外文翻译---遗传算法在非线性模型中的应用

英文翻译

2011 届电气工程及其自动化专业 0706073 班级

题目遗传算法在非线性模型中的应用

姓名学号070607313

英语原文：

Application of Genetic Programming to Nonlinear

Modeling

Introduction

Identification of nonlinear models which are based in part at least on the underlying physics of the real system presents many problems since both the structure and parameters of the model may need to be determined. Many methods exist for the estimation of parameters from measures response data but structural identification is more difficult. Often a trial and error approach involving a combination of expert knowledge and experimental investigation is adopted to choose between a number of candidate models. Possible structures are deduced from engineering knowledge of the system and the parameters of these models are estimated from available experimental data. This procedure is time consuming and sub-optimal. Automation of this process would mean that a much larger range of potential model structure could be investigated more quickly.

Genetic programming (GP) is an optimization method which can be used to optimize the nonlinear structure of a dynamic system by automatically selecting model structure elements from a database and combining them optimally to form a complete mathematical model. Genetic programming works by emulating natural evolution to generate a model structure that maximizes (or minimizes) some objective function involving an appropriate measure of the level of agreement between the model and system response. A population of model structures evolves through many generations towards a solution using certain evolutionary operators and a “survival-of-the-fittest”selection scheme. The parameters of these models may be estimated in a separate and more conventional phase of the complete identification process.

Application

Genetic programming is an established technique which has been applied to several nonlinear modeling tasks including the development of signal processing algorithms and the identification of chemical processes. In the identification of continuous time system models, the application of a block diagram oriented simulation approach to GP optimization is discussed by Marenbach, Bettenhausen and Gray, and the issues involved in the application of GP to nonlinear system identification are discussed in Gray ?s another paper. In this paper, Genetic programming is applied to the identification of model structures from experimental data. The systems under investigation are to be represented as nonlinear time domain continuous dynamic models.

The model structure evolves as the GP algorithm minimizes some objective function involving an appropriate measure of the level of agreement between the model and system responses. One examples is

∑==n i e J 121 (1)

Where 1e is the error between model output and experimental data for each of N data points. The GP algorithm constructs and reconstructs model structures from the function library. Simplex and simulated annealing method and the fitness of that model is evaluated using a fitness function such as that in Eq.(1). The general fitness of the population improves until the GP eventually converges to a model description of the system.

The Genetic programming algorithm

For this research, a steady-state Genetic-programming algorithm was used. At each generation, two parents are selected from the population and the offspring resulting from their crossover operation replace an existing member of the same population. The number of crossover operations is equal to the size of the population i.e. the crossover rate is 100℅. The crossover algorithm used was a subtree crossover with a limit on the depth of the resulting tree.

Genetic programming parameters such as mutation rate and population size

varied according to the application. More difficult problems where the expected model structure is complex or where the data are noisy generally require larger population sizes. Mutation rate did not appear to have a significant effect for the systems investigated during this research. Typically, a value of about 2℅was chosen.

The function library varied according to application rate and what type of nonlinearity might be expected in the system being identified. A core of linear blocks was always available. It was found that specific nonlinearity such as look-up tables which represented a physical phenomenon would only be selected by the Genetic Programming algorithm if that nonlinearity actually existed in the dynamic system.

This allows the system to be tested for specific nonlinearities.

Programming model structure identification

Each member of the Genetic Programming population represents a candidate model for the system. It is necessary to evaluate each model and assign to it some fitness value. Each candidate is integrated using a numerical integration routine to produce a time response. This simulation time response is compared with experimental data to give a fitness value for that model. A sum of squared error function (Eq.(1)) is used in all the work described in this paper, although many other fitness functions could be used.

The simulation routine must be robust. Inevitably, some of the candidate models will be unstable and therefore, the simulation program must protect against overflow error. Also, all system must return a fitness value if the GP algorithm is to work properly even if those systems are unstable.

Parameter estimation

Many of the nodes of the GP trees contain numerical parameters. These could be the coefficients of the transfer functions, a gain value or in the case of a time delay, the delay itself. It is necessary to identify the numerical parameters of each nonlinear model before evaluating its fitness. The models are randomly generated and can

therefore contain linearly dependent parameters and parameters which have no effect on the output. Because of this, gradient based methods cannot be used. Genetic Programming can be used to identify numerical parameters but it is less efficient than other methods. The approach chosen involves a combination of the Nelder-Simplex and simulated annealing methods. Simulated annealing optimizes by a method which is analogous to the cooling process of a metal. As a metal cools, the atoms organize themselves into an ordered minimum energy structure. The amount of vibration or movement in the atoms is dependent on temperature. As the temperature decreases, the movement, though still random, become smaller in amplitude and as long as the temperature decreases slowly enough, the atoms order themselves slowly enough, the atoms order themselves into the minimum energy structure. In simulated annealing, the parameters start off at some random value and they are allowed to change their values within the search space by an amount related to a quantity defined as system …temperature?. If a parameter change improves overall fitness, it is accepted, if it reduces fitness it is accepted with a certain probability. The temperature decreases according to some predetermined …cooling? schedule and the parameter values should converge to some solution as the temperature drops. Simulated annealing has proved particularly effective when combines with other numerical optimization techniques.

One such combination is simulated annealing with Nelder-simplex is an (n+1) dimensional shape where n is the number of parameters. This simples explores the search space slowly by changing its shape around the optimum solution .The simulated annealing adds a random component and the temperature scheduling to the simplex algorithm thus improving the robustness of the method .

This has been found to be a robust and reasonably efficient numerical optimization algorithm. The parameter estimation phase can also be used to identify other numerical parameters in part of the model where the structure is known but where there are uncertainties about parameter values.

Representation of a GP candidate model

Nonlinear time domain continuous dynamic models can take a number of different forms. Two common representations involve sets of differential equations or block diagrams. Both these forms of model are well known and relatively easy to simulate .Each has advantages and disadvantages for simulation, visualization and implementation in a Genetic Programming algorithm. Block diagram and equation based representations are considered in this paper along with a third hybrid representation incorporating integral and differential operators into an equation based representation.

Choice of experimental data set——experimental design The identification of nonlinear systems presents particular problems regarding experimental design. The system must be excited across the frequency range of interest as with a linear system, but it must also cover the range of any nonlinearities in the system. This could mean ensuring that the input shape is sufficiently varied to excite different modes of the system and that the data covers the operational range of the system state space.

A large training data set will be required to identify an accurate model. However the simulation time will be proportional to the number of data points, so optimization time must be balanced against quantity of data. A recommendation on how to select efficient step and PRBS signals to cover the entire frequency rage of interest may be found in Godfrey and Ljung?s texts.

Model validation

An important part of any modeling procedure is model validation. The new model structure must be validated with a different data set from that used for the optimization. There are many techniques for validation of nonlinear models, the simplest of which is analogue matching where the time response of the model is compared with available response data from the real system. The model validation

results can be used to refine the Genetic Programming algorithm as part of an iterative model development process.

Selected from “Control Engineering Practice, Elsevier Science Ltd. ,1998”

中文翻译：

遗传算法在非线性模型中的应用

导言：

非线性模型的辨识，至少是部分基于真实系统的基层物理学，自从可能需要同时决定模型的结构和参数以来，就出现了很多问题。尽管从测量的响应数据来估计模型参数有很多方法，但是结构的辨识却更为棘手。选择模型通常是通过专家知识和实验研究结合的试验和误差逼近法从大量的候选模型中去选择的。可能的模型结构是从系统的工程知识演绎出来的，而这些模型的参数是从现有的实验数据得来的。这样的方法是如此耗时却未达到最佳标准，可能只有这个过程的自动控制才能更快地从更大范围的可能模型结构中去研究。

遗传算法（GP）是一种最优化的方法，它可以通过从数据库自动选择模型结构元件用来使动态系统的非线性结构及元件之间的结合最优化，然后形成一个完善的数学模型。遗传算法是通过效仿自然界的进化去产生一个使一些目标函数最大化（或最小化）的模型结构，这些目标函数包括模型和系统响应之间的协调水平的适当测量。一些模型结构通过很多代向着一种解决方案而发展,这种方案是利用可靠的进化操作者和“适者生存”的选择规则进行。这些模型的参数可能通过被分离和更多完全的辨识过程的传统状态而估计出来。

应用：

遗传算法是一种早已投入使用的技术，这种技术已经在一些包括信号处理运算规则和化学加工辨识在内的非线性建模任务中得到应用。在连续时间系统模型的辨识中，玛伦巴赫、贝特哈慈和格雷研究了应用方框图导向仿真以达到遗传算法最优化问题，另外关于遗传算法在非线性系统辨识中的应用问题在格雷的另一片论文中得以讨论。在这篇文章中，遗传算法是应用在从实验数据得来的模型结构的辨识中，其中被研究的系统是用来代表非线性连续时域动态模型的。

这些模型结构逐渐发展成为遗传算法运算规则，使得包括模型和系统响应之

间的协调水平的适当测量在内的目标函数最小化。举例说明：

∑==n i e J 12

1 （1）

在此式子中，1e 是指N 次数据点中每一次模型输出和实验数据之间的误差。遗传算法运算规则是在函数库的基础上实现构造和重建的，那种模型的单一和模仿的及恰当的退火方法是用来估计一个合适的函数如同方程（1）所示。通常遗传算法是在不断的完善，直到这个遗传算法最后汇聚到这个系统的模型描述。

遗传算法运算规则

在这个研究中,应用了一个比较稳定的遗传算法运算规则。对于每一代，父母代都是从库里挑选出来的，下一代则是由他们的作用交叉而产生的代替了现有库中的成员。作用交叉的数量是和库的总类相等的，也就是说交叉率是百分之百。交叉运算法则是一种限定了作为结果的树的深度的子树交叉法。

遗传算法参数比如转换率和群体大小要依据应用而改变。更难的问题在于期望的模型结构是联合体或者数据是聒噪的，这时通常需要更大的群体大小。在这个研究中转换率不会出现对系统调查很明显的影响。通常只有2℅的受到影响。

函数库根据应用率和可能在这个系统辨识中期望的非线性模型的类型而改变。处理线性系统的核心方法经常是非常有用的。结果发现，具体的非线性系统比如查表，如果非线性存在于动态系统中，那么其中所代表的物理现象只有被遗传算法运算法则所选定。

这将允许系统，以测试具体的非线性系统。

程序模型结构辨识

遗传算法的库中的每个成员代表这个系统的候选人模型。评估每个模型并给定它一些合适的价值是必要的。每名候选人是综合采用数值积分例行制作时间响应。这个仿真时间响应，是比较实验数据为这个模型以提供一个合适的价值。在这个论文中平方误差函数的和（等式（1））是用来描述所有工作的，虽然可以用很多其他的合适的函数来描述。

仿真例子必须鲜明有力。无可避免地，有些候选模式会不稳定，因此，仿真

程序必须防止溢出的错误。此外,如果GP算法能正常工作，即使这些系统是不稳定的，所有系统也必须返回一个合适的价值。

参数估计

许多遗传算法树的节点包含有数值参数。这些传递函数的系数、增益值或是在时间延迟的情况下，将会使自身延迟。在评估它的适当的价值前，有必要查明每一个非线性模型的数值参数。模型是随机产生的，因此，可以线性地包含相关参数和参数，并不会影响产量。正因为如此，基于梯度的方法也就不能使用了。虽然遗传算法可以用于识别数值参数，但比起其他方法它的效率较低。而选定的做法是Nelder-Simplex和模拟退火方法的联合，模拟退火的最优化方法是类似于金属冷却的过程。作为金属冷却过程，原子组织起来形成一个有序的最低能源结构，而数额振动或运动中的原子是依赖于温度的。随着温度的降低，运动虽然是任意的，成为较少的振幅，并且只要温度足够慢地缓慢减少，原子就能使自己向最低的能源结构运动。在模拟退火过程中，参数估计是从一些随机值中开始的，并让他们改变他们的价值，这个搜索空间是由一个金额于数量界定为系统的“温度”。如果一个参数变化，全面提升性能，它是能被接受的，如果它降低了性能，也是有一定概率的被接受的。温度下降根据一些预定的“冷却”附表，参数值也应随着温度的降低收敛到一些解决方法。当其他的数字优化技术结合起来时，模拟退火方法是特别有效的。

这样一个模拟退火技术和Nelder-Simplex技术的组合是（n+1）的空间形状，其中n是参数的数量。这个简单的探讨搜索空间慢慢改变其形状靠近了最佳的解决方案。模拟退火以单纯的算法增加了随机性成分和温度调度，提高了方法的可靠性。

这已经被发现是一个稳健而合理的有效率的数值优化算法，参数估计阶段可以被用来确定模型的其他部分的数值参数。该模式的结构是众所周知的，但有不确定性参数值。

遗传算法候选模型的代表性

非线性连续时域动态模型可以采取一些不同的形式。微分方程和方框图是两

种普通代表，这两种形式的模型是众所周知的，并且是比较容易模拟的，对于仿真、可视化和在遗传算法运算规则的施行各有其优缺点。方框图和基于表示法的方程在本文中被考虑随着第三种混合表示法纳入微分和差分算子成为一个基于代表性的方程式。

实验数据集的选择——实验设计

非线性系统辨识提出了关于实验设计的特殊问题。该系统必须对于线性系统在整个频率范围内被激励，但是它也一定要涵盖系统中的任何非线性范围。这可能意味着，输入形式充分的多样化刺激着系统的不同模态并且数据覆盖了系统状态空间的运作范围。

识别一个准确的模型需要打的训练数据集。然而仿真时间将会和数据点的数量成正比，因而最优时间必须兼顾数据的数量。一项建议，就如何选择有效的步骤和 PRBS信号以覆盖整个频率范围，这个方法可能在高德费和刘佳的论文中有所体现。

模型验证

任何建模程序的一个重要组成部分是模型验证。新的模型结构必须同不同的数据集予以审定，从而用于优化过程。有许多非线性模型验证的技术，其中最简单的就是模拟匹配模型的时间响应和从实际系统中来的现有响应数据相比较的技术。该模型验证的结果可以用来改进作为反复的模型发展过程的一部分的遗传算法。