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


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.

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

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


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


Antlion optimizer


Accelerated opposition-based ALO


Opposition-based Laplacian ALO


Genetic algorithm


Differential evolution


Particle swarm optimization


Artificial bee colony


Gravitation search algorithm


Harmony search algorithm


Biogeography-based optimization


Ant colony optimization


Cuckoo search algorithm


Bat algorithm


Sine–cosine algorithm


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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).

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