本研究論文之主旨為探討里亞普諾夫(Lyapunov)穩態理論運用於離散(Discrete-time)及連續(Continuous-time)非線性不確定系統的直接適應控制(direct adaptive control)。對於系統理論,離散系統均有其對應之連續系統,然而我們發現許多適應控制結果已在連續性系
統應用之同時,相關離散系統之結果卻相對較少,尤其相關里亞普諾夫理論在離散直接適應控制上的應用,其主因在於無合適之里亞普諾夫候選方程式(Lyapunov candidate),導致無法有效運用亞普諾夫理論；另外有鑒於電腦應用的日益普及化,吾人認為這些訊號通常是由系
統量測或經由資料蒐集而得，故直接處理離散訊號更為真切。

本文提出以里亞普諾夫理論為架構下離散非線性不確定系統的直接適應控制理論，並說明此為對應連續系統的解，首先我們採用系統軌跡相依里亞普諾夫候選方程式
(trajectory dependent Lyapunov candidate; tdLC)求得所謂系統軌跡相依的解後，進而尋求解除系統軌跡相依的限制，並獲得直接適應控制理論發展，此一求解之過程不僅顯示出離散系統在證明里亞普諾夫差(Lyapunov difference) 的特性與連續系統在證明里亞普諾夫微分(Lyapunov difference)之不同，因為只有離散系統會出現系統軌跡相依的解，同時在對應連續系統的解時(continuous-time counterpart)顯然並非唯一，即便控制理論一般均有相對應連續與離散的解。

對於離散系統,因為里亞普諾夫備選方程式是由系統軌跡x(k)在k與k-1的組合,故結果稱之為系統軌跡相依,所求得之適應性回饋控制可由庫聶可算法(Kronecker calculus)進行描述,並確保符合系統的里亞普諾失穩態,
所謂解除系統軌跡相依的限制,即里亞普諾夫備選方程式是由系統軌跡x(k)的組合,同樣證明可獲得相同之結果,進而相對應的連續非線性不確定系統的直接適應控制,在此本研究推論系統軌跡相依里亞普諾夫備選方程 (tdLC) 僅出現在離散系統, 且當|t(k)-t(k-1) | ≤ δ及|x(k)-x(k-1)| ≤ ε, δ, ε均足夠小，這是系統軌跡相依解的限制條件,同時亦完成系統受外力(exogenous disturbances)或ι2能量受限的條件下的解，強鍵(robust)理論在本研究的應用，是除了前述證明系統迴路穩態之外，更進一步提升研究應用範疇的必要方向之ㄧ，僅針對系統不確定量於一已知限度內|A-Ac| ≡ |B Kg| ≤ |ΔA|，達到控制目標。

總之，直接適應控制理論在離散非線性不確定系統發展，運用里亞普諾夫理論可以求得全域的解，同時
可以達到參數預測與迴路系統的穩態。

The dissertation addresses direct adaptive control frameworks for Lyapunov stabilization of the MIMO nonlinear uncertain systems for both uncertain
discrete-time and continuous-time systems. For system theory, the development of continuous-time theory always comes along with its discrete-time counterpart. However, for direct adaptive control frameworks we find relative few Lyapunov-based results published, which is mainly due to difficulty to find feasible Lyapunov candidates and to prove negative definiteness of the Lyapunov difference.
Furthermore, digital computer is widely used in
all fields. Most of time, we have to deal with the direct source of discrete-time signals, even the discrete-time signals arise from continuous-time settings as results of measurement or data collection process. These motivate our study in this field.

For discrete-time systems, we have investigated the results with trajectory dependent hypothesis, where the Lyapunov candidate function V combines the information from the current state k and one step ahead k-1 along the track x(k), for k≥0. The proposed frameworks guarantee partial stability
of the closed-loop systems, such that the feedback gains stabilize the closed-loop system without the knowledge of the system parameters. In addition,
our results show that the adaptive feedback laws can be characterized by Kronecker calculus.

Later, we release this trajectory dependent hypothesis
for normal discrete-time nonlinear systems. At the same time, the continuous-time cases are also studied when system with matched disturbances, where the disturbances can be characterized by
known continuous function matrix and unknown parameters. Here, the trajectory dependent Lyapunov candidates (tdLC), so long as the time step
|t(k)-t(k-1) | ≤ δ and the corresponding track |x(k)-x(k-1)| ≤ ε are sufficiently small, only exist in discrete-time case. In addition, we have extended the above control designs to systems with exogenous disturbances and
ι2 disturbances. Finally, we develop a robust direct adaptive control framework for linear uncertain
MIMO systems under the variance of unknow system matrix from given stable solution is bounded, that is |A-Ac| ≡ |B Kg| ≤ |ΔA|.

In general, through Lyapunov-based design we can obtain the global solutions and direct adaptive control design can simultaneously achieve parameter estimation and closed-loop stability.

Acknowledgement i
List of Figures vi
Notations viii
Chinese Abstract ix
English Abstract x
1 Introduction 1
1.1 Goals andMotivation . . . . . . . . . . . . . . 1
1.2 Background and Paper Review . .. . . . 2
2 Mathematical Background 4
2.1 Discrete-time Partial Stability Theory .4
2.2 Kronecker Calculus . . . . . . . . . . . . . . . . 6
2.3 Positive Real Discrete-time Systems . 7
2.4 Stability and Dissipative Discrete-time Systems . 8
2.5 Trajectory Dependent Hypothesis . .. . . . 9
3 ANovelDirectAdaptive Design for Uncertain Continuous time Systems 12
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 12
3.2 System Stability withMatched Disturbances . .. . 13
3.3 Numerical Examples . . . . . . . . . . . . . . . . . . 17
3.3.1 The vanderPolOscillator . . . . . . . . . . . . . 17
3.3.2 One-Link Rigid Robot under Gravitation Field . . . . . . . . . . . . . . . 18
3.3.3 Flexible Joint Robot . . . . . . . . . . . . . . . . . . 20
3.3.4 Beam-and-Ball (BNB) System . . . . . . . . . 22
3.3.5 Active Suspension System. . . . . . . . . . . . 24
3.4 Note and References . . . . . . . . . . . . . . . . . . . 26

4 Direct Adaptive Control for Uncertain Linear Discrete time Systems 28
4.1 Introduction . .. . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.2 Systems with Reachability Hypothesis . . . . 28
4.3 Systems with Exogenous Disturbances . . .. 32
4.4 Systems with l2 Disturbances . . . . . . . . . . . . 35
4.5 A Class of Systems with Exogenous Disturbances . 38
4.6 Numerical Examples . . . . . .. . . . . . . . . . . . . . 42
4.6.1 MIMO case adopted from [39] . . . . . . . .. . . 43
4.6.2 MIMO case adopted from [8] . . . . . . . . . . . . 45
4.6.3 Aircraft Stabilization Problem . . . . . . . . . . . . 46
4.7 Notes and References . . . . . . . . . . . . . . . . . 51
5 DirectAdaptive Designs for Uncertain Normal Discretetime Nonlinear Systems 53
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.2 Systems with Trajectory Dependent Hypothesis. 53
5.2.1 Nonlinear Uncertain Systems . . . . . . . . . . 53
5.2.2 Specialization to Systems with Uncertain Dynamics . . . . . . . . . . 57
5.3 Systems without Trajectory Dependent Hypothesis . . . . . . . . . . . . . . . 59
5.3.1 Nonlinear Uncertain Systems . . . . . . . . 59
5.3.2 Uncertain Systems with l2 Disturbances . . . . 63
5.3.3 Systems with Disturbance Measurement . . . . 68
5.4 Numerical Examples . . . . . . . . . . . . . . . . . . 73
5.4.1 Uncertain Nonlinear SystemAdopted from [33] .74
5.4.2 Nonlinear SystemAdopted from [56] . . . . . 77
5.4.3 Discrete-timeActive SuspensionSystem . . . .80
5.5 Notes and References . . . . . . . . . . . . . . . . . . . . . 82
6 On Robust Direct Adaptive Designs 83
6.1 Introduction . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6.2 RobustDirect Adaptive ControlDesigns for Uncertain Linear Discretetime Systems 83
6.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . 85
6.3.1 MIMO case adopted from [39] . .. . . . . . . . . 85

6.3.2 MIMO case adopted from [8,39] . . . . . . . . . 87
6.4 Notes and References . . . . . . . . . . . . . . . . . . . 88
7 Conclusion 89
7.1 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . 89
7.2 Future Research Topics . . . . . .. . . . . . . . . . 90
References 91

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