模糊PID控制器的鲁棒性研究外文文献翻译.doc

毕 业 设 计（论 文）外 文 文 献 译 文 及 原 文基于内模控制的模糊PID参数的整定Xiao-Gang Duan, Han-Xiong Li,and Hua DengSchool of Mechanical and Electrical Engineering, Central South UniVersity, Changsha 410083, China, and Department of Manufacturing Engineering and Engineering Management, City UniVersity of Hong Kong, Hong Kong摘要:在本文中将利用内模控制的整定方法实现模糊PID控制。此种控制方式首次应用于模糊PID控制器，它包括一个线性PID控制器和非线性补偿部分。非线性补偿部分可视为一个干扰过程，模糊PID控制器的参数可在分析的基础上确定内模结构。模糊PID控制系统利用李亚谱诺夫稳定性理论进行稳定性分析。仿真结果表明利用内模控制整定模糊PID控制参数是有效的。1 引言一般而言，传统的PID控制器对于十分复杂的被控对象控制效果不太理想, 如高阶时滞系统。在这种复杂的环境下, 众所周知,模糊控制器由于其固有的鲁棒性可以有更好的表现，因此，在过去30年中，模糊控制器，特别是,模糊PID控制器因其对于线性系统和非线性系统都能进行简单和有效的控制，已被广泛用于工业生产过程1-4。 模糊PID控制器有多种形式5,如单输入模糊PID控制器,双输入模糊PID控制器和三个输入的模糊PID控制器。一般情况下，没有统一的标准。单输入可能会丢失派生信息, 三输入模糊PID控制器会产生按指数增长的规则。在本文中所采用的双输入模糊PID控制器有一个适当的结构并且实用性强，因此在各种研究和应用中,是最流行的模糊PID 类型。尽管业界对于应用模糊PID有越来越大的兴趣，但从控制工程的主流社会的角度来看,它仍然是一个极具争议的话题。原因之一是模糊PID参数整定的基本理论分析方法至今仍不明确。因此，模糊 PID控制器不得不进行两个级别的整定。在较低层次上， 该整定是由调整增益获得线性控制性能。在更高层次上的调整，是由改变知识库参数以提高控制性能, 然而调整知识库参数很难,此外，很难通过改变参数特性改善瞬态响应。根据知识库传达一般控制规则倾向于保持成员函数不变，通过离线设计和调试工作扩大增益，然而，由于由模糊PID控制器生成非线性控制表面的复杂性，调整机制的衡量因素和稳定性分析仍然是艰巨的任务。如果非线性能得到适当的利用，模糊PID控制器可能得到比传统PID控制器更好的系统性能。一些非常规的调整方法已进行了介绍9-12。虽然非线性被认为是在增益裕度和相位裕度基础上获得的，但是由于非线性因素，模糊PID控制器可能会产生比常规PID控制器较高的增益。而高增益可能使控制系统的稳定性变差。常规PID控制器很容易实现，大量的整定规则可以涵盖广泛的进程规格。在常规PID控制器的整定方法中，内模控制基础整定是在商业PID控制软件包中流行的方法之一，因为只需调整一个参数，便可以生产更好的设置点响应15。本文提出了一种基于内模控制的PID控制器的整定分析方法，模糊PID 控制器可分解为线性PID控制器加上非线性补偿部分的控制器。把非线性补偿部分近似看作一个过程干扰，模糊PID参数就可以分析设计使用内模控制。模糊PID控制器的稳定性分析是根据李亚谱诺夫稳定性理论。最后，通过仿真来证明此种调整方法是有效的。2 问题的提出2.1 常规PID控制器常规PID 控制器通常被描述为下列方程8-10：= （1） 其中E是跟踪误差，kp 是比例增益，ki是积分增益，kd是微分增益，Ti和TD分别是积分时间常数和微分时间常数，这些控制参数的关系是KI =KP/Ti 和KD =KPTd。PID控制器的传递函数可以表示如下： （2）在根轨迹中，PID控制器有两个零点和,一个极点是原点。条件是两个零点满足大于4。CP+udey+_yr 图1 内模控制配置图(a) +yedurP_图2 内模控制配置图(b)2.2 内模控制原则基本的内模控制原则如图1所示，其中P是被控对象，P是名义上的模型对象，C是控制器，r和d是设置点和干扰，y 和 yk分别是被控对象的输出和模型对象的输出。内模控制结构相当于古典单闭环反馈控制器如图1（b）所示，如果单闭环控制器如下： （3）及 （4）其中(s)是被控模型的最小相位部分， 包含任何时间延迟和右零点，f(s)是一个低通滤波器，一般形式是： （5） 调整参数tc是理想闭环时间常数n是一个待定的正整数。KiKdRuleBasesERu 图3 模糊PID控制器结构2.3 模糊PID控制器模型模糊PID控制器如图2所示，形式为：及 （6） 是一种非线性的时间变量参数(), A和B分别是每个输入和输出的成员函数一半的外延。模糊PID控制实际上有两个层次的增益。扩大增益(Ke, Kd, K0, 和K1)处于较低的水平。扩大增益的调整将会影响模糊PID控制器效果，造成控制参数的不断变化。作为控制行为的模糊耦合控制， Ke, Kd, K0, 和 K1以何种不同的控制行动仍然没有非常清楚，这使得实际设计和调试过程相当困难。3 基于内模控制的模糊PID整定在模糊PID控制器整定的基础上的内模控制方法，通过分析模糊PID控制模型得到第一个简单推导。然后，参数模糊PID 控制器可在内模控制的基础上确定参数。假设一个工业过程可以模仿成一阶加上延迟（ FOPDT ）环节，传递函数如下： (7) 其中K、T和 L分别是稳态增益，时间常数，和延迟时间，这些参数通过阶跃响应法，频率响应，和闭环继电反馈等方法来描述的，FOPDT是一种最常见最实用的模型，尤其是在过程控制中18。通过式（6）可以得到： （8） （9） （10）是一个非线性项，没有明确的分析表达。显然，模糊PID控制可视为常规PID的非线性补偿。常规PID控制部分是UPID(s)， 非线性补偿部分是UN(s)。基于内模控制的模糊PID整定。如果我们考虑非线性补偿UN(s)作为一个过程的干扰，并设置为Gf(s)如图3，基于内模控制的模糊PID控制器可简化如下： (11)因此，为 可以分解为= ，其中 (12)从而得到 （13）模糊PID在第k水平上的带宽可以通过适合的来控制。带宽和快速的反应，的值越小可得到较大的带宽和较快的响应速度，否则带宽变小 ，响应缓慢，因此，为了提高上升时间，的值应该小，所以，两个参数和可得到确定。备注：模糊PID控制实际上是一个传统PID控制器uPID加上滑动控制。由于滑模控制是一种鲁棒控制所以模糊PID控制是力的比传统的PID控制有更好的鲁棒性。4 控制仿真在这一节中， 通过上述方法进行模糊PID整定的控制性能与常规PID的比较，选择IEA和ITAE作为标准，数值越小意味着控制性能越好。 （14） 在所有控制仿真中常规PID控制参数是由内模控制方法决定的，模糊PID控制参数是由上述整定方法确定的。范例1 考虑一个工业过程，所描述的一阶延迟环节，模型函数如下： （15）线性部分在过程中占主导地位。小延迟时间意味着弱非线性特性。由图5可以看出，由于延迟时间小，常规PID控制和模糊PID控制差异不大。然而，当延迟时间增加至L= 0.6，如图6 ，模糊PID控制实现了优于常规PID控制控制性能。此外模糊PID控制器增益低于常规PID控制器。 图4 范例1中模糊PID控制（实线）和常规 图5 延迟时间增加至L= 0.6，模糊PID控PID控制（虚线）性能比较 制（实线）和常规PID控制（虚线）性能比较 范例2 假设一工业过成描述如下： （16）其中a=1，假设不存在建模误差，在阶跃响应和奈奎斯特工业过程曲线基础上可获得逼近模型如下： （17）如图7所示，常规PID控制和模糊PID控制差异不大。因为该模型是正确的。但是，假设有建模误差和参数a的实际值是0.95 。如图8，模糊PID控制比常规PID控制实现更好的控制性能。此外，由图8可以看出模糊PID 控制器增益低于常规PID控制器。 图6 a=1时，模糊PID控制（实线）和常规 图7 a=0.95时，模糊PID控制（实线）和常规PID控制（虚线）性能比较 PID控制（虚线）性能比较5 结论本文介绍了一种基于内模控制的模糊PID控制器的整定分析方法。解析模型是第一次应用于模糊PID控制器的整定。分析模型包括一个线性PID控制及非线性补偿部分。在内模控制方法基础上， 模糊PID控制器的参数可由过程干扰的补偿部分来分析确定。虽然扩大收益 和是耦合的，这一程序是在解耦基础上的滑动模型控制。稳定性分析表明，该控制系统是全局渐近稳定的。 模糊PID控制器采用此种整定方法比传统的PID控制器有更的鲁棒性强大。仿真结果表明，模糊PID控制器通过此种整定方法，与传统的PID控制器相比在动态和静态上都实现更好的控制性能和更好的鲁棒性。参考文献(1) Sugeno. 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Syst., Man, Cybernetics, Part B 1997, 27 (2), 306312.Effective Tuning Method for Fuzzy PID with Internal Model ControlXiao-Gang Duan, Han-Xiong Li, and Hua DengSchool of Mechanical and Electrical Engineering, Central South UniVersity, Changsha 410083, China, and Department of Manufacturing Engineering and Engineering Management, City UniVersity of Hong Kong, Hong KongAn internal model control (IMC) based tuning method is proposed to auto tune the fuzzy proportional integral derivative (PID) controller in this paper. An analytical model of the fuzzy PID controller is first derived, which consists of a linear PID controller and a nonlinear compensation item. The nonlinear compensation item can be considered as a process disturbance, and then parameters of the fuzzy PID controller can be analytically determined on the basis of the IMC structure. The stability of the fuzzy PID control system is analyzed using the Lyapunov stability theory. The simulation results demonstrate the effectiveness of the proposed tuning method.1. IntroductionGenerally speaking, conventional proportional integral derivative (PID) controllers may not perform well for the complex process, such as the high-order and time delay systems. Under this complex environment, it is well-known that the fuzzy controller can have a better performance due to its inherent robustness. Thus, over the past three decades, fuzzy controllers, especially, fuzzy PID controllers have been widely used for industrial processes due to their heuristic natures associated with simplicity and effectiveness for both linear and nonlinear systems.1-4 There are too many variations of fuzzy PID controllers,such as, one-input, two-input, and three-input PID type fuzzy controllers. In general, there is no standard benchmark. The one-input may miss more information on the derivative action, and the three-input fuzzy PID controllers may cause exponential growth of rules. The two-input fuzzy PID, as we used in the paper, has a proper structure and the most practical use, and thus is the most popular type of fuzzy PID used in various research and application. Despite the fact that industry shows greater and greater interest in the applications of fuzzy PID, it is still a highly controversial topic from the point of view of the mainstream control engineering community. One reason is that the fundamental theory for the analytical tuning methods of fuzzy PID is still missing. Therefore, fuzzy PID controllers had to be tuned qualitatively by two-level tuning. At a lower level, the tuning is performed by adjusting the scaling gains to obtain overall linear control performance. At a higher level, the tuning is performed by changing the knowledge base parameters to enhance the control performance. However, it is difficult to tune the knowledge base parameters. Moreover, it is hard to improve the transient response by changing the member function.As the knowledge base conveys a general control policy, it is preferred to keep the member function unchanged and to leave the design and tuning exercises to scaling gains. However, the tuning mechanism of scaling factors and the stability analysis are still difficult tasks due to the complexity of the nonlinear control surface that is generated by fuzzy PID controllers. If the nonlinearity can be suitably utilized, fuzzy PID controllers may pose the potential to achieve better system performance than conventional PID controllers. Some nonanalytical tuning methods were introduced.9-12 Although the nonlinearity was considered on the basis of gain margin and phase margin specifications, the fuzzy PID controller may produce higher gains than conventional PID controllers due to the nonlinear factor. A high gain could deteriorate the stability of the control system.15 The conventional PID controller is easy to implement, and lots of tuning rules are available to cover a wide range of process specifications. Among tuning methods of the conventional PID controller, the internal model control (IMC) based tuning is one of the popular methods in commercial PID software packages because only one tuning parameter is required and better set point response can be produced.17An analytical tuning method based on IMC to tune fuzzy PID controllers is proposed in this paper. The fuzzy PID controller is first decomposed as a linear PID controller plus an onlinear compensation item. When the nonlinear compensation item is approximated as a process disturbance, the fuzzy PID scaling parameters can then be analytically designed using the IMC scheme. The stability analysis of the fuzzy PID controllers is given on the basis of the Lyapunov stability theory. Finally, the effectiveness of the tuning methodology is demonstrated by simulations.2 Problem Formulation2.1 Conventional PID ControllerThe conventional PID controller is often described by the following equation:20,21= （1）where e is the tracking error, KP is the proportional gain, KI is the integral gain, KD is the derivative gain, and Ti and Td are the integral time constant and the derivative time constant, respectively. The relationships between these control parameters are KI = KP/Ti and KD= KPTd. The transfer function of the PID controller (1) can be expressed as follows: （2）On the root-locus plane, the PID controller has two zeros ti and td, and one pole at the origin. The condition to have real zeros is that Ti 4Td.CP+udey+_yr_Figure 1 IMC configuration（a）Pre_+udyFigure 2 IMC configuration (b)2.2 Principle of IMC The basic IMC principle is shown in Figure 1a, where P is the plant, P is a nominal model of the plant, C is a controller; r and d are the set point and the disturbance, and y and yk are the outputs of the plant and its nominal model, respectively.The IMC structure is equivalent to the classical single-loop feedback controller shown in Figure 1b. If the single-loop controller CIMC is given by（3）with （4）where P (s)=P -(s)P +(s), P -(s) is the minimum phase part of the plant model, P +(s) contains any time delays and right-half plane zeros, and f(s) is a low-pass filter with a steady-state gain of one, which typically has the form: （5） The tuning parameter tc is the desired closed-loop time constant, and n is a positive integer to be determined.KiKdRuleBasesERuFigure Figure 3 Fuzzy-PID controller structure2.3 Model of Fuzzy PID ControllerThe fuzzy PID controller, as shown in Figure 2, is described as follows: (6)with is a nonlinear time varying parameter(), A and B are half of the spread of each input and out member function, respectively.The fuzzy PID control actually has two levels of gains.6 The scaling gains (Ke, Kd, K0, and K1) are at the lower level. The tuning of these scaling gains will affect the gains of fuzzy PID The fuzzy PID control actually has two levels of gains.6 The scaling gains (Ke, Kd, K0, and K1) are at the lower level. The tuning of these scaling gains will affect the gains of fuzzy PID controlle