具降階估測器之可變結構控制器設計

(1)

國

立

交

通

大

學

電機學院電機與控制學程

碩

士

論

文

具降階估測器之可變結構控制器設計

Reduced-Order Observer-Based Sliding Mode

Controller Design

研究生：魏吉佑

指導教授：陳永平教授

(2)

具降階估測器之可變結構控制器設計

Reduced-Order Observer-Based Sliding Mode

Controller Design

研究生：魏吉佑 Student：Chi-Yu Wei

指導教授：陳永平 Advisor：Professor Yon-Ping Chen

國立交通大學

電機學院電機與控制學程

碩士論文

A Thesis

Submitted to College of Electrical and Computer Engineering National Chiao Tung University

In partial Fulfillment of the Requirements For the Degree of

Master of Science In

Electrical and Control Engineering June 2009

Hsinchu, Taiwan, Republic of China

(3)

具降階估測器之可變結構控制器設計

學生：魏吉佑指導教授

：

陳永平教授

國立交通大學電機學院電機與控制學程碩士班

摘

要

本論文主要在設計可變結構控制器用來控制部份狀態變數無法量測之系統，針對無法量測之狀態變數，此控制器結合 Kudva 所使用的 Luenberger 降階估測器，此估測器能在雜訊干擾的情況下準確估測狀態變數，不受雜訊的影響，此外此控制器也使用順滑層來減抑不當的切跳現象並且消除匹配式雜訊。本論文針對三種不同的狀況來設計可變結構控制法則，包括具匹配式雜訊系統之穩定性控制，具非匹配式雜訊系統之穩定性控制，以及具非匹配式雜訊系統之追蹤控制，除了詳細列出控制法則的推導過程外，最後還利用六個不同的範例來進行數值模擬驗證，分別探討雜訊維度對降階估測器的影響，匹配式雜訊與非匹配式雜訊對受控系統穩定性與效能的影響，以及雜訊對輸出追蹤控制的影響，根據模擬結果，此可變結構控制器確實能達到所預設的控制目的。

(4)

Reduced-Order Observer-Based Sliding Mode

Controller Design

Student：Chi-Yu Wei Advisors：Prof. Yon-Ping Chen

Degree Program of Electrical and Computer Engineering

National Chiao-Tung University

ABSTRACT

This thesis presents reduced-order observer-based sliding mode controller (ROSMC) design. The reduced-order observer, a kind of Luengerger observer and designed by Kudva, is used to accurately estimate the unmeasurable state variables, even under the influence of undesirable disturbance. Besides, the ROSMC also employs a sliding layer to reduce the chattering phenomenon and eliminate the matched disturbance. There are three cases discussed in this thesis, including stability control of system with matched disturbance, stability control of system with mismatching disturbance, and output tracking control of system with disturbance. To demonstrate the usefulness of the ROSMC, there are six examples given and simulated by the software package MATLAB. The simulation results are mainly used to show the effect on the observer caused by the dimension of unmeasurable state variables, the effect on the system stability and performance caused by matched and mismatching disturbance, the effect on the output tracking control caused by disturbance. Form the simulation results, the developed ROSMC is indeed able to achieve the desired control goal.

(5)

ACKNOWLEDGMENT

本論文能順利完成，首先感謝指導老師陳永平教授這段時間來孜孜

不倦的指導，讓作者在研究方法及英文寫作上有著長足的進步，在為學處

事的態度上亦有相當的成長，謹向老師致上最高的謝意；此外，感謝桓展、

世宏學長平日在攻讀博士學位之餘，不吝傳授知識與經驗及給予建議；最

後，感謝口試委員梁耀文老師以及張浚林老師提供寶貴意見，使得論

文能臻於完整。

另外，感謝可變結構控制實驗室的承育、新光、楊庭以及學弟們對作

者的照顧與陪伴，讓作者在實驗室的研究生活充滿溫馨與快樂；還要感謝

竹工同事在此段時間的協助與幫忙，讓我能同時兼顧工作與學業。最後，

感謝爸爸、媽媽在精神上的支持及感謝岳父、岳母的鼓勵與支持，還有親

愛的老婆宜慧在我面對論文壓力時，給予體諒與包容，對於未出生的寶貝

兒子－小檸檬，謝謝你沒有在爸爸壓力大時搗蛋，讓媽媽不舒服。

兩年碩士生活隨著論文結束而終止，一路走來憑著自己的努力及許多

貴人的協助，使我能順利至此。此段旅程已結束，象徵著另一個階段的到

來，期許自己在下階段能夠更加精進。

謹以此篇論文獻給所有關心我、照顧我的人

魏吉佑 2009.6

(6)

LIST OF FIGURES

Figure 4.1 The MATLAB slimulink connection diagram……….……. 21

Figure 4.2 The observer state error ……… 21

Figure 4.3 The sliding surface s

( )

t ………. 21

( )

t bound in s ≤ε ………….……… 22

Figure 4.5 The control input u

( )

t ………….……… 22

Figure 4.6 The disturbance d

( )

t ………... 22

Figure 4.7 The system statex₁

( )

t ……….. 23

Figure 4.8 The system state x₂

( )

t and observer state xˆ₂

( )

t ………... 23

Figure 4.9 The MATLAB slimulink connection diagram……….. 27

Figure 4.10 The observer state error………….……… 27

Figure 4.11 The observer state error of convergence speed………….……… 27

( )

t ………..………. 28

( )

t bound in s ≤ε …………..………...……… 28

( )

t ……….……… 28

( )

t ……….……….. 29

Figure 4.16 The system state x₁

( )

t ……… 29

( )

t ……… 29

Figure 4.18 The system state x₃

( )

t and observer statexˆ₃

( )

t ……….. 30

(9)

Figure 4.20 The observer state error……….……… 34

( )

t ……….………. 34

( )

t bound in s ≤ε ………..………...… 35

( )

t ……… 35

( )

t ……….. 35

( )

t ……… 36

( )

t ……..………..……… 36

Figure 4.27 The system state x3

( )

t and observer state xˆ3

( )

t ...………... 36

Figure 4.28 The MATLAB slimulink connection diagram……….. 39

Figure 4.29 The observer state error……….……… 39

( )

t ……….. 39

( )

t bound in s ≤ε ……….……… 40

( )

t ……….……… 40

( )

t ………... 40

Figure 4.34 The output tracking control y

( )

t to y_d

( )

t ……… 41

Figure 4.35 The output tracking control error……….. 41

Figure 4.36 The MATLAB slimulink connection diagram….………... 44

Figure 4.37 The observer state error………. 44

( )

t ……….. 44

(10)

( )

t ……… 45

( )

t ……….. 45

( )

t to y_d

( )

t ……… 46

Figure 4.43 The output tracking control error……….. 46

Figure 4.44 The observer state error………. 48

( )

t ……….. 48

( )

t bound in s ≤ε …..………..…… 48

( )

t ………. 49

( )

t ………... 49

( )

t to y_d

( )

t ……….... 49

(11)

Chapter 1 Introduction

1.1 Motivation

This thesis will focus on the sliding mode controller design for a system with unmeasurable state variables. Unfortunately, a sliding mode control is commonly based on the feedback of full state variables. That means it is difficult to design a sliding mode control when unmeasurable state variables exist. To solve this problem, it is required for the sliding mode control to combine an observer to estimate the unmeasurable state variables. Hence, how to choose an appropriate observer for the sliding mode control becomes an important task in this thesis.

There are several observers proposed to estimate these unmeasurable state variables, including the sliding observer [1] and the reduced order observer [2]. The sliding observer is proposed by Utkin for systems without noise and achieves good performance. However, the sliding observer is only limited to noiseless systems and not extendable to general systems, which often inevitably suffer from noises. Hence, in order to deal with noises existing in general systems, Kudva presented the reduced-order Luenberger observer [2], which has been shown robust to noises. In addition, it has been found that both the sliding mode observer and the reduced-order Luenberger observer are constrained to the same conditions. Clearly, the reduced-order Luenberger observer is indubitably better than the sliding mode observer and more suitable for systems encountering noises. Therefore, this thesis will employ the reduced-order Luenberger observer to estimate the unmeasurable state variables while design the sliding mode control, which is called the reduced-order observer-based sliding mode controller, or ROSMC in short.

(12)

1.2 Research background

The feature of sliding mode control system is claimed to result in superb system performance, which includes insensitivity to parameter variations and complete rejection of matched disturbances [3]. The sliding mode control research community has risen to respond to the critical challenge, chattering phenomenon, which is the only obstacle for sliding mode to become one of the most significant discoveries in modern control theory [4-7]. Many analytical design methods were proposed to reduce the affects of chattering [8-12] and the use of sliding layer is the commonest one, which, however, inevitably decreases the control precision [13]. In general, the design of a sliding mode control is composed of two basic steps. For the first step, design a sliding function to guarantee the desired system performance in the sliding mode. For the second step, develop the control law to drive the system trajectories into the sliding layer in a finite time and stay thereafter [1, 13-15].

When unmeasurable state variables exist, the use of observer to estimate is required. The sliding mode observer design method has been proposed by Utkin, where some sufficient conditions of uncertain input need to be satisfied for the asymptotic convergence of the estimated state [16]. By selecting appropriate gains for the sliding mode observer, it can be guaranteed that the convergence of any initial estimated state to its true state in a finite time [17]. Once the estimated state reaches the true state, it will remain on the trajectory of the true state or within a very small region around the true state. However, the sliding mode observer is designed under the assumption that the measurement is not corrupted by noise, so it may not perform well for a system with noise [1].

Recently, many investigators have focused on the reduced-order Luenberger observers for the state estimation of linear systems subject to unknown inputs. Besides the observability conditions, some other conditions in terms of the zeros of the system characteristic polynomial are also needed for the observer to be designable [2, 18-21]. This thesis had tried to develop sliding mode observer to LTI system with disturbance, but found that it required the same restricted conditions as those of the reduced-order Luenberger observer proposed by Kudva. Clearly, the reduced-order Luenberger observer is indubitably better than the sliding mode observer and more suitable for systems encountering disturbance. Hence, this thesis will adopt the observer proposed by Kudva to deal with the unknown-input [2].

(13)

1.3 Thesis organization

The thesis is organized as follows. The mathematical model of observer design is introduced in Chapter 2, including the reduced order observer and the sliding mode observer. In Chapter 3, the sliding mode controller is designed based on the reduced-order observer for three cases, which are regulation control for matched disturbance, regulation control for mismatching disturbance, and output tracking control for disturbance system. In chapter 4, the simulation and results will be shown to demonstrate the usefulness of the reduced-order observer-based sliding mode controller (ROSMC). Form the simulation results, the developed ROSMC is indeed able to achieve the desired control goal. Finally, the conclusions and future research will be proposed in Chapter 5.

(14)

Chapter 2 The Mathematical Model of Observer Design

In this chapter, two kinds of observers, reduced order observer [2] and sliding mode observer [1], are introduced in Section 2.1 and 2.2 respectively for a class of linear time-invariant systems with matched and mismatched uncertainties and disturbances. Both observers are found restricted to the same conditions within the design process. In fact, the reduced order observer is more convenient when applying to system control problems. Therefore, only the reduced order observer will be adopted in Chapter 3, combined into the sliding mode control design.

2.1 Reduced order observer design

In general, an observer is required when a controller design faces insufficient information of system states. Here, introduce the reduced order observer proposed by Kudva et al. [2] for the linear time-invariant system expressed as

( )

t =

( )

t +

( )

t +

( )

t

x Ax Bu Ed (2-1)

where x

( )

t ∈ℜn is the system state, u

( )

t ∈ℜm is the control input, and d

( )

t ∈ℜq is the

disturbance. Without loss of generality, let the system state be decomposed into two parts, x1(t)

and x2(t), i.e.,

( )

1

( )

2

( )

T T T t = ⎣⎡ t t ⎤_⎦ x x x , where 1

( )

p t ∈ℜ x is measurable and 2

( )

n p t ∈ℜ − x

is not obtainable. Hence, (2-1) can be rewritten as

( )

1 11 12 1 1 1 2 21 22 2 2 2 t t t t t t ⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤ = + + ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ x A A x B E u d x A A x B E (2-2) where 11 12 1 21 22 1 , ⎛⎡ ⎤ ⎡ ⎤⎞ ⎜⎢ ⎥ ⎢ ⎥⎟ ⎣ ⎦ ⎣ ⎦ ⎝ ⎠ A A B

A A B is controllable since

(

A B,

)

is controllable, and

(

A A12, 22

)

is

an observable pair. Then, an observer of order of (n-p) to estimate x₂

( )

t is constructed as

(15)

( )

t =

( )

t + 1

( )

t +

( )

t z Fz Jx Nu (2-3) where 22 12 21 11 2 1 − + − − F = A LA J = FL A LA N = B LB (2-4)

Since

(

A A₂₂, ₁₂

)

is an observable pair, there exists an L∈ℜ(n p− ×) p such that F is of Hurwitz

and contain desired stable eigenvalues. Further choose the estimated state of x₂

( )

t as

( ) ( )

( )

2 1

ˆ t = t + t

x z Lx (2-5)

then find the derivative of xˆ₂

( )

t from (2-2) and (2-3) as

( ) ( )

( )

(

( )

)

2 1 1 11 1 12 2 1 1 ˆ t t t t t t t t t t = + = + + + + + + x z Lx Fz Jx Nu L A x A x B u E d (2-6)

Define the estimated error as

( )

2 t = ˆ2 t − 2 t

x x x (2-7)

and then achieve the derivative of x₂

( )

t from (2-2) and (2-6) as

( )

( ) (

) ( )

2 t = ˆ2 t − 2 t = 2 t + 2− 1 t

x x x Fx E LE d (2-8)

where F = A - LA has been chosen to contain stable eigenvalues. In order to eliminate ₂₂ ₁₂

the effect caused by the disturbance d

( )

t , the term E - LE must vanish to achieve ₂ ₁

( )

2 t →0

x as t→ ∞ , i.e., xˆ2

( )

t →x2

( )

t as t→ ∞ . Clearly, the reduced order observer

should be designed under the following two conditions:

i) F = A₂₂−LA has stable eigenvalues ₁₂ (2-9)

ii) E₂−LE₁=0 (2-10)

In fact, it is not easy to choose L satisfying (2-9) and (2-10) simultaneously. To deal with such problem, an algorithm is given next to suitably design L.

The existence condition of L to satisfy (2-9) and (2-10) has been introduced by Kudva, et. al. as below: [2]

[

]

(

)

1 1 11 12 2

rank =rank⎡ ⎤_{⎢ ⎥}=rank =q

⎣ ⎦

E

E A A E

E , and p≥ (2-11) q

(16)

not less than that of the disturbance d

( )

t . Once (2-11) is guaranteed, L can be selected as

(

)

2 1 p 1 1 = ++ − + L E E Γ I E E (2-12) where 1

(

1 1

)

1 1 T − T +

E = E E E is the generalized inverse of E , then 1 1 1 q

+ ₌

E E I , and

(n p− ×) p ∈ℜ

Γ is designed to satisfy (2-9). Consider two conditions of p= and p qq > to

choose matrix L. For the first condition p= , q E should be square and invertible, which ₁

leads to I_p−E E₁ ₁+ =0. Then, (2-12) becomes

1

2 1

−

L = E E (2-13)

and (2-9) can be rewritten as

1

22 12 22 2 1 12

−

− = −

F = A LA A E E A (2-14)

Clearly, the stability of F can not be determined by Γ . Hence, if F in (2-14) is not stable, then the observer (2-3) is unable to estimate the system state x. On the other hand, if F in (2-14) is stable, then the observer (2-3) is available for estimating x. For the second condition

p> , q E is no more square and invertible, i.e., ₁ I_p−E E₁ ₁+ ≠0. By substituting (2-12) into

(2-9), F can be rearranged as

(

)

(

)

(

)

22 2 1 1 1 12 22 2 1 12 1 1 12 1 2 p p − + − = − + − = + + + + + F = A E E Γ I E E A A E E A Γ I E E A Φ ΓΦ (2-15) where 1 = 22− 2 1 12 + Φ A E E A and 2 =

(

p− 1 1

)

12 +

Φ I E E A . Clearly, it is required to find a matrix Γ which guarantees F is stable. If such a matrix Γ does not exist, it is not possible to

design the reduced order observer (2-3). In case that Γ exists, then L can be chosen as (2-12) accordingly and the design of the observer (2-3) is completed which can successfully estimate

2

x as given in (2-5).

2.2 Sliding mode observer design

This section will introduce the sliding mode observer proposed by Utkin et al. [1] for the linear time-invariant system, which is a full order observer and more complicated than the reduced order observer introduced in Section 2.1. In addition, this sliding mode observer and the reduced order observer are restricted to the same conditions given (2-9) and (2-10). Hence,

(17)

this thesis will only employ the reduced order observer and the sliding mode observer is introduced in this section just for reference.

The sliding mode observer design is applied to the same system (2-2), shown in Section 2.1 and rewritten as

( )

1 11 12 1 1 1 2 21 22 2 2 2 t t t t t t ⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤ = + + ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ x A A x B E u d x A A x B E (2-16)

where x₁

( )

t ∈ℜp is obtainable and x₂

( )

t ∈ℜn p− is not measurable. Note that

(

A A₁₂, ₂₂

)

is an observable pair. The sliding mode observer proposed by Utkin is constructed as below[1]:

( )

( ) ( )

( )

1 11 1 12 2 1 2 21 1 22 2 2 ˆ ˆ ˆ ˆ ˆ ˆ t t t t t t t t t t ⎧ = + + − ⎪ ⎨ = + + + ⎪⎩ x A x A x B u v x A x A x B u Lv (2-17)

where

(

xˆ₁

( ) ( )

t ,xˆ₂ t

)

represents the state estimate for

(

x₁

( ) ( )

t ,x₂ t

)

, L∈ℜ(n p− ×) p is a

constant feedback gain matrix and v∈ℜp is a discontinuous vector defined component wise

by

( )

t =M sgn⋅

(

1

( )

t

)

v x (2-18)

where M∈R₊ will be determined later. From (2-16) and (2-17), the error dynamics are

given by:

( )

( ) ( )

( )

1 1 1 11 1 12 2 1 2 2 2 21 1 22 2 2 ˆ ˆ t t t t t t t t t t t t t t ⎧ = − = + − − ⎪ ⎨ = − = + + − ⎪⎩ x x x A x A x v E d x x x A x A x Lv E d (2-19) Let

( )

2 t = 2 t + 1 t e x Lx (2-20)

then the derivative of e₂

( )

t is

( )

2 t = 2 t + 1 t = 22 2 t + 21 1 t − 2 t

e x Lx A x A x E d (2-21)

(18)

22 22 12 21 21 11 22 2 2 1 = + = + − = + A A LA A A LA A L E E LE (2-22)

Further substituting (2-18) into (2-19) of x1

( )

t yields

( )

( ) ( )

( )

1 t = 11 1 t + 12 2 t − t − 1 t

x A x A x v E d (2-23)

where A₁₁ =A₁₁−A L₁₂ . By premultiplying x₁T

( )

t into (2-23), it can be attained that

( ) ( )

( )

(

( )

(

( )

)

( )

)

( )

(

( )

)

1 1 1 11 1 12 2 1 1 1 11 1 12 2 1 T T T t t t t t Msgn t t t t t M δ = + − − < + − + x x x A x A x x E d x A x A x E (2-24) Clearly, if M is chosen as

( )

(

11 1 12 2 1

)

M =max A x t +A x t + E δ +σ (2-25)

where σ >0, then (2-24) becomes

( ) ( )

( )

1 1 < 1

T

t t −σ t

x x x (2-26)

which guarantees x₁

( )

t reaches zero in a finite time as t→ ∞ , i.e., xˆ t₁

( )

→x₁

( )

t .as

t→ ∞ . Hence (2-21) can be rewritten as

( )

2 t = 22 2 t − 2 t

e A e E d (2-27)

where the truth of e₂

( )

t = x₂

( )

t has been adopted from (2-20). There are two important

conditions listed as below:

i) The eigenvalues of A₂₂ = A₂₂+LA₁₂ are stable. (2-28)

ii) E₂ =E₂+LE₁=0 (2-29)

With these two conditions, the estimation error e₂

( )

t will approach zero as t→ ∞ , i.e.,

( )

2 2

ˆ t → t

x x as t→ ∞ . This confirms the success of the sliding mode observer (2-17).

(19)

for the reduced order observer. Since the sliding mode observer is much more complicated, the reduced order observer will be used in the sliding mode controller design in the next chapter.

(20)

Chapter 3 Reduced-Order Observer-Based Sliding

Mode Controller Design

In this Chapter, the sliding mode controller is designed based on the reduced-order observer for three cases, including regulation control for matched disturbance in Section 3.1, regulation control for mismatching disturbance in Section 3.2, and tracking control for mismatching disturbance in Section 3.3.

3.1 Reduced-order observer-based sliding mode controller

design for matched disturbance

In this section, consider reduced-order observer-based sliding mode controller design for matched disturbance system, expressed as

( )

t =

( )

t +

(

( ) ( )

t + t

)

x Ax B u d (3-1)

wherex

( )

t ∈ℜm is the control input, d

( )

t ∈ℜq is the

disturbance,

(

A B,

)

is controllable, and that can transform into (2-2) with E replaced by B.

The reduced-order observer in (2-5) will be used here to estimate the unmeasurable state variables. As usual, the first step in the sliding mode controller design is choosing an appropriate sliding function, such that the system trajectory is steered to the control goal in the sliding mode. Let the sliding function be

( )

1

( )

_{( )}

2 ˆ ˆ t t t t ⎡ ⎤ = _{= ⎢} _⎥ ⎣ ⎦ x s Cx C x (3-2)

where det

( )

CB ≠0. The matrix C can be determined by the transformation matrix method

[13], to guarantee the system stability in the sliding mode s

( )

t =0. For the second step, the

control algorithm is designed to drive the system into the sliding mode based on the approaching condition [13]. To derive the control algorithm, differentiating (3-2) yields

(21)

( )

t = ˆ

( )

t = ˆ

( )

t +

(

( ) ( )

t + t

)

s Cx CAx CB u d (3-3)

Then, the equivalent control input u_eq

( )

t in the sliding mode can be found from

( )

ˆ

( )

(

( ) ( )

)

0 eq eq u u t ₌ = t + t + t = s CAx CB u d (3-4) which leads to

( )

1

( ) ( )

1

( )

_o

( )

ˆ eq t t t eq t eq t − − = − − = + u CB CAx CB CBd u u (3-5)

Note that the equivalent control input is partitioned into two parts shown as

( )

1

( )

o ˆ eq t t − = − u CB CAx (3-6)

( )

1

( )

eq t t − = − u CB CBd (3-7)

where ueqo

( )

t is the dominant part and ueq

( )

t is related to the disturbance d

( )

t . Clearly,

because of the existence of d

( )

t in u_eq

( )

t , the control input u can not directly adopt the

equivalent control u_eq

( )

t , instead it is set as

( )

o

( )

(

( )

)

( )

max

eq eq

t = t − t +σ sgn

u u u s (3-8)

where the upper bound of u_eq

( )

t is selected as

( )

1

max

eq t γ

−

=

u CB with γ ≥ CBd

( )

t

and σ is a positive constant. By substituting (3-8) into (3-3), the derivative of s

( )

t

becomes

( )

t = −

(

γ σ+

)

sgn

( )

+

( )

t

s s CBd (3-9)

Further premultiplying s results in T

( ) ( )

( )

1

( )

T t t t t γ σ t t σ γ γ ⎛ ⎞ = − − + = − − _⎜_⎜ − _⎟_⎟ ⎝ ⎠ s CBd s s s s s CBd s s s (3-10)

Since γ ≥ CBd

( )

t and σ >0, (3-10) becomes

( ) ( )

T

t t < −σ

s s s (3-11)

which evidently guarantee the reach and sliding condition. As a consequence, s

( )

t →0 in a

finite time. It is well known that the chatting exists due to the use of sgn s

( )

in the control

algorithm (3-8). Hence, to ameliorate such undesired chatting, the switching function sgn s

( )

(22)

( )

, = > sat ε ε ε ε ⎧ ⎪⎪ ⎨ ⎪ _≤ ⎪⎩ s s s s s s (3-12)

where ε >0 is the thickness of the sliding layer s ≤ε . Therefore, the control algorithm

(3-8) is changed into

( )

1

( ) ( ) (

1

) ( )

ˆ - sat ,

u t = − CB − CAx t CB − γ σ+ sε (3-13)

which will drive the system trajectory into the sliding layer, not in the sliding mode, in a finite time and force it to stay within there. Once the system trajectory is bounded in the sliding layer, it will move toward the control goal and then around there. In other words, the control goal can not be precisely attained and the matched disturbance is not completely eliminated since the system trajectory is not restricted in the sliding mode. The errors are caused by the use of saturation function and depend on the scale of the matched disturbance and the eigenvalues chosen for the sliding mode. The effect of the matched disturbance and the eigenvalues will be demonstrated later by the simulation results in Section 4.1.

3-2 Reduced-order observer-based sliding mode controller

design for mismatching disturbance

In this section, the reduced order observer is used to design the mismatching disturbance of LTI system, described as

( )

t =

( )

t +

( )

t +

( )

t

x Ax Bu Ed (3-14)

wherex

( )

t ∈ℜq is the

mismatching disturbance since E≠BQ ,

(

A B,

)

is controllable, and that can transform into

(2-2). The reduced-order observer (2-5) will be used here to estimate the unmeasurable state variables. The first step in the sliding mode controller design is to choose an appropriate sliding function, such that the system trajectory will be moved to the control goal in the sliding mode. Let the sliding function be

( )

1

( )

_{( )}

2 ˆ ˆ t t t t ⎡ ⎤ = _{= ⎢} _⎥ ⎣ ⎦ x s Cx C x (3-15)

(23)

where det

( )

CB ≠0. The matrix C can be determined by the transformation matrix method

[13], such that the system is stabilized in the sliding mode s t

( )

=0. For the second step, the

control algorithm is designed to drive the system into the sliding mode based on the approaching condition. In the design process, the derivative of (3-15) is obtained as

( )

t = ˆ

( )

t = ˆ

( )

t +

( )

t +

( )

t

s Cx CAx CBu CEd (3-16)

Then, the equivalent control input u_eq

( )

t in the sliding mode can be found by

( )

ˆ

( )

0

eq eq

u u

t ₌ = t + t + t =

s CAx CBu CEd (3-17)

which leads to

( )

1

( ) ( )

1

( )

o

( )

ˆ eq t t t eq t eq t − − = − − = + u CB CAx CB CEd u u (3-18)

Note that the equivalent control input is partitioned into two parts, shown as

( )

1

( )

o _ˆ eq t t − = − u CB CAx (3-19)

( )

1

( )

eq t t − = − u CB CEd (3-20)

where u_eqo

( )

t is the dominant part and u_eq

( )

t is related to the disturbance d

( )

t . Clearly,

because of the existence of d

( )

t in ueq

( )

t , the control input u can not directly adopt the

equivalent control u_eq

( )

t , instead it is set as

( )

o

( )

(

( )

)

( )

max

eq eq

t = t − t +σ sgn

u u u s (3-21)

where the upper bound of ueq

( )

t is

( )

1 max

eq t γ

−

=

u CB with γ ≥ CEd

( )

t and σ is a

positive constant. Further substituting (3-21) into (3-16) yields

( )

t = −

(

γ σ+

)

sgn

( )

+

( )

t

s s CEd (3-22)

and premultiplying s results in T

( ) ( )

( )

1 T t t t t t t γ σ σ γ γ = − − + ⎛ ⎞ = − − _⎜_⎜ − _⎟_⎟ ⎝ ⎠ s s s s s CEd s CEd s s s (3-23)

Since γ ≥ CEd

( )

( ) ( )

T

t t < −σ

s s s (3-24)

which evidently guarantees the reaching and sliding condition. As a consequence, s

( )

t =0

(24)

ameliorate such undesired chattering, the switching function sgn s

( )

is often replaced by the saturation function, expressed as

( )

, = > sat ε ε ε ε ⎧ ⎪⎪ ⎨ ⎪ _≤ ⎪⎩ s s s s s s (3-25)

where ε >0 is the thickness of the sliding layer s ≤ε . Therefore, the control algorithm

(3-21) is changed into

( )

1

( ) ( ) (

1

) ( )

ˆ ,

u t = − CB − CAx t − CB − γ σ+ sat sε (3-26) and the system trajectory will be steered into the sliding layer, not in the sliding mode, during a finite time and then stays therein. Once the system trajectory is bounded in the sliding layer,

it will move toward the control goal x→0 and finally go around there. As a result, the

control goal can not be precisely attained and the mismatching disturbance can not be eliminated in the sliding layer. The errors are caused by the use of saturation function, the scale of the mismatching disturbance and the eigenvalues chosen for the sliding mode. The effect of the mismatching disturbance and the eigenvalues will be demonstrated later by the simulation results in Section 4.2.

3-3 Reduced-order observer-based sliding mode controller

design output tracking control for disturbance

In this section, the reduced-order observer is used to design tracking control for disturbance with LTI system as

( )

t =

( )

t +

( )

t +

( )

t

x Ax Bu Ed (3-27)

wherex

( )

t ∈ℜq is the

disturbance, and

(

A B,

)

is controllable. Without loss of generality, the system state x(t) is

preprocessed to contain measurable part x₁

( )

t and unmeasurable part x₂

( )

t , i.e.,

( )

1

( )

2

( )

T

t = ⎡_⎣ t t ⎤_⎦

x x x . Here, the reduced-order observer (2-5) will be used here to estimate

(25)

( )

t =

( )

t

y Gx (3-28)

where G is full rank. That means the controller is designed to drive the output to follow the

desired trajectory y_d

( )

t , i.e., the tracking error

( )

t =

( )

t − d

( )

t

e y y (3-29)

will vanish as t→ ∞ .

For the sliding mode tracking controller design, first define a new state concerning the tracking error, expressed as

( )

₀t

( )

t =

∫

τ τd h e (3-30) then

( ) ( )

t = t =

( )

t − d

( )

t h e y y (3-31)

Clearly, by combining (3-27) and (3-31), the system can be reconstructed as

( )

t = p

( )

t + p

( )

t + p

( )

t +

( )

t p A p B u E d f (3-32) where

( )

t t t ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ x p h , p ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ A 0 A G 0 , p ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ B B 0 , p ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ E E 0 ,

( )

d

( )

t t ⎡ ⎤ = ⎢₋ ⎥ ⎣ ⎦ 0 f y . Note that

(

Ap B is controllable since the pair p

)

(

A B

)

is controllable. The first step in the sliding

mode controller design is to choose an appropriate sliding function, such that the system trajectory can trace the control goal in the sliding mode. Let the sliding function be

( )

t ˆ

( )

t ˆ

_{( )}

( )

t t ⎡ ⎤ = _{= ⎢} _⎥ ⎣ ⎦ x s Cp C h (3-33)

where det

( )

CB_p ≠0 and C can be determined by the transformation matrix method [13],

such that the system is stabilized in the sliding mode s t

( )

=0. For the second step, the control

algorithm is designed to drive the system into the sliding mode based on the approaching condition.

Once C is determined in the first step, then the control algorithm is derived by differentiating (3-33) as

( )

t = ˆ

( )

t = _pˆ

( )

t + _p

( )

t + _p

( )

t +

( )

t

s Cp CA p CB u CE d Cf (3-34)

The equivalent control u_eq

( )

t can be found from

( )

ˆ

( )

0

eq p p eq p

u u

t ₌ = t + t + t + t =

(26)

which leads to

( )

0

( )

eq t = eq t + eq t

u u u (3-36)

with nominal part

( )

1

( )

1

( )

o ˆ eq t p p t p t − − = − − u CB CA p CB Cf (3-37) and disturbance

( )

1

( )

eq t p p t − = − u CB CE d (3-38)

Because of the existence of d

( )

t in ueq

( )

t , the control input u is generally designed as

( )

o

( )

(

( )

)

( )

max ,

eq eq

t = t − t +σ sat s ε

u u u (3-39)

where σ is a positive constant and

( )

(

)

1

max

eq t p γ

−

=

u CB with γ ≥ CE dp

( )

t is the

upper bound of u_eq

( )

t . The saturation function is given as

( )

, = > sat ε ε ε ε ⎧ ⎪⎪ ⎨ ⎪ _≤ ⎪⎩ s s s s s s (3-40)

where ε >0 is the thickness of the sliding layer s ≤ε . By substituting (3-39) into (3-34),

the derivative of s

( )

t becomes

( )

t = −

(

γ σ+

)

⋅sat s

( )

,ε + _p

( )

t

s CE d (3-41)

Further premultiplying s results in T

( ) ( )

( )

1

( )

p

( )

T p t t t t γ σ t t σ γ γ ⎛ ⎞ = − − + = − − _⎜_⎜ − _⎟_⎟ ⎝ ⎠ s CE d s s s s s CE d s s s (3-42)

Since γ ≥ CE d_p

( )

( ) ( )

T

t t < −σ

s s s (3-43)

which evidently guarantees the reaching and sliding condition of the sliding mode s

( )

t =0

in a finite time. By substituting (3-36) into (3-32), obtained as

( )

(

1

)

( )

(

( )

1

)

(

( )

)

p p eq p p p p p p p t t t t t t t t − − = + + + = − + − + p A p B u E d f I B CB C A p I B CB C E d f (3-44)

(27)

p = p

E B Q is matched disturbance and yd

( )

t is constant, then restructures (3-44) as

( )

(

( )

1

)

(

( )

)

p p p

t = − − t + t

p I B CB C A p f (3-45)

The design eigenvalues of C can be chosen such that

(

I B CB− _p

( )

_p −1C A p

)

(

_p

( )

t + f

( )

t

)

is

Hurwitz, and then p

( )

t =_⎣⎡x

( ) ( )

t h t ⎤_⎦T =0 as t→ ∞ , hence

( ) ( )

∞ = ∞ =

( )

∞ − _d

( )

∞ =0

h e y y , and then attains the control goal y t

( )

to y_d

( )

t , that

will simulate in Section 4.3 case 1.

If Ep ≠B Q is mismatching disturbance and p yd

( )

t is constant, then restructure (3-44)

as

( )

(

( )

1

)

(

( )

)

(

( )

1

)

( )

p p p p p p

t = − − t + t + − − t

p I B CB C A p f I B CB C E d (3-46)

The mismatching disturbance is not remove, and then will make control error, such that

output tracking control not exact tracks y

( )

t to y_d

( )

t . In order to reduce tracking error that

can chooses the design eigenvualues of C distant from origin in left phase plane, but that can make control input have high gain, that will be simulate in Section 4.3 case 2.

If E_p ≠B Q is mismatching disturbance and _p y_d

( )

t is not constant i.e., y_d =cos t

( )

,

then must be affected with mismatching disturbance and y , and then the output tracking d

control not complete tracks y

( )

t to y_d

( )

t . In order to reduce tracking error that can

chooses the design eigenvualues of C distant from origin in left phase plane, but that can make control input have high gain, that will be introduce in Section 4.3 case 3.

(28)

Chapter 4 Simulation and Result

In this chapter, consider six case of reduced-order observer-based sliding mode

controller design. There are the matched disturbance system and the dimension p= , the q

matched disturbance system and the dimension p> , the mismatching disturbance system q

for controller stability, the output tracking with matched disturbance system and tracking trajectory is constant, the output tracking with mismatching disturbance and tracking trajectory is constant, and the output tracking with mismatching disturbance system and tracking trajectory is not constant.

4.1 Observer-based sliding mode controller design for

matched disturbance

In this section, using reduced-order observer-based sliding mode controller design to simulates two case, there are matched disturbance system that dimension of measurable equal dimension of disturbance, and matched disturbance system that dimension of measurable greater than dimension of disturbance.

In Case 1, consider a LTI system (3-1) suffering from the matched disturbance, the

system state be decomposed into two parts

( )

1

( )

2

( )

T

T T

t = ⎣⎡ t t ⎤_⎦

x x x , where x1

( )

t is

measurable and x₂

( )

t is not obtainable. Then the system reconstruct as:

( )

(

( ) ( )

)

1 1 2 2 1 1 1 2 3 1 t t t t t t ⎡ ⎤ ⎡− ⎤⎡ ⎤ ⎡ ⎤ = + + ⎢ ⎥ ⎢ ₋ ⎥⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ x x u d x x (4-1)

The matched disturbance is assumed as

( )

0.5 sin 2

( )

cos 0.1sin

( )

1

4

t

d t = ⎛_⎜ πt + ⎛_⎜π ⎞_⎟⎞_⎟+ t x

⎝ ⎠

⎝ ⎠ (4-2)

(29)

( )

+1

d t ≤ x t (4-3)

The system dimension of measurable part and the dimension of disturbance represent as

(

11 12

)

(

[

]

)

( )

1

1 1 1

1

rank =rank − = =p rank =rank_⎜⎛⎡ ⎤_{⎢ ⎥}⎞_⎟= =q

⎣ ⎦ ⎝ ⎠

A A B (4-4)

The first step is to design the reduced order observer, and the observer (2-3) represent as

( )

t =

( )

t + 1

( )

t +

( )

t z Fz Jx Mu (4-5) where

( )

22 12 21 11 2 1 3 1 2 1 1 1 − = − − × + − = + − × − − = − × F = A LA L J = FL A LA FL L M = B LB L (4-6)

And the reduced order observer should be designed under two conditions of (2-9) and (2-10).

Because the dimension p= , hence, the matrix L can be chosen as (2-13) by q

1

2 1 1

− _{= −}

L = B B (4-7)

where L is not chosen by Γ . By substituting (4-7) into (4-6) and the observer (4-5) rewrite as

( )

t = −2

( )

t +3 1

( )

t

z z x (4-8)

In this case, the eigenvalue of F is stable, then it is completed estimate x as given in (4-8) ₂

successfully. The second step is using this information to design the sliding mode controller, Let the sliding function be

( )

t = ˆ

( )

t

s Cx (4-9)

Because the eigenvalues of A are -0.2679 and -3.7321, then the matrix C can be determined by the transformation matrix method and via the pole-assignement method to appoint the

eigenvalue is -4, and CB I= , hence, the design C as

[

]

= 1 0

C (4-10)

The design input u

( )

t as (3-12) by

( )

t = − ˆ

( ) (

t − γ σ+

)

sgn

( )

= − −

[

1 1

]

ˆ

( )

t −

(

ˆ

( )

t +1

)

sat

( )

,ε

u CAx s x x s (4-11)

where σ =0 and sliding layer −0.01≤ ≤ε 0.01. The simulation results as:

Figure 4.1 to Figure 4.8 are simulation results with initial condition x

( )

0 =

[

10 7

]

T and

( )

0 =0

z . Figure 4.1 shows the MATLAB simulink connection diagram, that contains system,

reduced-order observer, sliding mode controller, and disturbance. Figure 4.2 shows the observer state error, that at 4.87s convergence to approach of zero, that after into sliding layer,

(30)

and the convergence speed from (17, 0s) to (2.3, 1s), conform the eig

( )

F = −2. Figure 4.3 shows the sliding surface, that at 3.115s into the sliding layer, and then it effect the control input u that has a disjunctive part that from -0.9769 to 0.08283 at this time shows in Figure 4.5, it possible two reason that the observer does not completely estimated, and sliding surface must bound in sliding layer, hence, must has higher gain instant to conform two

possible reasons. Figure 4.4 shows the sliding surface bound into s ≤ε after complete

estimate, Figure 4.6 shows the disturbance, that the range between -1 and 1.35. Figure 4.7 and

(31)

Figure 4.1 The MATLAB slimulink connection diagram

Figure 4.2 The observer state error

( )

t

Disturbanc

Sliding Mode Controller Reduced Order Observer System y x_h2 x_d2 x_d1 _x2 _x1 u ex2 d Saturati on S Product w_h y sat u x_h2 S fcn M AT LAB Fun_S_u 1 s Integrator3 1 s Integrator1 1 s Integrator -K-Gai n z y ud z_d w_h fcn Em bedded M AT LAB Fun_w_h x1 x2 y fcn Em bedded M AT LAB Fun_Gx d u Bu_d1 Bu_d2 fcn Em bedded M AT LAB Fun_Bu_d x2 x1 Ax1 Ax2 fcn Em bedded M AT LAB Fun_Ax 0.5si n(2pi t) 0.5cos(pi t/4) 0.1si n(t) t:3.115s s:0.01 t:4.87ss:-0.007645 t:3.115s ex2:0.03348 t:4.87s ex2:0.001 t:1s ex2:2.3 t:0s ex2:17

(32)

( )

t bound in s ≤ε

( )

t

( )

t

t:4.87s u:0.7718 t:3.115s u:-0.9769 t:3.151s u:0.08283 t:3.115s s:0.01 t:4.87s s:-0.007645

(33)

( )

t

( )

t and observer state xˆ₂

( )

t

t:3.115s x2:-0.1065 t:4.87s x2:-0.006052 t:3.115s x1:0.009881 t:4.87s x1:-0.007645

(34)

In case 2, consider a LTI system (3-1) suffering from the matched disturbance, the

system state be decomposed into three parts

( )

₁T

( )

₂T

( )

₃T

( )

T

t = ⎣⎡ t t t ⎤_⎦

x x x x , where x₁

( )

t

and x₂

( )

t are measurable and x₃

( )

t is not obtainable. Then the system reconstruct as:

( )

(

( ) ( )

)

1 1 2 2 3 3 -0.277 1 -0.002 1 -1.71 -0.178 -12.2 0 0 0 -6.67 1 t t t t t t t t ⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎢ ⎥₌⎢ ⎥⎢ ⎥₊⎢ ⎥ ₊ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢_⎣ ⎥_⎦⎢ ⎥ ⎢ ⎥_{⎣ ⎦} ⎣ ⎦ ⎣ ⎦ x x x x u d x x (4-12)

The matched disturbance is assumed as

( )

0.5 sin 2

( )

cos 0.1sin

( )

1

4

t

d t = ⎛_⎜ πt + ⎛_⎜π ⎞_⎟⎞_⎟+ t x

⎝ ⎠

⎝ ⎠ (4-13)

Apparently, the upper bound of the matched disturbance is obtained as

( )

+1

d t ≤ x t (4-14)

The system dimension of measurable part and the dimension of disturbance represent as

(

)

( )

11 12 0.277 1 -0.002 2 1.71 0.178 12.2 1 0 1 1 rank rank p rank rank q ⎛ −⎡ ⎤⎞ = _⎜_⎢ _⎥_⎟= = − − − ⎣ ⎦ ⎝ ⎠ ⎛⎡ ⎤⎞ ⎜⎢ ⎥⎟ = _⎜_{⎢ ⎥}_⎟= = ⎜_{⎢ ⎥}_{⎣ ⎦}⎟ ⎝ ⎠ A A B (4-15)

where the dimension p> . The first step is to design the reduced order observer, and the q

observer (2-3) represent as

( )

t =

( )

t + 1

( )

t +

( )

t z Fz Jx Mu (4-16) where

[

]

22 12 21 11 2 1 0.002 6.67 12.2 0.277 1 0 0 1.71 0.178 1 1 0 − ⎡ ⎤ − = − _{− × ⎢} _⎥ − ⎣ ⎦ − ⎡ ⎤ + − = + _{− × ⎢} _⎥ − − ⎣ ⎦ ⎡ ⎤ − _{= − × ⎢ ⎥} ⎣ ⎦ F = A LA L J = FL A LA FL L M = B LB L (4-17)

And the reduced order observer should be designed under two conditions of (2-9) and (2-10).

Because the dimension p> , hence, the matrix L can be chosen as (2-12) by q

(

)

[

]

2 1 1 1 0 0 1 0 0 1 p ⎡ ⎤ = + − = _{+ ⎢} _⎥ ⎣ ⎦ + + L B B Γ I B B Γ (4-18)

(35)

determine for eigenvalues of F. By substituting (4-18) into (4-16) and design the eigenvalues of F are -1, -5, and -10 respectively, then the L represent separately as

[

]

[

]

[

]

1 0.4648 1 0.1369 1 0.273 = − = − = − − L L L (4-19)

By substituting (4-19) into (4-17), then the observer (4-16) rewrite as

( )

[

]

( )

_{( )}

( )

[

]

( )

_{( )}

( )

[

]

( )

_{( )}

1 2 1 2 1 2 1 1.5192 0.6173 5 4.9589 0.3396 10 9.2588 3.6821 t t t t t t t t t t t t ⎡ ⎤ = − − _⎢ _⎥ ⎣ ⎦ ⎡ ⎤ = − − _⎢ _⎥ ⎣ ⎦ ⎡ ⎤ = − − _⎢ _⎥ ⎣ ⎦ x z z x x z z x x z z x (4-20)

Therefore, it is completed which can successfully estimate x as given in (4-20). The second ₃

step is using this information to design the sliding mode controller, let the sliding function be

( )

t = ˆ

( )

t

s Cx (4-21)

Because the eigenvalues of A are -0.2275±1.3067i and -6.67, then the matrix C can be determined by the transformation matrix method and via the pole-assignment method to

appoint the eigenvalues are -1 and -4, andCB I= , hence, the design C as

[

]

= 0.1323 -0.2659 0.8677

C (4-22)

The design input u

( )

t u as (3-12) by

( )

( ) (

)

( )

[

]

( )

(

( )

)

( )

ˆ ˆ ˆ 0.4180 0.1796 -2.5438 1 , t t sgn t t sat γ σ ε = − − + = − − + u CAx s x x s (4-23)

where σ =0 and sliding layer −0.01≤ ≤ε 0.01. The simulation results as:

Figure 4.9 to Figure 4.18 are simulation results with initial condition x

( )

0 =

[

10 7 3

]

T

and z

( )

0 =5. Figure 4.9 shows the MATLAB simulink connection diagram, that contains

system, reduced-order observer, sliding mode controller, and disturbance. Figure 4.10 shows the observer state error, that at 8.453s, 1.771s and 0.9199s convergence to approach of zero respectively, therefore, Figure 4.11 shows the convergence speed from (4.746, 0s) to (1.744,

1s), (7.042, 0s) to (0.04736, 1s), (9.911, 0s) to (0.0004488, 1s), conform theeig

( )

F = −1,

( )

5

(36)

left phase plane, the observer accurate estimate soon. Figure 4.12 shows the sliding surface, that at 0.3244s, 0.191s, and 0.2413s into the sliding layer respectively, and then it lead to the control input have disjunctive parts shows in Figure 4.14 at these time, and the sliding surface guarantee bound into sliding layer after accurate estimate show in Figure 4.13. Figure 4.15 shows the disturbance, that the range between -1 and 1.3. Figure 4.16, 4.17 and 4.18 illustrate

(37)

Figure 4.9 The MATLAB slimulink connection diagram

Figure 4.10 The observer state error

Figure 4.11 The observer state error of convergence speed

y2 y1 x_h3 x_d3 x_d2 x_d1 x3 x2 x1 u ex3 d Saturati on S Product w_h y 2 y 1 sat u x_h3 S fcn MAT LAB Fun_S_u

1 s Integrator3 1 s Integrator2 1 s Integrator1 1 s Integrator -K-Gai n z y 2 y 1 ud z_d w_h fcn Embedded M AT LAB Fun_z_d x1 x2 x3 y 1 y 2 fcn Em bedded M AT LAB Fun_Gx d u Bu_d1 Bu_d2 Bu_d3 fcn Em bedded

MAT LAB Fun_Bu_d x3 x2 x1 Ax1 Ax2 Ax3 fcn Embedded MAT LAB Fun_Ax 0.5si n(2pi t)

0.5cos(pi t/4) 0.1sin(t)

Disturbanc

Sliding Mode Controller Reduced Order Observer System t:8.453s ex3:0.001 t:1.771s ex3:0.001 t:0.9199s ex3:0.001 t:0s ex3:4.746 t:0s ex3:7.042 t:0s ex3:9.91 t:1s ex3:0.0004488 t:1s ex3:0.04736 t:1s ex3:1.744

具降階估測器之可變結構控制器設計

國

立

交

通

大

學

電機學院 電機與控制學程

碩

士

論

文

具降階估測器之可變結構控制器設計

Reduced-Order Observer-Based Sliding Mode

Controller Design

研 究 生：魏吉佑

指導教授：陳永平 教授

具降階估測器之可變結構控制器設計

Reduced-Order Observer-Based Sliding Mode

Controller Design

研 究 生：魏吉佑 Student：Chi-Yu Wei

指導教授：陳永平 Advisor：Professor Yon-Ping Chen

國 立 交 通 大 學

電機學院 電機與控制學程

碩 士 論 文

具降階估測器之可變結構控制器設計

學生：魏吉佑 指導教授

陳永平 教授

國立交通大學 電機學院 電機與控制學程碩士班

摘

要

Reduced-Order Observer-Based Sliding Mode

Controller Design

Student：Chi-Yu Wei Advisors：Prof. Yon-Ping Chen

Degree Program of Electrical and Computer Engineering

National Chiao-Tung University

ABSTRACT

ACKNOWLEDGMENT

本論文能順利完成，首先感謝指導老師 陳永平教授這段時間來孜孜

不倦的指導，讓作者在研究方法及英文寫作上有著長足的進步，在為學處

事的態度上亦有相當的成長，謹向老師致上最高的謝意；此外，感謝桓展、

世宏學長平日在攻讀博士學位之餘，不吝傳授知識與經驗及給予建議；最

後，感謝口試委員 梁耀文老師以及 張浚林老師提供寶貴意見，使得論

文能臻於完整。

另外，感謝可變結構控制實驗室的承育、新光、楊庭以及學弟們對作

者的照顧與陪伴，讓作者在實驗室的研究生活充滿溫馨與快樂；還要感謝

竹工同事在此段時間的協助與幫忙，讓我能同時兼顧工作與學業。最後，

感謝爸爸、媽媽在精神上的支持及感謝岳父、岳母的鼓勵與支持，還有親

愛的老婆宜慧在我面對論文壓力時，給予體諒與包容，對於未出生的寶貝

兒子－小檸檬，謝謝你沒有在爸爸壓力大時搗蛋，讓媽媽不舒服。

兩年碩士生活隨著論文結束而終止，一路走來憑著自己的努力及許多

貴人的協助，使我能順利至此。此段旅程已結束，象徵著另一個階段的到

來，期許自己在下階段能夠更加精進。

謹以此篇論文獻給所有關心我、照顧我的人

魏吉佑 2009.6

CONTENTS

LIST OF FIGURES

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

電機學院電機與控制學程

研究生：魏吉佑

指導教授：陳永平教授

研究生：魏吉佑 Student：Chi-Yu Wei

國立交通大學

電機學院電機與控制學程

碩士論文

學生：魏吉佑指導教授

陳永平教授

國立交通大學電機學院電機與控制學程碩士班

本論文能順利完成，首先感謝指導老師陳永平教授這段時間來孜孜

後，感謝口試委員梁耀文老師以及張浚林老師提供寶貴意見，使得論