本文選題：非線性隨機(jī)系統(tǒng) + PDF形狀控制　； 參考：《西安理工大學(xué)》2016年博士論文

【摘要】：當(dāng)系統(tǒng)具有隨機(jī)輸入、隨機(jī)干擾或隨機(jī)特性(參數(shù))時(shí),系統(tǒng)的狀態(tài)、輸出和控制量都是隨機(jī)過(guò)程。在過(guò)去的幾十年里,隨機(jī)動(dòng)態(tài)系統(tǒng)是以性能指標(biāo)均值和方差為目標(biāo)進(jìn)行控制器的設(shè)計(jì),而這種方法常常以考慮閉環(huán)系統(tǒng)的穩(wěn)定性和關(guān)于某個(gè)目標(biāo)函數(shù)的最優(yōu)為出發(fā)點(diǎn),在理論和應(yīng)用中都存在一定的局限性,極大地限制了控制器的性能。因此,控制泛函概率密度函數(shù)(Probability Density Function,簡(jiǎn)稱PDF)的形狀引起了人們的關(guān)注,并且已成為控制領(lǐng)域當(dāng)前研究的一個(gè)熱點(diǎn)。本文針對(duì)當(dāng)前國(guó)內(nèi)外對(duì)于隨機(jī)系統(tǒng)中存在的幾個(gè)熱點(diǎn)問題進(jìn)行了深入研究,其研究的主要內(nèi)容包括多項(xiàng)式非線性隨機(jī)系統(tǒng)概率密度形狀的非線性控制器設(shè)計(jì)方法和分段線性控制器設(shè)計(jì)方法,基于FPK方程近似解的PDF形狀控制方法,和先進(jìn)智能算法在非線性隨機(jī)系統(tǒng)概率密度函數(shù)形狀控制中的應(yīng)用。對(duì)解決以上幾個(gè)所研究問題的關(guān)鍵技術(shù),本文給出了比較新穎和實(shí)用的算法,取得了具有一定參考價(jià)值的研究成果。具體研究工作包括以下幾個(gè)方面的內(nèi)容:1.針對(duì)具有加性高斯白噪聲并且具有多項(xiàng)式形式的一類非線性隨機(jī)系統(tǒng),研究了非線性控制器的設(shè)計(jì)方法。PDF形狀最優(yōu)控制的目標(biāo)是設(shè)計(jì)的控制器使得狀態(tài)變量的PDF形狀以最優(yōu)的方式逼近期望形狀。這樣將PDF形狀控制問題轉(zhuǎn)化為一個(gè)控制器參數(shù)的優(yōu)化問題。設(shè)計(jì)多項(xiàng)式型的非線性控制器,通過(guò)求FPK方程的精確解獲得了狀態(tài)變量的穩(wěn)態(tài)PDF的表達(dá)式,當(dāng)狀態(tài)變量的PDF形狀逼近期望的PDF形狀時(shí),采用線性最小二乘法獲得控制器增益的最優(yōu)值。仿真結(jié)果表明,設(shè)計(jì)的非線性控制器可以有效地控制狀態(tài)變量的PDF形狀。2.針對(duì)多項(xiàng)式形式的非線個(gè)性隨機(jī)系統(tǒng)的PDF形狀控制問題,多項(xiàng)式非線性性控制器由于其在數(shù)學(xué)上的靈活性和連續(xù)性備受喜愛而經(jīng)常采用。然而,由于其形式上的復(fù)雜性增加了整個(gè)算法的復(fù)雜度和計(jì)算量.此提,因此提出了分段線性控制器。分段線性控制器包括兩個(gè)比例系數(shù)和一個(gè)分段點(diǎn)共有三個(gè)參數(shù),參數(shù)較少,并且在一定范圍內(nèi)是線性的,在數(shù)學(xué)上更易處理。當(dāng)狀態(tài)變最的PDF形狀以最優(yōu)方式逼近目標(biāo)形狀時(shí),我們是把PDF形狀的控制問題轉(zhuǎn)化為一個(gè)數(shù)學(xué)規(guī)劃問題。對(duì)解決這類問題,現(xiàn)有的工具非常多。因?yàn)槲覀兯芯康膯栴}的結(jié)構(gòu)是線性或者多項(xiàng)式形式,當(dāng)解決這類優(yōu)化問題時(shí),采用共軛梯度法在有限步內(nèi)就可以搜索到最優(yōu)值,搜索效率較高。因此,采用非線性共軛梯度法獲得控制器中的三個(gè)參數(shù)。將提出的分段線性控制器與二階非線性控制器、三階非線性控制器進(jìn)行了對(duì)比實(shí)驗(yàn),體現(xiàn)了線性控制器的優(yōu)勢(shì)和和特點(diǎn)。3.隨機(jī)系統(tǒng)的結(jié)構(gòu)模型決定了控制器的結(jié)構(gòu)。對(duì)不同結(jié)構(gòu)的隨機(jī)系統(tǒng),控制器的控制規(guī)律也不相同。一個(gè)隨機(jī)系統(tǒng)的非線性不僅可以用多項(xiàng)式形式來(lái)表示,也可能體現(xiàn)為三角函數(shù),指數(shù)函數(shù)等多種形式。因此,我們需要研究適合各類非線性系統(tǒng)的PDF形狀的控制方法。眾所周知,狀態(tài)變量的穩(wěn)態(tài)PDF對(duì)應(yīng)于隨機(jī)動(dòng)態(tài)系統(tǒng)的FPK方程的解,但是,FPK方程的精確解很難解得。本文提出了一種求FPK方程近似解的方法。首先,找到一個(gè)含有參數(shù)的特殊函數(shù),該函數(shù)具有PDF的性質(zhì)。將該函數(shù)作為FPK方程的近似穩(wěn)態(tài)解,然后推導(dǎo)出含有參數(shù)的PDF控制器的表達(dá)式。FPK方程的近似解也就是狀態(tài)變量的PDF,再去跟蹤期望的PDF。通過(guò)非線性最小二乘法求出函數(shù)中的相關(guān)參數(shù),也就得到了 FPK方程的近似解,同時(shí)得到了不同目標(biāo)分布的PDF形狀控制器。并將該P(yáng)DF形狀控制方法用到磨礦系統(tǒng)中,對(duì)礦粒分布進(jìn)行控制。通過(guò)優(yōu)化磨礦機(jī)新添礦料量,使水力旋流器溢流礦粒的分布滿足后續(xù)選別工序要求的分布指標(biāo),控制效果說(shuō)明該P(yáng)DF形狀控制方法在實(shí)際應(yīng)用中的有效性。4.對(duì)于非線性隨機(jī)動(dòng)態(tài)系統(tǒng),通過(guò)求解FPK方程得到狀態(tài)響應(yīng)的概率密度函數(shù)是比較復(fù)雜的,甚至是不可能的。根據(jù)狀態(tài)變量的穩(wěn)態(tài)PDF、各階統(tǒng)計(jì)矩及各階累積量的關(guān)系,用埃德沃斯(Edgeworth)漸進(jìn)展開式近似表示狀態(tài)變量PDF。首先得到非線性隨機(jī)動(dòng)態(tài)系統(tǒng)響應(yīng)PDF的統(tǒng)計(jì)矩微分方程,進(jìn)而研究PDF的控制問題。群體智能算法因其強(qiáng)大的問題求解能力被應(yīng)用到諸多領(lǐng)域,本文提出了一種新的智能算法—煙花算法,詳細(xì)介紹了其算法原理,并將其應(yīng)用到隨機(jī)系統(tǒng)的PDF形狀控制中,優(yōu)化求解控制器的增益。最后通過(guò)實(shí)例將煙花算法與遺傳算法和粒子群算法進(jìn)行了比較,充分展現(xiàn)了煙花算法在PDF形狀控制中的優(yōu)越性。最后,對(duì)全文進(jìn)行了概括性總結(jié),并指出有待進(jìn)一步研究和完善的問題。
[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.
【學(xué)位授予單位】：西安理工大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2016
【分類號(hào)】：O231

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