

Cuckoo search algorithm

BA

Bat algorithm

SCA

Sine–cosine algorithm

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Mirjalili, S.: SCA: a sine cosine algorithm for solving optimization problems. Knowl. Based Syst. 96, 120–133 (2016)Google Scholar
  2. 2.
    Boussaï, D.I.; Lepagnot, J.; Siarry, P.: A survey on optimization metaheuristics. Inf. Sci. 237, 82–117 (2013)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Eftimov, T.; Korošec, P.; Seljak, B.K.: A Novel Approach to statistical comparison of meta-heuristic stochastic optimization algorithms using deep statistics. Inf. Sci. 417, 186–215 (2017)Google Scholar
  4. 4.
    Mousavi-Avval, S.H.; Rafiee, S.; Sharifi, M.; Hosseinpour, S.; Notarnicola, B.; Tassielli, G.; Renzulli, P.A.: Application of multi-objective genetic algorithms for optimization of energy, economics and environmental life cycle assessment in oilseed production. J. Clean. Prod. 140, 804–815 (2017)Google Scholar
  5. 5.
    Chou, J.S.; Pham, A.D.: Nature-inspired metaheuristic optimization in least squares support vector regression for obtaining bridge scour information. Inf. Sci. 399, 64–80 (2017)Google Scholar
  6. 6.
    Ghaedi, A.M.; Ghaedi, M.; Pouranfard, A.R.; Ansari, A.; Avazzadeh, Z.; Vafaei, A.; Gupta, V.K.: Adsorption of Triamterene on multi-walled and single-walled carbon nanotubes: artificial neural network modeling and genetic algorithm optimization. J. Mol. Liq. 216, 654–665 (2016)Google Scholar
  7. 7.
    Wang, Z.; Xing, H.; Li, T.; Yang, Y.; Qu, R.; Pan, Y.: A modified ant colony optimization algorithm for network coding resource minimization. IEEE Trans. Evol. Comput. 20(3), 325–342 (2016)Google Scholar
  8. 8.
    Dinkar, S.K.; Deep, K.: Opposition based Laplacian antlion optimizer. J. Comput. Sci. 23, 71–90 (2017)MathSciNetGoogle Scholar
  9. 9.
    Holland, J.H.: Adaptation in Natural and Artificial System. The University of Michigan Press, Ann Arbor (1975)Google Scholar
  10. 10.
    Storn, R.; Price, K.: Differential evolution-a simple and efficient heuristic for global optimization over continuous spaces. J. Glob. Optim. 11(4), 341–359 (1997)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Kennedy, J.; Eberhart, R.: Particle swarm optimization. Proc. IEEE Int. Conf. Neural Netw. 4, 1942–1948 (1995)Google Scholar
  12. 12.
    Karaboga, D.: An Idea Based on Honey Bee Swarm for Numerical Optimization vol. 200. Technical Report-tr06, Erciyes University, engineering faculty, Computer Engineering Department (2005)Google Scholar
  13. 13.
    Mirjalili, S.; Mirjalili, S.M.; Lewis, A.: Grey wolf optimizer. Adv. Eng. Softw. 69, 46–61 (2014)Google Scholar
  14. 14.
    Rashedi, E.; Nezamabadi-Pour, H.; Saryazdi, S.: GSA: a gravitational search algorithm. Inf. Sci. 179(13), 2232–2248 (2009)zbMATHGoogle Scholar
  15. 15.
    Formato, R.A.: Central force optimization: a new metaheuristic with applications in applied electromagnetics. Prog. Electromagn. Res. 77, 425–491 (2007)Google Scholar
  16. 16.
    Geem, Z.W.; Kim, J.H.; Loganathan, G.V.: A new heuristic optimization algorithm: harmony search. Simulation 76, 60 (2001)Google Scholar
  17. 17.
    Simon, D.: Biogeography-based optimization. IEEE Trans. Evol. Comput. 12(6), 702–713 (2008)Google Scholar
  18. 18.
    Dorigo, M.; Di Caro, G.: Ant colony optimization: a new meta-heuristic. In: Proceedings of the 1999 Congress on Evolutionary Computation, 1999. CEC 99, vol. 2, pp. 1470–1477. IEEE (1999)Google Scholar
  19. 19.
    Yang, X. S.; Deb, S.: Cuckoo search via Lévy flights. In: World Congress on Nature & Biologically Inspired Computing, 2009. NaBIC 2009, pp. 210–214. IEEE (2009)Google Scholar
  20. 20.
    Yang, X.S.: A new metaheuristic bat-inspired algorithm. In: Nature Inspired Cooperative Strategies for Optimization (NICSO 2010), vol. 2010, pp. 65–74 (2010)Google Scholar
  21. 21.
    Wolpert, D.H.; Macready, W.G.: No Free Lunch Theorems for Search. Technical Report SFI-TR-95-02-010 (Santa Fe Institute) (1995)Google Scholar
  22. 22.
    Mirjalili, S.: The antlion optimizer. Adv. Eng. Softw. 83, 80–98 (2015)Google Scholar
  23. 23.
    Deep, K.; Thakur, M.: A new crossover operator for real coded genetic algorithms. Appl. Math. Comput. 188(1), 895–911 (2007)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Tizhoosh, H.R.: Opposition-based learning: a new scheme for machine intelligence. In: International Conference on Computational Intelligence for Modelling, Control and Automation, 2005 and International Conference on Intelligent Agents, Web Technologies and Internet Commerce, vol. 1, pp. 695–701. IEEE (2005)Google Scholar
  25. 25.
    Gupta, E.; Saxena, A.: Performance evaluation of antlion optimizer based regulator in automatic generation control of interconnected power system. J. Eng. 2016, 4570617 (2016).  https://doi.org/10.1155/2016/4570617 Google Scholar
  26. 26.
    Balachandar, P.; Ganesan, S.; Jayakumar, N.; Subramanian, S.: Multi-fuel power dispatch in an interconnected power system using antlion optimizer: multi-fuel dispatch considering tie-line limits. Int. J. Energy Optim. Eng: IJEOE 6(3), 29–54 (2017)Google Scholar
  27. 27.
    Trivedi, I.N.; Jangir, P.; Parmar, S.A.: Optimal power flow with enhancement of voltage stability and reduction of power loss using ant-lion optimizer. Cogent Eng. 3(1), 1208942 (2016)Google Scholar
  28. 28.
    Ali, E.S.; Elazim, S.A.; Abdelaziz, A.Y.: Antlion optimization algorithm for renewable distributed generations. Energy 116, 445–458 (2016)Google Scholar
  29. 29.
    Satheeshkumar, R.; Shivakumar, R.: Antlion optimization approach for load frequency control of multi-area interconnected power systems. Circuits Syst. 7(09), 2357 (2016)Google Scholar
  30. 30.
    Mouassa, S.; Bouktir, T.; Salhi, A.: Antlion optimizer for solving optimal reactive power dispatch problem in power systems. Eng. Sci. Technol. Int. J. 20(3), 885–895 (2017)Google Scholar
  31. 31.
    Sam’on, I.N.; Yasin, Z.M.; Zakaria, Z.: Antlion optimizer for solving unit commitment problem in smart grid system. Indones. J. Electr. Eng. Comput. Sci. 8(1), 129–136 (2017)Google Scholar
  32. 32.
    Ali, E.S.; Elazim, S.A.; Abdelaziz, A.Y.: Improved harmony algorithm and power loss index for optimal locations and sizing of capacitors in radial distribution systems. Int. J. Electr. Power Energy Syst. 80, 252–263 (2016)Google Scholar
  33. 33.
    Abdelaziz, A.Y.; Ali, E.S.; Elazim, S.A.: Flower pollination algorithm and loss sensitivity factors for optimal sizing and placement of capacitors in radial distribution systems. Int. J. Electr. Power Energy Syst. 78, 207–214 (2016)Google Scholar
  34. 34.
    Yao, P.; Wang, H.: Dynamic Adaptive Ant Lion Optimizer applied to route planning for unmanned aerial vehicle. Soft Comput. 21(18), 5475–5488 (2017)Google Scholar
  35. 35.
    Emary, E.; Zawbaa, H.M.; Hassanien, A.E.: Binary antlion approaches for feature selection. Neurocomputing 213, 54–65 (2016)Google Scholar
  36. 36.
    Zawbaa, H.M.; Emary, E.; Grosan, C.: Feature selection via chaotic antlion optimization. PLoS ONE 11(3), e0150652 (2016)Google Scholar
  37. 37.
    Mafarja, M.; Eleyan, D.; Abdullah, S.; Mirjalili, S.: S-shaped vs. V-shaped transfer functions for antlion optimization algorithm in feature selection problem. In: Proceedings of the International Conference on Future Networks and Distributed Systems, p. 14. ACM (2017)Google Scholar
  38. 38.
    Mirjalili, S.; Jangir, P.; Saremi, S.: Multi-objective antlion optimizer: a multi-objective optimization algorithm for solving engineering problems. Appl. Intell. 46(1), 79–95 (2017)Google Scholar
  39. 39.
    Saha, S.; Mukherjee, V.: A novel quasi-oppositional chaotic antlion optimizer for global optimization. Appl. Intell. (2017).  https://doi.org/10.1007/s10489-017-1097-7
  40. 40.
    Yogarajan, G.; Revathi, T.: Improved cluster based data gathering using antlion optimization in wireless sensor networks. Wirel. Pers. Commun. 98(3), 2711–2731 (2018)Google Scholar
  41. 41.
    Babers, R.; Ghali, N.I.; Hassanien, A.E.; Madbouly, N.M.: Optimal community detection approach based on Antlion optimization. In: 2015 11th International Computer Engineering Conference (ICENCO), pp. 284–289. IEEE. (2015)Google Scholar
  42. 42.
    Petrović, M.; Petronijević, J.; Mitić, M.; Vuković, N.; Plemić, A.; Miljković, Z.; Babić, B.: The antlion optimization algorithm for flexible process planning. J. Prod. Eng. 18(2), 65–68 (2015)Google Scholar
  43. 43.
    Tizhoosh, H.R.: Opposition-based reinforcement learning. J. Adv. Comput. Intell. Intell. Inform. 10(3), 578–585 (2006)Google Scholar
  44. 44.
    Shokri, M.; Tizhoosh, H.R.; Kamel, M.: Opposition-based Q (\(\lambda )\) algorithm. In: International Joint Conference on Neural Networks. IJCNN’06, pp. 254–261. IEEE (2006)Google Scholar
  45. 45.
    Ventresca, M.; Tizhoosh, H.R.: Improving the convergence of backpropagation by opposite transfer functions. In: International Joint Conference on Neural Networks. IJCNN’06, pp. 4777–4784. IEEE (2006)Google Scholar
  46. 46.
    Rahnamayan, S.; Tizhoosh, H.R.; Salama, M.M.: Opposition versus randomness in soft computing techniques. Appl. Soft Comput. 8(2), 906–918 (2006)Google Scholar
  47. 47.
    Rahnamayan, S.; Tizhoosh, H.R.; Salama, M.M.: Opposition-based differential evolution. IEEE Trans. Evol. Comput. 12(1), 64–79 (2008)Google Scholar
  48. 48.
    Chen, K.; Zhou, F.; Yin, L.; Wang, S.; Wang, Y.; Wan, F.: A hybrid particle swarm optimizer with sine cosine acceleration coefficients. Inf. Sci. 422, 218–241 (2018)MathSciNetGoogle Scholar
  49. 49.
    Ahandani, M.A.; Alavi-Rad, H.: Opposition-based learning in the shuffled differential evolution algorithm. Soft. Comput. 16(8), 1303–1337 (2012)Google Scholar
  50. 50.
    Saremi, S.; Mirjalili, S.; Lewis, A.: Grasshopper optimisation algorithm: theory and application. Adv. Eng. Softw. 105, 30–47 (2017)Google Scholar
  51. 51.
    Digalakis, J.G.; Margaritis, K.G.: On benchmarking functions for genetic algorithms. Int. J. Comput. Math. 77(4), 481–506 (2001)MathSciNetzbMATHGoogle Scholar
  52. 52.
    Yao, X.; Liu, Y.; Lin, G.: Evolutionary programming made faster. IEEE Trans. Evol. Comput. 3(2), 82–102 (1999)Google Scholar
  53. 53.
    Van Den Bergh, F.; Engelbrecht, A.P.: A study of particle swarm optimization particle trajectories. Inf. Sci. 176(8), 937–971 (2006)MathSciNetzbMATHGoogle Scholar
  54. 54.
    Biradar, S.; Hote, Y.V.; Saxena, S.: Reduced-order modeling of linear time invariant systems using big bang big crunch optimization and time moment matching method. Appl. Math. Model. 40(15), 7225–7244 (2006)MathSciNetGoogle Scholar
  55. 55.
    Aoki, M.: Control of large-scale dynamic systems by aggregation. IEEE Trans. Autom. Control 13(3), 246–253 (1968)Google Scholar
  56. 56.
    Shamash, Y.: Stable reduced-order models using Padé-type approximations. IEEE Trans. Autom. Control 19(5), 615–616 (1974)zbMATHGoogle Scholar
  57. 57.
    Hutton, M.; Friedland, B.: Routh approximations for reducing order of linear, time-invariant systems. IEEE Trans. Autom. Control 20(3), 329–337 (1975)MathSciNetzbMATHGoogle Scholar
  58. 58.
    Sinha, N.K.; Kuszta, B.: Modelling and Identification of Dynamic Systems. Springer, Berlin (1983)Google Scholar
  59. 59.
    Krishnamurthy, V.; Seshadri, V.: Model reduction using the Routh stability criterion. IEEE Trans. Autom. Control 23(4), 729–731 (1978)Google Scholar
  60. 60.
    Sharma, H.; Bansal, J.C.; Arya, K.V.: Fitness based differential evolution. Memet. Comput. 4(4), 303–316 (2012)Google Scholar
  61. 61.
    Sikander, A.; Prasad, R.: Linear time-invariant system reduction using a mixed methods approach. Appl. Math. Model. 39(16), 4848–4858 (2015)MathSciNetGoogle Scholar
  62. 62.
    Desai, S.R.; Prasad, R.: A novel order diminution of LTI systems using Big Bang Big Crunch optimization and Routh Approximation. Appl. Math. Model. 37(16), 8016–8028 (2013)MathSciNetzbMATHGoogle Scholar
  63. 63.
    Smamash, Y.: Truncation method of reduction: a viable alternative. Electron. Lett. 17(2), 97–99 (1981)MathSciNetGoogle Scholar
  64. 64.
    Krishnamurthy, V.; Seshadri, V.: Model reduction using the Routh stability criterion. IEEE Trans. Autom. Control 23(4), 729–731 (1978)Google Scholar
  65. 65.
    Pal, J.: An algorithmic method for the simplification of linear dynamic scalar systems. Int. J. Control 43(1), 257–269 (1986)zbMATHGoogle Scholar
  66. 66.
    Singh, N.: Reduced order modeling and controller design. PhD thesis. Indian Institute of Technology Roorkee, India (2007)Google Scholar

Copyright information

© King Fahd University of Petroleum & Minerals 2018

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology RoorkeeRoorkeeIndia

Personalised recommendations