混沌数据恢复与非线性系统的模型参考控制
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摘要
本文的主要工作分为两个部分：第一部分(包括第二章和第三章),主要研究了混沌时间序列的缺失数据问题；第二部分(第四章),主要研究了非线性系统的模型参考控制.
在第二章,我们针对已知终端数据型混沌缺失数据问题,提出了终端控制方法；在跟踪控制变量所组成向量的模最小的意义下,我们给出了带有跟踪控制变量的最优预测函数；最后,我们以经济学中的混沌缺失数据为例,验证了该方法的有效性.在第三章,我们针对截断型混沌缺失数据问题,提出了预测混沌时间序列的新算法,即辅助变量法；最后,我们以Lorenz系统为例,检验了该方法的效率.在第四章,我们将工程学中的模型参考控制方法应用于非线性系统,并以Goodwin增长循环模型为例,首次将模型参考控制方法引入到经济系统的控制问题中.在第五章,我们对本文的工作进行了总结,并对未来的工作进行了展望.
Chaotic Data Recovery and Model Reference Control in Nonlinear System
In this paper, we have discussed two part:the first part (including Chapter 2 and 3) is the missing data problem of chaotic time series, the second part (including Chapter 4) is model reference control of nonlinear dynamical systems.
In the first chapter, we have introduced the definition of chaos and some results of delay embedding theorem.
Chaos is a most important part of nonlinear sciences. It has attractted many attentions of mathematicians, physicists, chemists and experts in some engineering disciplines since 1970s. Chaos, theory of relativity, and quantum mechanics have even been considered as the most important discovery of science in 20th century.
In 1975, Li and Yorke gave the definition of chaos for the first time[2].
Definition 0.1 We say continuous map f in [a,b] is chaotic, if:
(1) There are no upper boundary of the periodic points of f;
(2) There exists an uncountable subset S (?) [a, b], and there are no periodic points in S, satisfied as below:
(i)for every x,y∈S, we have
(ii)for every x,y∈S, we have x≠y,
(iii)for every x,y∈S and every periodic point y of f, we have 0.
For disordered data, to verify chaos is equivalent to distinguish between chaos and random.
Considering the initial value sensitivity of chaotic systems, we usually use Lya-punov exponent to verify chaos. Lyapunov exponent describes the divergence index of the nearby orbits.
Except for Lyapunov exponent, Kolmogorov information entropy is also an efficient tool for describing chaos.
The models of chaotic dynamics are Logistic mapping, Henon mapping, Lorenz family and so on.
Related researches of embedding theorem began with Whitney delay embedding theorem in the 1930s [29].
With the discovery of Takens delay embedding theorem, it was not difficult for one to describe the complex dynamical behavior. System theory becomed a novel tool for time series analysis.
Takens delay embedding theorem is as follows:
Theorem 0.2 Let M be a compact manifold of dimension m.φ:M→M is a smooth diffeomorphism and y:M→Ris a smooth function, it is a geneic property that the mapΦ(φ,y):M→R2m+1, defined by is an embedding; by smooth we mean at least C2.
Later, people found that the information in 2d+1 dimensions is sometimes redun-dancy. In 1991, T. Sauer, J. Yorke and M. Casdagli introduced a new form theorem based on 2dbox+1 dimensions, called prevalent delay embedding theorem. Here, dbox is box-counting dimension.
We have also introduce delay embedding theorems of infinite dimensional, forced and stochastic system.
We have also introduced the fundamental theory of chaotic time scries prediction. From the Takens delay embedding theorem, we can define the delay reconstruction vector x0 is initial value, and Thus, F inΦ(M) have results as below:
Here, F is the prediction function. The general method for chaotic time series prediction is as below:
1. Pretreatment of data;
2. Suppose F is a function with some unkown parameters;
3. Determine the parameters;
4. Check out the parameters:predict the time series, if the result is good; repeat the Step 2 and 3.
Phase space reconstruction method is firstly proposed by Packard, and its theoret-ical foundation is Takens time delay embedding theorem. The method is to use a single scalar time series vector in delay form, and to reconstruct the topological structure of the original system.
Missing data problem is that a part or all of the target data can not be observed, and these missing data may seriously affect the statistical analysis result of the data set. In recent years, the problem of missing data in various disciplines has become a difficult data analysis and statistical work.
In Chapter 2, to solve the terminal-known type chaotic missing data problem, we propose the tracking control method, and some numerical simulation results is given.
In the sense of module minimization of tracking control variable, we construct the optimal prediction function.
In the first section, we have introduced the background of missing data problem. Generally speaking, data is generated in two ways:experiment data and survey data. Missing data problem is more important than general data analysis. It is because that the accurate statistical results depend on the true, accurate and comprehensive data.
We also introduce the different kinds of missing data problem, and the special sense of chaotic missing data problem.
In the second section, to solve the terminal-known type chaotic missing data prob-lem, we propose the tracking control method. In the sense of module minimization of tracking control variable, we construct the optimal prediction function.
We can describe the terminal-known type chaotic missing data problem as follow: when a sequence of data{x(n)}, n≤N, the first m time series value is observable, the following N-m-1 data is missing, and the terminal data is observable. Here is the data set X:
To recovery the N-m-1 data in the middle of X, we describe the method as be-low. Firstly, we use the least-square method to determine the parameter in the d-step auto regression model. This is to say that data x(1),x(2),...,x(m) is the training set, determine the regression parameter of the model below based on least-square sense: Here, d andτare optimal embedding dimension and delay based on phase space reconstruction,{x(n)} is the auro-regression estimation value. Therefore, we obtain the substitute values x{m+1), x(m+2),...,x(N-1). Meanwhile, we also obtain the approximate value x (N) of the true terminal data value x(N).
We say the least-squared regression value is reliable, if the approximation x(N) and x(N) are equal. However, there is always some error between the two values.
Therefore, we consider the introduction of tracking control variables{r(n)|n= m+1,m+2,...,N},making Here,when i=1,2,...,m,x(n)=x(n);when n=m+1,m+2,...,N,x(n)is the prediction value.
In the following,we determine the value of{r(n)}.
Our aim is to make the module of tracking control variable minimal.
To determine{r(n)}turns out to be a optimization problem:
Definition 0.3 We define the revised error s(n)between x(n)and x(n) as
Theorem 0.4 For missing data problem X,when m+1≤n≤N,s(n)can be expressed as the linear conmbination of which is egual to Here,bn(i) are constants.
Then we have the orollary,
Corollary 0.5
Theorem 0.6 For missing data problem X,when m and N are deterministic integer, and d andτare also deterministic, Here,{bN(i)|i=m+1,m+2,...,N}are determinstic constants.
Whenτ≥N-m-1,we have shown the method is available.
The minimal module theorom of control variable is
Theorem 0.7 For missing data problem X,whenτ≥N-m-1,use to regress the model.here oi is the regresszon parameter,x(n)is the regression value, x(n) is prediction value.{r(n)|n=m+1,m+2,...,N}is the tracking control variable. we have the optimal prediction function:
In common instance,we can also obtain the optimal prediction function.
Theorem 0.8 For missing data problem X,the optimal prediction function Here,{bN(i)|i=m+1,m+2,…,N)are determinstic constants with regard to{ai|i= 1,2,...,d}.
In the third section,we apply the optimal prediction function on the real economic problems,and recover the chaotic missing data.The simulation result is great.
In Chapter 3,to solve the truncated type chaotic missing data problem,we propose the assistant variable method,and simulation examples of Lorenz system are given in detail.
In the first section,we discuss the existence of prediction function and the predic-tion theory of neighbors in space.
In the second section, we introduce the assistant variable method for the truncated type chaotic missing data problem.
We propose the local linear prediction model based on the assistant variable method, Here, A2 depends on Y(n), and its first p nearest neighbors are Y(ni)(i=1,2,...,p). We can obtain A2 by solving the equations A2B=D. Here, B is a (d+1)×p matrix, and its ith rank is B2(ni); D is a 1×p vector, and its ith row is x(ni+1).
In the third section, we chek out the efficiency of our method in Lorenz system.
In Chapter 4, we focus on the traditional control method applied in the new disciplines in recent years, such as economics, biology and so on. We put forward the application of model reference control method to nonlinear systems for solving practical problems in some general way. Then, based on the Goodwin growth cycle model between two countries, we show that one can make the employment rates stable at a high level by adjusting the workers' share in the national income.
In the first section, we introduce the two parts of MRC:system identification and controller generation. We also give the frame of MRC, and introduce the work of Shui-Nee Chow on the application of MRC in SIRS and SIS disease model.
In the second section, we apply the MRC to the Goodwin growth cycle model. Our idea is to make the employment rates stable at a high level by adjusting the workers' share in the national income.
In Chapter 5, we summarize the work, and propose some new expection for the future.
在第二章,我们针对已知终端数据型混沌缺失数据问题,提出了终端控制方法；在跟踪控制变量所组成向量的模最小的意义下,我们给出了带有跟踪控制变量的最优预测函数；最后,我们以经济学中的混沌缺失数据为例,验证了该方法的有效性.在第三章,我们针对截断型混沌缺失数据问题,提出了预测混沌时间序列的新算法,即辅助变量法；最后,我们以Lorenz系统为例,检验了该方法的效率.在第四章,我们将工程学中的模型参考控制方法应用于非线性系统,并以Goodwin增长循环模型为例,首次将模型参考控制方法引入到经济系统的控制问题中.在第五章,我们对本文的工作进行了总结,并对未来的工作进行了展望.
Chaotic Data Recovery and Model Reference Control in Nonlinear System
In this paper, we have discussed two part:the first part (including Chapter 2 and 3) is the missing data problem of chaotic time series, the second part (including Chapter 4) is model reference control of nonlinear dynamical systems.
In the first chapter, we have introduced the definition of chaos and some results of delay embedding theorem.
Chaos is a most important part of nonlinear sciences. It has attractted many attentions of mathematicians, physicists, chemists and experts in some engineering disciplines since 1970s. Chaos, theory of relativity, and quantum mechanics have even been considered as the most important discovery of science in 20th century.
In 1975, Li and Yorke gave the definition of chaos for the first time[2].
Definition 0.1 We say continuous map f in [a,b] is chaotic, if:
(1) There are no upper boundary of the periodic points of f;
(2) There exists an uncountable subset S (?) [a, b], and there are no periodic points in S, satisfied as below:
(i)for every x,y∈S, we have
(ii)for every x,y∈S, we have x≠y,
(iii)for every x,y∈S and every periodic point y of f, we have 0.
For disordered data, to verify chaos is equivalent to distinguish between chaos and random.
Considering the initial value sensitivity of chaotic systems, we usually use Lya-punov exponent to verify chaos. Lyapunov exponent describes the divergence index of the nearby orbits.
Except for Lyapunov exponent, Kolmogorov information entropy is also an efficient tool for describing chaos.
The models of chaotic dynamics are Logistic mapping, Henon mapping, Lorenz family and so on.
Related researches of embedding theorem began with Whitney delay embedding theorem in the 1930s [29].
With the discovery of Takens delay embedding theorem, it was not difficult for one to describe the complex dynamical behavior. System theory becomed a novel tool for time series analysis.
Takens delay embedding theorem is as follows:
Theorem 0.2 Let M be a compact manifold of dimension m.φ:M→M is a smooth diffeomorphism and y:M→Ris a smooth function, it is a geneic property that the mapΦ(φ,y):M→R2m+1, defined by is an embedding; by smooth we mean at least C2.
Later, people found that the information in 2d+1 dimensions is sometimes redun-dancy. In 1991, T. Sauer, J. Yorke and M. Casdagli introduced a new form theorem based on 2dbox+1 dimensions, called prevalent delay embedding theorem. Here, dbox is box-counting dimension.
We have also introduce delay embedding theorems of infinite dimensional, forced and stochastic system.
We have also introduced the fundamental theory of chaotic time scries prediction. From the Takens delay embedding theorem, we can define the delay reconstruction vector x0 is initial value, and Thus, F inΦ(M) have results as below:
Here, F is the prediction function. The general method for chaotic time series prediction is as below:
1. Pretreatment of data;
2. Suppose F is a function with some unkown parameters;
3. Determine the parameters;
4. Check out the parameters:predict the time series, if the result is good; repeat the Step 2 and 3.
Phase space reconstruction method is firstly proposed by Packard, and its theoret-ical foundation is Takens time delay embedding theorem. The method is to use a single scalar time series vector in delay form, and to reconstruct the topological structure of the original system.
Missing data problem is that a part or all of the target data can not be observed, and these missing data may seriously affect the statistical analysis result of the data set. In recent years, the problem of missing data in various disciplines has become a difficult data analysis and statistical work.
In Chapter 2, to solve the terminal-known type chaotic missing data problem, we propose the tracking control method, and some numerical simulation results is given.
In the sense of module minimization of tracking control variable, we construct the optimal prediction function.
In the first section, we have introduced the background of missing data problem. Generally speaking, data is generated in two ways:experiment data and survey data. Missing data problem is more important than general data analysis. It is because that the accurate statistical results depend on the true, accurate and comprehensive data.
We also introduce the different kinds of missing data problem, and the special sense of chaotic missing data problem.
In the second section, to solve the terminal-known type chaotic missing data prob-lem, we propose the tracking control method. In the sense of module minimization of tracking control variable, we construct the optimal prediction function.
We can describe the terminal-known type chaotic missing data problem as follow: when a sequence of data{x(n)}, n≤N, the first m time series value is observable, the following N-m-1 data is missing, and the terminal data is observable. Here is the data set X:
To recovery the N-m-1 data in the middle of X, we describe the method as be-low. Firstly, we use the least-square method to determine the parameter in the d-step auto regression model. This is to say that data x(1),x(2),...,x(m) is the training set, determine the regression parameter of the model below based on least-square sense: Here, d andτare optimal embedding dimension and delay based on phase space reconstruction,{x(n)} is the auro-regression estimation value. Therefore, we obtain the substitute values x{m+1), x(m+2),...,x(N-1). Meanwhile, we also obtain the approximate value x (N) of the true terminal data value x(N).
We say the least-squared regression value is reliable, if the approximation x(N) and x(N) are equal. However, there is always some error between the two values.
Therefore, we consider the introduction of tracking control variables{r(n)|n= m+1,m+2,...,N},making Here,when i=1,2,...,m,x(n)=x(n);when n=m+1,m+2,...,N,x(n)is the prediction value.
In the following,we determine the value of{r(n)}.
Our aim is to make the module of tracking control variable minimal.
To determine{r(n)}turns out to be a optimization problem:
Definition 0.3 We define the revised error s(n)between x(n)and x(n) as
Theorem 0.4 For missing data problem X,when m+1≤n≤N,s(n)can be expressed as the linear conmbination of which is egual to Here,bn(i) are constants.
Then we have the orollary,
Corollary 0.5
Theorem 0.6 For missing data problem X,when m and N are deterministic integer, and d andτare also deterministic, Here,{bN(i)|i=m+1,m+2,...,N}are determinstic constants.
Whenτ≥N-m-1,we have shown the method is available.
The minimal module theorom of control variable is
Theorem 0.7 For missing data problem X,whenτ≥N-m-1,use to regress the model.here oi is the regresszon parameter,x(n)is the regression value, x(n) is prediction value.{r(n)|n=m+1,m+2,...,N}is the tracking control variable. we have the optimal prediction function:
In common instance,we can also obtain the optimal prediction function.
Theorem 0.8 For missing data problem X,the optimal prediction function Here,{bN(i)|i=m+1,m+2,…,N)are determinstic constants with regard to{ai|i= 1,2,...,d}.
In the third section,we apply the optimal prediction function on the real economic problems,and recover the chaotic missing data.The simulation result is great.
In Chapter 3,to solve the truncated type chaotic missing data problem,we propose the assistant variable method,and simulation examples of Lorenz system are given in detail.
In the first section,we discuss the existence of prediction function and the predic-tion theory of neighbors in space.
In the second section, we introduce the assistant variable method for the truncated type chaotic missing data problem.
We propose the local linear prediction model based on the assistant variable method, Here, A2 depends on Y(n), and its first p nearest neighbors are Y(ni)(i=1,2,...,p). We can obtain A2 by solving the equations A2B=D. Here, B is a (d+1)×p matrix, and its ith rank is B2(ni); D is a 1×p vector, and its ith row is x(ni+1).
In the third section, we chek out the efficiency of our method in Lorenz system.
In Chapter 4, we focus on the traditional control method applied in the new disciplines in recent years, such as economics, biology and so on. We put forward the application of model reference control method to nonlinear systems for solving practical problems in some general way. Then, based on the Goodwin growth cycle model between two countries, we show that one can make the employment rates stable at a high level by adjusting the workers' share in the national income.
In the first section, we introduce the two parts of MRC:system identification and controller generation. We also give the frame of MRC, and introduce the work of Shui-Nee Chow on the application of MRC in SIRS and SIS disease model.
In the second section, we apply the MRC to the Goodwin growth cycle model. Our idea is to make the employment rates stable at a high level by adjusting the workers' share in the national income.
In Chapter 5, we summarize the work, and propose some new expection for the future.
引文
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