Fractional Differential Equations

Swarm intelligence (SI) is briefly defined as the collective behaviour of decentralized and self-organized swarms. The well known examples for these swarms are bird flocks, fish schools and the colony of social insects such as termites, ants and bees. In 1990s, especially two approaches based on ant colony and on fish schooling/bird flocking introduced have highly attracted the interest of researchers. Although the self-organization features are required by SI are strongly and clearly seen in honey bee colonies, unfortunately the researchers have recently started to be interested in the behaviour of these swarm systems to describe new intelligent approaches, especially from the beginning of 2000s. During a decade, several algorithms have been developed depending on different intelligent behaviours of honey bee swarms. Among those, artificial bee colony (ABC) is the one which has been most widely studied on and applied to solve the real world problems, so far. Day by day the number of researchers being interested in ABC algorithm increases rapidly. This work presents a comprehensive survey of the advances with ABC and its applications. It is hoped that this survey would be very beneficial for the researchers studying on SI, particularly ABC algorithm.

A comprehensive survey: artificial bee colony (ABC) algorithm and applications

Differential Evolution (DE) is arguably one of the most powerful and versatile evolutionary optimizers for the continuous parameter spaces in recent times. Almost 5 years have passed since the first comprehensive survey article was published on DE by Das and Suganthan in 2011. Several developments have been reported on various aspects of the algorithm in these 5 years and the research on and with DE have now reached an impressive state. Considering the huge progress of research with DE and its applications in diverse domains of science and technology, we find that it is a high time to provide a critical review of the latest literatures published and also to point out some important future avenues of research. The purpose of this paper is to summarize and organize the information on these current developments on DE. Beginning with a comprehensive foundation of the basic DE family of algorithms, we proceed through the recent proposals on parameter adaptation of DE, DE-based single-objective global optimizers, DE adopted for various optimization scenarios including constrained, large-scale, multi-objective, multi-modal and dynamic optimization, hybridization of DE with other optimizers, and also the multi-faceted literature on applications of DE. The paper also presents a dozen of interesting open problems and future research issues on DE.

Recent advances in differential evolution – An updated survey

Particle swarm optimization (PSO) is a population-based stochastic optimization algorithm motivated by intelligent collective behavior of some animals such as flocks of birds or schools of fish. Since presented in 1995, it has experienced a multitude of enhancements. As researchers have learned about the technique, they derived new versions aiming to different demands, developed new applications in a host of areas, published theoretical studies of the effects of the various parameters and proposed many variants of the algorithm. This paper introduces its origin and background and carries out the theory analysis of the PSO. Then, we analyze its present situation of research and application in algorithm structure, parameter selection, topology structure, discrete PSO algorithm and parallel PSO algorithm, multi-objective optimization PSO and its engineering applications. Finally, the existing problems are analyzed and future research directions are presented.

Particle swarm optimization algorithm: an overview

In nature, the eastern North American monarch population is known for its southward migration during the late summer/autumn from the northern USA and southern Canada to Mexico, covering thousands of miles. By simplifying and idealizing the migration of monarch butterflies, a new kind of nature-inspired metaheuristic algorithm, called monarch butterfly optimization (MBO), a first of its kind, is proposed in this paper. In MBO, all the monarch butterfly individuals are located in two distinct lands, viz. southern Canada and the northern USA (Land 1) and Mexico (Land 2). Accordingly, the positions of the monarch butterflies are updated in two ways. Firstly, the offsprings are generated (position updating) by migration operator, which can be adjusted by the migration ratio. It is followed by tuning the positions for other butterflies by means of butterfly adjusting operator. In order to keep the population unchanged and minimize fitness evaluations, the sum of the newly generated butterflies in these two ways remains equal to the original population. In order to demonstrate the superior performance of the MBO algorithm, a comparative study with five other metaheuristic algorithms through thirty-eight benchmark problems is carried out. The results clearly exhibit the capability of the MBO method toward finding the enhanced function values on most of the benchmark problems with respect to the other five algorithms. Note that the source codes of the proposed MBO algorithm are publicly available at GitHub (
 https://github.com/ggw0122/Monarch-Butterfly-Optimization
 , C++/MATLAB) and MATLAB Central (
 http://www.mathworks.com/matlabcentral/fileexchange/50828-monarch-butterfly-optimization
 , MATLAB).

Monarch butterfly optimization

Fractional-order PID (FOPID) controller is a generalization of standard PID controller using fractional calculus. Compared to PID controller, the tuning of FOPID is more complex and remains a challenge problem. This paper focuses on the design of FOPID controller using chaotic ant swarm (CAS) optimization method. The tuning of FOPID controller is formulated as a nonlinear optimization problem, in which the objective function is composed of overshoot, steady-state error, raising time and settling time. CAS algorithm, a newly developed evolutionary algorithm inspired by the chaotic behavior of individual ant and the self-organization of ant swarm, is used as the optimizer to search the best parameters of FOPID controller. The designed CAS-FOPID controller is applied to an automatic regulator voltage (AVR) system. Numerous numerical simulations and comparisons with other FOPID/PID controllers show that the CAS-FOPID controller can not only ensure good control performance with respect to reference input but also improve the system robustness with respect to model uncertainties.

Optimum design of fractional order PIλDµ controller for AVR system using chaotic ant swarm

Graphical abstractA new particle swarm optimization algorithm based on the Bayesian techniques(BPSO) is proposed. Fig. 1 is the comparisons between different inertia weight strategies for f5 on 10 dimensions. Fig. 2 is comparisons between different PSO methods for f5 on 10 dimensions. Parameter s is the interval of the adjacent two inertia weight change in all iterations. As shown in Fig. 3, different values of s affect the convergence rate in the test function. Fig. 4 is the change of ω in the iterations. Display Omitted HighlightsWhy BPSO can achieve the excellent balance between exploration and exploitation in optimization processing is explained.To overcome the defect of ordinary PSO, a new algorithm with adaptive inertia weight based on Bayesian techniques is proposed.Analysis of parameters s and ω in the BPSO. Particle swarm optimization is a stochastic population-based algorithm based on social interaction of bird flocking or fish schooling. In this paper, a new adaptive inertia weight adjusting approach is proposed based on Bayesian techniques in PSO, which is used to set up a sound tradeoff between the exploration and exploitation characteristics. It applies the Bayesian techniques to enhance the PSO's searching ability in the exploitation of past particle positions and uses the cauchy mutation for exploring the better solution. A suite of benchmark functions are employed to test the performance of the proposed method. The results demonstrate that the new method exhibits higher accuracy and faster convergence rate than other inertia weight adjusting methods in multimodal and unimodal functions. Furthermore, to show the generalization ability of BPSO method, it is compared with other types of improved PSO algorithms, which also performs well.

A new particle swarm optimization algorithm with adaptive inertia weight based on Bayesian techniques

Fractional order sliding mode controller design for antilock braking systems

The knowledge about time delays and parameters is very important for control and synchronization of time-delay chaotic system. In this paper, parameter estimation for time-delay chaotic system is given by treating the time delay as an additional parameter. The parameter estimation is converted to an optimization problem, which finds a best parameter combination such that an objective function is minimized. Particle swarm optimization (PSO) is used to optimize the objective function through particles’ cooperation and evolution. Two illustrative examples are given to show the validity of the proposed method.

Parameter estimation for time-delay chaotic system by particle swarm optimization

Graphical abstractDisplay Omitted HighlightsA new cooperative learning strategy is hybridized with DMS-PSO.Information can be exchanged among sub-swarms before the regrouping process.Experimental results show that DMS-PSO-CLS has a superior performance. In this article, the dynamic multi-swarm particle swarm optimizer (DMS-PSO) and a new cooperative learning strategy (CLS) are hybridized to obtain DMS-PSO-CLS. DMS-PSO is a recently developed multi-swarm optimization algorithm and has strong exploration ability for the use of a novel randomly regrouping schedule. However, the frequently regrouping operation of DMS-PSO results in the deficiency of the exploitation ability. In order to achieve a good balance between the exploration and exploitation abilities, the cooperative learning strategy is hybridized to DMS-PSO, which makes information be used more effectively to generate better quality solutions. In the proposed strategy, for each sub-swarm, each dimension of the two worst particles learns from the better particle of two randomly selected sub-swarms using tournament selection strategy, so that particles can have more excellent exemplars to learn and can find the global optimum more easily. Experiments are conducted on some well-known benchmarks and the results show that DMS-PSO-CLS has a superior performance in comparison with DMS-PSO and several other popular PSO variants.

Yinggan Tang

Papers

Optimum design of fractional order PIλDµ controller for AVR system using chaotic ant swarm

A new particle swarm optimization algorithm with adaptive inertia weight based on Bayesian techniques

Fractional order sliding mode controller design for antilock braking systems

Parameter estimation for time-delay chaotic system by particle swarm optimization

Dynamic multi-swarm particle swarm optimizer with cooperative learning strategy