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研究生：	陳志豪 Chen, Zhi-Hao
論文名稱：	H∞-ERL滑動控制器設計 The Composite Design of H∞-ERL Sliding-Mode Controller
指導教授：	黃正能 Hwang, Cheng-Neng
學位類別：	碩士 Master
系所名稱：	工學院 - 系統及船舶機電工程學系 Department of Systems and Naval Mechatronic Engineering
論文出版年：	2014
畢業學年度：	102
語文別：	英文
論文頁數：	162
中文關鍵詞：	滑動控制、H_∞控制理論、落後領先補償器、Popov準則
外文關鍵詞：	Sliding mode control, H_∞ control theory, Lag-Lead compensator, Popov criterion
相關次數：	點閱：157 下載：0
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在一個多輸入多輸出非線性系統中，由於系統受到外部干擾和參數不確定性的影響，將會使輸出響應無法達到所希望的規格，甚至使系統不穩定。本篇論文提出H∞-ERL滑動控制器來解決這些問題。
此控制器使用ERL型滑動控制來當作主要架構，並運用Lyapunov穩定性理論來確保系統在所預設的外部干擾與參數不確定性的範圍內為漸進穩定。為了最佳化ERL型滑動控制器中的可調參數，我們結合H_∞控制理論與落後領先補償器來找到最佳控制參數，而此參數可將外部干擾和參數不確定性所造成的不良影響壓至最低。我們指定擴增系統的閉迴路極點落在目標區域，而來滿足所希望的性能。最後使用Popov準則處理無法對消的系統不確定性，來確保系統的強健穩定性。
最後，將此篇論文所提出的控制器運用於機器手臂與水下載具上來進行電腦模擬。模擬結果展現出此控制器在滿足使用者所希望的規格時，對於外部干擾與參數不確定性有一定的強健性。

In a multi-input multi-output nonlinear system, because the system subjects to the impacts of external disturbances and parametric uncertainties, its output response may not be able to satisfy the desired specification or even may make the system unstable. The H∞-ERL sliding mode controller proposed in this thesis is motivated to solve these problems.
This controller utilizes the concept of sliding mode controller with ERL (Exponential Reaching Law) as its major framework, and then uses Lyapunov stability theorem to ensure the closed-loop stability when the system encounters prescribed external disturbances and parametric uncertainties. For optimal selection of the adjustable parameters in the proposed sliding mode control with ERL, the H_∞ control methodology and the lag-lead compensator are formulated together in the proposed control scheme to find optimal control gains, which are used to minimize the ill-effect of external disturbances and plant parametric uncertainties on the controlled output. The closed-loop poles of the augmented system are then placed on the specified region to match the desired performance. The Popov criterion is then applied to handle of the uncanceled dynamics caused by the unmodeled uncertainties so that the system robustness can be guaranteed.
Finally, a robot manipulator and a ROV are controlled and simulated respectively by the proposed controller. The simulation results reveal that the proposed control law is robust to plant uncertainties and disturbances while the desired specification assigned by users is matched.

摘要	I
Abstract	II
Acknowledgement	III
Contents…………………………………………………………………….......IV
List of tables	VI
List of figures	VI
Chapter 1: Introduction	1
1.1	Motivation	1
1.2	Literature Reviews	3
1.3	Thesis Outline	4
Chapter 2: Sliding mode control theory	5
2.1 Introduction	5
2.2 The concept of sliding mode controller	5
2.3 The utilities of sliding surface	8
2.4 The Design of Traditional Sliding Mode Control	11
2.5 Sliding mode control with Exponential Reaching Law	13
2.6 The robustness of sliding mode control with exponential reaching law	16
Chapter 3: H_∞ control theory	19
3.1 Introduction	19
3.2 The basic concept of H_∞ control theory	20
3.3 Variation approach to H_∞ control problem for linear time invariant system	21
3.3.1 Augmented system matrix	21
3.3.2 State feedback controller	24
3.3.3 State observer	32
3.3.4 Algorithm for the H_∞-optimal control problem	34
Chapter 4: The composite design of H_∞-ERL sliding mode controller	36
4.1 Introduction	36
4.2 System description	37
4.3 The design of sliding mode control with ERL (Exponential Reaching Law)	39
4.3.1 Define the sliding surfaces and the desired form that sliding surfaces differentiate with respect to time	39
4.3.2 The design of sliding mode control with ERL	40
4.3.3The stability proof of sliding mode control with ERL	41
4.4 The design of H_∞-ERL sliding mode controller	44
4.4.1 The design of equivalent control input U_H∞i	47
4.4.2 The stability proof of H_∞-ERL sliding mode controller	59
4.5 The design procedures of lag-lead compensator	62
4.6 The Composite Design of H∞-ERL Sliding-Mode Controller   	……………………………………………………………….67
4.7 The design procedures and design flowchart of H_∞-ERL sliding mode controller	76
Chapter 5: Computer simulation	79
5.1 Simulation 1: Two degree of freedom manipulator	79
5.1.1 System description	79
5.1.2	The design of H_∞-ERL sliding mode controller for two degree of freedom manipulator	82
5.1.3 Conclusion	126
5.2 Simulation 2: Depth control of ROV	127
5.2.1 System description	127
5.2.2	The design of H_∞-ERL sliding mode controller for depth control system of ROV	129
5.2.3 Conclusion	159
Chapter 6: Conclusion	160
References	161

List of tables
Table 5-1 System parameters of manipulator	80
Table 5-2 System parameters of depth control system	128
List of figures
Figure 2-1 Concept diagram of sliding surface	5
Figure 2-2 Phase plan of each structure	7
Figure 2-3 Phase plan with switching structure	7
Figure 2-4 Computing bounds on e(t)	9
Figure 2-5 Computing bounds on e^i (t)	10
Figure 3-1 The concept of H_∞ control problem in Bode plot	20
Figure 3-2 The concept of H_∞ control problem in Nyquist plot	21
Figure 3-3 Block diagram of plant G(s)	22
Figure 3-4 Structure diagram of standard form of H_∞ control problem	23
Figure 3-5 Block diagram of standard H_∞ control problem	24
Figure 4-1 The diagram of system structure with compensators	47
Figure 4-2 Augmented system diagram	48
Figure 4-3 Skeleton diagram of transfer function G_(xcd_ybar_i)	51
Figure 4-4 System structure with lag-lead compensator	53
Figure 4-5-a System structure with observer (Without lag-lead compensator)………………………………………………………………………….56
Figure 4-5-b System structure with observer and lag-lead compensator	57
Figure 4-6 Bode plot of k G(s)	65
Figure 4-7 The Bode plot of G_c (s)	66
Figure 4-8 The Bode plot of G_c (s)  G(s)	66
Figure 4-9 System structure in absolute stability problems	72
Figure 4-10 System structure in absolute stability problems	73
Figure 4-11 Loop transformation diagram	75
Figure 4-12 Design flow chart of H_∞-ERL sliding mode controller	78
Figure 5-1 System diagram of two degree of freedom manipulator	79
Figure 5-2 Augmented system structure diagram	86
Figure 5-3 Bode plot of G_(xcd_ybar_i)	91
Figure 5-4-a Response of x axis	94
Figure 5-4-b Tracking error of x axis	94
Figure 5-4-c Response of y axis	95
Figure 5-4-d Tracking error of y axis	95
Figure 5-4-e Control force τ_1	96
Figure 5-4-f Control torque τ_2	96
Figure 5-4-g Tracking trajectory of x-y plane	97
Figure 5-5-a Response of x axis	98
Figure 5-5-b Tracking error of x axis	98
Figure 5-5-c Response of y axis	99
Figure 5-5-d Tracking error of y axis	99
Figure 5-5-e Control force τ_1	100
Figure 5-5-f Control torque τ_2	100
Figure 5-5-g Tracking trajectory of x-y plane	101
Figure 5-6-a Response of x axis	102
Figure 5-6-b Tracking error of x axis	102
Figure 5-6-c Response of y axis	103
Figure 5-6-d Tracking error of y axis	103
Figure 5-6-e Control force τ_1	104
Figure 5-6-f Control torque τ_2	104
Figure 5-6-g Tracking trajectory of x-y plane	105
Figure 5-7-1 Popov plot of (G_i (jw))/(1-〖α_i G〗_i (jw) )	106
Figure 5-7-2 Nonlinearity ϕ_1 (t,y_1 ) belongs to the sector [-α_i,β_i]	107
Figure 5-7-3 Nonlinearity ϕ_2 (t,y_2 ) belongs to the sector [-α_i,β_i]	107
Figure 5-8 Bode plot of G_(xcd_ybar_i)	111
Figure 5-9-a Response of x axis	112
Figure 5-9-b Tracking error of x axis	112
Figure 5-9-c Response of y axis	113
Figure 5-9-d Tracking error of y axis	113
Figure 5-9-e Control force τ_1	114
Figure 5-9-f Control torque τ_2	114
Figure 5-9-g Tracking trajectory of x-y plane	115
Figure 5-10-a Response of x axis	116
Figure 5-10-b Tracking error of x axis	116
Figure 5-10-c Response of y axis	117
Figure 5-10-d Tracking error of y axis	117
Figure 5-10-e Control force τ_1	118
Figure 5-10-f Control torque τ_2	118
Figure 5-10-g Tracking trajectory of x-y plane	119
Figure 5-11-a Response of x axis	120
Figure 5-11-b Tracking error of x axis	120
Figure 5-11-c Response of y axis	121
Figure 5-11-d Tracking error of y axis	121
Figure 5-11-e Control force τ_1	122
Figure 5-11-f Control torque τ_2	122
Figure 5-11-g Tracking trajectory of x-y plane	123
Figure 5-12-1 Popov plot of (G_i (jw))/(1-〖α_i G〗_i (jw) )	124
Figure 5-12-2 Nonlinearity ϕ_1 (t,y_1 ) belongs to the sector [-α_i,β_i]	125
Figure 5-12-3 Nonlinearity ϕ_2 (t,y_2 ) belongs to the sector [-α_i,β_i]	125
Figure 5-13 System diagram of depth control of ROV	127
Figure 5-14 Augmented system structure diagram	132
Figure 5-15 Bode plot of G_(xcd_ybar)	137
Figure 5-16-a Depth z response	139
Figure 5-16-b Error of depth z	139
Figure 5-16-c Control input u	140
Figure 5-16-d Disturbance (d_sea)	140
Figure 5-17-a Depth z response	141
Figure 5-17-b Error of depth z	141
Figure 5-17-c Control input u	142
Figure 5-17-d Disturbance (d_sea)	142
Figure 5-18-a Depth z response	143
Figure 5-18-b Error of depth z	143
Figure 5-18-c Control input u	144
Figure 5-18-d Disturbance (d_sea)	144
Figure 5-19-1 Popov plot of G(jw)/(1-αG(jw) )	146
Figure 5-19-2 Nonlinearity ϕ(t,y_popov ) belongs to the sector [-α,β]	146
Figure 5-20 Bode plot of G_(xcd_ybar)	150
Figure 5-21-a Depth z response	151
Figure 5-21-b Error of depth z	151
Figure 5-21-c Control input u	152
Figure 5-21-d Disturbance (d_sea)	152
Figure 5-22-a Depth z response	153
Figure 5-22-b Error of depth z	153
Figure 5-22-c Control input u	154
Figure 5-22-d Disturbance (d_sea)	154
Figure 5-23-a Depth z response	155
Figure 5-23-b Error of depth z	155
Figure 5-23-c Control input u	156
Figure 5-23-d Disturbance (d_sea)	156
Figure 5-24-1 Popov plot of G(jw)/(1-αG(jw) )	158
Figure 5-24-2 Nonlinearity ϕ(t,y_popov ) belongs to the sector [-α,β]	158
                                    

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