On Dynamic Data Reconciliation with Discrete Wavelet Transform

本研究利用離散小波轉換與數據調和理論處理動態系統之數據調和問題。在處理動態數據調和問題時，可考慮以下列兩種方式進行。第一種方式為先將系統動態方程式轉換成代數方程式，接著以數學規劃法方式求得調和估測值；第二種方法則是藉由卡門濾波器(Kalman Filter)相關之估測法同步處理系統動態與狀態估測問題。 研究中提出以不同方式應用離散小波轉換於上述之動態數據調和方法，藉此達到數據調和之目的。利用離散小波轉換主要目的為分析、過濾量測訊號，並解決在處理動態調和問題時所遇到之缺點與不足，如多項式方法(Polynomial Approach)不適於處理變動較大之動態訊號等等。 文章主要分成三個部分。在第一部分，考慮線性動態系統情況下，以辛普森積分法將系統動態方程式轉換成代數方程式，接下來可利用穩態數據調和理論解決此動態數據調和的問題。應用離散小波轉換可以降低轉換時所產生之誤差，並可以預先過濾充滿雜訊之量測訊號。 在第二部份，利用卡門濾波估測的方法具有時間遞迴方式演算式的優點，適合應用於線上即時動態數據調和的問題，但在利用卡門濾波方法估測時，在擴增的狀態變數(Augmented State Variables)常會帶有較大的誤差，故在此部分提出以離散小波轉換過濾為基礎之線上過濾方法應用於卡門濾波估測法，藉此獲得更佳之估測效果。 第三部分中以離散小波理論中之尺度函數（Scaling Function）去近似量測訊號及其微分，藉此將系統動態方程式轉換成代數方程式後藉由數學規劃法處理動態數據調和之問題。在最佳化參數之過程中，搜尋之值域為在尺度函數之係數所在之值域，如此可減少最佳化搜尋變數之數目，並可達到動態數據調和之目的。 研究中並提出針對單一重大錯誤之偵測與隔離方法，可有效地偵測出系統中之重大錯誤。在文中將以範例說明所提出之各項理論，最後並討論與比較所提出三種方法之優缺點。In this dissertation, methodologies are proposed to solve dynamic data reconciliation problems by using discrete wavelet transform. As dealing with the dynamic data reconciliation problems, two approaches are usually considered: By the first approach, system differential equations must be transformed to algebraic equations in advance and the problems are solved by the mathematical programming method in the following; by the second approach, Kalman filter method is used to deal with system dynamic equations along with the estimation simultaneously. In the research, discrete wavelet transform theories are applied to the above-mentioned two approaches to accomplish the reconciliation. Wavelets theories are aimed to analyze and filter the measurement signals, which can deal with some shortcomings and deficiencies, e.g. the polynomial approach has difficulty as encountering the tortuous signals, in the data reconciliation problems. There are three main parts in the research. In the first part, considering a linear dynamic system, system’s differential algebraic equations are converted into algebraic equations using Simpson’s integration rule, which can be solved easily by the steady-state data reconciliation theories for this dynamic data reconciliation problem. Discrete wavelets transform is used to filter the measurement signals to decrease the errors from the transformation. In the second part, on-line dynamic data reconciliation approach is proposed by using the Kalman filter estimation which has time-recursive formulations. Augmenting the states is always needed in this approach which may result significant errors for reconciliation. Thus, an on-line filtering using the discrete wavelets transform is applied to filter the measurements from a real time process for reconciliation. Some difficulties, e.g. the boundary problems, encountered in the wavelets filtering are overcome by a proposed robust algorithm. A single gross error detection and isolation method is also proposed which can isolate the fault successfully. In the third part, through using the scaling functions in the wavelets theories, the measurement signals and their derivatives are approximated and the dynamic data reconciliation problem is formulated and solved by optimization via mathematical programming method. As optimization, the searching space for is the domain of the scaling function coefficients, which can reduce the number of the searching variables. Examples are performed to illustrate the proposed approaches and summaries are given for the comparisons of the proposed three methods.致 謝 I 摘 要 III Abstract V Contents VII List of Figures XI List of Tables XIII Acronyms XV Chapter 1 Introduction 1 1.1. Overview 1 1.2. Filtering Methods without Models 3 1.3. Filtering Methods with Models 7 1.4. Dynamic Data Reconciliation (DDR) by Wavelets Approaches 12 1.5. Research Objectives and Motivations 13 1.6. Organization 14 Chapter 2 Concepts of Data Reconciliation (DR) 15 2.1. Overview 15 2.1.1. Types of errors 15 2.1.2. Statistical basis of DR problem 17 2.2. Linear Steady-State Data Reconciliation (SSDR) 19 2.2.1. Formulation 19 2.2.2. Lagrange multipliers 20 2.3. Redundancy and Observability 21 2.4. Variance-covariance Calculation for SSDR 23 2.5. Non-linear Data Reconciliation 24 2.6. Dynamic Data Reconciliation (DDR) 25 2.6.1. Kalman filtering (KF) 25 2.6.2. Constrained Kalman filtering 27 2.6.3. Extended Kalman filtering (EKF) 28 2.6.4. DDR based on mathematical programming 29 2.7. Gross Error Detection (GED) 31 Chapter 3 Concepts of Wavelet Analysis 35 3.1. Overview 35 3.1.1. Multi-scale time-frequency representations 35 3.1.2. Short term Fourier transform 37 3.1.3. Types of wavelets 37 3.1.4. Two-parameter functions 39 3.1.5. Properties of wavelets 40 3.2. Discrete Wavelet Transform (DWT) and Inverse Discrete Wavelet Transform (IDWT) 42 3.3. DWT for Finite Length Signal 46 3.4. DWT and IDWT in Matrix Form 48 3.5. Scaling Mother Function 50 3.6. The Derivative of Scaling Mother Function 52 Chapter 4 Dynamic Data Reconciliation by Integral Approach 57 4.1. Introduction 57 4.2. Wavelets Filtering 58 4.2.1. DWT analysis and filtering 58 4.2.2. Band-pass frequency of DWT 59 4.2.3. Determining the DWT analysis level 61 4.3. Integral Approach by Simpson’s Rule 65 4.4. Gross-error Detection and Fault Isolation 69 4.5. Examples 70 4.5.1. Example 1 73 4.5.2. Example 2 80 4.6. Conclusions 82 Chapter 5 Constrained Kalman Filtering for Dynamic Data Reconciliation 85 5.1. Introduction 85 5.2. On-line Robust Wavelets Filtering 86 5.2.1. DWT and IDWT analysis for filtering 87 5.2.2. Determination the wavelets filtering level 88 5.2.3. Robust wavelets filtering method 90 5.2.4. Filtering examples 93 5.3. KF Approach for Reconciliation 98 5.3.1. Data reconciliation using constrained KF 98 5.3.2. Detection and isolation of gross errors 101 5.4. Examples 104 5.4.1. Illustration example of a four-tank system 104 5.4.2. Bias, leak detection and isolation 110 5.5. Conclusions 114 Chapter 6 Dynamic Data Reconciliation by Mathematical Programming 119 6.1. Introduction 119 6.2. Approximation by Scaling Functions 121 6.2.1. Expansion signal by scaling function 121 6.2.2. Expanding the derivative of signal 122 6.3. The Comparisons of Two Approaches 123 6.4. Reconciliation 124 6.4.1. Linear system 125 6.4.2. Non-linear system 130 6.5. Examples 130 6.5.1. Linear four-tank system 131 6.5.2. Continuous stirred-tank reactor (CSTR) system 136 6.6. Conclusions 143 Chapter 7 Conclusions and Future Works 147 7.1. Summary and Comparisons 147 7.2. Contributions 148 7.3. Future Works 150 A. Wavelets Decomposition and Reconstruction Matrices 151 B. Wavelets Differential Operator 154 Bibliography 15