在許多工程應用中，我們經常需要知道系統中某些狀態訊號或是這些狀態訊號的組合以完成診斷系統是否異常、監督系統或是控制系統的運作之用途。當這些訊號無法由感測器直接量測，這時就必須透過其它可量測的訊號和描述系統動態行為的數學模型並應用特定的演算法來估測這些訊號值。在許多實際的應用中這是一項重要問題，因此「估測技術」是一個由來已久的研究課題並且在系統理論界得到相當多的關注。由其是當系統模型具有不確定性時，估測的強韌性就必須要加以考慮這項課題。

為了處理系統模型包含有不同類型的不確定性，如參數不確定性、未知的動態模型、非線性操作子等，我們將採用二次積分限制(IQC)架構來設計強韌估測器。在 IQC 架構中，藉由二次積分限制不等式來描述各個不確定性的特性；然後匯整這些不等式獲得設計準則。所以，此架構有處理一個接一個「難以處理」操作的優點，因此更適合處理大型複雜的動態系統。

在本論文中，我們考慮系統含有穩定或不穩定標稱系統和能以 IQC 描述的不確定性之連結系統的強韌估測器設計問題。對於系統含穩定的標稱系統，文獻 [1] 已經考慮系統為連續時間設定的設計問題。在本論文中，我們擴展問題到系統為離散時間的設定，藉由兩種不同的方法來獲得兩種不同的設計準則。對於系統含不穩定的標稱系統，此問題雖然有少數的研究文獻，但現有的結果並非最佳。對於這個問題，我們將應用近年來所開發的強韌穩定性分析定理：「結合新間隙尺度與二次積分限制的定理」來獲得基於線性矩陣不等式(LMI)的設計準則。

我們以幾個學術和實際例子來測試所獲得的結果，這些數值例子驗證了我們提出的設計方法的有效性。

In many applications, we often require the knowledge of certain state variables of a system, or specification combinations of these, for diagnosis, supervision, or control of the systems operation. When these signals cannot be measured directly by sensors, it would be necessary to estimate their values by certain algorithms based on other measured signals and a mathematical model that describes the dynamical behavior of the system. Due to its important role in many practical applications, "estimation technique" has been a long standing research topic and received considerable attention in the systems theory community. When the model of the system exhibits uncertainty, the robustness of the estimator has to be taken into account.

In order to handle systems with various types of uncertainties, such as parametric uncertainties, unmodelled dynamics, nonlinear operators, we adopt the Integral Quadratic Constraint (IQC) framework for designing robust estimators. In the IQC framework, the properties of individual uncertainties are described by integral quadratic inequalities; the inequalities are then aggregated to form the design criteria. As such, the framework has the advantage of allowing one to tackle "trouble-making" blocks in the system one-by-one, and thus is better suited for handeling large-scale complex dynamical systems.

In this thesis, we consider the robust estimation problem for systems with stable or unstable nominal part in feedback interconnection with a structured uncertainty that is described by IQC. For systems with stable nominal parts, the problem in the continuous-time setting has been considered in [1]. In this thesis, we extend the result to the discrete-time setting by two different approaches and obtain two different design criteria. For systems with unstable nominal parts, the problem has only been considered by a few researchers and the existing results are sparse at best. For this problem, we apply a recently-developed robust stability result arising from a blended nu-gap(ν-gap) metric and Integral Quadratic Constraint based analysis to obtain design criteria, posed as Linear Matrix Inequalities (LMI).

The results we obtained were tested by several numerical examples, some academic and some practical. The numerical experiments verify the effectiveness of the design approach we propose.

中文摘要 i
英文摘要 ii
目錄 iii
圖目錄 iv
表目錄 v
第 1 章 緒論 1
1.1 文獻回顧 1
1.2 研究動機、目的與貢獻 7
1.3 論文大綱 8
1.4 符號說明 8
第 2 章 動態 IQC 之強韌估測器設計 10
2.1 問題描述 10
2.2 IQC 定理 10
2.3 LMI 強韌估測器設計準則 16
2.4 主要定理 18
2.5 數值例子 32
第 3 章 新間隙尺度/動態 IQC 之強韌估測器設計 43
3.1 問題描述 43
3.2 新間隙尺度/IQC 定理 43
3.3 LMI 強韌估測器設計準則 50
3.4 主要定理 52
3.5 數值例子 57
第 4 章 結論與未來展望 61
4.1 結論 61
4.2 未來展望 61
參考文獻 62

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