粒子群优化与差分进化算法研究及其应用
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摘要
粒子群优化(PSO)与差分进化(DE)是两种基于种群的现代随机优化算法,都具有良好的优化性能。本文对PSO与DE算法进行了分析与研究,从不同角度提出了几种改进算法,并将PSO与DE算法应用于自适应滤波器与天线阵列综合。
在PSO算法研究方面,首先利用离散时间线性动力系统理论,推导了确定性PSO算法收敛、临界稳定与发散的充分必要条件。根据理论分析结果,给出了PSO算法的参数选择指导方法,讨论了随机性与粒子交互作用对算法性能的影响。然后对PSO算法的信息共享机制进行了研究,以标准PSO为原型设计了4种使用不同信息共享策略的PSO算法,测试比较了它们的性能,归纳总结了有效信息共享策略应满足的一些条件
在对PSO算法理论与信息共享机制研究的基础上,并借鉴社会学的一些思想,提出了几种改进的PSO算法。主要包括：
(1)基于分工合作的思想,设计了一种新的自适应PSO算法。新算法对不同性能粒子分配不同任务并采取相应的惯性权,粒子的加速系数根据惯性权自适应调整。
(2)基于“精英领导多数”与“分工合作”的思想,借鉴人类社会中的等级结构组织方式,构建了一种分等级的多子群PSO (HSPSO)算法。HSPSO算法将整个粒子群划分为多个子群并将其以等级结构形式组织,上层粒子由下层不同子群的精英粒子组成。HSPSO可对不同层粒子分配不同任务,较好地平衡了探索与开发能力。
(3)研究了两种利用有效信息的PSO (EIPSO)算法形式。EIPSO算法中粒子有选择地共享所有不差于它本身的优秀邻域粒子的信息,既充分利用了优秀邻域粒子的信息,又避免了较差邻域粒子的负面影响。
(4)设计了一种基于粒子群本质特征的混沌PSO算法。新算法使用混沌搜索方法代替Kennedy的随机数产生器在较好区域内进行局部搜索,混沌搜索区域半径根据粒子个体最优位置间的距离自适应调整。本文还研究了几种改进的自适应滤波算法,并将PSO算法思想应用于自适应滤波器的优化。主要内容包括：
(1)提出了一种用零阶Sugeno模糊推理系统自适应调整步长的模糊步长LMS算法,并从理论上分析了新算法的计算复杂度及其收敛性能。
(2)将变阶数自适应滤波器抽头长度与权值调整问题归结为单一的权值调整问题,研究了抽头长度一般更新公式及新的变抽头长度LMS算法,分析了新算法的合理性与收敛性。
(3)设计了一种能在不同大小噪声条件下都收敛到最优阶数的变抽头长度新算法,并将之应用于变阶数自适应格型RLS滤波器的阶数更新,讨论了格型滤波器阶数更新时相关参数的调整方法。
(4)根据PSO算法的社会心理学指导思想并结合自适应FIR滤波器的特点,设计合适的惯性项、认知项与社会项表达式更新组合自适应滤波器,提出了基于PSO算法思想的组合自适应滤波算法。仿真结果表明新算法在不同环境下都可以较好地平衡稳态失调与跟踪能力。
在DE算法研究与改进方面,首先将DE的差分变异理解为局部搜索操作,设计了一种新的差分变异策略：DE/BoR/*/*。DE/BoR/*/*每次先从种群中随机选出若干个个体,然后将其中的最优个体作为差分变异基,剩下的个体构成差分向量。这样,差分变异基可同时具有较好的质量与多样性,更好地平衡了算法的探索与开发能力。
然后对DE算法中的交叉操作进行了比较研究。为了公平地比较DE中常用的两种交叉方法,即二项式交叉与指数交叉,并研究交叉长度概率分布与交叉连续性的影响,设计了两种新的交叉方法：连续二项式交叉与非连续指数交叉。从理论上分析了文中所用二项式交叉与指数交叉方法的交叉长度概率分布与期望值,综合比较了几种使用不同交叉方法的DE算法性能。根据理论分析与仿真结果讨论了交叉对DE算法可靠性与效率的影响,加深了对交叉在DE中作用的理解。
最后,本文将DDE/BoR/1/bin与另一种新的改进PSO算法,即利用有效信息的高斯粒子群(EIGPS)算法,应用于非等间距线性天线阵列综合,最小化天线阵的峰值旁瓣电平(PSLL)。研究了入射角分辨率对PSLL计算值的影响。仿真结果表明DDE/BoR/1/bin与EIGPS都具有良好的综合能力,可以得到比一些已有文献所报道结果更小的PSLL值。
Both particle swarm optimization (PSO) and differential evolution (DE) are population-based stochastic optimization algorithms with good performance. In this thesis, PSO and DE are studied and several improved algorithms have been presented. Then, PSO and DE are applied to adaptive filter and antenna arrays synthesis.
On PSO study, the formal sufficient and necessary condition for the deterministic standard PSO to converge, diverge or oscillate within a range, is first derived by using discrete time linear dynamic system theory. Based on the theory analysis, a general guideline for parameters selection is provided and the effects of randomness and interaction between particles are discussed. Then the information sharing mechanism of PSO is studied. Four kinds of PSO algorithms with different information sharing strategies are designed and compared. Some conditions for a good information sharing strategy are summarized.
Based on the study of PSO theory and information sharing mechanism, also inspired by some effective sociological ideas, several improved PSO algorithms are proposed. These include:
(1) Inspired by the idea of specialization and cooperation, a new adaptive particle swarm optimization (APSO) algorithm is proposed. In APSO, particles with different performance are assigned with different tasks and accordingly different inertial weights. And the acceleration coefficients are adaptively adjusted according to the inertial weight.
(2) Based on the metaphor of "specialization and cooperation" and "the elites lead the majority" in hierarchical social organization, hierarchical subpopulation particle swarm optimization (HSPSO) algorithm is presented. In HSPSO, the entire population is divided into several subpopulations arranged in a hierarchy. The particles at higher level of the hierarchy are composed of the elite particles from different subpopulations of the lower level. Particles at different levels are assigned different tasks thus a good balance of exploration and exploitation can be obtained.
(3) Two kinds of effectively informed particle swarm optimization (EIPSO) algorithms are presented. In EIPSO, the particle selectively shares the information of all its neighbors better than itself. Thus it can not only make the best of the information from its outstanding neighbors but also avoid the negative influence of its inferior neighbors.
(4) A chaotic particle swarm optimization (CPSO) algorithm based on the essence of particle swarm is proposed. CPSO uses chaotic search rather than Kennedy's random number generator to search a promising region. The radius of the chaotic searching region is adaptively adjusted according to the distances between the personal best positions of particles.
Then several improved adaptive filtering algorithms are studied and the idea of PSO is applied to the optimization of adaptive filter. Those improved adaptive filtering algorithms include:
(1) A fuzzy step size least mean square (FSS-LMS) algorithm is proposed, in which the step size of LMS is adaptively adjusted by a zero-order Sugeno fuzzy inference system. The complexity and convergence performance of FSS-LMS algorithm are also analyzed.
(2) The tap-length and tap-weight adjusting problems of variable order adaptive filter are converted into a single tap-weight adjusting problem. Then a general tap-length updating formula and a new variable tap-length LMS algorithm are presented. The rationality and convergence property of the new algorithm are analyzed.
(3) A new tap-length algorithm is presented, which can converge to optimal order or tap-length under noises of different magnitudes. It is then applied to variable order adaptive lattice recursive least square (RLS) filter. The adjustment of correlative parameters when updating the lattice filter order is also discussed.
(4) Based on the social psychology idea behind PSO and the feature of adaptive FIR filter, the proper expressions for the "inertial", "cognitive" and "social" parts are designed to update the combined adaptive filter. Thus a combined adaptive filtering algorithm based on the idea of PSO is presented. The theory analysis and simulation results show that the new algorithm can well balance the steady state misadjustment and tracking ability under different environments.
On DE study, a new differential mutation strategy for DE, namely DE/BoR/*/*, is first presented by regarding differential mutation as a local search. DE/BoR/*/* uses the best one of several randomly chosen individuals as differential mutation base while the rest ones as donors for vector differences. Hence, both good diversity and quality of mutation bases are obtained, leading to better balance of exploration and exploitation.
Then a comparative study of crossover in DE is carried out. In order to fairly compare two most often implemented crossover methods in DE, namely, binomial crossover and exponential crossover, and study the effect of prcbability distribution of crossover length and crossover continuity, two new crossover methods, namely consecutive binomial crossover and non-consecutive exponential crossover, are designed. The probability distribution and expectation of crossover length for binomial and exponential crossover used in this paper are derived. Various DE algorithms with different crossover methods including mutation-only DE are comprehensively compared. Based on the theoretical analysis and simulation results, the effect of crossover on the reliability and efficiency of DE is discussed. Some insights are revealed.
Finally, DDE/BoR/1/bin and another new PSO algorithm, namely, effectively informed Gaussian particle swarm (EIGPS), are applied to unequally spaced antenna array synthesis to minimize the peak sidelobe level (PSLL). Effect of angle resolution on the PSLL has also been investigated. The simulation results show that both DDE/BoR/1/bin and EIGPS have good synthesis capacity and are able to obtain better results than those reported in some existing literatures.
在PSO算法研究方面,首先利用离散时间线性动力系统理论,推导了确定性PSO算法收敛、临界稳定与发散的充分必要条件。根据理论分析结果,给出了PSO算法的参数选择指导方法,讨论了随机性与粒子交互作用对算法性能的影响。然后对PSO算法的信息共享机制进行了研究,以标准PSO为原型设计了4种使用不同信息共享策略的PSO算法,测试比较了它们的性能,归纳总结了有效信息共享策略应满足的一些条件
在对PSO算法理论与信息共享机制研究的基础上,并借鉴社会学的一些思想,提出了几种改进的PSO算法。主要包括：
(1)基于分工合作的思想,设计了一种新的自适应PSO算法。新算法对不同性能粒子分配不同任务并采取相应的惯性权,粒子的加速系数根据惯性权自适应调整。
(2)基于“精英领导多数”与“分工合作”的思想,借鉴人类社会中的等级结构组织方式,构建了一种分等级的多子群PSO (HSPSO)算法。HSPSO算法将整个粒子群划分为多个子群并将其以等级结构形式组织,上层粒子由下层不同子群的精英粒子组成。HSPSO可对不同层粒子分配不同任务,较好地平衡了探索与开发能力。
(3)研究了两种利用有效信息的PSO (EIPSO)算法形式。EIPSO算法中粒子有选择地共享所有不差于它本身的优秀邻域粒子的信息,既充分利用了优秀邻域粒子的信息,又避免了较差邻域粒子的负面影响。
(4)设计了一种基于粒子群本质特征的混沌PSO算法。新算法使用混沌搜索方法代替Kennedy的随机数产生器在较好区域内进行局部搜索,混沌搜索区域半径根据粒子个体最优位置间的距离自适应调整。本文还研究了几种改进的自适应滤波算法,并将PSO算法思想应用于自适应滤波器的优化。主要内容包括：
(1)提出了一种用零阶Sugeno模糊推理系统自适应调整步长的模糊步长LMS算法,并从理论上分析了新算法的计算复杂度及其收敛性能。
(2)将变阶数自适应滤波器抽头长度与权值调整问题归结为单一的权值调整问题,研究了抽头长度一般更新公式及新的变抽头长度LMS算法,分析了新算法的合理性与收敛性。
(3)设计了一种能在不同大小噪声条件下都收敛到最优阶数的变抽头长度新算法,并将之应用于变阶数自适应格型RLS滤波器的阶数更新,讨论了格型滤波器阶数更新时相关参数的调整方法。
(4)根据PSO算法的社会心理学指导思想并结合自适应FIR滤波器的特点,设计合适的惯性项、认知项与社会项表达式更新组合自适应滤波器,提出了基于PSO算法思想的组合自适应滤波算法。仿真结果表明新算法在不同环境下都可以较好地平衡稳态失调与跟踪能力。
在DE算法研究与改进方面,首先将DE的差分变异理解为局部搜索操作,设计了一种新的差分变异策略：DE/BoR/*/*。DE/BoR/*/*每次先从种群中随机选出若干个个体,然后将其中的最优个体作为差分变异基,剩下的个体构成差分向量。这样,差分变异基可同时具有较好的质量与多样性,更好地平衡了算法的探索与开发能力。
然后对DE算法中的交叉操作进行了比较研究。为了公平地比较DE中常用的两种交叉方法,即二项式交叉与指数交叉,并研究交叉长度概率分布与交叉连续性的影响,设计了两种新的交叉方法：连续二项式交叉与非连续指数交叉。从理论上分析了文中所用二项式交叉与指数交叉方法的交叉长度概率分布与期望值,综合比较了几种使用不同交叉方法的DE算法性能。根据理论分析与仿真结果讨论了交叉对DE算法可靠性与效率的影响,加深了对交叉在DE中作用的理解。
最后,本文将DDE/BoR/1/bin与另一种新的改进PSO算法,即利用有效信息的高斯粒子群(EIGPS)算法,应用于非等间距线性天线阵列综合,最小化天线阵的峰值旁瓣电平(PSLL)。研究了入射角分辨率对PSLL计算值的影响。仿真结果表明DDE/BoR/1/bin与EIGPS都具有良好的综合能力,可以得到比一些已有文献所报道结果更小的PSLL值。
Both particle swarm optimization (PSO) and differential evolution (DE) are population-based stochastic optimization algorithms with good performance. In this thesis, PSO and DE are studied and several improved algorithms have been presented. Then, PSO and DE are applied to adaptive filter and antenna arrays synthesis.
On PSO study, the formal sufficient and necessary condition for the deterministic standard PSO to converge, diverge or oscillate within a range, is first derived by using discrete time linear dynamic system theory. Based on the theory analysis, a general guideline for parameters selection is provided and the effects of randomness and interaction between particles are discussed. Then the information sharing mechanism of PSO is studied. Four kinds of PSO algorithms with different information sharing strategies are designed and compared. Some conditions for a good information sharing strategy are summarized.
Based on the study of PSO theory and information sharing mechanism, also inspired by some effective sociological ideas, several improved PSO algorithms are proposed. These include:
(1) Inspired by the idea of specialization and cooperation, a new adaptive particle swarm optimization (APSO) algorithm is proposed. In APSO, particles with different performance are assigned with different tasks and accordingly different inertial weights. And the acceleration coefficients are adaptively adjusted according to the inertial weight.
(2) Based on the metaphor of "specialization and cooperation" and "the elites lead the majority" in hierarchical social organization, hierarchical subpopulation particle swarm optimization (HSPSO) algorithm is presented. In HSPSO, the entire population is divided into several subpopulations arranged in a hierarchy. The particles at higher level of the hierarchy are composed of the elite particles from different subpopulations of the lower level. Particles at different levels are assigned different tasks thus a good balance of exploration and exploitation can be obtained.
(3) Two kinds of effectively informed particle swarm optimization (EIPSO) algorithms are presented. In EIPSO, the particle selectively shares the information of all its neighbors better than itself. Thus it can not only make the best of the information from its outstanding neighbors but also avoid the negative influence of its inferior neighbors.
(4) A chaotic particle swarm optimization (CPSO) algorithm based on the essence of particle swarm is proposed. CPSO uses chaotic search rather than Kennedy's random number generator to search a promising region. The radius of the chaotic searching region is adaptively adjusted according to the distances between the personal best positions of particles.
Then several improved adaptive filtering algorithms are studied and the idea of PSO is applied to the optimization of adaptive filter. Those improved adaptive filtering algorithms include:
(1) A fuzzy step size least mean square (FSS-LMS) algorithm is proposed, in which the step size of LMS is adaptively adjusted by a zero-order Sugeno fuzzy inference system. The complexity and convergence performance of FSS-LMS algorithm are also analyzed.
(2) The tap-length and tap-weight adjusting problems of variable order adaptive filter are converted into a single tap-weight adjusting problem. Then a general tap-length updating formula and a new variable tap-length LMS algorithm are presented. The rationality and convergence property of the new algorithm are analyzed.
(3) A new tap-length algorithm is presented, which can converge to optimal order or tap-length under noises of different magnitudes. It is then applied to variable order adaptive lattice recursive least square (RLS) filter. The adjustment of correlative parameters when updating the lattice filter order is also discussed.
(4) Based on the social psychology idea behind PSO and the feature of adaptive FIR filter, the proper expressions for the "inertial", "cognitive" and "social" parts are designed to update the combined adaptive filter. Thus a combined adaptive filtering algorithm based on the idea of PSO is presented. The theory analysis and simulation results show that the new algorithm can well balance the steady state misadjustment and tracking ability under different environments.
On DE study, a new differential mutation strategy for DE, namely DE/BoR/*/*, is first presented by regarding differential mutation as a local search. DE/BoR/*/* uses the best one of several randomly chosen individuals as differential mutation base while the rest ones as donors for vector differences. Hence, both good diversity and quality of mutation bases are obtained, leading to better balance of exploration and exploitation.
Then a comparative study of crossover in DE is carried out. In order to fairly compare two most often implemented crossover methods in DE, namely, binomial crossover and exponential crossover, and study the effect of prcbability distribution of crossover length and crossover continuity, two new crossover methods, namely consecutive binomial crossover and non-consecutive exponential crossover, are designed. The probability distribution and expectation of crossover length for binomial and exponential crossover used in this paper are derived. Various DE algorithms with different crossover methods including mutation-only DE are comprehensively compared. Based on the theoretical analysis and simulation results, the effect of crossover on the reliability and efficiency of DE is discussed. Some insights are revealed.
Finally, DDE/BoR/1/bin and another new PSO algorithm, namely, effectively informed Gaussian particle swarm (EIGPS), are applied to unequally spaced antenna array synthesis to minimize the peak sidelobe level (PSLL). Effect of angle resolution on the PSLL has also been investigated. The simulation results show that both DDE/BoR/1/bin and EIGPS have good synthesis capacity and are able to obtain better results than those reported in some existing literatures.
引文
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