輸出迴授型順滑控制於機械系統之應用

行政院國家科學委員會專題研究計畫 成果報告

輸出迴授型順滑控制於機械系統之應用

研究成果報告(精簡版)

計 畫 類 別 ： 個別型

計 畫 編 號 ： NSC 96-2221-E-161-005-

執 行 期 間 ： 96 年 08 月 01 日至 97 年 07 月 31 日

執 行 單 位 ： 亞東技術學院電機工程系

計 畫 主 持 人 ： 張浚林

計畫參與人員： 大專生-兼任助理人員：鄒孟哲

處 理 方 式 ： 本計畫可公開查詢

中 華 民 國 97 年 08 月 14 日

輸出回授型順滑控制於機械系統之應用

亞東技術學院電機系

張浚林 副教授

台北縣板橋市四川路二段 58 號

E-mail:jlchang@ee.oit.edu.tw 中文摘要 本計劃主要式針對具有非匹配型干擾的機械系統，首先 提出具有系統狀態以及干擾估測能力的估測器設計方法， 當干擾量在兩個取樣時間內的變化不要太厲害的話，我們 提出的設計方法可以達到相當準確的估測能力。接著使用 command generator tracker 的方式產生想要的參考模式， 這些估測的狀態與干擾將直接使用在順滑控制器的設計當 中，最後我們可以經由理論證明系統的追蹤誤差將被逼進 在一個很小的範圍內，也會利用一個馬達帶動的機械系統 做相關理論的驗證的工作。 關鍵詞：順滑模態，干擾估測，模式追蹤，輸出回授 Abstract

In response to a MIMO discrete-time linear system with mismatched disturbance, an algorithm capable of performing estimated system state and disturbance is proposed first and then proceeded with the design of controller. Provided the variation of disturbance in the two consecutive sampling instances is not changed significantly, both the system state and disturbance can be simultaneously estimated by our proposed observer algorithm with the estimation error being constrained in a small bounded region. Then, a method utilizing command generator tracker is designed to generate the reference model. The estimations of the system state and disturbance are then to be used in the sliding mode controller where it can cause the tracking error to be constrained in a small bounded region with the guaranteed system stability. Finally, a numerical example is presented to demonstrate the applicability of the proposed control scheme.

Index Terms—Discrete-time, sliding mode, disturbance

estimation, model reference.

I. INTRODUCTION

Sliding-mode control has been widely studied and well established with respect to continuous-time systems [1]. Recently, the use of PC-based controllers is becoming popular; and many investigators have paid more attentions to the design of discrete-time sliding mode controller. However, due to the characteristics of a finite sampling rate, some of the properties that are available to be applied to the continuous-time sliding mode control could be considered inappropriate for a discrete-time system [2]. More importantly, a sliding mode controller to be developed mainly for continuous-time systems may be unstable when implemented with direct digital applications [3]. Hence, many researchers [4-15] have focused their efforts on investigation into the design of time sliding mode control (DSMC) or discrete-time variable structure control.

Milosavljevic [4] was one of the first researchers among others to formally propose that the sampling process in discrete-time systems can limit the existence of

sliding mode. Later on, Sapturk et al. [5] suggested that a reaching condition should be widely used in current DSMC systems. Bartoszewicz [6] proposed a control algorithm that guarantees finite-time convergence of the state trajectory to the sliding surface. Koshkouei and Zinober [7] clarified the concept of DSMC by presenting several new sufficient conditions to illustrate the necessity of the existence of using discrete-time sliding mode. Su et al. [8] used the time delay approach to designing perturbation estimation that the upper bound of perturbation is not required. Similarly, such similar methods can be found in other papers [9-10]. However, in these papers [4-12], the controller design is based on the assumption that all of the system state is available, whereas in most physical systems this premise is not always the case. With emphasis given to a robust output feedback scheme, Edwards and Spurgeon [1] have given a very detailed discussion for the design of output feedback sliding mode controller in continuous-time systems. In light of designing direct torsion control of flexible shaft, Korondi et al. [13] proposed a scheme implementing an observe-based DSMC. Tang and Misawa [14] used output feedback and made use of a single sliding surface in the controller design for MIMO systems. Janardhanan and Bandyopadhyay [15] used fast output sampling technique to avoid measuring the system state.

When designing control systems, unmeasurable disturbances not exactly known to a plant can affect the system performance. The disturbance observer (DOB), which has been formulated in frequency domain, is known to be very effective in compensating disturbances and it has become very popular for robust motion control [16-20]. Combining DOB with DSMC design can improve the tracking accuracy of servo systems [19-20]. Conventionally, based on the transfer function approach, the DOB is designed using the inverse dynamics of a plant and the so-called Q-filter is to be used to determine the performance of robustness and disturbance rejection. The main problems with this approach are that it is difficult for DOB to be designed into MIMO systems and the system must be in minimum phase (with respect to the relation between the output and the disturbance. Unknown input observer (UIO), which is another formulation related to the unknown input estimation, can be used to estimate the system state and disturbance. In several literatures [21-24], various approaches have been proposed to design an UIO for the discrete-time systems in the presence of unknown inputs. Two conditions presented here are used for checking for the existence of a stable UIO. The first is called the matching condition of which it is a rank condition. The second is that the system (with respect to

the relation between the output and the disturbance) must be minimum phase. If one of the two conditions mentioned above is not satisfied, there exists no method to be found to design a stable UIO. As proposed by Jin et al. [22], it was demonstrated that the matching condition can be relaxed by allowing delays of the system output in the observer, but no design method was provided. Sundaram and Hadjicostis [23] extended this method and proposed a streamlined design procedure. Of considerable significance, Floquet and Barbot [24] used an output canonical transformation to design an UIO with the matching condition being invalid.

In this paper, a design technique using DSMC by employing output information only for a class of MIMO systems with mismatched disturbance is proposed. First, we introduce the integral term of the estimation output error in the observer design to propose certain degrees of freedom. Using this technique and assuming that the disturbance does not vary too much in between two consecutive sampling instances, we shall demonstrate that the proposed algorithm in digital implementations can not only reduce the state estimation errors but also restrict the disturbance estimation errors to be smaller than a size of

( )

O T where T is the sampling interval. Hence, the

conventionally-assumed upper bound restriction on the disturbance is relaxed to the restriction of the rate of the disturbance, of which it is considerably slower than the sampling rate. In a like manner, using the estimation state and the estimation disturbance, the control law is then designed and the closed-loop stability is analyzed. Together, some significant features are discussed including the selection of the sliding surface, the performance of estimation disturbance and the convergence rate to sliding mode.

This paper is organized as follows. The system description and the problem formulation are given in Section 2. Section 3 presents the state estimator and the disturbance observer. In Section 4, the sliding surface design that can effectively reduce the influence of mismatched disturbance during sliding motion is demonstrated. In Section 5, we develop the controller algorithm using the estimation state and disturbance to stabilize the system with system stability analysis being demonstrated. Section 6 gives a numerical example to exhibit the effectiveness of the proposed controller. Finally, Section 7 presents the concluding remarks.

II. PROBLEM FOMULATION

Consider a continuous-time MIMO linear system with mismatched disturbance described by

( ) ( ) ( ) ( ) ( ) ( ) t t t t t t = + + = x Hx Du Gd y Cx  (1) where _x∈_\n _{is the system state, u}∈_\m _{is the control}

input, _∈ l

d \ is mismatched disturbance, and y∈\ is m

the measurable output of the system. For system (1), the system matrices have appropriate dimensions and the matrix G is of full rank. Suppose that the sampling interval is T and a zero-order-hold is adopted for the above-mentioned continuous-time model. Denoting

( )k = (kT)

x x , y( )k =y(kT), and u( )k =u(kT) where

0

k≥ is an integer, the discrete-time model can be then

given by ( 1) ( ) ( ) ( ) ( ) ( ) k k k k k k + = + + = x Ax Bu f y Cx (2)

where these matrices A and B can be determined as

( )

( )

( )

2 2 2 0 exp 2 1 exp 2! n T T T T d T T O T τ τ = = + + + =

_∫

= + + = H A H I H B H D D HD " " (3) Moreover, the disturbance term ( )f k in (2) can be

written as

(

)

( )

0 2 ( ) ( 1) ( ) T k e k T d k O T = + − = +

∫

H f Gd Ed τ _{τ τ} (4) where

( )

0exp T d =

_∫

E H Gτ τ and the magnitudes of

( )k

f is on the order of O T( ). If d( )t is smooth and the

sampling rate is small enough, Adridi et al. [10] have shown that

( )

( )

2 3 ( 1) ( ) ( 1) 2 ( ) ( 1) k k O T k k k O T + − = + − + − = f f f f f (5) The magnitude of g is said to be

( )

n

O T = g if 0 lim _n 0 T→ _T ≠ g and ₁ 0 lim _n 0 T→ _T − = g (6) where n is an integer and we denote _{O T}

( )

0 _=O

( )

_{1 . In}

this paper, a state and unknown disturbance estimation algorithm developed for system (2) was first proposed in Section 3. When the sampling period is small enough and the disturbance is smooth enough, the proposed design method can constrain the estimation error to be on the order of O T( ). Using the estimation information, the

discrete-time sliding mode controller, which can track the desired signal with the tracking error being constrained in a small region on the order of O T( ), is then designed in

Section 5. Before introducing the main results, the following four assumptions are made throughout this paper.

Assumption 1: System (2) is controllable and observable. Assumption 2: These matrices A, G and C can satisfy

n rank⎛⎜_⎜⎡_⎢ − ⎤_⎥⎞⎟_⎟= +n l ⎣ ⎦ ⎝ ⎠ A I G C 0

The assumption follows that the dimension of disturbance is smaller than the dimension of output.

Assumption 3: The matrices A, B and C satisfy n rank⎛⎜_⎜⎡_⎢ − ⎤_⎥⎞⎟_⎟= +n m ⎣ ⎦ ⎝ ⎠ A I B C 0 and rank

( )

CB =m.

This condition implies that system (2) has no transmission zeros at one.

Assumption 4 [8-10]: The sampling interval T is

sufficiently small such that the disturbance does not vary too much between two consecutive sampling instances. Moreover, the following relations can be effectively approximated as

( ) ( ) ( )

( ) ( ) (

)

( )

( )

( )

1 , 1 n n n n m n m n n O T O T O T n O T O T O T n m O T O O T n + + + ≈ ∀ ∈ℵ ≈ ∀ ∈ℵ ≈ ∀ ∈ℵ

where ℵ is the set consisting of all integers. III. STATE AND DISTURBANCE ESTIMATION

Before proceeding any further, there exists an observer design method to be called as the proportional integral observer (PIO). This PIO structure, which uses an additionally introduced integral term of the output estimation error in the observer design, can offer certain degrees of freedom. Beale and Shafai [25] reported this additional freedom in the observer-based controller design, as a result of which PIO becomes less sensitive to the variations of the system parameters. Busawon and Kabore [26] demonstrated that the PIO design is capable of effectively reducing the effect of noises as opposed to the proportional observer. Shafai et al. [27] used the PIO method to study the loop transfer recovery problem in discrete-time systems. In particular, Soffker et al. [28] used the PIO technique to simultaneously estimate the system state and system nonlinearities. In this work, we shall employ the PIO structure to simultaneously estimate the system state and unknown disturbance.

Let xˆ( )k and ˆ( )yk be the estimations of x( )k and ( )k

y , respectively. Design state and disturbance

observers as

(

)

(

)

1 2 ˆ( 1) ˆ( ) ( ) ( ) ˆ( ) ( ) ˆ ( 1) ( ) ( ) ( ) ˆ( ) ˆ( ) k k k k k k k k k k + = + + − + + = + − = x Ax Bu L y k y k Gq q q L y y y Cx (7) where _∈ l

q \ , L₁∈\n m× and L₂∈\l m× are matrices designed by the latter. Define the errors of the estimation state and the estimation output as x( )k =x( )k −xˆ( )k and

ˆ

( )k = ( )k − ( )k = ( )k

y y y Cx

  , respectively. From (2) and

(7) we have

(

1

)

(k+ =1) − ( )k + ( )k − ( )k

x A L C x f Gq (8) Since the matrix G is of full rank, the estimation error of the disturbance is defined as

( ) ( ) ( ) f k k k + =G f −q

e

(9) where + ₌

(

T

)

−1 T_∈ l n× G G G G \ and e_f ∈\l represents

the estimation error of unknown disturbance. Equation (8) can be then rewritten as

(

1

)

(

)

(k 1) ( )k f( )k n ( )k

+

+ = − + + −

x A L C x Ge I GG f (10) Moreover, the dynamic equation of e is given by _f

(

)

(

)

2 2 ( 1) ( 1) ( 1) ˆ ( 1) ( ) ( ) ( ) ( 1) ( ) ( ) ( ) f f k k k k k k k k k k k + + + + = + − + = + − − − = + − + − e G f q G f q L y Cx G f f e L Cx (11) Introducing an augmented state vector

( ) T( ) T( ) T n l f k =⎡_⎣ k k ⎤_⎦ ∈ + w x e \ and augmenting (10) and (11) obtain

(

)

(

)

(

)

[

]

( ) ( 1) ( ) ( 1) ( ) ( ) ( ) ( ) ( ) n e e e f k k k k k k k k k + + ⎡ − ⎤ ⎢ ⎥ + = − + ⎢ + − ⎥ ⎣ ⎦ ⎡ ⎤ = _{⎢ ⎥}= ⎣ ⎦ I GG f w A LC w G f f x y C 0 C w e   (12) where _e l ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ A G A 0 I , 1 2 T T T ⎡ ⎤ = ⎣ ⎦ L L L and C_e =

[

C 0

]

. From (12), we know that, if

(

A Ce. e

)

is observable, the

matrix Ae−LCe can be stabilized.

Lemma 1: If the pair

(

A C,

)

is observable, then the following statements are equivalent:

(1) rank n n l ⎛⎡ − ⎤⎞ = + ⎜⎢ ⎥⎟ ⎜_{⎣ ⎦}⎟ ⎝ ⎠ A I G C 0

(2) The pair

(

A C_e, _e

)

is observable.

Proof: From linear system, the pair

(

A C,

)

is observable in terms of which it is equivalent to

n rank⎛⎜_⎜⎡_⎢λ − ⎤_⎥⎞⎟_⎟=n ⎣ ⎦ ⎝ ⎠ I A C ∀ ∈λ C Hence, statement 2 holds if and only if

(

)

rank rank 1 n n l e l e n l C λ λ λ λ + ⎛⎡ − − ⎤⎞ ⎛⎡ − ⎤⎞₌ ⎜⎢ ₋ ⎥⎟ ⎜⎢ ⎥⎟ ⎜⎢ ⎥⎟ ⎜_{⎣ ⎦}⎟ ⎝ ⎠ ⎜_⎝⎢_⎣ ⎥_⎦⎟_⎠ = + ∀ ∈ I A G I A 0 I C C 0

There are two cases to be discussed: (i) λ≠1 and (ii)

1

λ= in the following.

(i) When λ≠1 , the above-mentioned equation holds because the pair

(

A C,

)

is observable.

(ii) When λ=1, it follows that

rank n l e rank n e n l λ ₊ ⎛⎡ − ⎤⎞ ⎛⎡ − ⎤⎞ = = + ⎜_{⎢ ⎥}⎟ ⎜_{⎜⎢ ⎥}⎟_⎟ ⎜_{⎣ ⎦}⎟ _{⎝⎣ ⎦⎠} ⎝ ⎠ I A A I G C C 0

Hence, two statements (1) and (2) are equivalent. This

completes the proof of lemma. …

Lemma 2: If the sampling period is small enough and the

disturbance does not vary too much between two consecutive sampling instances, then it follows that

(

)

_{( )}

( )

2

n k O T

+

− =

I GG f .

Proof: From (4),

(

I−GG+

)

f( )k can be written as

(

)

(

)

(

)

(

)

(

)

( )

0 2 2 0 2 ( ) ( 1) ( 1) 2 T T k e k T d k T d O T τ _{τ τ} τ τ τ τ + + + − = − + − ⎛ ⎞ = − _⎜ + + + _⎟ + − ⎝ ⎠ =

∫

∫

H I GG f I GG Gd H I GG I H " Gd

Hence, we complete this lemma. …

From (5) and Lemma 2, equation (12) can be rewritten as

(

)

( 1) ( ) ( ) ( ) ( ) e e e k k k k k + = − + = w A LC w h y C w  (13) where

( )

2 2 ( )k = =γ O T

h . Since the pair

(

A C_e. _e

)

is

be used to decide the gain matrix L. Another alternative method is given by

(

)

1 T T e e e e e e e − = + L A P C R C P C (14) where (n l) (n l) 0 e + × + ∈ >

P \ is obtained from the following

discrete algebraic Riccati equation

(

)

1 T T T T e e e e e e e e e e e e e e e − + + − = P A P C C P C R C P A A P A Q (15) where (n l) (n l) 0 e + × + ∈ > Q \ and R_e∈\m m× >0. We know

that the matrix A_e−LC_e can be decomposed as 1

e e

−

− =

A LC MJM (16)

where J is the Jordan matrix of the eigenvalues of

e− e

A LC and M is the corresponding eigenvector matrix. Then the solution of (13) is

1 1 1 0 ( ) k (0) k i ( 1) i k k i − − − = = +

∑

− − w MJ M w MJ M h (17) Since the matrix Ae−LCe is stable, it is easily shown

that lim k 0 k→∞J = and 1 1 0 lim ( ) limk i ( 1) k k i k k i − − →∞w = →∞

∑

₌ MJ M h − − (18) Taking 2-norm in both sides of (18) obtains

1 1 2 2 2 2 2 0 lim ( ) limk i ( 1) k k i k k i − − →∞ w ≤ →∞

∑

₌ M J M h − − 1 2 0 limk i k i aγ − →∞ ₌ ≤

∑

J (19) where 1 2 2

a= M M− . The performance of the

proposed estimator satisfies the property of Theorem 1.

Theorem 1: Consider the dynamic system (1) and its

corresponding discrete-time model can be described by (2). If the state and disturbance observers are designed as (7), then it follows from (5), (12) and Lemma 2 that

(

)

( )

2 ( 1) ( ) ( 1) e e ( ) f f k k O T k k + ⎡ ⎤ ⎡ ⎤ = − + ⎢ ₊ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ x x A LC e e  

Since a matrix L from Lemma 1 should be found such that A_e−LC is stable, the following statement can be _e

obtained: (1) lim ( )₂

( )

k→∞ x k ≤O T (2)

( )

2 lim f( ) k→∞ e k ≤O T

Proof: Since the matrix Ae−LC is stable, it follows e

that J ₂ = <β 1. Using Tustin’s approximation, Abidi et

al. [10] have shown that 2

2 Tp Tp + = − β where p is the

corresponding eigenvalue in the continuous-time case to be assumed as O

( )

1 . Hence, we have

( )

1 1 2 0 1 2 lim 1 2 k i k i Tp O T Tp β − − →∞ ₌ − = = = − −

∑

J and

( ) ( )

1 2

( )

2 lim ( ) 1 k a k O T− O T O T →∞ w ≤ −β = = γ . Since ₂ 2₂ 2

( )

2 lim ( ) lim ( ) f( ) k→∞ w k =k→∞ x k + e k ≤O T  _{, it} follows that (1) lim ( )₂

( )

k→∞ x k ≤O T (2)

( )

2 lim f( ) k→∞ e k ≤O T

The proof of this theorem is completed. …

Remark 1: As can be seen from the above-mentioned

analysis, if the disturbance is smooth enough and if T is small enough, the proposed observer method can be used to accurately perform the operation with well estimation. This assumption, which indicates the insignificant variation of the disturbance between the two sampling points, points out the fact that the unknown input bandwidth is far smaller than the control bandwidth.

Remark 2: In response to a system incapable of

satisfying the matching condition, the matrix Ae−LCe is

stable and then our proposed algorithm can obtain a stable estimator as long as Assumption 2 is valid. In comparison with the traditional UIO design methods [21-24], the algorithm we propose is capable of not only dealing with an unsatisfied matching condition system but also being implemented in specific nonminimum phase systems.

Remark 3: In these papers [17-18], a constant disturbance

model has been introduced into system (2). From (4) and (5), the augmented system model is then given by

( )

( )

2 ( 1) ( ) ( ) ( 1) l ( ) O T k k k k k _{O T} ⎡ ⎤ + ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ _{⎢ ⎥} =_{⎢ ⎥} + + ⎢ ₊ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦ _{⎣ ⎦} A E x x B u 0 I d d 0

Based on the above system to design the observers of the state the disturbance, the dynamic equation of the estimation error can be written as

[

]

( )

1 2 ( 1) ( ) ( 1) l ( ) k k O T k k + ⎛ ⎞ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ =⎜ − ⎟ + ⎢ ₊ ⎥ _⎜⎢ ⎥ ⎢ ⎥ _⎟⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎝ ⎠⎣ ⎦ x A E L x C 0 0 I L d d    

It follows from Theorem 1 and the above equation that our proposed observer design method can obtain the better performance.

IV. COMMAND GENERATOR TRACKER

Command generator tracker is used to construct a paradigm developed for the purpose of establishing the desired reference model. Broussard and O’Brien [29] proposed the command generator tracker methodology and this method was originally developed for continuous-time flight control. Basically, the fundamental model of command generator is a linear system excited by a constant input. The most unique feature of this method is that there do not exist any restrictions of imposing the order of the model to be the same as the plant; thus, the conventional perfect model following constraint can be replaced. When the dimension of the system is increased, the command generator tracker method can be utilized in improving the calculation time quite significantly.

Let the reference output ym be generated by the

following the reference model:

(

)

( 1) ( ) ( ) ( ) ( ) m m k m m m m m m k k k k k + = + + = x A I x B u y C x (20) where k m∈

p m∈

u \ is the reference input, and ym having the same

dimension as y is the reference output. Note that the order of x_m is allowed to be unequal to the order of system state x. Hence, the model (20) can be of any order that is sufficiently large to maintain the command of the plant. For simplicity, we assume here that the command generator uses the constant um to generate the desired

output y_m . It follows that u_m( )k =u_m(k+1) . Let

y = − m

e y y and e_x = −x F x_{1 m}−F u_{2 m}. Then we have the following lemma.

Lemma 3: If there exist parameter matrices 1

n k×

∈

F \ ,

2∈ n p×

F \ , F3∈\m k× and F4∈\m p× such that

(

)

(

)

1 1 3 2 1 4 1 2 , , , and n m n m m − − = − − − = − = = A I F F A BF A I F F B BF CF C CF 0 (21) then it follows that

( 1) ( ) ( ) ( ) ( ) ( ) x x y x k k k k k k + = + + = e Ae B f e Ce τ (22) where τ( )k =u( )k −F x3 m( )k −F u4 m( )k .

Proof: From (2) and (20), the dynamic equation of e is x

given by

(

)

(

1 2

)

1 1 ( 1) ( ) ( ) ( ) ( ) ( ) x m m m m k k k k k k + = + − + − + + e Ax Bu F A F x F B F u f

Substituting the matrix relation equations (21) into the above equation yields

(

)

(

)

(

)

1 3 2 4 1 2 1 ( 1) ( ) ( ) ( ) + ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = ( ) x x m m x y x m m m x k k k k k k k k k k k k k k k + = + + + + + = + + = + + − e Ae Bu AF BF x AF BF u f Ae B f e C e F x F u CF x Ce τ

This completes the proof. …

Lemma 4 [30]: Consider a Sylvester equation of the form s s s

+ =

X A XB C (23)

where _∈ r s×

X \ is an unknown matrix, and A , s B , and s s

C are constant matrices of appropriate dimensions. Let

( )

i s

λ A represent the ith eigenvalue of A and _s vec X

[ ]

denote a column vector obtained by stacking all column vectors of X together. Equation (23) has a unique solution if and only if λi

( ) ( )

As λj Bs ≠ −1 for any i and j; in this

case the unique solution is written as

[ ]

(

)

1

[ ]

T

s s pq s

vec X =⎡_⎣ B ⊗A +I ⎤_⎦− vec C (24)

where ⊗ represents the Kronecker product. When

s =

C 0 , the homogenous equation also has a unique

solution X =0 . …

Rewrite (21) as a matrix equation form

1 2 1 1 3 4 m m n m − ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ F F F A F B A I B F F C 0 C 0 . (25)

Since Assumption 4 holds, we have

1 11 12 21 22 n − − ⎡ ⎤ ⎡ ⎤ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ A I B C 0 Ω Ω Ω Ω (26) where _{Ω11∈ \}n n× _, 12 n m× ∈ \ Ω , _{Ω21∈ \}m n× _{, and} 22 m m× ∈ \

Ω . It follows from (25) and (26) that

1 2 11 12 1 1 3 4 21 22 m m m ⎡ ⎤ ⎡ ⎤⎡ ⎤ = ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ F F F A F B F F C 0 Ω Ω Ω Ω , (27) or in equation forms 1= 11 1 m+ 12 m F Ω F A Ω C (28) 3 = 21 1 m+ 22 m F Ω F A Ω C (29) 2 = 11 1 m F Ω F B (30) 4 = 21 1 m F Ω F B (31)

Given the known matrices A , m C , m Ω , and 11 Ω , and 12 applying Lemma 4 to (28), if λ_i

( ) ( )

Ω₁₁ λ_j A_m ≠1 we can

obtain the solution for the matrix F as ₁

[ ]

(

(

)

)

1

[

]

1 mT 11 nk 12 m

vec F =_⎣⎡ A ⊗ −Ω +I ⎤_⎦− vec Ω C (32)

Then substituting it into equations (29)-(31), the total solutions for the matrices F , 1 F , 2 F , and 3 F can be 4 obtained. Once these parameter matrices for system (22) have been derived, a control law is then designed such that the system output y( )k can approximately track the

reference output ym( )k .

V. OBSERVER BASED SLIDING MODE CONLLER

DESIGN

After our having established the reference model and obtained the associated matrices, in this section, we proceed with the design of controller using the estimations of the state and disturbance. In order to guarantee that the system output in steady state to be equal to given the desired value as it should be, we introduce a feedforward term and the integral action in the controller. It is shown that the integrator can be used not only to eliminate steady state error but also to reduce the effect of disturbance.

First the sliding surface is designed as

( )

1 ( )k y( )k (k 1) − = − − s CB e v (33) where _∈ m

v \ , designed in the latter, is used to be stabilized the reduced-order system. The dynamic equation of the sliding surface is

( )

( )

1 3 1 4 ( 1) ( ) ( ) ( ) ( ) ( ) ( ) x m m k k k k k k k − − + = + − − + − s CB CAe u F x F u CB Cf v (34) Let diag w

(

1, ,wm

)

m m × = " ∈\ Φ be a chosen matrix

where it must satisfy Φ ₂ <1 . Design the control algorithm u( )k as

( )

(

)

(

)

(

)

1 3 4 2 2 2 ˆ ( ) ( ) ( ) ( ) ( ) ( ), ( ) ( ) ( ) for ( ) ( ) ( ), ( ) for ( ) x m m k k k k k sat k k k k k k sat k k k − = − + + + − + + ⎧ _> ⎪ ⎪ = ⎨ ⎪ _≤ ⎪⎩ u CB CAe CGq v s s F x F u s s s s s s ρ ρ ρ ρ ρ ρ Φ Φ (35)

where eˆ_x = −xˆ F x_{1 m}−F u_{2 m} and ρ >0 is a design

parameter chosen in the latter. Let

( )

1

(

)

f − = + n CB C Ax Ge . Substituting (35) into (34) obtains

(

)

( )

1

(

)

( 1) ( ) ( ) ( ), ( ) k k k sat k k − ₊ + = + − + − s s n s CB C I GG f ρ ρ Φ Φ (36) From Theorem 1, we have

( )

( )

( )

2 2 2 2 1 lim ( ) lim ( ) 1 1 k k a k k O T O T O γ η β →∞ →∞ − ≤ = = − = = n Γ w Γ (37) where Γ= CB

( )

−1C A G

[

]

and 2

( )

1 1 a O γ η β = = − Γ . Since Assumption 4 holds and

( )

1

(

)

( )

O T − ₋ + ₌ CB C I GG f , (36) can be approximated as

(

)

(k+ ≈1) ( )k + ( )k −ρ sat ( ),k ρ s Φs n Φ s (38)

Theorem 2: Consider system (22) with the chosen

sliding surface (33). Let

(

2

)

2 1 aγ ε ρ η β + = > − Γ Φ where 0

ε > is a chosen constant and design the control input as (35). When the system satisfies s( )k ₂ >ρ , then its

trajectories will get into the region s( )k ₂ ≤ρ in a finite

step, where the region s( )k ₂ ≤ρ is called to be the

attraction region. In addition, the system is finally restricted in the region s( )k ₂ ≤ =η O

( )

1 as k→ ∞ ,

which is named as the sliding region.

Proof: When s( )k ₂ >ρ, it follows from (38) that

(

)

2 2 2 ( ) ( 1) ( ) ( ) ( ) ( ) ( ) ( ) ( ) k k k k k k k k k + = − + = − + s s s n s s s n s ρ ρ Φ Φ Φ

Taking the norm of two sides of the above equation results in

(

)

(

)

2 2 2 2 2 2 2 2 ( 1) ( ) ( ) ( ) ( ) 1 k k k a k k + ≤ − + + ≤ − + − s s n s n ρ γ ε β Φ Γ Φ

Since ε > , from (37) there exists a non-negative integer 0

1 k such that

(

2

)

2 2 ( ) 1 a k γ ε β + ≤ − n Γ

Φ for k≥ , and then k1

the above equation becomes

2 2 2

(k+1) ≤ ( )k

s Φ s for k≥ k₁

In the worst case, we assume that the condition 2

( )k >ρ

s holds when k≥ . From k₁ 0< Φ ₂ <1, we

should be able to find a non-negative integer k such that 2

( )

2 2 1 ₂ k k ρ < s

Φ . It follows that s( )k ₂ ≤ρ when

1 2

k≥ + . Hence, the system trajectory will reach and k k

enter the region s( )k ₂ ≤ρ in a finite number of steps.

When the system enters in the attraction region, the control law (35) is switched as

( )

1

(

)

3 4 ˆ ( ) ( ) ( ) ( ) ( ) ( ) x m m k k k k k k − = − + + + + u CB CAe CGq v F x F u

The dynamics of s( )k can be then rewritten as

( )

1

(

)

( 1) ( ) ( ) ( ) k k k k − ₊ + = + − ≈ s n CB C I GG f n

As k→ ∞ , it is obvious that s( )k ₂ ≤η . That means

that the system is finally restricted in the region 2

( )k ≤η

s , which is to be called as the sliding region. Hence, the system trajectory first gets into the attraction region within finite steps and finally stays in the sliding

region. …

Finally, we shall analyze the system stability when the system stays in the sliding region. From (33) and (34), the control input in the sliding region becomes

( )

1

(

)

3 4 ˆ ( ) ( ) ( ) ( ) ( ) ( ) x m m k k k k k k − = − + + + + u CB CAe CGq v F x F u (39) Substituting (39) into (22) obtains the system dynamic in the sliding layer as

(

)

( 1) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) x x x k k k k k k k k k k + + = + + + + − ≈ + + + e Ae Bv Rw r I GG f Ae Bv Rw r (40) where _r=

(

_{I−B CB}

( )

−1_{C GG f}

)

+ _{, R=BΓ , and}

( )

−1 = −

A A B CB CA. In order to reduce the tracking

error, it is desired to introduce integrators to the control algorithm To this end, we employ the linear quadratic integral technique to design v( )k . First, introduce the

integral vector of the tracking error as (k+ =1) ( )k +ey( )k = ( )k +Cex( )k η η η (41) where η∈ \m _{. Let} T T T x ⎡ ⎤ = ⎣ ⎦ z e η and T T z = ⎣⎡ ⎤⎦ r r 0 . Augmenting (41) with system (40) will yield

( 1) ( ) ( ) ( ) ( ) ( ) ( ) z z z z y z k k k k k k k + = + + + = z A z B v R w r e C z (42) where

[

]

, z z , z , z m ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ =_{⎢ ⎥} =_{⎢ ⎥} =_{⎢ ⎥} = ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ B R A 0 A B R C C 0 0 0 C I

Lemma 5 [31]: If the pair

(

A B,

)

is controllable and Assumption 4 holds, then the pair

(

A Bz, z

)

is

controllable. …

Consider the following quadratic cost to be minimized as 0 ( ) ( ) ( ) ( ) T T k J k k k k ∞ = =

∑

z Qz +v Rv (43) where _∈ m m× _>0

R \ and Q∈\(n m+ × +) (n m)≥0 are the given matrices. Since system (42) is controllable, we can design the feedback gain as

(

)

1

[

]

1 2 T T z z z z − = − + = − K B PB R B PA K K (44) where 1 m n× ∈ K \ and 2 m m× ∈

K \ . Moreover, the matrix

(n m+ × +) (n m) ₀

∈ >

P \ can be obtained by the following

algebraic Riccati equation:

(

)

1 T T T T z z z z z z z z − + + − = P A PB B PB R B PA A PA Q (45)

Since ex is not obtainable, ( )v k can be designed as

1ˆ 2

( )k = − x( )k − ( )k

v K e Kη (46)

where eˆ_x = −xˆ F x_{1 m}−F u_{2 m}

.

Substitute (39) into (36) to obtain

(

)

( 1) ( ) ( ) ( ) ( ) ( ) z z z y z k k k k k k + = − + + = z A B K z Nw r e C z (47)

where N =B Kz

[

1 0

]

+Rz . Combining (47) and (13)

obtains the overall closed-loop system as

( 1) ( ) ( 1) ( ) ( ) ( ) z z e e z k k k k k k − + ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ = ⎢ ⎥ ⎢ ₊ ⎥ ₋ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎡ ⎤ + ⎢ ⎥ ⎣ ⎦ A B K N z z 0 A LC w w r h (48)

From (48), the stability of the overall system can be accomplished by designing Az−B Kz and Ae−LCe

separately. Therefore, one can conclude that the robust stability is guaranteed under the proposed control algorithm. Once having guaranteed the system stability, we obtain the system performance in the following.

Theorem 3: Consider the dynamic system (22) in

conjunction with the estimator model (7). The sliding surface is designed as

1ˆ 2

( )k = _y( )k + _x(k− +1) (k−1)

s e K e Kη , (49)

and the control algorithm is given by

( )

(

)

(

)

1 1 2 3 4 ˆ ( ) ( ) ( ) ( ) ˆ ( ) ( ) ( ), ( ) ( ) x x m m k k k k k k sat k k k ρ ρ − = − + + − − − + + u CB CAe CGq s K e K s F x F u Φ Φ η (50)

If the disturbance is smooth enough between two consecutive sampling rates, then, when the system is in the sliding region, the tracking error has the following property:

( )

2

lim y( )

k→∞ e k ≤O T

Proof: Let the operator Δ be defined as

( )k ( )k (k 1)

Δz =z −z − and then the following

representation can be obtained

(

)

(k 1) z z ( )k ( )k z( )k Δz + = A −B K Δz + ΔN w + Δr

From (13), the dynamic equation of Δw can be written as

(

)

(k 1) _{e e} ( ) k ( )k Δw + = A −LC Δw + Δh It follows from (5) that

( )

(

1

)

(

)

( ) ( 1) ( ) z k k k − ₊ ⎡ _{− − −} ⎤ ⎢ ⎥ Δ = ⎢ ⎥ ⎣ ⎦ I B CB C GG f f r 0

Hence, we from (5) have

( )

2

2 ( ) z k O T Δr = and

( )

3 2 ( )k =O T h

Δ . Combine the dynamic equations of

( )k Δz and Δw( )k to obtain

( )

2 ( 1) ( ) ( 1) ( ) z z e e k k O T k k − Δ + ⎡ ⎤ Δ ⎡ ⎤ ⎡ ⎤ =_{⎢ ⎥} + ⎢_{Δ +} ⎥ ₋ ⎢_Δ ⎥ ⎣ ⎦ ⎣ ⎦⎣ ⎦ A B K N z z 0 A LC w w

Similar to the work of Theorem 1, we obtain

( )

2

lim ( )

k→∞ Δz k ≤O T

It follows from (ey k− = Δ1) η( )k that

( )

2

2 2

lim y( ) lim y( 1) lim ( )

k→∞ e k =k→∞ e k− ≤k→∞ Δz k =O T

We complete this proof. …

Remark 4: When the sliding surface is designed as

( )k =Sx( )k =0

σ , these papers [6, 8-10] proposed the

equivalent control law u( )k = −

( )

SB −1SAx( )k to control

the system. In this case, the dynamics of ( )σ k can be

obtained as

(k+ = Sf1) ( )k

σ .

From the above equation, the system will enter the sliding region in one step. However, from A=exp

(

HT

)

this

control law can be written as a linear combination of the state ( )x k and ( )s k as follows:

( )

( )

( )

1 2 2 1 1 ( ) ( ) ( ) ( ) 2 k k T k T k − − − = − ⎛ ⎞ = − − _⎜ + + _⎟ ⎝ ⎠ u SB SAx H SB s SB S H " x Since from (3) the order of B is O T

( )

, the first term of

the above equation has no elements to cancel the effect of

( )

−1

SB unless ( )σ k is in the sampling time order or its

trajectory is in the vicinity of ( )σ k = 0 . Therefore,

without considering the structural properties of A and B, this control maybe results in inadmissible control input, which is called the high-gain control problem. Our proposed control algorithm can avoid this problem.

VI. SIMULATION RESULTS

In this section, we consider the velocity control problem for a two-mass spring system shown in Figure 1. The system has a flexible coupling between the motor and the load. Let w and M w denote angular velocities of L

motor and load, respectively. We assume that only w _M

can be measured. The equations of the two-mass spring system are

[

]

21 22 2 31 33 3 0 1 1 0 0 0 0 0 0 0 1 0 S S M M L L L S M L w a a w b u w a a w g y w w ⎡ ⎤ ⎡ − ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎢ _{⎥ ⎢= − −} ⎥ ⎢ ⎥ ⎢ ⎥_{+ +}⎢ ⎥ ⎢ _{⎥ ⎢} ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ _{⎥ ⎢⎣ − ⎥ ⎢⎦ ⎣ ⎥ ⎢ ⎥⎦ ⎣ ⎦ ⎢ ⎥⎣ ⎦} ⎣ ⎦ ⎡ ⎤ ⎢ ⎥ = _{⎢ ⎥} ⎢ ⎥ ⎣ ⎦    θ θ τ θ

where τ is unknown torque which is imposed on the _L load side and the elements in the matrix are defined as follows: 2 21 22 31 33 2 3 1 , , 1 , , S M S M M M M L L M L M M L K K K a a B a J J R J B K a b g J J R J ⎛ ⎞ = = ⎜ + ⎟ = ⎝ ⎠ = = =

Table I gives the physical parameters of the two-mass spring system [32]. The system does not satisfy the matching condition. This system is sampled by

0.005

T = s and then its discrete-time model will lead to

the following system matrices as

0.9999 0.0050 0.0049 0 0.0047 0.9817 0 and 0.0114 0.0192 0 0.9631 0 − ⎡ ⎤ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ = −_{⎢ ⎥} =_{⎢ ⎥} ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ A B

Following the design procedure in section 3 and choosing 3

e=

Q I and R_e=1, we can obtain the system matrices 1 L and L₂as

[

]

1 6.3115 0.65 29.0189 and 2 0.6067 T = − = L L

The reference model is given by

[

]

2 1 1 ( 1) ( ) 1 0.9996 ( ) 1 0 ( ) m m m m k k k k ⎛⎡− ⎤ ⎞ + =⎜_⎜⎢₋ ⎥+ ⎟_⎟ ⎣ ⎦ ⎝ ⎠ = x I x y x

, and the parameter matrices F1 and F3 can be calculated by 1 13.0906 13.0651 1 0 5.5646 5.6110 − ⎡ ⎤ ⎢ ⎥ = ⎢ ⎥ ⎢ − ⎥ ⎣ ⎦ F , F₃ = −

[

80.9194 82.5359

]

Setting Q=I4 and R=0.01 and solving the algebraic Riccati equation (45), we can obtain

[

]

1= 0.0255 9.3909 0.0102

K and K₂ =9.3908

Design the parameters Φ=0.96 and ρ=1 in the controller law (50). Under the initial state Under initial state T(0)₌

[

0 ₋0.5 0.3

]

x , xˆ (0)T =

[

0 0 0

]

, and

[

]

(0) 0 0.02 T m =

x , the estimation error of x( )k is shown

in Figure 2. In spite of the fact that the system does not satisfy the matching condition, the system state is well estimated. Hence, the estimation method we propose can still be functioning normally to obtain valid estimation under the disturbance is smooth enough. The response of

( )

s k shown in Figure 3 can reach and enter in the first

layer s k( ) 1< and then it is constrained in the second

layer. Figure 4 shows the control input response. And

trajectories of the system output y k and the reference ( ) ( )

m

y k are given in Figure 5. Based on these figures,

although the system is in the presence of unknown disturbance, the system performance is excellent using the proposed algorithm.

VII. CONCLUSIONS

An algorithm of utilizing output feedback method in performing sliding mode controller design is proposed in regard to a MIMO discrete-time system with mismatched disturbance. At first, an observer design method with estimating both the system state and disturbance simultaneously is presented. Although the perfect estimation is not achievable, when the disturbance in between the two consecutive sampling points is not changed significantly, our proposed algorithm is able to render the estimation error to be restricted within O T

( )

.

Moreover, our proposed estimator method should be implemented in some nonminimum phase systems. The command generator tracker technique is then used to establish the reference model. Due to the fact that the estimated system state and disturbance are to have been implemented in the controller, the closed-loop stability is guaranteed and the tracking error is to be constrained in a small region. Finally, our control law is feasible arising from the simulation results.

u S K y M w S θ , L M w τ

Figure 1. Two mass spring system TABLE I PGYSICAL PARAMETERS JM 1.88 10 kgm× −4 2 JL 0.46 10 kgm× −4 2 BM 6.90 10 Nms rad× −4 BL 4 3.45 10 Nms rad_× − KM 4 4.96 10 Nm A_× − RM 1.15 Ω KS 4 1.8 10 Nm rad× −

(a) x k and ₁( ) x k ˆ ( )1

(b) x k₂( ) and x kˆ ( )2

(c) x k and₃( ) x kˆ ( )3

Figure 2. Estimation performance of the system state

Figure 3. Trajectory of s .

Figure 4. Response of the control input.

Figure 5. Response of y and y . m

REFERENCES

[1] C. Edwards and S. K. Spurgeon, Sliding Mode Control: Theory

and Applications. Taylor and Francis Ltd, London, 1998.

[2] Y. P. Chen, J. L. Chang, and S.R. Chu, “PC-based sliding-mode control applied to parallel-type double inverted pendulum system,”

Mechatronics, vol. 9, pp. 553-564, 1999.

[3] V.I. Utkin, “Sliding Mode Control in Discrete-Time and Difference Systems,” in Variable Structure and Lyapunov Control, A.S. Zinober, Ed., London: Springer Verlag, pp.87-107, 1993. [4] D. Milosavljevic, “General conditions for the existence of a

quasi-sliding mode on the switching hyperplane in discrete variable structure systems,” Automation Remote Control, vol. 46, pp. 307-314, 1985.

[5] S. Z. Sarpturk, Y. Istefabopulos, and O. Kaynak, “On the stability of discrete-time sliding mode control systems,” IEEE Trans.

Automatic Control, vol. 32, pp. 930-932, 1987.

[6] A. Bartoszewicz, “Discrete time quasi-sliding-mode control strategies,” IEEE Trans. Industrial Electronics, vol. 45, pp. 633-637, 1998.

[7] A. J. Koshkouei and A. S. I. Zinober, “Sliding mode control of discrete-time Systems,” ASME Trans. Dynamic Systems,

Measurement, and Control, vol. 122, pp. 793-802, 2000.

[8] W. C. Su, S. V. Drakunov, and U. Ozguner, “An o(T2_{) boundary} layer in sliding mode for sampled–data systems,” IEEE Trans.

Automatic Control, vol. 45, pp. 482-485, 2000.

[9] K. Abidi and A. Sabanovic, “Sliding-mode control for high-precision motion of a piezostage,” IEEE Trans. Industrial

Electronics, vol. 54, pp. 629-637, 2007.

[10] K. Abidi, J. X. Xu, and Y. Xinghuo, “On the discrete-time integral sliding-mode control,” IEEE Trans. Automatic Control, vol. 52, pp. 709-715, 2007.

[11] W. Gao, Y. Wang, and A. Homaifa, “Discrete-time variable structure control systems,” IEEE Trans. Industrial Electronics, vol. 42, pp. 117-122, 1995.

[12] M. Sun, Y. Wang, and D. Wang, “Variable structure repetitive control: a discrete-time strategy,” IEEE Trans. Industrial Electronics, vol. 52, pp. 610-616, 2005.

[13] P. Korondi, H. Hashimoto, and V. Utkin, “Direct torsion control of flexible shaft in an observer-based discrete-time sliding mode,” IEEE

Trans. Industrial Electronics, vol. 45, pp. 291-296, 1998.

[14] C. Y. Tang, and E. A. Misawa, “Discrete variable structure control for linear multivariable systems,” ASME Trans. Dynamic Systems,

Measurement, and Control, vol. 122, pp. 783-792, 2000.

[15] S. Janardhanan and B. Bandyopadhyay, “Output feedback sliding-mode control for uncertain systems using fast output sampling technique,”

IEEE Trans. Industrial Electronics, vol. 53, pp. 1677-1682, 2006.

[16] H. Kobayashi, S. Katsura, and K. Ohnishi, “An analysis of parameter variations of disturbance observer for motion control,” IEEE Trans.

Industrial Electronics, vol. 54, pp. 3413-3421, 2007.

[17] K. Ohnishi, M. Shibata, and T. Murakami, “Motion control for advanced mechatronics,” IEEE/ASME Trans. Mechatronics, Vol. 1, pp. 56-67, 1996.

[18] T. Mita, M. Hirata, K. Murata, and H. Zhang, “H∞ control versus disturbance-observer-based control,” IEEE Trans. Industrial Electronics, vol. 45, pp. 488-495, 1998.

[19] A. Hace, K. Jezernik, and A. Sabanovic, “SMC with disturbance observer for a linear belt drive,” IEEE Trans. Industrial Electronics, vol. 54, pp. 3402-3412, 2007.

[20] B. Veselic, D. Milosavljevic, B. Perunicic-Drazenovic, and D. Mitic, “Digitally controlled sliding mode based servo-system with active disturbance estimator,” in Proc. International Workshop Variable

Structure, Alghero, Italy, 2006, pp. 51-56.

[21] M. E. Valcher, “State observers for discrete-time linear systems with unknown inputs,” IEEE Trans. Automatic Control, vol. 42, pp. 397-401, 1999.

[22] J. Jin, M. J. Tahk, and C. Park, “Time-delay state and unknown input observation,” International Journal of Control, vol. 66, pp. 733-745, 1994.

[23] S. Sundaram and C. N. Hadjicostis, “Delay observers for linear systems with unknown inputs,” IEEE Trans. Automatic Control; vol. 52, pp. 334-339, 2007.

[24] T. Floquet and J. P. Barbot, “State and unknown input estimation for linear discrete-time systems,” Automatica, vol. 42, pp. 1883-1889, 2006. [25] S. Beale and B. Shafai, “Robust control design with a proportional integral observer,” International Journal Control, vol. 50, pp. 97-111, 1989.

[26] K. K. Busawon and P. Kabore, “Disturbance attenuation using proportional integral observers,” International Journal of Control, vol. 74, pp. 618-627, 2001.

[27] B. Shafai, S. Beale, H. H. Niemann, and J L. Stoustrup, “LTR design for discrete-time proportional-integral observers,” IEEE Trans. Automatic

Control, vol. 41, pp. 1056-1062, 1996.

[28] D. Soffker, T. J. Yu, and P. C. Muller, “State estimation of dynamical systems with nonlinearities by using proportional-integral observer,”

International Journal Systems Science, vol. 26, pp. 1571-1582, 1995.

[29] J. Broussard and M. O'Brien, “Feedforward control to track the output of a forced model,” IEEE Trans. Automatic Control, vol. 25, pp. 851-853, 1980.

[30] F. Ding and T. Chen, “Gradient based iterative algorithms for solving a class of matrix equation,” IEEE Trans. Automatic Control, vol. 50, pp. 1216-1221, 2005.

[31] L. B. Jemaa and E. Davison, “Performance limitations in the robust servomechanism problem for discrete-time lti systems,” IEEE Trans.

Automatic Control, vol. 48, pp.1299-1311.

[32] K. Fuwa, T. Narikiyo, and Y. Funahashi, “Construction of a control system for disturbance rejection using a dual observer,” Electrical

Engineering in Japan, vol. 138, pp. 333-341, 2002.

成果自評

首 先 此 計 劃 ， 非 常 感 謝 國 科 會 工 程 處 補 助 此 計 劃 (NSC962221-E-161-005)，目前此計劃已有兩篇論文被接 受在國際著名期刊（IEEE Trans. Industrial Electronics 和

International Journal of Robust and Nonlinear Control），可 見所提出的方法相當具有創新性，主持人目前正積極朝向 將此理論實際實現在真實的系統上面，相信不久之後又會 有不錯的成果產出。