二阶非线性系统的最小规则库模糊推理非线性控制器设计(IJISA-V6-N7-10)

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本刊核心层次论文
非 线 性 时滞 系统 的模 糊 控 制 设 计
张
摘
伟 ，佟 绍成
（ 宁工 业大 学 理 学 院， 辽宁 锦 州 １１０ ） 辽 ２０ １
要 ：针 对一类非线性 时滞 系统 ，采 用模糊 Ｔ Ｓ模型对系统进行逼近 ，应用平 行分布补偿算法（ＤＣ 设计 － Ｐ ）
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辽 宁工 业大学学报 （ 自然科 学版）
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模糊模型PID优化控制在非线性系统中的应用
李园;李平
【期刊名称】《江南大学学报（自然科学版）》
【年(卷),期】2007(006)004
【摘要】通过提出一种基于T-S模糊模型的非线性系统PID优化控制算法,将该算法应用于具有强非线性与不稳定性的填料塔热交换装置,为非线性控制系统的实际应用提供一种方案.在机理分析的基础上建立了分段非线性模型,设计了该装置的模糊模型PID优化控制器,对系统温度进行控制.实验结果与传统的PID控制结果对比表明,采用文中提出的模糊模型PID优化控制方法能够有效解决非线性控制问题.【总页数】5页(P414-418)
【作者】李园;李平
【作者单位】浙江大学,工业控制技术研究所,浙江,杭州,310027;浙江大学,工业控制技术研究所,浙江,杭州,310027
【正文语种】中文
【中图分类】TP271.72;TP273.3
【相关文献】
1.粒子群优化变论域模糊PID控制在风电机组变桨距中的应用 [J], 党存禄;杜小波
2.一种优化fuzzy-PID控制在MPPT中的应用 [J], 王严龙;杨旭红;王军成;薛阳
3.PSO-BP算法优化PID控制在光照系统中的应用 [J], 王雷
4.基于遗传算法优化的模糊PID控制在粮食干燥中的应用 [J], 聂放;周晓光;代爱妮;
邢颖;王海霞
5.粒子群优化模糊PID控制在SCR系统中的应用 [J], 陈彦豪;朱志宇
因版权原因，仅展示原文概要，查看原文内容请购买。

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机电工程学院课程设计报告课程题目二阶系统模糊控制算法的研究专业电气工程及其自动化姓名指导教师学期2015-2016二阶系统的模糊控制算法的研究学生指导老师:摘要:模糊控制是以模糊数学为基础发展的，为一些无法建立数学模型或者数学模型相当粗糙的系统提供的一种非线性的控制方法。

对于这些系统，模糊控制可以得到比较满意的控制效果，并且能够解决一些无法通过传统方法解决的问题。

本文利用MATLAB模糊控制工具箱设计的模糊控制器来控制一个二阶系统，由给定的控制器的输入和输出变量，输入和输出变量的隶属函数，分析了输入和输出变量之间的关系，设计了模糊控制规则库，并通过SIMULINK仿真将模糊控制方法与经典的PID控制方法进行对比，分析仿真结果，探讨模糊控制器的隶属函数，控制规则，以及量化因子和比例因子在模糊控制中所起到的作用。

关键字：模糊控制；MATLAB；SIMULINK；PIDResearch of fuzzy control algorithm of second ordersystemsUndergraduate:Supervisor:Abstract：Fuzzy control, which is based on the fuzzy mathematics, is a new way of nonlinearity control system in which the mathematical model is unable established or the mathematical model is very rough. For these systems, fuzzy control offers users a satisfied control result, and settles down some problems which cannot be solved by traditional methods.This paper aims to introduce how to use a fuzzy controller which is based on the MATLAB fuzzy control toolbox to control a second-order system. In order to fulfill this target, the author firstly defines the input variables, output variables and their membership functions. Then, the author analyzes the relationship between the input variables and output variables, and designs the fuzzy control rule bank. Finally, the author makes a difference between the methods of the classic PID control and the fuzzy control by SIMULINK. Membership function of fuzzy controller, control rules, and the function of quantizes and scale factor in the fuzzy control process are also discussed in this paper.Key words: MATLAB; Fuzzy control; PID;SIMULINK simulation目录绪论 (1)1控制理论算法 (5)3.1 PID控制规律 (5)3.1PID控制器原理 (5)3.1 (5)3.1.3微分（D）控制 (5)3.2传统PID控制过程 (5)1模糊控制 (1)1.1模糊控制的背景及意义 (1)1.2模糊控制的基本理论 (1)1.3模糊控制的基本结构 (1)1.4模糊控制的组成 (1)1.5模糊控制的运行模型 (1)1.6模糊控制与SIMULINK的 (1)2基于MATLAB的模糊控制仿真 (2)2.1模糊控制器的设计 (2)2.1模糊集合及论域的定义 (2)2.2模糊控制规则确定 (4)2.3仿真曲线 (5)4比较 (7)参考文献 (8)致 (9)绪论模糊控制器由三个环节组成：用于输入信号处理的模糊量化和模糊化环节，模糊控制算法功能单元，以及用于输出解模糊化的模糊判决环节。

非线性系统系统辨识与控制研究引言：非线性系统是指系统在其输入与输出之间的关系不符合线性关系的系统。

这种系统具有复杂的动态行为和非线性特性，使得其辨识与控制变得非常具有挑战性。

然而，非线性系统在现实生活中的应用非常广泛，例如电力系统、机械系统和生物系统等。

因此，对非线性系统的系统辨识与控制研究具有重要意义。

一、非线性系统辨识方法研究1. 仿射变换法仿射变换法是一种常用的非线性系统辨识方法之一。

它通过将非线性系统进行仿射变换，将其转化为线性系统的形式，从而利用线性系统辨识的方法进行处理。

该方法适用于具有输入输出非线性关系的系统，但对于参数模型的选择和计算量较大的问题需要进一步研究。

2. 基于神经网络的方法神经网络作为一种强大的表达非线性关系的工具，被广泛应用于非线性系统辨识。

基于神经网络的方法可以通过训练神经网络模型，从大量的输入输出数据中学习非线性系统的映射关系。

该方法的优点是可以逼近任意非线性函数，但对于网络结构的选择和训练过程中的收敛性等问题还需深入研究。

3. 基于系统辨识方法的非线性系统辨识传统的系统辨识方法主要适用于线性系统的辨识，但其在非线性系统辨识中也有应用的价值。

通过对非线性系统进行线性化处理，可以将其转化为线性系统的辨识问题。

同时，利用最小二乘法、频域法等常用的系统辨识方法对线性化后的系统进行辨识。

这种方法的优势在于利用了线性系统辨识的经验和技术，但对于线性化的准确性和辨识结果的合理性需要进行评估。

二、非线性系统控制方法研究1. 反馈线性化控制反馈线性化是一种常用的非线性系统控制方法。

该方法通过在非线性系统中引入反馈控制器，将非线性系统转化为可控性的线性系统。

然后，利用线性系统控制方法设计控制器，并通过反馈线性化控制策略实现对非线性系统的控制。

该方法的优点在于简化了非线性系统控制的设计和分析过程，但对于系统的稳定性和性能等问题还需要进行进一步的研究。

2. 自适应控制自适应控制是一种针对非线性系统的适应性控制方法。

文献综述模糊PID控制器的研究与应用学院自动化与电子信息学院二O一四年四月四川理工学院毕业(设计)论文文献综述0 前言PID控制作为一种典型的传统反馈控制器，以其结构简单，易于实现和鲁棒性好等特点在工业过程控制中广泛应用。

但是传统PID控制器的参数需要被控对象的数学模型来进行调整，而控制过程中的滞后性、控制参数的非线性和高阶性增加了对Kp、Ki、Kd三个参数的调整难度。

所以对确定的控制系统通过复杂的计算后，其三个参数的值在控制运行中一般是固定的，不易进行在线的调整。

而在实际的工业生产过程中，许多被控对象受到负荷变化和干扰因素的作用，其对象参数的特征和结构易发生改变，这就需要对参数进行动态的调整。

同样因为被控系统的复杂性和不确定性，其精确的数学模型难以建立，甚至无法建立模型，所以需要利用模糊控制技术等方法来解决。

模糊PID无需考虑被控系统的模型，而只根据其误差e 和误差变化ec等检测数据来自适应调整Kp、Ki、Kd的值，最终使被控系统处于稳定工作态。

1 国外研究现状ŞabanÇetin,AliVolkanAkkaya[1](2010)表示准确度和精密度液压系统的位置控制是为了设置更经济和高质量系统的关键参数。

在此背景下，他们提出了由一个非对称液压缸由一个四通、三位比例阀驱动的液压驱动系统的建模与位置控制。

在此系统模型中，体积弹性模量被认为是一个变量。

此外，基于规则的混合型模糊 PID控制器（H F P I DC R）提出了液压系统的位置控制，并对其性能进行了仿真研究测试。

这种控制器的新颖方面是模糊逻辑和PID 控制器结合在一个开关条件。

该HFPIDCR 基于控制器的模拟结果与经典PID、模糊逻辑控制器（FLC）和混合模糊PID 控制器（HFPID）的结果进行了比较。

因此，它被证明了混合型模糊PID控制器加上规则比其他的控制器更有效。

IndranilPana[ 2] 等（2011）通过减少积分时间降低最优PID 和最优模糊PID的绝对误差（ITAE）和平方控制器输出的网络控制系统（NCS）的响应速度。

I.J. Intelligent Systems and Applications, 2013, 03, 74-82Published Online February 2013 in MECS (/)DOI: 10.5815/ijisa.2013.03.08A Type-2 Fuzzy Logic Based Framework forFunction PointsAnupama KaushikDept. of IT, Maharaja Surajmal Institute of Technology, GGSIP University, Delhi, Indiaanupama@msit.inA.K. SoniDept. of IT, School of Engineering and Technology, Sharda University, Greater Noida, Indiaak.soni@sharda.ac.inRachna SoniDept. of Computer Science and Applications, DAV College, Yamuna Nagar, Haryana, Indiasonirachna67@Abstract —Software effort estimation is very crucial in software project planning. Accurate software estimation is very critical for a project success. There are many software prediction models and all of them utilize software size as a key factor to estimate effort. Function Points size metric is a popular method for estimating and measuring the size of application software based on the functionality of the software from the user‘s point of view. While there is a great advancement in software development, the weight values assigned to count standard FP remains the same. In this paper the concepts of calibrating the function point weights using Type-2 fuzzy logic framework is provided whose aim is to estimate a more accurate software size for various software applications and to improve the effort estimation of software projects. Evaluation experiments have shown the framework to be promising.Index Terms —Project management, Software Effort Estimation, Type-2 Fuzzy Logic System, Function Point AnalysisI.IntroductionSoftware development has become an important activity for many modern organizations. Software engineers have become more and more concerned about accurately predicting the cost and quality of software product under development. Consequently, many models for estimating software cost have been proposed such as Constructive Cost Model(COCOMO) [1],Constructive Cost Model II (COCOMO II) [2], Software Life Cycle Management (SLIM) [3] etc. These models identify key contributors to effort and use historical organizational projects data to generate a set of mathematical formulae that relates these contributors to effort. Such a set of mathematical formulae are often referred to as parametric model because alternative scenarios can be defined by changing the assumed values of a set of fixed coefficients (parameters) [4]. All these models use the software size as the major determinant of effort. Function Points is an ideal software size metric to estimate cost since it can be used in the early development phase, such as requirement, measures the software functional size from user‘s view, and is programming language independent [5].Today the scenario of software industry has changed from what it has many years ago. Now-a-days the object oriented paradigm has incorporated into the software development which leads to the creation of object oriented function points [6]. All the traditional cost estimation models are limited by their inability to cope with vagueness and imprecision in the early stages of the software life cycle. So, a number of soft computing approaches like fuzzy logic (FL), artificial neural networks (ANN), evolutionary computation (EC) etc. are incorporated to make rational decisions in an environment of uncertainty and vagueness. The first realization of the fuzziness of several aspects of COCOMO was that of Fei and Liu [7] called F-COCOMO. Jack Ryder [8] investigated the application of fuzzy modelling techniques to COCOMO and the Function Points models, respectively. Venkatachalam [9] investigated the application of artificial neural network (ANN) to software cost estimation. Many researchers have applied the evolutionary computation approach towards cost estimation [10, 11].1.1 Background and related workOsias de Souza Lima Junior et al. [12] have worked on trapezoidal fuzzy numbers to model function point analysis for the development and enhancement projectassessment. Ho Leung [13] has presented a case study for evaluation of function points. Finnie et al. [14] provided the combination of machine learning approach with FP. They compared the three approaches i.e. regression analysis, artificial neural networks and case based reasoning using FP as an estimate of software size. The authors observed that both artificial neural networks and case based reasoning performed well on the given dataset in contrast to regression analysis. They concluded that case based reasoning is appealing because of its similarity to the expert judgement approach and for its potential in supporting human judgement. Al-Hajri et al. [15] establish a new FP weight system using artificial neural network. Lima et al. [16] proposed the concepts and properties from fuzzy set theory to extend FP analysis into a fuzzy FP analysis and the calibration was done using a small database comprised of legacy systems developed mainly in Natural 2, Microsoft Access and Microsoft Visual Basic. Yau and Tsoi [17] introduced a fuzzified FP analysis model to help software size estimators to express their judgement and use fuzzy B-spline membership function to derive their assessment values. The weak point of their work is that they use limited in-house software to validate the model. Abran and Robillard‘s empirical study [18] demonstrates the clear relationship between FPA‘s primary component and work-effort. Kralj et al. [19] identified the function point analysis method deficiency of upper boundaries in the rating complexity process and proposed an improved FPA method. Wei Xia et al. [20] proposed a Neuro-Fuzzy calibration approach for function point complexity weights. Their model provided an equation between Unadjusted Function Points and work effort which is used to train the neural network and estimated the effort. Moataz A. Ahmed and Zeeshan Muzaffar [4] provided an effort prediction framework that is based on type-2 fuzzy logic to allow handling imprecision and uncertainty present in the effort prediction. Mohd. Sadiq et al. [21] developed two different linear regression models using fuzzy function point and non fuzzy function point in order to predict the software project effort.The above researches have concluded that the combination of soft computing approaches and the traditional cost estimation models yields a more accurate prediction of software costs and effort. All the earlier work on software cost estimation using fuzzy logic incorporated type-1 or type-2 fuzzy framework for effort prediction. This paper proposes an improved FPA method by calibrating the function point‘s weight using type-2 fuzzy logic framework.1.2 Function Point Analysis: A short description Function point analysis is a process used to calculate the software size from the user‘s point of view, i.e. on the basis of what the user requests and receives in return from the system. Allan J Albrecht [22] of IBM proposed Function Point Count (FPC) as a size measure in the late 1970s. Albrecht had taken up the task of arriving at size measures of software systems to compute a productivity measure that could be used across programming languages and development technologies. The current promoter of Albrecht‘s function point model is the International Function Point User‘s Group (IFPUG). IFPUG evolves the FPA method and periodically releases the Counting Practices Manual for consistent counting of function points across different organizations. In FPA, a system is decomposed into five functional units: Internal Logical Files (ILF), External Interface Files (EIF), External Inputs (EI), External Outputs (EO) and External Inquiry (EQ). These functional units are categorized into data functional units and transactional function units. All the functions do not provide the same functionality to the user. Hence, the function points contributed by each function varies depending upon the type of function (ILF, EIF, EI, EO or EQ) and complexity (Simple, Average or Complex) of the function. The data functions complexity is based on the number of Data Element Types (DET) and number of Record Element Types (RET). The transactional functions are classified according to the number of file types referenced (FTRs) and the number of DETs. The complexity matrix for all the five components is given in Table 1, Table 2 and Table 3. Table 4 illustrates how each function component is then assigned a weight according to its complexity.The actual calculation process of FPA is accomplished in three stages: (i) determine the unadjusted function points (UFP); (ii) calculate the value adjustment factor (VAF); (iii) calculate the final adjusted function points.The Unadjusted Function Points (UFP) is calculated using ―(1)‖, where W ij are the complexity weights and Z ij are the counts for each function component.∑∑ (1) The second stage, calculating the value adjustment factor (VAF), is derived from the sum of the degree of influence (DI) of the 14 general system characteristics (GSCs). The DI of each one of these characteristics ranges from 0 to 5 as follows: (i) 0 – no influence; (ii) 1 –incidental influence; (iii) 2 –moderate influence; (iv) 3 – average influence; (v) 4 – significant influence; and (vi) 5 – strong influence.The general characteristics of a system are: (i) data communications; (ii) distributed data processing; (iii) performance; (iv) heavily used configuration; (v) transaction rate; (vi) online data entry; (vii) end-user efficiency; (viii) on-line update; (ix) complex processing; (x) reusability; (xi) installation ease; (xii) operational ease; (xiii) multiple sites; and (xiv) facilitate change. VAF is then computed using ―(2)‖:∑ (2)x i is the Degree of Influence (DI) rating of each GSC. Finally, the adjusted function points are calculated as given in ―(3)‖.(3)Table 1: Complexity Matrix of ILF/EIFTable 2: Complexity Matrix of EITable 3: Complexity Matrix of EO/EQTable 4: Functional Units with weighting factorsII.Type 2 Fuzzy Logic SystemsFuzzy Logic is a methodology to solve problems which are too complex to be understood quantitatively. It is based on fuzzy set theory and introduced in 1965 by Prof. Zadeh in the paper fuzzy sets [23]. It is a theory of classes with unsharp boundaries, and considered as an extension of the classical set theory [24]. The membership µA(x) of an element x of a classical set A, as subset of the universe X, is defined by:µA(x) = {That is, x is a member of set A (µA (x) = 1) or not (µA (x) = 0). The classical sets where the membership value is either zero or one are referred to as crisp sets. Fuzzy sets allow partial membership. A fuzzy set A is defined by giving a reference set X, called the universe and a mapping;µA : X []called the membership function of the fuzzy set A µA(x), for x X is interpreted as the degree of membership of x in the fuzzy set A. A membership function is a curve that defines how each point in the input space is mapped to a membership value between 0 and 1. The higher the membership x has in the fuzzy set A, the more true that x is A. The membership functions (MFs) may be triangular, trapezoidal, Gaussian, parabolic etc.Fuzzy logic allows variables to take on qualitative values which are words. When qualitative values are used, these degrees may be managed by specific inferential procedures. Just as in fuzzy set theory the set membership values can range (inclusively) between 0 and 1, in fuzzy logic the degree of truth of a statement can range between 0 and 1 and is not constrained to the two truth values {true, false} as in classic predicate logic.Fuzzy Logic System (FLS) is the name given to any system that has a direct relationship with fuzzy concepts. The most popular fuzzy logic systems in the literature may be classified into three types [25]: pure fuzzy logic systems, Takagi and Sugeno‘s fuzzy system and fuzzy logic system with fuzzifier and defuzzifier also known as Mamdani system. As most of the engineering applications use crisp data as input and produce crisp data as output, the Mamdani system [26] is the most widely used one where the fuzzifier maps crisp inputs into fuzzy sets and the defuzzifier maps fuzzy sets into crisp outputs.Zadeh [27], proposed more sophisticated kinds of fuzzy sets, called type-2 fuzzy sets (T2FSs). A type-2 fuzzy set lets us incorporate uncertainty about the membership function into fuzzy set theory. In order to symbolically distinguish between a type-1 fuzzy set and a type-2 fuzzy set, a tilde symbol is put over the symbol for the fuzzy set; so, A denotes a type-1 fuzzy set, whereas à denotes the comparable type-2 fuzzy set. Mendel and Liang [28, 29] characterized T2FSs using the concept of footprint of uncertainty (FOU), and upper and lower MFs. To depict the concept, let us consider type-1 gauss MF shown in ―Fig. 1‖.As can be seen from the figure type-1 gaussian membership function is constrained to be in between 0 and 1 for all x X, and is a two dimensional function. These types of membership don‘t carry any uncertainty. There exists a clear membership value for every input data point.If the Gaussian function in ―Fig.1‖ is blurred ―Fig. 2‖can be obtained. The FOU represents the bounded region obtained by blurring the boundaries of type-1 MF. The upper and lower MFs represent the upper and lower boundaries of the FOU, respectively. In this case, for a specific input value, there is no longer a single certain value of membership; instead the MF takes on values wherever the vertical line intersects the blur. Those values do not have to be all weighted the same; hence, an amplitude distribution can be assigned to those points. Doing this for all input values x, a three dimensional MF is created, which is a type-2 MF. In this, the first two dimensions allow handlingimprecision via modelling the degree of membership of x; while the third dimension allows handling uncertainty via modelling the amplitude distribution of the degree of membership of x. Here also, like in type-1 MFs the degree of membership along the second dimension and the amplitude distribution values along the third dimension is always in the interval [0, 1]. Clearly, if the blur disappears; then a type-2 MF reduces to a type-1 MF.A general architecture of type-2 fuzzy logic system (T2FL) as proposed by Mendel is depicted in ―Fig. 3‖.Fig. 1: A Gaussian Type-1 membership functionFig. 2: A Gaussian Type-2 membership functionFig. 3: A typical type-2 fuzzy logic system [29]Table 5: Example on FP complexity classificationT2FL systems contain five components –rules, fuzzifier, inference engine, type reducer, and defuzzifier. Rules are the heart of a T2FL system, and may be provided by experts or can be extracted from numerical data. These rules can be expressed as a collection of IF-THEN statements. The IF part of a rule is its antecedent, and the THEN part of the rule is its consequent. Fuzzy sets are associated with terms that appear in the antecedents or consequents of rules, and with inputs to and output of the T2FL system. The inference engine combines rules and gives mapping from input type-2 fuzzy sets to output type-2 fuzzy set. The fuzzifier converts inputs into their fuzzy representation. The defuzzifier converts the output of the inference engine into crisp output. The type reducer transforms the type-2 fuzzy output set into type-1 fuzzy set to be processed by the defuzzifier. A T2FL system is very similar to a T1FL system; the major difference being that the output processing block of T1FL system is just a defuzzifier while the output processing block of a T2FL system contains the type reducer as well. III.Problem Description and AnalysisIn cost estimation process, the primary input is the software size and the secondary inputs are the various cost drivers. There is a significant relationship between the software size and cost. There are mainly two types of software size metrics: lines of code (LOC) and Function Point (FP). Size estimation is best done when there is complete information about the system; but this is not available till the system is actually built. The challenge for the estimator is therefore to arrive at a reasonable estimate of the size of the system with partial information.LOC is usually not available until the coding phase, so FP has gained popularity because it can be used at an earlier stage of software development.In our work, we are using type-2 based fuzzy logic approach to calibrate the function point weight values which provides an improvement in the software size estimation process. There are 15 parameters in the FP complexity weight system to calibrate. These parameters are low, average and high values of External Inputs, External Outputs, Internal Logical Files, External Interface Files and External Inquiries respectively. A fuzzy based approach is chosen since it can capture human‘s judgement with ease and instead of giving an exact number to all 15 function points parameters we can define fuzzy linguistic terms and assign a fuzzy set within numeric range. This provides an ability to cope up with the vagueness and imprecision present in the earlier stages of software development.In Function Point Analysis (FPA) method each component is classified to a complexity level determined by the number of its associated files such as DET, RET or FTR as given in Table 4. If we determine the FPA complexity of a particular software application, in some cases it may not correctly reflect the complexity for its components.Table 5 shows a software project with three EIF‘s A, B and C. According to the complexity matrix, A and B are classified as having the same complexity and are assigned the same weight value of 10. However, A has 19 more DET than B and is certainly more complex. But both of them are assigned the same complexity. Also, EIF C is having only one DET less than EIF B and it is classified as average and assigned a weight value of 7. From the above example it is concluded that there is a huge scope of improvement in the FPA complexity classification. Processing the number of FP component associated files such as DET, RET and FTR using fuzzy logic can provide an exact complexity degree.IV.Fuzzy Logic calibration to improve FPAType-2 fuzzy inference system is developed for all the five FPA components (ILF, EIF, EI, EO, EQ) using the Mamdani approach. We define three new linguistic terms: small, medium and large, to express the inputs qualitatively. Also we use linguistic terms: simple, average and complex for the output. To fuzzify the inputs and outputs, we define fuzzy sets to represent the linguistic terms [30]. The fuzzy membership grade is captured through the membership functions of each fuzzy set. The inputs and outputs are represented using gaussian igaussstype2 membership which is represented in ―Fig. 4‖. It has certain mean m, and an uncertain standard deviation that takes on values in [σ1, σ2]. The shaded area represents the FOU. Using interval type-2 Gaussian MF‘s makes it easier to build T2FL systems since the mathematics behind the corresponding inferential procedures and training algorithms are less complicated [29]. ―Fig.5 (a)‖and ―Fig.5 (b)‖ shows how the inputs of EIF are assigned the membership functions and represented using linguistic variables of fuzzy sets. ―Fig. 6‖ depicts the output of EIF using membership functions. After representing the inputs and output of EIF using membership functions nine fuzzy rules are defined using rule editor based on the original complexity matrices and illustrated in Table 6. Each rule has two parts in its antecedent linked with an ‗AND‘ operatorand one part in its consequence. These fuzzy rules define the connection between the input and output fuzzy variables. A fuzzy rule has the form: IF <Antecedent> THEN <Consequent>, where antecedent is a compound fuzzy logic expression of one or more simple fuzzy expressions connected with fuzzy operators; and the consequent is an expression that assigns fuzzy values to output variables. The inference system evaluates all the rules of the rule base and combines the weights of the consequents of all relevantrules in one fuzzy set using the aggregate operation. Finally, the output fuzzy set is defuzzified to a crisp single number.Fig. 4: FOU for Gaussian MFFig. 5 (a): Input fuzzy set DET for EIFFig. 5 (b): Input fuzzy set RET for EIFFig. 6: Output fuzzy set Complexity for EIFTable 6: Truth table of fuzzy logic rule setFig. 7: Type-2 Fuzzy Inference process of Function Points Model Table 7: Calibration using type-2 fuzzy logicAn example of the complete fuzzy inference process is shown in ―Fig. 7‖. Input values are set to DET 51 and RET 5. These are represented using the antecedent part of the fuzzy rules. Finally, the consequent part isdefuzzified and the output is achieved as a single value of 7.63.A fuzzy logic system for each FPA element (ILF, EIF, EI, EO, EQ) is constructed. A fuzzy complexity measurement system that takes into account all five Unadjusted Function Points function components is built after the fuzzy logic system for each function component is established as shown in ―Fig. 8‖. The calibrated values for EIF A, EIF B and EIF C is listed in Table 7 and it is found that these calibrated weight values are more convincing than the original weight values.Fig. 8: Fuzzy complexity measurement system for Type-2 Fuzzyfunction points modelTable 8: Calculation of t2UFFP and UFP for ILFV.Experimental Methodology and ResultsWe have conducted some experiments to develop a type-2 fuzzy system for function points analysis using our framework as depicted in ―Fig. 8‖. Our model has been implemented in Matlab(R2008a). As it is the case with validating any prediction model, real industrial data necessary to use our framework to develop and tune the parameters of prediction models were not available. To get around this data scarcity problem for the sake of showing the validity of our framework for the industry where organizations have their own data available, we generated artificial datasets consisting of 20 projects. A complexity calculation for all the five components for each project is done using the type-2 fuzzy framework. The Tables (8, 9, 10, 11, 12) lists the complexity values for all the five components for the first project using type-2 fuzzy framework (t2UFFP) and conventional method i.e.UFP.Using ―(1)‖ total unadjusted function points from the type-2 technique and the conventional technique is calculated and listed in Table 13. It is found that the type-2 technique is at par than the conventional technique.Table 9: Calculation of t2UFFP and UFP for EIFTable 10: Calculation of t2UFFP and UFP for EITable 11: Calculation of t2UFFP and UFP for EOTable 12: Calculation of t2UFFP and UFP for EQTable 13: Comparison of t2UFFP and UFPTable 14: Comparison of type-2 fuzzy FP and conventional FPIn order to compute the value of the conventional function point and type-2 fuzzy function point, we have treated all the 14 general system characteristics as average. Using ―(2)‖and ―(3)‖VAF and FPA is calculated and listed in Table 14.From the above results it is concluded that the calibrated function points using type-2 fuzzy yields better results than conventional function points.VI.ConclusionsFP as a software size metric is an important topic in the software engineering domain. The use of type2 fuzzy logic to calibrate FP weight values further improves the estimation of FP. This in turn will improve the cost estimation process of software projects. Empirical evaluation has shown that T2FL is promising. But there are potentials for improvements when the framework is deployed in practice. As all the experiments were conducted using artificial datasets, a need to evaluate the prediction performance of the framework on real data still persists. Some future work can be directed towards developing inferential procedures using various other membership functions present in type-2 fuzzy systems. This work can also be extended using Neuro Fuzzy approach. AcknowledgementThe authors would like to thank the anonymous reviewers for their careful reading of this paper and for their helpful comments.References[1] B.W. Boehm. Software Engineering Economics.Prentice Hall, Englewood Cliffs, NJ, 1981.[2] B. Boehm, B. Clark, E. Horowitz, R. Madachy, R.Shelby, C. Westland. Cost models for future software life cycle processes: COCOMO 2.0.Annals of Software Engineering, 1995.[3]L.H. Putnam. A general empirical solution to themacro software sizing and estimation problem.IEEE Transactions on Software Engineering, vol.4, 1978, pp 345-361.[4]Moataz A. Ahmed, Zeeshan Muzaffar. Handlingimprecision and uncertainty in software development effort prediction: A type-2 fuzzylogic based framework. Information and Software Technology Journal. vol. 51, 2009, pp. 640-654. [5]Function Point Counting Practices Manual, fourthedition, International Function Point Users Group, 2004.[6]G. Antoniol, C. Lokan, G. Caldiera, R. Fiutem. Afunction point like measure for object oriented software. Empirical Software Engineering. vol. 4, 1999, pp. 263-287.[7]Fei. Z, X. Liu. f-COCOMO-Fuzzy ConstructiveCost Model in Software Engineering. Proceedings of IEEE International Conference on Fuzzy System. IEEE Press, New York, 1992, pp. 331-337.[8]J. Ryder. 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International Journal of computer, Internet and management vol. 13, 2005, pp. 31-40.[14]G.R. Finnie, G.E. Wittig, J.M. Desharnais. Acomparison of software effort estimation techniques: using function points with neural networks, case-based reasoning and regression models. Journal of Systems Software, Elsevier.vol. 39, 1977, pp. 281-289.[15]M.A. Al-Hajri, A.A.A Ghani, M.S. Sulaiman,M.H. Selamat. Modification of standard function point complexity weights system. Journal of Systems and Software, Elsevier,vol. 74, 2005, pp.195-206.[16]O.S. Lima, P.F.M. Farias, A.D. Belchior. Fuzzymodeling for function point analysis. Software Quality Journal, vol. 11, 2003, pp. 149-166. [17]C. Yau, H. L. Tsoi. Modelling the probabilisticbehavior of function point analysis. Journal ofInformation and Software Technology, Elsevier.vol. 40, 1998, pp. 59-68.[18]A. Abran, P. Robillard. Function Points Analysis:An empirical study of its measurement processes.IEEE Transactions on Software Engineering, vol.22, 1996, pp.895-910.[19]T. Kralj, I. Rozman, M. Hericko, A. Zivkovic.Improved standard FPA method- resolving problems with upper boundaries in the rating complexity process. Journal of Systems and Software, Elsevier, vol. 77, 2005, pp. 81-90. [20]Wei Xia, Luiz Fernando Capretz, Danny Ho,Faheem Ahmed. A new calibration for function point complexity weights. Journal of Information and Software Technology, Elsevier. vol. 50, 2008 pp.670-683.[21]Mohd. Sadiq, Farhana Mariyam, Aleem Ali,Shadab Khan, Pradeep Tripathi. Prediction of Software Project Effort using Fuzzy Logic.Proceedings of IEEE International Conference on Fuzzy System, 2011, pp. 353-358.[22]A. Albrecht. Measuring application developmentproductivity. Proceedings of the Joint SHARE/GUIDE/IBM Application Development Symposium, 1979, pp. 83-92.[23] L. A. Zadeh. Fuzzy Sets. Information and Control,vol. 8, 1965, pp. 338-353.[24]M. Wasif Nisar, Yong-Ji Wang, Manzoor Elahi.Software Development Effort Estimation using Fuzzy Logic – A Survey. Fifth International Conference on Fuzzy Systems and Knowledge Discovery, 2008, pp 421-427.[25]L. Wang. Adaptive Fuzzy System and Control:Design and Stability Analysis. Prentice Hall, Inc., Englewood Cliffs, NJ 07632, 1994.[26]E.H. Mamdani. Applications of fuzzy algorithmsfor simple dynamic plant. Proceedings of IEEE, vol. 121, 1974, pp. 1585-1588.[27]L. A. Zadeh. The Concept of a Linguistic Variableand Its Application to Approximate Reasoning–1. Information Sciences, vol. 8, 1975, pp. 199-249.[28]J.M. Mendel, Q. Liang. Pictorial comparison ofType-1 and Type-2 fuzzy logic systems.Proceedings of IASTED International Conference on Intelligent Systems and Control, Santa Barbara, CA, October 1999.[29]J.M. Mendel. Uncertain Rule-Based Fuzzy LogicSystems, Prentice Hall, Upper Saddle River, NJ 07458, 2001.[30]E.H. Mamdani. Application of fuzzy logic toapproximate reasoning using linguistic synthesis.IEEE transactions on computers, vol. 26, 1977, pp.1182-1191. Anupama Kaushik is an Assistant Professor at Maharaja Surajmal Institute of Technology, New Delhi, India. Her research area includes Software Engineering, Object Oriented Software Engineering and Soft Computing.Dr. A.K Soni has done his Ph.D. and M.S.(Computer Science) both from Bowling Green State University in Ohio, USA . He is the Professor and Head, Department of Information Technology, Sharda University, Greater Noida, India. His research area includes Software Engineering, Datamining, Database Management Systems and Object Oriented Systems.Dr. Rachna Soni did her M. Phil from IIT Roorkee and Ph.D. from Kururukshetra University, Kurukshetra. She is the Associate Professor and Head, Dept. of Computer Science and Applications, D.A.V. College, Yamunanagar, India. Her area of interest includes Software Risk Management, Project Management, Requirement Engineering, Simulation and Component based Software Engineering.。

几类非线性系统的自适应Backstepping模糊控制研究几类非线性系统的自适应Backstepping模糊控制研究摘要：自适应控制是一种有效的控制方法，能够适应系统的变化并保持控制性能。

Backstepping控制是一种常用的非线性控制方法，可以通过逐步构造Lyapunov函数来实现系统的稳定控制。

在本文中，我们通过将自适应控制与Backstepping控制相结合，并引入模糊逻辑，研究了几类非线性系统的控制问题。

我们首先介绍了自适应控制和Backstepping控制的基本原理，然后通过数学推导得出了自适应Backstepping模糊控制器的设计方法，并通过仿真实例验证了该方法的有效性。

关键词：自适应控制；Backstepping控制；模糊逻辑；非线性系统1. 引言随着科技的不断进步，控制系统的应用范围越来越广。

然而，由于现实世界中的许多系统具有复杂的非线性特性，传统的线性控制方法往往无法满足控制要求。

因此，研究非线性控制方法成为当前的热点问题之一。

2. 自适应控制的基本原理自适应控制是一种根据系统的状态变化来自动调整控制参数的控制方法。

其基本原理是通过观测系统的输出和状态，利用适应法则来实时调整参数，以达到控制系统稳定性和性能的要求。

自适应控制可以根据系统模型的误差进行参数调整，从而保持控制系统的性能。

3. Backstepping控制的基本原理Backstepping控制是一种逐步构造Lyapunov函数的非线性控制方法。

其基本思想是将控制问题分解为多个子问题，并逐步设计反馈控制律。

通过反复迭代设计控制器，每次迭代都使系统的Lyapunov函数下降，最终达到系统的稳定控制。

这种方法能够有效处理非线性系统控制问题。

4. 自适应Backstepping模糊控制器设计在研究过程中，我们发现将自适应控制与Backstepping控制相结合可以提高系统的鲁棒性和控制性能。

同时，引入模糊逻辑可以处理系统中的不确定性和模糊性，进一步提高控制器的鲁棒性。

兵工自动化Ordnance Industry Automation 2021-0140(1)・27・doi: 10.7690/bgzdh.2021.01.007基于永磁调速直流电机模糊PID 伺服控制系统李广伟J 张红萍J 刘涵睿2,周进良2,陈松波2(1.陆军装备部驻重庆地区军事代表局驻广元地区军事代表室，四川广元628017；2.中国兵器装备集团火控技术中心，四川 成都611731)摘要：为改善永磁调速直流电机调速系统的性能，提岀一种模糊PID 控制策略。

根据模糊理论的基本规则，使 用Matlab 生成模糊控制器，选取电机的实际参数，依据伺服控制策略，将模糊理论与PID 控制技术相结合，使用重 心法解模糊化，得到PID 的控制参数，并通过Matlab 仿真对比模糊PID 控制与单独PID 控制对信号的影响。

仿真结 果表明：模糊PID 控制性能要优于单独PID 控制，可满足直流电机的高精度平稳运行。

关键词：直流电机；PID 控制；模糊理论中图分类号：TP273 文献标志码：AFuzzy PID Servo Control System Based on Permanent Magnet Speed-regulating DC MotorLi Guangwei 1, Zhang Hongping 1, Liu Hanrui 2, Zhou Jinliang 2, Chen Songbo 2(1. Military Representative Office in Guangyuan District, Military Representative Bureau ofArmy Equipment Department in Chongqing, Guangyuan 628017, China;2. Fire Control Technology Center of China South Industries Group Corporation, Chengdu 611731, China)Abstract: In order to improve the performance of the speed control system of permanent magnet speed-regulated DC motor, a fuzzy PID control strategy is proposed. According to the basic rules of fuzzy theory, use Matlab to generate a fuzzy controller, select the actual parameters of the motor, combine the fuzzy theory with PID control technology according to the servo control strategy, use the center of gravity method for fuzzification, get the PID control parameters, and pass Matlab simulation contrasts the influence of fuzzy PID control and individual PID control on the signal. The simulation results show that the fuzzy PID control performance is better than the single PID control, which can meet the high precision and smooth operation of the DC motor.Keywords: DC motor; PID control; fuzzy theory0引言永磁调速直流电机作为调速系统具有结构简单、体积小、质量轻、效率高、功率因数高等优势。

机电工程学院课程设计报告课程题目二阶系统模糊控制算法的研究专业电气工程及其自动化姓名指导教师学期2015-2016二阶系统的模糊控制算法的研究学生指导老师:摘要:模糊控制是以模糊数学为基础发展的，为一些无法建立数学模型或者数学模型相当粗糙的系统提供的一种非线性的控制方法。

对于这些系统，模糊控制可以得到比较满意的控制效果，并且能够解决一些无法通过传统方法解决的问题。

本文利用MATLAB模糊控制工具箱设计的模糊控制器来控制一个二阶系统，由给定的控制器的输入和输出变量，输入和输出变量的隶属函数，分析了输入和输出变量之间的关系，设计了模糊控制规则库，并通过SIMULINK仿真将模糊控制方法与经典的PID控制方法进行对比，分析仿真结果，探讨模糊控制器的隶属函数，控制规则，以及量化因子和比例因子在模糊控制中所起到的作用。

关键字：模糊控制；MATLAB；SIMULINK；PIDResearch of fuzzy control algorithm of secondorder systemsUndergraduate:Supervisor:Abstract：Fuzzy control, which is based on the fuzzy mathematics, is a new way of nonlinearity control system in which the mathematical model is unable established or the mathematical model is very rough. For these systems, fuzzy control offers users a satisfied control result, and settles down some problems which cannot be solved by traditional methods.This paper aims to introduce how to use a fuzzy controller which is based on the MATLAB fuzzy control toolbox to control a second-order system. In order to fulfill this target, the author firstly defines the input variables, output variables and their membership functions. Then, the author analyzes the relationship between the input variables and output variables, and designs the fuzzy control rule bank. Finally, the author makes a difference between the methods of the classic PID control and the fuzzy control by SIMULINK. Membership function of fuzzy controller, control rules, and the function of quantizes and scale factor in the fuzzy control process are also discussed in this paper.Key words: MATLAB; Fuzzy control; PID;SIMULINK simulation目录绪论 (1)1控制理论算法 (5)3.1 PID控制规律 (5)3.1PID控制器原理 (5)3.1 (5)3.1.3微分（D）控制 (5)3.2传统PID控制过程 (5)1模糊控制 (1)1.1模糊控制的背景及意义 (1)1.2模糊控制的基本理论 (1)1.3模糊控制的基本结构 (1)1.4模糊控制的组成 (1)1.5模糊控制的运行模型 (1)1.6模糊控制与SIMULINK的 (1)2基于MATLAB的模糊控制仿真 (2)2.1模糊控制器的设计 (2)2.1模糊集合及论域的定义 (2)2.2模糊控制规则确定 (4)2.3仿真曲线 (5)4比较 (7)参考文献 (8)致 (9)绪论模糊控制器由三个环节组成：用于输入信号处理的模糊量化和模糊化环节，模糊控制算法功能单元，以及用于输出解模糊化的模糊判决环节。

现代控制理论_哈尔滨工程大学中国大学mooc课后章节答案期末考试题库2023年1.已知线性定常系统如下所示，下面说法错误的是（）【图片】参考答案:引入状态反馈后，不改变系统的能观测性。

2.串联组合系统的传递函数矩阵为各串联子系统的传递函数矩阵之和。

参考答案:错误3.在最优控制问题中，如果系统的性能指标是状态变量和控制变量的二次型函数，则称为线性二次型最优控制问题，简称LQ（Linear Quadratic）问题。

参考答案:错误4.用不大的控制能量，使系统输出尽可能保持在零值附近，这类问题称为输出调节器问题。

参考答案:正确5.研究系统控制的一个首要前提是建立系统的数学模型，线性系统的数学模型主要有两种形式，即时间域模型和频率域模型。

参考答案:正确6.现代控制理论以多变量线性系统和非线性系统作为研究对象，以时域法，特别是状态空间方法作为主要的研究方法。

参考答案:正确7.1892年俄国数学家李亚普诺夫发表了论文《运动稳定性的一般问题》，用严格的数学分析方法全面地论述了稳定性问题。

参考答案:正确8.经典控制理论以单变量线性定常系统作为主要的研究对象，以时域法作为研究控制系统动态特性的主要方法。

参考答案:错误9.下述描述中哪些作为现代控制理论形成的标志（）参考答案:用于系统的整个描述、分析和设计过程的状态空间方法._最优控制中的Pontriagin极大值原理和Bellman动态规划。

_随机系统理论中的Kalman 滤波技术。

10.内部稳定性表现为系统的零初态响应，即在初始状态恒为零时，系统的状态演变的趋势。

参考答案:错误11.系统矩阵A所有特征值均具有负实部是线性时不变系统渐近稳定的充要条件。

参考答案:正确12.从物理直观性看，能观测性研究系统内部状态“是否可由输入影响的问题”。

参考答案:错误13.由系统结构的规范分解所揭示，传递函数矩阵一般而言只是对系统结构的不完全描述，只能反映系统中的能控能观测部分.参考答案:正确14.下面论述正确的是（）参考答案:李亚普诺夫意义下渐近稳定等同于工程意义下稳定。

Leena G.2 , Professor
EEE Dept, Faculty of Engineering and Technology, ManavRachna International University , India Email: leenag.fet@.in M. K. Soni 3 , Exe. Director and Dean, Faculty of Engineering and Technology, ManavRachna International University , India Email: ed.fet@.in
I. INT RODUCT ION In the region of control systems [1], the inverted pendulum method is a normal problem. For assessing the justification of different kinds of control theories for mechanical systems, an inverted pendulum is familiar which is tough to be controlled by a valid time controller [2] [3]. An Inverted Pendulu m is aneasy pendulum that is apprehended in an erect vertical position. It has a cart and a simp le rod (with or without bob) [4]. Two equilibria, one of which is stable while the other is unstable [5] are found. Rotational single- arm pendulu m, the double arm pendulum and the cart inverted pendulum, etc. [6] are the most well-known kinds of the inverted pendulum. A mu ltivariable, nonlinear, fast reaction, unsteady and high order system [7] [8] is the cart type inverted pendulum system. The general cart-double-pendulum system (CDPS) contains two-link man ipulator mounted on a cart where only the cart is set in motion and the Copyright © 2014 MECS

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非线性系统智能控制算法的研究与应用随着科技的快速发展，非线性系统在现代工程技术中的应用越来越广泛。

非线性系统问题的解决需要相对复杂的数学和物理模型，加上非线性系统难以发现特征值，对传统控制方法的稳定性等方面的要求也更高。

智能控制算法的出现给非线性系统的问题提供了有效的解决方案，大大提高了非线性系统控制的精度和效率。

非线性系统智能控制算法主要包括模糊控制、神经网络控制、遗传算法控制等，这些算法的出现一定程度上弥补了传统线性控制系统在非线性控制领域的不足，也取得了很高的应用价值。

一、模糊控制模糊控制是通过提供符合人类的思维模式来实现自适应控制的一种方法，它可以在非线性系统控制中应用。

该控制方法的优势在于对于复杂问题建模能力强，适合应用于非线性的控制问题中。

在智能控制中，模糊控制器主要由何种控制规则构成，以及如何对它们进行分析来确定输出控制的变量值。

模糊控制器通常将小数、整数和语言描述转化为逻辑形式，这种方法可以避免特定条件的误导性对大多数系统的控制造成的影响，提高了非线性系统控制精度和效率。

二、神经网络控制神经网络控制是一种由多个神经元组成的复杂系统，结构类似于人类大脑的神经网络。

神经网络控制器可以对系统的非线性动态行为进行预测，通过学习和训练使控制策略不断优化，从而提高控制精度和效率。

与模糊控制相比，神经网络控制在设计时不需要任何的数学模型，更加适合复杂系统和噪声较大情况下的控制。

三、遗传算法控制遗传算法控制是一种基于遗传学原理的智能控制方法，该算法通过将控制参数进行编码，利用自然选择和遗传变异的机制进行控制策略的优化，达到提高非线性系统控制效率的目的。

对于非线性系统，遗传算法控制可以通过一定的迭代计算使得得到的控制策略得到优化，达到自适应控制达到优化效果。

综上所述，智能控制在非线性系统控制中发挥着重要作用。

模糊控制、神经网络控制和遗传算法控制在非线性系统控制应用中起着重要的促进作用，进一步推动了非线性系统控制技术的发展。

Solution of Weak Nonlinear Dynamic Equations based on Second Order Orthogonal Method
XIN Zhenfang 1, LU Hongya 1, SONG Yu 2, HAN Shuyong 1
(1. Beijing Institute of Mechanical Equipment, Beijing 100854, China; 2. The Military Office of Chinese People's Liberation Army for 206, 2nd Institute of CASIC,
第 38 卷 第 Z1 期 2018 年 4 月
噪声与振动控制 NOISE AND VIBRATION CONTROL
文章编号：1006-1355(2018)Z1-0100-04
非线性方程的 2 阶正交解法
Vol 38 No.Z1 Apr. 2018
辛振芳 1，陆红亚 1，宋 宇 2，韩书永 1
（1. 北京机械设备研究所，北京 100854； 2. 中国人民解放军 驻航天科工集团第二研究院 二〇六所军事代表室，北京 100854 ）
力学方程的频响函数表达通式以及动力学方程的响应，并通过对 Duffing 方程进行算例说明。
关键词：振动与波；非线性；动力学方程；2 阶正交法；动力学响应；频响函数
中图分类号：O322；TB123；231.92
文献标志码：A
DOI 编码：10.3969/j.issn.1006-1355.2018.Z1.021
其中：x 是 N × 1 位移向量；[M] 、[C] 、[K] 分别是系
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