處理強韌穩定性分析大致可分成兩種方式：輸入輸出及李亞普諾夫穩定性分析。通常這兩類方式都是獨立發展，除了少數特例外，少有文獻探討兩者之間的關聯，此為本論文欲探討此題目的動機。
廣義性能問題旨在判斷轉移函數是否滿足某些頻域條件；本論文證明在適當假設下廣義性能與強韌穩定性之間有一等價關係。然而廣義性能假設轉移函數必須是內部穩定，此非閉迴路系統穩定的必要條件，有鑑於此，本論文進一步推導不需要內部穩定的頻域條件，並證明所得的結果包含某一版本的圓環準則 (circle criterion)。
為了探討廣義性能問題，本論文提出某一版本的Kalman-Yakubovich-Popov (KYP) 引理將頻域條件轉成線性矩陣不等式，並建立此線性矩陣不等式與二次穩定性之間的連結。綜合上述所得結果可讓我們對強韌穩定性、廣義性能、二次穩定性及KYP引理之間的關聯更加清楚，本論文不但整合了之前文獻的結果，並將這些結果推廣到更廣義的穩定區域及不確定量描述。
除了穩定性分析，本論文同時探討控制器設計問題，即強韌極點放置。強韌極點放置的區域可為簡單區域的交集或是連集。(簡單區域指的是半平面、圓的內部或圓的外部)關於控制器設計部分本論文的主要貢獻是極點放置區域可以是非凸集合—大部分現有文獻都只探討凸集合。本論文並以戰鬥機的縱向控制及衛星的姿態控制驗證所提結果的可行性。

There are two main approaches to robust stability analysis:
the input-output stability framework with scaling or multiplier, and the Lyapunov functions.
Analysis methods in these two directions are usually developed independently,
and the relationship between the two is not clear except for some special cases.
This motivates us to study the relationship between the two approaches.

The generalized performance problem refers to certain frequency-domain conditions on a transfer matrix.
We prove the equivalent relationship between generalized performance and robust stability under certain assumptions.
The definition of generalized performance requires the internal stability of a transfer matrix,
which is not a necessity for robust stability.
In view of this, we derive new frequency-domain conditions for robust stability without this requirement.
Our result contains a version of the circle criterion as a special case.

To tackle the generalized performance problem, we propose a version of the Kalman-Yakubovich-Popov (KYP) lemma to
transform the frequency-domain conditions into linear matrix inequalities (LMIs).
The proposed LMI condition is then connected to the quadratic stability of an uncertain linear system.
Combining the derived results gives a clear picture of
the relationships between robust stability, generalized performance, quadratic stability, and KYP lemma.
The connections not only unify some previous results
but also extend those results to more general stability regions and types of uncertainty.

In addition to robust stability analysis,
we also tackle the corresponding synthesis problem, i.e. robust pole placement.
The desired region for robust pole placement can be the intersection or the union of simple regions.
(Simple regions are the half plane, the disk, and the outside of a disk.)
One contribution of our synthesis result is that
the desired region can be non-convex—most results on robust pole placement focus on convex regions only.
Two examples of the longitudinal control of a combat aircraft and
the attitude control of a satellite demonstrate the effectiveness of our result.

1 Introduction 1
1.1 Literature Overview and Thesis Outline . . . . . . . . . . . . . . . . 1
1.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Well-Posedness of Feedback Loops 7
2.1 Properties of the Uncertainty Set . . . . . . . . . . . . . . . . . . . . 7
2.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3 Frequency-Domain Inequalities and RobustMatrix Root-Clustering 14
3.1 Generalized Performance and Robust Stability . . . . . . . . . . . . . 15
3.1.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . 15
3.1.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.1.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.1.4 Weak Generalized Performance . . . . . . . . . . . . . . . . . 23
3.2 Loop Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2.1 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2.2 Circle Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4 KYP Lemmas for Simple Regions 40
4.1 KYP Lemma with Positive Definite P . . . . . . . . . . . . . . . . . 41
4.1.1 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.1.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.2 Quadratic Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
iii
4.2.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . 51
4.2.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.2.3 Equivalent Relationships Between Generalized Performance,
Robust Stability, and Quadratic Stability . . . . . . . . . . . . 56
5 KYP Lemmas for Arbitrary Intersections and Unions of Simple
Regions 58
5.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.2 Intersection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.2.1 Simple Regions . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.2.2 DR Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.3 Union . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.4 Arbitrary Union of Intersections . . . . . . . . . . . . . . . . . . . . . 70
6 Controller Design 72
6.1 Introduction of Extra Variables . . . . . . . . . . . . . . . . . . . . . 72
6.2 Multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
6.3 Static State-Feedback Control . . . . . . . . . . . . . . . . . . . . . . 79
6.3.1 Robust Pole Placement . . . . . . . . . . . . . . . . . . . . . . 79
6.3.2 Robust Partial Pole Placement . . . . . . . . . . . . . . . . . 82
6.4 Dynamic Output-Feedback Control . . . . . . . . . . . . . . . . . . . 85
6.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
6.5.1 Longitudinal Control of a Combat Aircraft . . . . . . . . . . . 92
6.5.2 Attitude Control of a Satellite . . . . . . . . . . . . . . . . . . 97
7 Conclusions 102
7.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
7.2 Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
Appendix A 105
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