Skip to main content

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

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.

This is a preview of subscription content, access via your institution.

Abbreviations

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

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

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

References

  1. 1.

    Mirjalili, S.: SCA: a sine cosine algorithm for solving optimization problems. Knowl. Based Syst. 96, 120–133 (2016)

    Article  Google Scholar 

  2. 2.

    Boussaï, D.I.; Lepagnot, J.; Siarry, P.: A survey on optimization metaheuristics. Inf. Sci. 237, 82–117 (2013)

    MathSciNet  MATH  Article  Google 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)

    Article  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)

    Article  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)

    Article  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)

    Article  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)

    Article  Google Scholar 

  8. 8.

    Dinkar, S.K.; Deep, K.: Opposition based Laplacian antlion optimizer. J. Comput. Sci. 23, 71–90 (2017)

    MathSciNet  Article  Google 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)

    MathSciNet  MATH  Article  Google Scholar 

  11. 11.

    Kennedy, J.; Eberhart, R.: Particle swarm optimization. Proc. IEEE Int. Conf. Neural Netw. 4, 1942–1948 (1995)

    Article  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)

  13. 13.

    Mirjalili, S.; Mirjalili, S.M.; Lewis, A.: Grey wolf optimizer. Adv. Eng. Softw. 69, 46–61 (2014)

    Article  Google Scholar 

  14. 14.

    Rashedi, E.; Nezamabadi-Pour, H.; Saryazdi, S.: GSA: a gravitational search algorithm. Inf. Sci. 179(13), 2232–2248 (2009)

    MATH  Article  Google Scholar 

  15. 15.

    Formato, R.A.: Central force optimization: a new metaheuristic with applications in applied electromagnetics. Prog. Electromagn. Res. 77, 425–491 (2007)

    Article  Google Scholar 

  16. 16.

    Geem, Z.W.; Kim, J.H.; Loganathan, G.V.: A new heuristic optimization algorithm: harmony search. Simulation 76, 60 (2001)

    Article  Google Scholar 

  17. 17.

    Simon, D.: Biogeography-based optimization. IEEE Trans. Evol. Comput. 12(6), 702–713 (2008)

    Article  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)

  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)

  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)

  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)

  22. 22.

    Mirjalili, S.: The antlion optimizer. Adv. Eng. Softw. 83, 80–98 (2015)

    Article  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)

    MathSciNet  MATH  Google 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)

  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

    Article  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)

    Article  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)

    Article  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)

    Article  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)

    Article  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)

    Article  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)

    Article  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)

    Article  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)

    Article  Google Scholar 

  35. 35.

    Emary, E.; Zawbaa, H.M.; Hassanien, A.E.: Binary antlion approaches for feature selection. Neurocomputing 213, 54–65 (2016)

    Article  Google Scholar 

  36. 36.

    Zawbaa, H.M.; Emary, E.; Grosan, C.: Feature selection via chaotic antlion optimization. PLoS ONE 11(3), e0150652 (2016)

    Article  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)

  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)

    Article  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)

    Article  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)

  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)

    Article  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)

  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)

  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)

    Article  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)

    Article  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)

    MathSciNet  Article  Google 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)

    Article  Google Scholar 

  50. 50.

    Saremi, S.; Mirjalili, S.; Lewis, A.: Grasshopper optimisation algorithm: theory and application. Adv. Eng. Softw. 105, 30–47 (2017)

    Article  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)

    MathSciNet  MATH  Article  Google Scholar 

  52. 52.

    Yao, X.; Liu, Y.; Lin, G.: Evolutionary programming made faster. IEEE Trans. Evol. Comput. 3(2), 82–102 (1999)

    Article  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)

    MathSciNet  MATH  Article  Google 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)

    MathSciNet  Google Scholar 

  55. 55.

    Aoki, M.: Control of large-scale dynamic systems by aggregation. IEEE Trans. Autom. Control 13(3), 246–253 (1968)

    Article  Google Scholar 

  56. 56.

    Shamash, Y.: Stable reduced-order models using Padé-type approximations. IEEE Trans. Autom. Control 19(5), 615–616 (1974)

    MATH  Article  Google 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)

    MathSciNet  MATH  Article  Google 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)

    Article  Google Scholar 

  60. 60.

    Sharma, H.; Bansal, J.C.; Arya, K.V.: Fitness based differential evolution. Memet. Comput. 4(4), 303–316 (2012)

    Article  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)

    MathSciNet  Article  Google 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)

    MathSciNet  MATH  Article  Google Scholar 

  63. 63.

    Smamash, Y.: Truncation method of reduction: a viable alternative. Electron. Lett. 17(2), 97–99 (1981)

    MathSciNet  Article  Google Scholar 

  64. 64.

    Krishnamurthy, V.; Seshadri, V.: Model reduction using the Routh stability criterion. IEEE Trans. Autom. Control 23(4), 729–731 (1978)

    Article  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)

    MATH  Article  Google Scholar 

  66. 66.

    Singh, N.: Reduced order modeling and controller design. PhD thesis. Indian Institute of Technology Roorkee, India (2007)

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Shail Kumar Dinkar.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Dinkar, S.K., Deep, K. Accelerated Opposition-Based Antlion Optimizer with Application to Order Reduction of Linear Time-Invariant Systems. Arab J Sci Eng 44, 2213–2241 (2019). https://doi.org/10.1007/s13369-018-3370-4

Download citation

Keywords

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