流場在離心式壓縮機內的不穩定現象，如喘振是在發動機運轉下所造成主要的不穩定，由於這種不穩定會因造成渦輪發動機能量損失而導致運作效能降低，嚴重的話會因震動而造成結構上的破壞。此研究的目的是在離心式發動機中，利用軸承扭力的輸入來做控制，藉此用函數近似法控制器來控制上述的不穩定現象，並藉由壓縮機的數學模型來模擬真實的壓縮機系統。

Compressor instabilities in fluid field such as surge is the main instabilities phenomena in operation of jet engine.
It reduces the performance by energy loss and causes structural damage by vibration due to the instabilities.
In this study, we use drive torque actuation in active surge control of centrifugal compressor by function approximation approach to design. The proposed method is simulated on a compressor model using a real compression system.

Contents
Acknowledgement i
Chinese Abstract ii
Abstract iii
Nomenclature iv
1 Introduction 1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 The Centrifugal Compressor . . . . . . . . . . . . . . . . . . . . . . 1
1.2.1 Impeller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.2 Diffuser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.3 Collector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Previous Work on Modelling of Compression Systems . . . . . . . . 3
1.4 Stability of Compression Systems . . . . . . . . . . . . . . . . . . . 5
1.4.1 Surge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4.2 Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . 6
2 Mathematical Model 7
2.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Nondimensional Plant . . . . . . . . . . . . . . . . . . . . . . . . . 9
3 A Review of the Function Approximation Technique 13
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.1.1 Function Approximation with the Orthonormal Series . . . . 13
3.1.2 Function Approximation with the Fourier Series . . . . . . . 15
4 Adaptive Controller for the Compression System 17
4.1 Parameterized System Equilibrium . . . . . . . . . . . . . . . . . . 17
4.2 Surge Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.3 Velocity Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
5 Numerical Results 32
5.1 case 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
5.2 case 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
6 Conclusion 39
A Modelling for a Centrifugal Compressor 40
A.1 Impeller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
A.2 Diffuser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
A.3 Energy Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
A.3.1 Ideal Energy Transfer . . . . . . . . . . . . . . . . . . . . . . 42
A.3.2 Incidence Losses . . . . . . . . . . . . . . . . . . . . . . . . . 43
A.3.3 Frictional Losses . . . . . . . . . . . . . . . . . . . . . . . . 45
A.3.4 Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
A.4 Energy Transfer and Pressure Rise . . . . . . . . . . . . . . . . . . 47
A.5 Dynamic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
B Nondimensional Plant 49
B.1 Conservation of Mass in the Plenum . . . . . . . . . . . . . . . . . 49
B.2 Momentum Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
B.3 Shafted Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
C Find c and  Values 54
C.1 when r2 + eq > 0 ( > 0) . . . . . . . . . . . . . . . . . . . . . . . 54
C.2 When r2 + eq < 0 ( < 0) . . . . . . . . . . . . . . . . . . . . . . . 55
Bibliography 56

List of Tables
1.1 Type of machine described by the model. A: Axial type compressor,
C: Centrifugal compressor, S: Surge and R: Rotating stall . . . . . . 4

List of Figures
1.1 Profile impeller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Diagram of centrifugal compressor fitted with a volute . . . . . . . 3
1.3 Compressor map with deep surge cycle . . . . . . . . . . . . . . . . 6
2.1 Equivalent compressor system . . . . . . . . . . . . . . . . . . . . . 7
5.1 Phase portrait of pressure-flow map . . . . . . . . . . . . . . . . . . 33
5.2 Pressure rise versus time . . . . . . . . . . . . . . . . . . . . . . . . 34
5.3 Mass flow versus time . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5.4 Angular velocity of compressor spool versus time . . . . . . . . . . 35
5.5 Control effort versus time . . . . . . . . . . . . . . . . . . . . . . . 35
5.6 Phase portrait of pressure-flow map . . . . . . . . . . . . . . . . . . 36
5.7 Pressure rise versus time . . . . . . . . . . . . . . . . . . . . . . . . 37
5.8 Mass flow versus time . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.9 Angular velocity of compressor spool versus time . . . . . . . . . . 38
5.10 Control effort versus time . . . . . . . . . . . . . . . . . . . . . . . 38
A.1 Velocity triangle at inducer, [19]. . . . . . . . . . . . . . . . . . . . 40
A.2 Velocity triangle at impeller tip, [15]. . . . . . . . . . . . . . . . . . 41
A.3 Incidence angles at inducer, [3]. . . . . . . . . . . . . . . . . . . . . 44
A.4 Incidence angles at diffuser, [3]. . . . . . . . . . . . . . . . . . . . . 45

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