本論文討論以矩陣型式表示多樣的複數平面圖形區域，以及正多項的低階控制器設計。本文中所討論穩定區域的圖形包含一維度、二維度以及多維度的組合圖形如：平移平面、圓形、橢圓、拋物線及其任意組成之區域圖形。在多項式方法的控制器設計中，我們以設定的區域為閉迴路極點放置的目標，並給定我們所想要的控制器階數以求解一組符合的控制器。最後我們以淡江大學航太系UAV實驗室所設計的無人飛行載具為例，做縱向運動之高度及姿態保持控制設計。

This thesis discusses the matrix representations of various complex stability regions and the designs of fixed-order controllers using positive polynomials. Stability regions presented in this thesis include one dimensional, two dimensional and their combinations. Regions such as shifted half plane, circle, ellipse, parabola, and union of regions are narrated and collated. A stabilizing control problem with low-order controller to satisfy additional constraints on the closed-loop pole location is explored in the thesis. A H-infinity control problem using positive polynomial concepts is also investigated. The longitudinal auto-pilot designs for a low-speed uninhabited experimental aircraft are presented to illustrate the fixed-order controller design using positive polynomials.

Contents
List of Figures	v
List of Tables	vii
Chapter 1 Introduction	1
I.	Norm	2
II.	Kroncker product	3
III.	Lyapunov stability for Linear system	4
Chapter 2 Stability region	6
I.	Normal asymptotic stable	7
II.	Circle region	8
III.	Parabola region	9
IV.	Elliptical region	11
V.	Union region	12
VI.	Rotated and shifted region	14
Chapter 3 Stabilizing controller	16
I.	Stabilizing problem formulation	17
II.	Numerical Example	21
Chapter 4 H-inifity controller	26
I.	H-infinity problem formulation	26
II.	Numerical Example	29
Chapter 5 Illustration	33
Chapter 6 Conclusion	44
Reference	45
List of Figures
Figure 1.1 H-inifity block diagram	3
Figure 2.1 Unit feedback diagram	7
Figure 2.2 Left half plane region	8
Figure 2.3 Circle centered on origin	8
Figure 2.4 Circle centered on (-5,0)	9
Figure 2.5 Parabolic region	10
Figure 2.6 Fat ellipse	11
Figure 2.7 Tall ellipse	12
Figure 2.8 Union region composed of circle and ellipse	13
Figure 2.9 Union region composed of circle and parabola	13
Figure 2.10 Union region composed of circles	14
Figure 2.11 Rotated ellipsoid	15
Figure 3.1 D-Region	23
Figure 3.2 Step response of feedback system.	24
Figure 3.3 Pole-zero map of the feedback system	24
Figure 4.1  control model	26
Figure 4.2  in bode plot	27
Figure 4.3 Unit feedback diagram	30
Figure 4.4 Initial design closed-loop step response	30
Figure 4.5 Closed-loop step response (a zero added)	31
Figure 4.6 Closed-loop step response(root-locus initial guess)	32
Figure 4.7 Closed-loop step response (redesigned)	32
Figure 5.1 Experimental aircraft developed by Tamkang University	33
Figure 5.2 Dynamic of the wing	34
Figure 5.3 Unit feedback diagram	36
Figure 5.4 Response without controller	36
Figure 5.5 Closed-loop with initial controller	37
Figure 5.6 Closed-loop with redesigned controller	37
Figure 5.7 Simulink of plant	38
Figure 5.8 Elevator input	38
Figure 5.9 Height of aircraft	39
Figure 5.10 Response of pitch angle 	39
Figure 5.11 Response of angle of attack 	40
Figure 5.12 Pitch hold	40
Figure 5.13 Unit feedback diagram	41
Figure 5.14 Initial step response	41
Figure 5.15 step response	42
Figure 5.16 Initial step response	43
Figure 5.17  step response	43
List of Tables
Table 5.1 Aero dynamic parameters	35

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