群体智能算法及其在数字滤波器优化设计中的研究
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摘要
群体智能是指无智能的主体通过合作表现出智能行为特性的系统,群体智能在没有集中控制并且不提供全局模型的前提下,为寻找复杂问题的解决方案提供了基础。群体智能算法是通过模拟社会性生物群体的群体行为,对给定的目标进行寻优的启发式搜索算法,其寻优过程体现了随机、并行和分布式等特点。群体智能算法的典型代表是模拟了鸟类群体行为的粒子群优化(Particle Swarm Optimization,PSO)算法,由Kennedy和Eberhart于1995年提出。PSO算法自提出以来,由于其计算简单、易于实现、控制参数少等特点,引起了国内外相关领域众多学者的关注和研究。具有量子行为的粒子群优化(Quantum-behaved Particle Swarm Optimization,QPSO)算法是在深入研究PSO算法单个粒子收敛行为的基础上,受量子物理学的启发而提出,QPSO算法具有控制参数更少,收敛速度快,全局搜索能力强等特点。
本文以PSO算法与QPSO算法的理论分析及改进方法研究为重点,系统的研究了QPSO算法及其改进算法在数字滤波器优化设计中的应用方法,具体内容如下:
(1)阐述了群体智能优化算法及数字滤波器优化设计的研究背景;介绍了两种典型的群体智能优化算法的研究现状,即蚁群优化算法与PSO算法;对多种不同类型的数字滤波器的优化设计方法作了详细介绍;提出了本课题的研究思路与方法。
(2)通过代数方法分析了PSO算法的收敛性,得出了PSO算法的收敛条件与发散条件,并通过仿真实验验证了分析结论的正确性;然后针对PSO算法在多峰优化问题中易于出现早熟收敛的现象,根据分析的结论提出了基于群体多样性信息控制的PSO算法,算法通过判断群体的多样性来设定群体的搜索状态,即当群体多样性超出设定的上限时,立刻将群体的状态更改为收缩状态,当群体多样性低于设定的下限时,立刻将群体的状态更改为发散状态,群体通过不断的收缩、发散操作完成寻优过程:对多个不同特征的标准测试函数的求解结果显示了算法在多峰优化问题中具有较强的优化能力。
(3)给出了QPSO算法的设计思路。分析了随机算法收敛的两个判断准则,即全局搜索算法的收敛准则与局部搜索算法的收敛准则,利用这两个收敛准则作为依据,证明了QPSO算法是一个全局搜索的随机算法,这为进一步研究QPSO算法的理论问题提供了基础。
(4)算法参数是影响算法性能和效率的关键,文中对QPSO算法中除群体规模和迭代次数外的唯一参数(扩张-压缩因子)的取值方式作了系统的研究,提出了该参数的四种控制策略,即固定取值策略,线性取值策略,非线性取值策略与自适应控制取值策略,通过对标准测试函数的求解分别研究了这四种控制策略,得出了具有指导意义的结论。
(5)针对QPSO算法在解决多峰优化问题中也可能出现局部收敛的现象,分析了出现局部收敛的主要原因在于群体多样性较低而使得群体失去了在大范围内进行搜索的能力,基于两种群体多样性的度量方式,提出了采用全局最优点变异策略对群体进行操作以避免群体的多样性过小,从而提高算法的全局搜索能力,通过对标准测试函数的求解结果表明改进算法的全局求解能力得到了提升。
(6)提出了基于QPSO算法与其他进化算法的混合算法以进一步提高QPSO算法的搜索效率及优化性能。第一种方式是采用进化算法中的变异操作与OPSO算法混合,研究了多种类型的变异操作算子对QPSO算法优化性能的影响,变异操作能够增加群体的多样性,使得算法具有突跳的能力,进入新的搜索区域;第二种方式是在OPSO算法中将交叉操作引入,使得粒子可以不按照算法本身的方式产生新的位置,从而增加群体的多样性,提高算法跳出聚集区域的能力。混合的QPSO算法通过在标准测试函数中的求解显示了较好的优化能力。
(7)分析了不同类型数字滤波器的数学模型及优化设计方法,包括FIR数字滤波器、IIR数字滤波器、自适应IIR数字滤波器与二维IIR数字滤波器。根据数字滤波器优化设计的实质,即全局优化问题,提出了通过QPSO算法及其相应改进算法来完成数字滤波器优化设计模型的求解;对应不同的滤波器类型利用多个设计实例来验证OPSO算法及改进算法的性能与设计效果;通过与其它优化算法的设计结果相比较表明OPSO算法及改进算法能够在各类型数字滤波器的设计中取得更好的设计效果。
论文最后对所做工作与主要研究成果进行了总结,并提出了进一步的研究方向。
Swarm Intelligence (SI) is the property of a system whereby the collective behaviors of agents interacting locally with their environment. SI provides a basis with which it is possible to explore collective problem solving without centralized control or the provision of a global model. SI algorithm is a kind of heuristic search method that can solve the specified problems by simulating the collective behaviors. The characteristic of SI algorithm is stochastic, parallel and distributed. Particle Swarm Optimization (PSO) algorithm, which inspired by social behavior of bird flocking or fish schooling, is one of the typical SI algorithms and it is developed by Eberhart and Kennedy in 1995. Since PSO algorithm was developed, it has attracted many researchers in the fields concerned as its characteristics of simple computation, easy realization, and few parameters. Based on the deep study of PSO algorithm and inspired by quantum physics, Quantum-behaved Particle Swarm Optimization (QPSO) algorithm is proposed by us. QPSO algorithm has much fewer parameters and much stronger global search ability than the PSO algorithm.
Theoretical analyses and algorithm improving on PSO algorithm and QPSO algorithm are mainly discussed in our work and the application of QPSO algorithm and its improvements in the optimal design of digital filters are also studied in the work. The main contents of this dissertation are as follows:
(1) Research background of SI algorithms and optimal design of digital filters are expatiated. The current research situations of two typical SI algorithms are detailed introduced, which are Ant Colony Optimization (ACO) algorithm and PSO algorithm. An introduction of optimal design of different types of digital filters is presented. Research methods and ideas in the work are proposed.
(2) Convergence of PSO algorithm is analyzed by algebraic method and then the conditions of convergence and repulse for PSO algorithm are reached. The conditions are shown to be true by the simulation experiments. According to the conditions a diversity-controlled PSO algorithm which is called DCPSO is proposed in order to solve the problem of premature convergence in PSO algorithm. In DCPSO algorithm, guided by the controlled swarm's diversity, the particles search in the attractive phase sufficiently and adjust them by moving away from the center of the swarm quickly in the repulsive phase once the diversity measure reach to a low bound. The attractive-repulsive procedure can guarantee the swarm search in a wide space and help to avoid trapping into the local minima. Experimental results on several well-known benchmark functions show that DCPSO has strong global optimization ability in solving the multimodal problems.
(3) The thought of QPSO algorithm is discussed. Convergence criteria of random search algorithms are studied, including global convergence criteria and local convergence criteria. Based on these two convergence criteria, QPSO algorithm is proven to be a global search stochastic algorithm, which can provide the foundation for further studying the theoretical problems in QPSO algorithm.
(4) The parameter of an algorithm is the key issue that affects the algorithm's performance and efficiency. The methods for taking values of Contraction-Expansion coefficient which is the only parameter in QPSO algorithm excluding the population size and iteration is analyzed systemically. Four strategies are proposed, including: setting the parameter as a fixed value, making the parameter take value linearly or nonlinearly according to the iteration, and letting the parameter take value adaptively according to the evolution results. Some guiding conclusions are summarized which can provide advantages for those algorithm users.
(5) Premature convergence is also appeared in QPSO algorithm when solving multimodal problems. The reason for premature convergence lies in the collections of swarm which makes the swarm diversity decline and the particles lose the ability of searching in a wide space. Based on two types of diversity measure, an improvement on QPSO algorithm is proposed to avoid the swarm diversity getting into a low level. The improvement is realized by mutating the swarm's global best particle. The improved QPSO algorithm shows preferable ability in solving the multimodal problems.
(6) By hybridizing QPSO algorithm and other evolutionary algorithms to improve the search efficiency and performance of QPSO algorithm. Firstly, mutation mechanism is introduced into QPSO algorithm. Mutation operator can increase the swarm's diversity and can put the particles into new search area. A set of mutation operators is used to compare the effects on QPSO algorithm. Secondly, crossover operator is used in QPSO algorithm and it could generate new positions for the particles instead of the origin method in the algorithm and then increase the swarm's diversity which can enhance the ability of jump out of the collective area. The hybrid QPSO algorithms show good performances by tested them on the benchmark functions.
(7) The mathematic model of different types of digital filters and optimal design methods are discussed. FIR digital filter, IER digital filter, adaptive IIR digital filter and two-dimensional digital filter are analyzed. The essential of optimal design digital filters can be attributed to the global optimization problem. QPSO algorithm and its corresponding improvements are proposed to solve the digital filters' optimal design model. Several different design examples are used to examine the design effect of QPSO algorithm and its improvements. The results show that QPSO algorithm and its improvements can design better filters than those designed by other optimization algorithms.
The main contributions in this work are summarized at last and further research considerations are put forward.
本文以PSO算法与QPSO算法的理论分析及改进方法研究为重点,系统的研究了QPSO算法及其改进算法在数字滤波器优化设计中的应用方法,具体内容如下:
(1)阐述了群体智能优化算法及数字滤波器优化设计的研究背景;介绍了两种典型的群体智能优化算法的研究现状,即蚁群优化算法与PSO算法;对多种不同类型的数字滤波器的优化设计方法作了详细介绍;提出了本课题的研究思路与方法。
(2)通过代数方法分析了PSO算法的收敛性,得出了PSO算法的收敛条件与发散条件,并通过仿真实验验证了分析结论的正确性;然后针对PSO算法在多峰优化问题中易于出现早熟收敛的现象,根据分析的结论提出了基于群体多样性信息控制的PSO算法,算法通过判断群体的多样性来设定群体的搜索状态,即当群体多样性超出设定的上限时,立刻将群体的状态更改为收缩状态,当群体多样性低于设定的下限时,立刻将群体的状态更改为发散状态,群体通过不断的收缩、发散操作完成寻优过程:对多个不同特征的标准测试函数的求解结果显示了算法在多峰优化问题中具有较强的优化能力。
(3)给出了QPSO算法的设计思路。分析了随机算法收敛的两个判断准则,即全局搜索算法的收敛准则与局部搜索算法的收敛准则,利用这两个收敛准则作为依据,证明了QPSO算法是一个全局搜索的随机算法,这为进一步研究QPSO算法的理论问题提供了基础。
(4)算法参数是影响算法性能和效率的关键,文中对QPSO算法中除群体规模和迭代次数外的唯一参数(扩张-压缩因子)的取值方式作了系统的研究,提出了该参数的四种控制策略,即固定取值策略,线性取值策略,非线性取值策略与自适应控制取值策略,通过对标准测试函数的求解分别研究了这四种控制策略,得出了具有指导意义的结论。
(5)针对QPSO算法在解决多峰优化问题中也可能出现局部收敛的现象,分析了出现局部收敛的主要原因在于群体多样性较低而使得群体失去了在大范围内进行搜索的能力,基于两种群体多样性的度量方式,提出了采用全局最优点变异策略对群体进行操作以避免群体的多样性过小,从而提高算法的全局搜索能力,通过对标准测试函数的求解结果表明改进算法的全局求解能力得到了提升。
(6)提出了基于QPSO算法与其他进化算法的混合算法以进一步提高QPSO算法的搜索效率及优化性能。第一种方式是采用进化算法中的变异操作与OPSO算法混合,研究了多种类型的变异操作算子对QPSO算法优化性能的影响,变异操作能够增加群体的多样性,使得算法具有突跳的能力,进入新的搜索区域;第二种方式是在OPSO算法中将交叉操作引入,使得粒子可以不按照算法本身的方式产生新的位置,从而增加群体的多样性,提高算法跳出聚集区域的能力。混合的QPSO算法通过在标准测试函数中的求解显示了较好的优化能力。
(7)分析了不同类型数字滤波器的数学模型及优化设计方法,包括FIR数字滤波器、IIR数字滤波器、自适应IIR数字滤波器与二维IIR数字滤波器。根据数字滤波器优化设计的实质,即全局优化问题,提出了通过QPSO算法及其相应改进算法来完成数字滤波器优化设计模型的求解;对应不同的滤波器类型利用多个设计实例来验证OPSO算法及改进算法的性能与设计效果;通过与其它优化算法的设计结果相比较表明OPSO算法及改进算法能够在各类型数字滤波器的设计中取得更好的设计效果。
论文最后对所做工作与主要研究成果进行了总结,并提出了进一步的研究方向。
Swarm Intelligence (SI) is the property of a system whereby the collective behaviors of agents interacting locally with their environment. SI provides a basis with which it is possible to explore collective problem solving without centralized control or the provision of a global model. SI algorithm is a kind of heuristic search method that can solve the specified problems by simulating the collective behaviors. The characteristic of SI algorithm is stochastic, parallel and distributed. Particle Swarm Optimization (PSO) algorithm, which inspired by social behavior of bird flocking or fish schooling, is one of the typical SI algorithms and it is developed by Eberhart and Kennedy in 1995. Since PSO algorithm was developed, it has attracted many researchers in the fields concerned as its characteristics of simple computation, easy realization, and few parameters. Based on the deep study of PSO algorithm and inspired by quantum physics, Quantum-behaved Particle Swarm Optimization (QPSO) algorithm is proposed by us. QPSO algorithm has much fewer parameters and much stronger global search ability than the PSO algorithm.
Theoretical analyses and algorithm improving on PSO algorithm and QPSO algorithm are mainly discussed in our work and the application of QPSO algorithm and its improvements in the optimal design of digital filters are also studied in the work. The main contents of this dissertation are as follows:
(1) Research background of SI algorithms and optimal design of digital filters are expatiated. The current research situations of two typical SI algorithms are detailed introduced, which are Ant Colony Optimization (ACO) algorithm and PSO algorithm. An introduction of optimal design of different types of digital filters is presented. Research methods and ideas in the work are proposed.
(2) Convergence of PSO algorithm is analyzed by algebraic method and then the conditions of convergence and repulse for PSO algorithm are reached. The conditions are shown to be true by the simulation experiments. According to the conditions a diversity-controlled PSO algorithm which is called DCPSO is proposed in order to solve the problem of premature convergence in PSO algorithm. In DCPSO algorithm, guided by the controlled swarm's diversity, the particles search in the attractive phase sufficiently and adjust them by moving away from the center of the swarm quickly in the repulsive phase once the diversity measure reach to a low bound. The attractive-repulsive procedure can guarantee the swarm search in a wide space and help to avoid trapping into the local minima. Experimental results on several well-known benchmark functions show that DCPSO has strong global optimization ability in solving the multimodal problems.
(3) The thought of QPSO algorithm is discussed. Convergence criteria of random search algorithms are studied, including global convergence criteria and local convergence criteria. Based on these two convergence criteria, QPSO algorithm is proven to be a global search stochastic algorithm, which can provide the foundation for further studying the theoretical problems in QPSO algorithm.
(4) The parameter of an algorithm is the key issue that affects the algorithm's performance and efficiency. The methods for taking values of Contraction-Expansion coefficient which is the only parameter in QPSO algorithm excluding the population size and iteration is analyzed systemically. Four strategies are proposed, including: setting the parameter as a fixed value, making the parameter take value linearly or nonlinearly according to the iteration, and letting the parameter take value adaptively according to the evolution results. Some guiding conclusions are summarized which can provide advantages for those algorithm users.
(5) Premature convergence is also appeared in QPSO algorithm when solving multimodal problems. The reason for premature convergence lies in the collections of swarm which makes the swarm diversity decline and the particles lose the ability of searching in a wide space. Based on two types of diversity measure, an improvement on QPSO algorithm is proposed to avoid the swarm diversity getting into a low level. The improvement is realized by mutating the swarm's global best particle. The improved QPSO algorithm shows preferable ability in solving the multimodal problems.
(6) By hybridizing QPSO algorithm and other evolutionary algorithms to improve the search efficiency and performance of QPSO algorithm. Firstly, mutation mechanism is introduced into QPSO algorithm. Mutation operator can increase the swarm's diversity and can put the particles into new search area. A set of mutation operators is used to compare the effects on QPSO algorithm. Secondly, crossover operator is used in QPSO algorithm and it could generate new positions for the particles instead of the origin method in the algorithm and then increase the swarm's diversity which can enhance the ability of jump out of the collective area. The hybrid QPSO algorithms show good performances by tested them on the benchmark functions.
(7) The mathematic model of different types of digital filters and optimal design methods are discussed. FIR digital filter, IER digital filter, adaptive IIR digital filter and two-dimensional digital filter are analyzed. The essential of optimal design digital filters can be attributed to the global optimization problem. QPSO algorithm and its corresponding improvements are proposed to solve the digital filters' optimal design model. Several different design examples are used to examine the design effect of QPSO algorithm and its improvements. The results show that QPSO algorithm and its improvements can design better filters than those designed by other optimization algorithms.
The main contributions in this work are summarized at last and further research considerations are put forward.
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