非线性随机系统概率密度函数形状控制研究

发布时间：2018-06-02 00:40

本文选题：非线性随机系统 + PDF形状控制　；参考：《西安理工大学》2016年博士论文

【摘要】：当系统具有随机输入、随机干扰或随机特性(参数)时,系统的状态、输出和控制量都是随机过程。在过去的几十年里,随机动态系统是以性能指标均值和方差为目标进行控制器的设计,而这种方法常常以考虑闭环系统的稳定性和关于某个目标函数的最优为出发点,在理论和应用中都存在一定的局限性,极大地限制了控制器的性能。因此,控制泛函概率密度函数(Probability Density Function,简称PDF)的形状引起了人们的关注,并且已成为控制领域当前研究的一个热点。本文针对当前国内外对于随机系统中存在的几个热点问题进行了深入研究,其研究的主要内容包括多项式非线性随机系统概率密度形状的非线性控制器设计方法和分段线性控制器设计方法,基于FPK方程近似解的PDF形状控制方法,和先进智能算法在非线性随机系统概率密度函数形状控制中的应用。对解决以上几个所研究问题的关键技术,本文给出了比较新颖和实用的算法,取得了具有一定参考价值的研究成果。具体研究工作包括以下几个方面的内容:1.针对具有加性高斯白噪声并且具有多项式形式的一类非线性随机系统,研究了非线性控制器的设计方法。PDF形状最优控制的目标是设计的控制器使得状态变量的PDF形状以最优的方式逼近期望形状。这样将PDF形状控制问题转化为一个控制器参数的优化问题。设计多项式型的非线性控制器,通过求FPK方程的精确解获得了状态变量的稳态PDF的表达式,当状态变量的PDF形状逼近期望的PDF形状时,采用线性最小二乘法获得控制器增益的最优值。仿真结果表明,设计的非线性控制器可以有效地控制状态变量的PDF形状。2.针对多项式形式的非线个性随机系统的PDF形状控制问题,多项式非线性性控制器由于其在数学上的灵活性和连续性备受喜爱而经常采用。然而,由于其形式上的复杂性增加了整个算法的复杂度和计算量.此提,因此提出了分段线性控制器。分段线性控制器包括两个比例系数和一个分段点共有三个参数,参数较少,并且在一定范围内是线性的,在数学上更易处理。当状态变最的PDF形状以最优方式逼近目标形状时,我们是把PDF形状的控制问题转化为一个数学规划问题。对解决这类问题,现有的工具非常多。因为我们所研究的问题的结构是线性或者多项式形式,当解决这类优化问题时,采用共轭梯度法在有限步内就可以搜索到最优值,搜索效率较高。因此,采用非线性共轭梯度法获得控制器中的三个参数。将提出的分段线性控制器与二阶非线性控制器、三阶非线性控制器进行了对比实验,体现了线性控制器的优势和和特点。3.随机系统的结构模型决定了控制器的结构。对不同结构的随机系统,控制器的控制规律也不相同。一个随机系统的非线性不仅可以用多项式形式来表示,也可能体现为三角函数,指数函数等多种形式。因此,我们需要研究适合各类非线性系统的PDF形状的控制方法。众所周知,状态变量的稳态PDF对应于随机动态系统的FPK方程的解,但是,FPK方程的精确解很难解得。本文提出了一种求FPK方程近似解的方法。首先,找到一个含有参数的特殊函数,该函数具有PDF的性质。将该函数作为FPK方程的近似稳态解,然后推导出含有参数的PDF控制器的表达式。FPK方程的近似解也就是状态变量的PDF,再去跟踪期望的PDF。通过非线性最小二乘法求出函数中的相关参数,也就得到了 FPK方程的近似解,同时得到了不同目标分布的PDF形状控制器。并将该PDF形状控制方法用到磨矿系统中,对矿粒分布进行控制。通过优化磨矿机新添矿料量,使水力旋流器溢流矿粒的分布满足后续选别工序要求的分布指标,控制效果说明该PDF形状控制方法在实际应用中的有效性。4.对于非线性随机动态系统,通过求解FPK方程得到状态响应的概率密度函数是比较复杂的,甚至是不可能的。根据状态变量的稳态PDF、各阶统计矩及各阶累积量的关系,用埃德沃斯(Edgeworth)渐进展开式近似表示状态变量PDF。首先得到非线性随机动态系统响应PDF的统计矩微分方程,进而研究PDF的控制问题。群体智能算法因其强大的问题求解能力被应用到诸多领域,本文提出了一种新的智能算法—烟花算法,详细介绍了其算法原理,并将其应用到随机系统的PDF形状控制中,优化求解控制器的增益。最后通过实例将烟花算法与遗传算法和粒子群算法进行了比较,充分展现了烟花算法在PDF形状控制中的优越性。最后,对全文进行了概括性总结,并指出有待进一步研究和完善的问题。
[Abstract]:When a system has random input, random interference or random characteristic (parameter), the state, output and control amount of the system are random processes. In the past few decades, the stochastic dynamic system is designed for the controller based on the mean and variance of the performance index, and this method often takes into account the stability of the closed loop system and about a certain one. The optimization of the objective function is the starting point. There are some limitations in both theory and application, which greatly restrict the performance of the controller. Therefore, the shape of the Probability Density Function (PDF) has attracted people's attention, and has become a hot spot in the current research in the control field. Several hot issues in stochastic systems are studied at home and abroad. The main contents of the research include the design method of nonlinear controller and the design method of piecewise linear controller for the probability density shape of the polynomial nonlinear stochastic systems, the PDF shape control method based on the near quasi solution of the FPK equation, and the advanced intelligence. The application of the algorithm in the shape control of the probability density function of a nonlinear stochastic system. In order to solve the key technology of the above problems, a new and practical algorithm is given in this paper, and the research results with certain reference value are obtained. The specific research work includes the following aspects: 1. for the additive Gauss A nonlinear stochastic system with white noise and polynomial form, the design method of the nonlinear controller is studied. The objective of.PDF shape optimal control is designed to make the PDF shape of the state variable approximate the desired shape in an optimal way. Thus, the PDF shape control problem is transformed into a controller parameter optimization. The nonlinear controller of a polynomial type is designed to obtain the expression of the steady-state PDF of the state variable by the exact solution of the FPK equation. The linear least square method is used to obtain the optimal gain of the controller when the PDF shape of the state variable approximated the desired PDF shape. The simulation results show that the nonlinear controller designed can be effective. The PDF shape.2. of the state variable is controlled for the PDF shape control problem of a polynomial nonlinear stochastic system. The polynomial nonlinear controller is often used because of its flexibility and continuity in mathematics. However, the complexity and computational complexity of the whole algorithm is increased because of its complexity in the form. A piecewise linear controller is proposed. The piecewise linear controller, which consists of two proportional coefficients and a piecewise point, has three parameters, which is less parameter and linear in a certain range. It is easier to handle in mathematics. When the PDF shape of the most state is optimal approach to the shape of the target, we are the control problem of the PDF shape. There are many existing tools for solving these problems. Because the structure of the problem we have studied is linear or polynomial. When solving this problem, the conjugate gradient method can be used in the finite step to search the optimal value, and the search efficiency is high. Therefore, the nonlinear conjugate gradient is used. The method obtains three parameters in the controller. The proposed piecewise linear controller is compared with the two order nonlinear controller and the three order nonlinear controller. The advantages of the linear controller and the structure model of the.3. random system determine the structure of the controller. The control rules for the random system of different structures and the controller are also shown. The law of a random system can not only be expressed in polynomial forms, but also can be embodied in a variety of forms, such as trigonometric and exponential functions. Therefore, we need to study the PDF shape control methods suitable for all kinds of nonlinear systems. It is known that the state variable steady-state PDF corresponds to the FPK square of the stochastic dynamic system. The exact solution of the FPK equation is difficult to solve. In this paper, a method for solving the approximate solution of the FPK equation is proposed. First, a special function with a parameter is found. The function has the property of PDF. The function is the approximate steady solution of the FPK equation, and then the approximation of the expression of the.FPK equation with the PDF controller with parameters is derived. The solution is the PDF of the state variable, and then the desired PDF. is traced by the nonlinear least square method to find the relevant parameters in the function, and the approximate solution of the FPK equation is obtained. At the same time, the PDF shape controller with different target distribution is obtained. And the PDF shape control method is used in the grinding system to control the distribution of the ore particles. The size distribution of the overflow ore particles in the hydrocyclone meets the requirements of the subsequent selection process. The control effect shows that the validity of the PDF shape control method in the practical application.4. is more complex for the nonlinear stochastic dynamic system, and the probability density function of the state response obtained by solving the FPK equation is more complex. It is impossible even. According to the steady-state PDF of the state variable, the relation between the statistical moments and the order cumulants of each order, the statistical moment differential equation of the nonlinear stochastic dynamic system response PDF is first obtained by the asymptotic expansion of Ed worth (Edgeworth) PDF., and then the control problem of PDF is studied. The swarm intelligence algorithm is due to its control problem. The powerful problem solving ability is applied to many fields. In this paper, a new intelligent algorithm, smoke algorithm is proposed, and its algorithm principle is introduced in detail. The algorithm is applied to the PDF shape control of random system to optimize the gain of the controller. Finally, the smoke flower algorithm is compared with the genetic algorithm and the particle swarm optimization algorithm by an example. In comparison, the advantages of the fireworks algorithm in PDF shape control are fully demonstrated. Finally, a general summary of the full text is made, and the problems to be further studied and perfected are pointed out.
【学位授予单位】：西安理工大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O231

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