系统辨识与参数估计

Chapter 3 System Identification and Parameter Estimation
第三章 系统辨识与参数估计 5
3.1.5 System identification methods 系统辨识的方法
Method: There are many system identification methods, but the least squares estimation is used most frequently. 法：有多种方法，其中最小二乘法最常用。 Off-line identification: complete estimation one time based on the data set in a long period. 离线辨识：将一定时间内积累的采样数据集中进行一次辨识计算. On-line identification : complete recursive estimation one time based on new data in every sampling interval. It is able to decrease calculating time spending and memory occupancy, and easy to find out system actuality. 在线辨识：每个采样周期都根据新的采样数据进行一次递推辨识计算，节省计算时 间和内存空间，便于及时掌握系统现状。
3.1.3 Development of System Identification

系统辨识的发展
Modern Control Theory is based on known a mathematic model of dynamic process. 现代控制理论建立在数学模型已知的前提下

The obstacle using Modern Control Theory in practice : It is not easy to obtain a mathematic model of dynamic process, thus the theory deviates from the practice. 实际应用中的障碍：数学模型并不容易获得，造成理论与实际脱节
A(q 1 ) y(k ) B(q 1 )u(k )
Here 其中
(3.1)
A 1 a1q 1 ana q na
B b0 b1q1 bnb qnb
3.2.2 The model of noise
(1)
(3.2) (3.3)
噪声模型
Random variable 随机变量 x Mathematic description: probability density function
能代表该系统的数学模型。 Three essentials: an input/output dada, a set of models, and an optimized criterion
三个要素：输入/输出数据、模型集、最优准则
Parameter Estimation: a simplified system identification problem when the
Chapter 3 System Identification and Parameter Estimation
第三章 系统辨识与参数估计
3.1 Introduction 概述
3.1.1 What is the Model of Dynamic System? 什么是模型？

Theory model and experiment model 理论模型与实验模型 Modeling from Theory and Analysis: educe system model according to physical, chemical or other natural rules. 理论（分析）建模：根据已知的物理、 化学规律推导 In practice, Theory Modeling is not easy. 现实中理论建模存在困难
Chapter 3 System Identification and Parameter Estimation
第三章 系统辨识与参数估计 9
3.2.3 Mathematical Model of the process with random disturbance 受随机干扰的过程数学模型（CARMA）
Auto regressive model
ew (k ) 为白噪声序列
自回归（AR）模型
(3.7)
e( k )

ew (k ) D(q 1 )
Auto regressive moving average model
自回归滑动平均（ARMA）模型
(3.8)
D(q 1 )e(k ) C(q1 )ew (k )
Chapter 3 System Identification and Parameter Estimation
第三章 系统辨识与参数估计 8
（4）Non-white noise: is formed when white noise goes through a linear filter. 非白噪声：白噪声经过一线性滤波器后形成非白噪声 Moving average model
model structure is known, only its parameters is unknown.
参数估计：结构已知。参数未知时，系统辨识问题的简化
Chapter 3 System Identification and Parameter Estimation
第三章 系统辨识与参数估计 3
数学描述：概率密度函数
Chapter 3 System Identification and Parameter Estimation
第三章 系统辨识与参数估计 7
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Mathematical expectation of random variable 数学期望（均值） E(x)
E[c] c
E (kx) kE( x)
第三章 系统辨识与参数估计 4
3.1.4 System identification includes the following steps 系统辨识的步骤
Experiment design: Its purpose is to obtain good experimental data, it includes the choice of the measured variables and of the character of the input signals. 实验设计：如何获取尽可能多的信息，包括检测信号和输入信号的选取。 Selection of model structure: A suitable model structure is chosen using prior knowledge and trial and error. 模型结构：根据先验知识和试凑确定模型的结构。 Choice of the criterion to fit: A suitable cost function is chosen, which reflects how well the model fits the experimental data. 最优准则：选择能反应模型对实验数据拟合程度的目标函数。 Parameter estimation: An optimization problem is solved to obtain the numerical values of the model parameters. 参数估计：得到模型参数数值解的优化问题。 Model validation: The model is tested in order to reveal any inadequacies. 校验与确认：测试模型以发现存在的问题。
which confirm a model in a set of models that presents the dynamic characteristics
of the system under an optimized criterion.
系统辨识：根据系统的输入、输出数据，从一类模型中确定出一个在某中意义下最

Chapter 3 System Identification and Parameter Estimation 估计 2
第三章 系统辨识与参数
3.1.2 System Identification and Parameter Estimation 系统辨识与参数估计
System Identification: is the experimental approach to process modeling, and the modeling method for identification of dynamical systems from input/output data,

Experiment Modeling: Fit the model to experimental data according to an optimized criterion. 实验建模：按一定准则的数据拟和 Experiment model: holistic approach, complemented by theory model 实验建模的特点：整体性、可用机理模型弥补（互补）
滑动平均（MA）模型
(3.5) (3.6)
e(k ) c(q 1 )ew (k )
C (q 1 ) 1 c1q 1 cnc q nc D(q 1 ) 1 d1q 1 d nd q nd
ew (k ) is a white noise sequence
E ( x y ) E ( x) E ( y )
Variance of random variable 方差（二阶中心距） D(x)
D( x) E{[ x E( x)]2 }
(3.4)
（2）Steady random sequence: Statistical character is independent of time 平稳随机序列：各个时刻随机变量的统计特征相同，即统计特征与时间无关 （3）White noise: is an independent steady random sequence. Random variable is independent of time, and can be described by E(x) and D(x). 白噪声：独立平稳随机序列。各个时刻随机变量独立，可由均值和方差两个特征描 述。 均值=0，方差=σ 2（常数） 因为其功率谱密度在整个频率范围内为常数，类 似白光的光谱，故称为白噪声。
方


Chapter 3 System Identification and Parameter Estimation
第三章 系统辨识与参数估计 6
3.2 Linear Difference Equation Model
线性差分方程模型
3.2.1 Difference equation model of linear constant SISO system 线性定常单输入单输出系统的差分方程模型

System Identification and Parameter Estimation just fill up this gap between the theory and the practice. 系统辨识/参数估计正是为了弥合这一差距
Chapter 3 System Identification and Parameter Estimation