functional observers for linear systems with unknown inputs

462 Asian Journal of Control, Vol. 6, No. 4, pp. 462-468, December 2004

Manuscript received July 1, 2003; revised October 3, 2003; accepted December 15, 2003.

The authors are with Department of Manufacturing Engi-neering and Engineering Management, City University ofHong Kong, Hong Kong.

FUNCTIONAL OBSERVERS FOR LINEAR SYSTEMS WITH UNKNOWN INPUTS

Hua Deng and Han-Xiong Li

ABSTRACT

In this paper, a straightforward approach is presented to design func-tional observers for linear systems with unknown inputs. Necessary and suf-ficient conditions are provided for the existence of the observers. Some il-lustrated examples are included.

KeyWords: Functional observer, disturbance decoupling, linear systems.

I. INTRODUCTION

It is well-known that only a single (but prespeci-fied) linear function of the system’s state is necessarily estimated for the purpose of implementing a feedback law [1,2]. In observer-based feedback control, optimal filtering and fault diagnosis, frequently, we only require some linear functions of the system’s state instead of all the state variables of a system [3-6]. Considerable atten-tion has been paid to the design of functional observers and various methods have been proposed for systems without unknown inputs (disturbances) [1-3,5-7]. How-ever, the functional observers for systems with unknown inputs need further investigation not only for their theo-retical importance but also for many related applications [8].

In [9], the design of functional observers, or distur-bance decoupled functional observers, was considered for systems with unknown inputs. A design method was provided with sufficient conditions. In a recent paper [8], the pencil matrix approach is used to design disturbance decoupled functional observers, and necessary and suf-ficient conditions are provided for the existence of the observers. The existence conditions can be used to check if a functional observer may exist. However, when the linear function of the system’s state to be es-timated, i.e., Kx, is specified a priori, they could not be used to test if the disturbance decoupled functional ob-

server exists. Furthermore, the design method may not be economic if Kx is specified a priori [8].

In this paper, a straightforward approach is intro-duced for the design of disturbance decoupled functional observers when Kx is specified a priori. The order of the observers is equal to the rank of K. Necessary and suffi-cient conditions for the existence of the observers are provided in terms of the original system matrices. It is interesting that, if matrix K is replaced by identity ma-trix I or any other nonsingular matrices with the dimen-sion of the original system, the proposed existence con-ditions are reduced to the known existence conditions of disturbance decoupled state observers given in [10,11]. Therefore the existence conditions for disturbance de-coupled state observers can be considered as a special case when compared with the proposed conditions for disturbance decoupled functional observers.

II. DESIGN OF FUNCTIONAL OBSERVER

Consider the linear time-invariant system described by

x Ax Bu Ed= + + (1a)

y Cx Dd= + (1b)

w Kx= (1c)

where x ∈ Rn is the state vector, u ∈ RP, is the known input vector, d ∈ Rq is the unknown input vector, y ∈ Rm is the output vector and w ∈ Rr is the vector to be estimated of the system. A, B, C, E, D, and K are known constant matrices of appropriate dimensions. Without

H. Deng and H.-X. Li: Functional Observers for Linear Systems with Unknown Inputs 463

loss of generality, K ∈ Rr × n, rank K = r, rank C = m, and

rank 0E

qD

⎡ ⎤=⎢ ⎥

⎣ ⎦ are assumed. Choose two nonsingular

matrices P1 and P2 such that

11

2

CPC

C⎡ ⎤

= ⎢ ⎥⎣ ⎦

, 11

2

yP y

y⎛ ⎞

= ⎜ ⎟⎝ ⎠

, 0 1

1 2

0 00 q q

P DPI −

⎡ ⎤= ⎢ ⎥

⎣ ⎦,

112

2

dP d

d− ⎛ ⎞

= ⎜ ⎟⎝ ⎠

, 2 1 2[ ]EP E E= (2)

where: E1 ∈ Rn × (q − q0 + q1), E2 ∈ Rn × (q0 − q1), rank E1 = q1, q0 ≤ q, and q0 − q1 ≤ m since rank D ≤ m. Using the above expressions, system (1) can be rewritten as

1 1 2 2x Ax Bu E d E d= + + + (3a)

1 1y C x= (3b)

2 2 2y C x d= + (3c)

or equivalently as

1 1x A x B u E d= + + (4a)

1 1y C x= (4b)

where: 2

uu

y⎡ ⎤

= ⎢ ⎥⎣ ⎦

, 2[ ]B B E= , 2 2A A E C= − .

Consider the n-dimensional dynamical equation (4a) with (4b). Define a functional observer having the following form [5]:

1 1( )z Fz Ly K MC B u= + + − (5a)

1w z My= + (5b)

where z ∈ R r, ŵ is an estimate of Kx, and F, L, and M are, respectively, r × r, r × (m − q0 + q1), and r × (m − q0 + q1) real constant matrices.

The following lemma gives the existence conditions for the functional observer above. Lemma 1. ŵ in (5b) is an estimate of Kx for a given K in the sense that, ŵ − Kx → 0 as time t → ∞, for any ini-

tial state of x and z and any 2

uu

y⎡ ⎤

= ⎢ ⎥⎣ ⎦

if and only if

F is stable (6a)

1 1 1( ) ( )K MC A LC F K MC− − = − (6b)

1 1( ) 0K MC E− = (6c)

Proof. Let the estimation error of the functional ob-server be

1 1ˆ ( )e w w Kx z My K MC x z= − = − − = − − .

Then

1 1 1ˆ ( )( )e w w K MC Ax B u E d= − = − + +

1 1( )Fz Ly K MC B u− − − −

1 1 1 1 1(( ) ) ( )K MC A LC x Fz K MC E d= − − − + − (7)

If the conditions in (6a)-(6c) are met, then ė = Fe and e(t) = eFte(0) → 0 as t → ∞ for any initial state of x and z and any ū. Thus ŵ is an estimate of Kx, i.e. ŵ → Kx as e(t) → 0. ■

Generally speaking, the conditions in Lemma 1 may not be useful because of the unknown matrices F, L and M. It is desirable to have only the original system matrices involved in the conditions.

For the solutions of Eqs. (6b) and (6c), we have the following theorem.

Theorem 1. Equations (6b) and (6c) have solutions if and only if

0

00

0 00 0

KA KECA CE D

CA CE Drank rank C D

C DK

K

⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥ = ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥ ⎣ ⎦

⎣ ⎦

(8)

Proof. Choosing nonsingular matrices

11

1

0 0 00 0 00 0 00 0 0

r

r

IP

UP

I

⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

, 1

2 1

0 00 00 0 r

PU P

I

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

,

2

2

0 00 00 0

nIV P

P

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

where P1 and P2 are two nonsingular matrices as in (2), then (8) can be written as

1 2

0

00

0 00 0

KA KECA CE D

CA CE Drank U V rank U C D V

C DK

K

⎛ ⎞⎡ ⎤⎛ ⎞⎡ ⎤⎜ ⎟⎢ ⎥ ⎜ ⎟⎢ ⎥⎜ ⎟⎢ ⎥ = ⎜ ⎟⎢ ⎥⎜ ⎟⎢ ⎥ ⎜ ⎟⎢ ⎥⎜ ⎟⎢ ⎥ ⎣ ⎦⎝ ⎠⎜ ⎟⎣ ⎦⎝ ⎠

(9) Using the expressions in (2), we have

464 Asian Journal of Control, Vol. 6, No. 4, December 2004

11 1 1

1 1 11

1

00

00

KA KEC A C E

C A C Erank rank C

CK

K

⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥ = ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥ ⎣ ⎦⎢ ⎥⎣ ⎦

(10)

where matrices Ā, C1, and E1 are the same as in (6b) and (6c). Thus (8) is equivalent to (10) and the problem is reduced to prove if (10) holds.

(6b) and (6c) can be rewritten as

1 1 1

1 1[ ] [ ] 00

C A C EKA KE M L F C

K

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

(11a)

with

L L FM= − (11b)

According to well-known matrix equation solution theory [12], (11a) holds if and only if (10) holds. This completes the proof of the theorem.

If (8) holds, (6b) and (6c) have solutions. However, F may not be stable and then the observer does not exist. Thus, when (8) holds, we need to investigate under what conditions F is stable. Using the general solution of lin-ear equations [12], the solution of (6c) can be written as

0 11 1 1 1 1 1 1( ) ( ( )( ) )m q qM KE C E Y I C E C E+ +− += + − (12a)

or equivalently

0 11 1 1 1 1( ( ) ( )) 0q q qKE I C E C E+− + − = (12b)

where (C1E1)

+ denotes the Moore-Penrose generalized inverse of matrix C1E1 [12] and Y is an arbitrary matrix of appropriate dimension.

Using (11b), (6b) can be written as

1 1KA LC MC A FK− − = (13)

Choosing full row rank matrix [k+ In − K+K], then (13)

is equivalent to

1 1( ) [ ] 0nKA LC MC A FK K I K K+ +− − − − = (14)

Using (12a), (14) becomes

1 1 1 1( ) [ ]F KAK KE C E C AK L Y CK+ + + += − − (15)

and

1 1 1 1( ( ) )( )nKA KE C E C A I K K+ +− −

[ ] ( )nL Y C I K K+= − (16)

where 0 1

1

1 1 1 1 1( ( )( ) )m q q

CC

I C E C E C A+− +

⎡ ⎤= ⎢ ⎥−⎣ ⎦

.

Similar to the solution of Eq. (6c), we obtain

0 12( )[ ] ( ) ( )n m q qL Y KA I K K G I+ + +− += − ∆ + − ∆∆ (17)

where Ã = Ā − Ε1(C1E1)+ C1 Ā, ( )nC I K K+∆ = − and G is an arbitrary matrix of appropriate dimension.

Replacing (17) in (15), we have

0 0F A G C= − (18)

where 0 ( ( ) )nA K A A I K K C K+ + += − − ∆ and 0C =

0 12( )( )m q qI C K+ +− + − ∆∆ .

The condition for a stable F, or the pair (C0 A0) to be detectable, is given by the following theorem.

Theorem 2. The pair (C0 A0) is detectable if and only if

00

0 0 0

sK KA KE CA CE Drank CA CE D rank C D

C D K

− +⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

, Re( ) 0s C s∀ ∈ ≥ (19)

Proof. Defining U2 and V as in the proof of Theorem 1, then (19) can be written as

2

0

0

sK KA KErank U CA CE D V

C D

⎛ − + ⎞⎡ ⎤⎜ ⎟⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟⎢ ⎥⎣ ⎦⎝ ⎠

2 00 0

CA CE Drank U C D V

K

⎛ ⎞⎡ ⎤⎜ ⎟⎢ ⎥= ⎜ ⎟⎢ ⎥⎜ ⎟⎢ ⎥⎣ ⎦⎝ ⎠

(20)

and is equivalent to

1 1 1 1

1 1 1 1

1

00 0

sK KA KE C A C Erank C A C E rank C

C K

⎡ ⎤ ⎡ ⎤− +⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

(21)

Note that KE1 (Iq − q0 + q1−(C1E1)+ (C1E1)) = 0 and KÃ(In

− Κ + Κ) (I2(m – q0 + q1) − ∆+∆) = 0. The left-hand side of (21) can be written as


1

1 1 1

1 0

sK KA KErank C A C E

C

⎡ ⎤− +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

0 1 0 1

0 1

11 1

1 1 11 1 1 1 1 1 1 1 1 1

1

0 00 ( ) 0 0 00 ( )( ) 0 0 ( ) ( ) ( )

00 0

r

n

m q q q q q

m q q

IsK KA KEC E I

rank C A C EI C E C E C E I C E C EC

I

+

+ + +− + − +

− +

⎛ ⎞⎡ ⎤⎡ ⎤− +⎜ ⎟⎢ ⎥

⎡ ⎤⎢ ⎥⎜ ⎟⎢ ⎥= ⎢ ⎥⎢ ⎥⎜ ⎟⎢ ⎥− −⎣ ⎦⎢ ⎥⎜ ⎟⎢ ⎥ ⎣ ⎦⎜ ⎟⎢ ⎥⎣ ⎦⎝ ⎠

1 1 1

1 1 1 1 1 1 1

( )( ) ( ) ( )

0

sK KA KE C EsK KA

rank C E C A C E rank rank C EC

C

+

+ + +

⎡ ⎤− +⎡ ⎤− +⎢ ⎥

= = +⎢ ⎥⎢ ⎥⎣ ⎦⎢ ⎥

⎣ ⎦

1 1[ ] ( )nsK KA

rank K I K K rank C EC

+ + +⎛ ⎞⎡ ⎤− += − +⎜ ⎟⎢ ⎥⎜ ⎟⎣ ⎦⎝ ⎠

1 1( ) ( )r nsI KAK KA I K Krank rank C E

CK

+ ++

+

⎡ ⎤− + −= +⎢ ⎥

∆⎣ ⎦

0 1

1 1

2( )

0( )0 ( )

0

rr n

m q q

IsI KAK KA I K Krank rank C E

CKI

+ ++ +

++

− +

⎛ ⎞⎡ ⎤⎜ ⎟⎡ ⎤− + −⎢ ⎥= ∆ +⎜ ⎟⎢ ⎥⎢ ⎥ ∆⎣ ⎦⎜ ⎟⎢ ⎥− ∆∆⎣ ⎦⎝ ⎠

0 1

1 1

2( )

( )( )

( ) 0

r n

m q q

sI KAK KA I K Krank CK rank C E

I CK

+ +

+ + + +

+ +− +

⎡ ⎤− + −⎢ ⎥

= ∆ ∆ ∆ +⎢ ⎥⎢ ⎥− ∆∆⎣ ⎦

0 1

1 1

2( )

( )( )

( ) 0

r n

m q q

sI KAK KA I K Krank CK rank C E

I CK

+ + +

+ + + +

+ +− +

⎡ ⎤− + − ∆ ∆⎢ ⎥

= ∆ ∆ ∆ +⎢ ⎥⎢ ⎥− ∆∆⎣ ⎦

0 01 1 1 1

0 0( ) ( ) ( )r rsI A sI A

rank rank rank C E rank rank rank C EC C

+ + +− + − +⎡ ⎤ ⎡ ⎤= + ∆ ∆ + = + ∆ +⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦

and the right-hand side is equal to

0 1

1 1 1 1 1 1

1 10

0 00 0

0 0

n

q q q

C A C E C A C EK I K K

rank C rank CI

K K

+ +

− +

⎛ ⎞⎡ ⎤ ⎡ ⎤⎡ ⎤−⎜ ⎟⎢ ⎥ ⎢ ⎥= ⎢ ⎥⎜ ⎟⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦⎜ ⎟⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎝ ⎠

1 1 1 1

1 1

( )( ) 0

0 0

n

n

r

C AK C A I K K C Erank C K C I K K

I

+ +

+ +

⎡ ⎤−⎢ ⎥

= −⎢ ⎥⎢ ⎥⎣ ⎦

1 1 1

1

( )( ) 0

n

n

C A I K K C Er rank

C I K K

+

+

⎡ ⎤−= + ⎢ ⎥−⎣ ⎦


0 1

0 1

1 11 1 1

1 1 1 11

( ) 0( )

( )( ) 0( ) 0

0

nm q q

nm q q

C EC A I K K C E

r rank I C E C EC I K K

I

++

+− + +

− +

⎛ ⎞⎡ ⎤⎜ ⎟⎢ ⎥ ⎡ ⎤−

= + −⎜ ⎟⎢ ⎥ ⎢ ⎥−⎜ ⎟⎣ ⎦⎢ ⎥⎜ ⎟⎢ ⎥⎣ ⎦⎝ ⎠

1 1 1 1 1 1 11 1 1 1

( ) ( ) ( ) ( ) ( ) ( )0

nC E C A I K K C E C Er rank r rank rank C E C E+ + +

+⎡ ⎤−= + = + ∆ +⎢ ⎥

∆⎣ ⎦

1 1( )r rank rank C E += + ∆ +

Thus we have 0

0

rsI Arank r

C− +⎡ ⎤

=⎢ ⎥⎣ ⎦

and hence the result.

Using Theorems 1 and 2, we have following result.

Theorem 3. The functional observer having the form of (5) for system (1) to estimate a linear function Kx exists if and only if

(1)

0

00

0 00 0

KA KECA CE D


C DK

K

⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥ = ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥ ⎣ ⎦

⎣ ⎦

(2) 0

0

sK KA KErank CA CE D

C D

− +⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

0 , Re( ) 00 0

CA CE Drank C D s C s

K

⎡ ⎤⎢ ⎥= ∀ ∈ ≥⎢ ⎥⎢ ⎥⎣ ⎦

.

Proof. According to the proofs of Theorems 1 and 2, this theorem can be easily proved and thus it is omitted. ■

Using the analyses above, the design procedure can be summarized as following: Step 1: Check if the conditions in Theorem 3 are satis-

fied. If they are not satisfied, the functional ob-server does not exist.

Step 2: Obtain G in (18) in terms of standard pole placement methods and then obtain F from (18).

Step 3: Obtain L and Y using (17) since G is available. Step 4: Obtain L and M from (11b) and (12a), respec-

tively.

Remarks. (a) If there are no unknown inputs, i.e. E and D are

equal to zero, the conditions of Theorem 3 are re-

duced to those given in [5], which are

KACA

CArank rank C

CK

K

⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥ = ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥ ⎣ ⎦

⎣ ⎦

and

, Re( ) 0sK KA CA

rank CA rank C s C sC K

− +⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥= ∀ ∈ ≥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

.

(b) If K = In or any other nonsingular matrices having the same dimension of the original system, this is the case in which a full-order observer is con-structed. For the conditions in Theorem 3, we have

00

00

0 0n

A EE

CA CE Drank n rank CE D

C DD

I

⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥ = + ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥ ⎣ ⎦

⎣ ⎦

0 00 0

0 0 0

m

n

m

C I En rank I CE D

I D

−⎛ ⎞⎡ ⎤ ⎡ ⎤⎜ ⎟⎢ ⎥ ⎢ ⎥= + ⎜ ⎟⎢ ⎥ ⎢ ⎥⎜ ⎟⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦⎝ ⎠

En rankD rank

D⎡ ⎤

= + + ⎢ ⎥⎣ ⎦

and

00

0 0n

CA CE DCE D

rank C D n rankD

I

⎡ ⎤⎡ ⎤⎢ ⎥ = + ⎢ ⎥⎢ ⎥ ⎣ ⎦⎢ ⎥⎣ ⎦

.

Thus


0CE D E

rank rankD rankD D

⎡ ⎤ ⎡ ⎤= +⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦ (R1)

On the other hand,

0

0

nsI A Erank CA CE D

C D

− +⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

0 0 0

0 0 0

n n

m m

m

I sI A Erank C I sI CA CE D

I C D

− +⎛ ⎞⎡ ⎤ ⎡ ⎤⎜ ⎟⎢ ⎥ ⎢ ⎥= − −⎜ ⎟⎢ ⎥ ⎢ ⎥⎜ ⎟⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦⎝ ⎠

nsI A Erank rankD

C D− +⎡ ⎤

= +⎢ ⎥⎣ ⎦

Therefore

nsI A E Erank n rank

C D D− +⎡ ⎤ ⎡ ⎤

= +⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

(R2)

(R1) and (R2) are just the standard existence conditions of disturbance decoupled state observers given in [10,11]. Thus the existence conditions of disturbance decoupled state observers can be regarded as a special case of the proposed conditions of disturbance decoup-led functional observers. (c) The existence conditions of functional observers for

discrete time systems with unknown inputs can be obtained by simply replacing the conditions in Theo-rem 3 by

(i)

0

00

0 00 0

KA KECA CE D


C DK

K

⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥ = ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥ ⎣ ⎦

⎣ ⎦

(ii)

0

0

zK KA KErank CA CE D

C D

− +⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

0 , | | 10 0

CA CE Drank C D z C z

K

⎡ ⎤⎢ ⎥= ∀ ∈ ≥⎢ ⎥⎢ ⎥⎣ ⎦

.

III. EXAMPLES

Example 1 [8]. Consider the system (1) having the fol-lowing coefficient matrices

1 1 1 0 0 00 1 0 1 1 0

, ,1 0 1 0 0 01 1 0 1 0 1

A E

−⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥= =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

1 0 0 0 0 0,

0 1 0 0 0 0C D⎡ ⎤ ⎡ ⎤

= =⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

.

Assume K = [1 0 0 0]. Since the condition (R1) in re-marks is not met, a disturbance decoupled state observer for this example does not exist. However, the conditions in Theorem 3 are satisfied and thus a first order distur-bance decoupled functional observer can be constructed as follows:

0 02 2 2 1, 03 9 9 9

T

A C ⎡ ⎤= = −⎢ ⎥⎣ ⎦

Simply choosing G = [l 0 0 0], we have F = 2/3 − 2l/9, where l > 3 and the pole of the observer can be arbitrar-ily assignable. Then, we have

2 4 1 4 2 2, 09 9 3 9 3 9

L l l Y l⎡ ⎤ ⎡ ⎤= + − + = −⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦,

22 4 4 1 2 2 2, 13 27 81 3 9 3 9

L l l l M l⎡ ⎤ ⎡ ⎤= + + + = −⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦.

If we choose − 2 as the pole of the observer, then l = 12. Thus the functional observer to estimate Kx is con-structed as follows:

862 39

z z y⎡ ⎤= − + ⎢ ⎥⎣ ⎦ ˆ [ 2 1]w z y= + −

Example 2. Assume the system (1) having the same coefficient matrices as example 1 but K = [0 0 1 1].

Since the condition (1) in Theorem 3 is not satisfied, the functional observer to estimate [0 0 1 1] x does not exist. Thus further computation is unnecessary and can be avoided.

IV. CONCLUSION

This paper presents a straightforward approach for the design of functional observers with order r for mul-tivariable systems with unknown inputs when the func-tion to be estimated is specified a priori. The existence conditions of the observers are provided only in terms of the original system matrices. The proposed existence conditions and design procedure can be directly ex-tended to discrete time systems. Also the known exis-tence conditions of disturbance decoupled state observ-ers can be considered as a special case of the proposed


existence conditions of disturbance decoupled func-tional observers.

ACKNOWLEDGEMENTS

The authors would like to thank the anonymous re-viewers for their valuable comments and suggestions. The work is fully supported by a project from RGC of Hong Kong SAR (project no: CityU 1086/01E).

REFERENCES

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Hua Deng received the B.Eng from Nanjing Aeronautical Institute, China, in 1983, and the M.Eng from Northwestern Polytechnical University, China, in 1988. Cur-rently, He is a Ph.D. candidate in the Department of Manufacturing Engineering and Engineering

Management, City University of Hong Kong, China. He is an Associate Professor at Changsha Univer-

sity of Science and Technology, China. His research interests include identification and control of distrib-uted parameter systems, intelligent control, fault diag-noses and computer control system design.

Han-Xiong Li received Ph.D. in electrical engineering from Univer-sity of Auckland, New Zealand, in January of 1997; M.E. in electrical engineering from Delft University of Technology in The Netherlands in 1991, and B.E. from National Uni-

versity of Defence Technology in China in 1982. Cur-rently, he is an Associate Professor in the Department of Manufacturing Engineering and Engineering Man-agement, City University of Hong Kong. In last twenty years, he has opportunities to work in different fields including, military service, industry and academia. He has gained industrial experience in IC packaging as a senior process engineer for die-bonding and dispensing from ASM ⎯ a leading supplier for semiconductor process equipment. His research interests include modelling and intelligent control for complex indus-trial process with special interest to electronic packag-ing. He serves as an associate editor for IEEE Transac-tions on Systems, Man & Cybernetics - part B.

functional observers for linear systems with unknown inputs

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