本論文基於李亞普諾夫理論 (Lyapunov Theorem)，提出n階適應性積分可變結構導數估測器(AIVSDE)之設計方法。文中所設計的導數估測器不僅改善現有的AIVSDE，而且可用來估測具有n+1階有界微分且連續之平滑訊號的n次微分。經由串接一個低通濾波器與調整系統的設計參數達成消除雜訊的影響。在控制結構中，我們使用適應性運算法則除去了需預知輸入訊號上界的要求。本論文可以保證導數估測器的穩定性，並比較近年來所提出之非線性適應可變結構導數估測器(NAVSDE)與AIVSDE之微分估測誤差的上界。最後，本論文提供一個數值範例以驗證AIVSDE的可行性。

Based on Lyapunov stability theorem, a design methodology of nth order adaptive integral variable structure derivative estimator (AIVSDE) is proposed in this thesis. The proposed derivative estimator not only is an improved version of the existing AIVSDE, but also can be used to estimate the nth differentiation of a smooth signal which has continuous and bounded derivatives up to n+1. A low pass filter is cascaded with AIVSDE so that the effects of noise can be alleviated by adjusting the designing parameters of filter and AIVSDE. The adaptive algorithm is incorporated in the control scheme for removing the a priori knowledge of the upper bound of the observed signal. The stability of the proposed derivative estimator is guaranteed, and the comparison of upper bound of derivative estimation error between recently proposed nonlinear adaptive variable structure derivative estimator (NAVSDE) and AIVSDE is also demonstrated. An example is given for showing the applicability of the proposed AIVSDE.

Contents
List of Figures iii
Chapter 1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Brief Sketch of the Thesis . . . . . . . . . . . . . . . . . . . . . 3
Chapter 2 Design of the First Order AIVSDE 4
2.1 Design of the First Order AIVSDE . . . . . . . . . . . . . . . . 4
2.2 Analysis of System’s Performance in the Presence of Noise . . 8
2.3 Comparison of upper bound of derivative estimation error of
NAVS and AIVS techniques . . . . . . . . . . . . . . . . . . . 14
Chapter 3 Design of the Second Order AIVSDE 21
Chapter 4 Design of the nth Order AIVSDE 30
Chapter 5 Simulations 41
Chapter 6 Conclusions 75
Bibliography 76

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