514 Asian Journal of Control, Vol. 6, No. 4, pp. 514-520, December 2004

Manuscript received July 1, 2003; revised October 23, 2003;accepted December 15, 2003.

H. Trinh and S. Nahavandi are with School of Engineeringand Technology, Deakin University, Geelong 3217, Austra-lia.

T. Fernando is with Department of Electrical and Elec-tronic Engineering, The University of Western Australia,Crawley, WA 6009, Australia.

－Brief Paper－

DESIGN OF REDUCED-ORDER FUNCTIONAL OBSERVERS FOR LINEAR SYSTEMS WITH UNKNOWN INPUTS

H. Trinh, T. Fernando, and S. Nahavandi

ABSTRACT

This brief paper presents new conditions for the existence and design of reduced-order linear functional state observers for linear systems with un-known inputs. Systematic procedures for the synthesis of reduced-order functional observers are given. Numerical examples are given to illustrate the attractiveness and simplicity of the new design procedures.

KeyWords: Unknown inputs, linear functional observers, stability, linear systems.

I. INTRODUCTION

In this paper, we present some new results on de-signing reduced-order functional observers for linear systems with unknown inputs. Consider a system de-scribed by

( ) ( ) ( ) ( )x t Ax t Bu t Dv t= + + (1a)

( ) ( )y t Cx t= (1b)

( ) ( )z t Fx t= (1c)

where x(t) ∈ ℜn, u(t) ∈ ℜk, v(t) ∈ ℜq and y(t) ∈ ℜr are the state, known input, unknown input and the output vectors, respectively. z(t) ∈ ℜm is the vector to be esti-mated. Matrices A, B, D, C and F are known constant matrices of appropriate dimensions. As in [1], it is as-sumed that r ≥ q and, without loss of generality, rank (D) = q, rank (C) = r and matrix C takes the following ca-nonical form

[ 0]rC I= . (1d)

In the literature, the state estimation problem (i.e. F = In) of system (1) is well researched and many well-known results are available for the design of full-order and reduced-order state observers (see, [1]-[5] and references therein). The problem of designing re-duced-order observers to estimate any given subset (z(t) = Fx(t)) of the state vector has not been widely studied [6]. As a result, conditions for the existence of a re-duced-order linear functional observer with order p, where m ≤ p < n − r, are not yet available.

The purpose of this brief paper is therefore to pre-sent new conditions for the existence and design of re-duced-order observers capable of asymptotically esti-mating any vector state functional. The aim is to design a pth-order (m ≤ p < n − r) linear functional observer of the form

( ) ( ) ( ) ( )w t Ew t Hu t Qy t= + + (2a)

ˆ( ) ( ) ( )z t Kw t My t= + (2b)

where w(t) ∈ ℜp, K = [Im 0], and matrices M, E, H and Q are to be designed so that ˆ( )z t asymptotically esti-mates z(t) ˆ( ( ) ( )).z t z t→ New and systematic proce-dures for the synthesis of reduced-order functional ob-servers are given. Numerical examples are given to il-lustrate the attractiveness and simplicity of the new de-sign procedures.

H. Trinh et al.: Design of Reduced-Order Functional Observers for Linear Systems with Unknown Inputs 515

II. MAIN RESULTS

Let us first verify the following proposition.

Proposition 1. ˆ( )z t in (2) is an asymptotic estimate of z(t) if there exists a matrix L ∈ ℜp×n such that the fol-lowing equations are satisfied

F KL MC= + (3)

0, is HurwitzQC LA EL E− + = (4)

0LD = (5)

.H LB= (6)

Proof. Let an error vector ε(t) ∈ ℜp be defined as ( ) ( ) ( ) .t w t Lx tε = −

Hence

( ) ( ) ( ), ort w t Lx tε = −

( ) ( ) ( ) ( )t E t QC LA EL x tε = ε + − +

( ) ( ) ( ) .H LB u t LDv t+ − −

Upon the satisfaction of the conditions (4)-(6) of the proposition 1, the above error dynamics equation is re-duced to ( ) ( )t E tε = ε , where matrix E is Hurwitz. This implies that ε(t) → 0 as t → ∞, and consequently w(t) → Lx(t). Now, by using the condition (3) of the Proposition 1, the error e(t) ∈ ℜm between ˆ( )z t and z(t) can be expressed as e(t) = Kε(t) (i.e. e(t) = [Im 0] ε(t)). Since ε(t) → 0 as t → ∞, it follows that e(t) → 0 as t → ∞, and hence ˆ( )z t → z(t). This completes the proof of the proposition 1.

Partition matrices F, L, A and D as follows

1 2 1 2[ ] , [ ] ,F F F L L L= =

11 12 1

21 22 2,

A A DA D

A A D⎡ ⎤ ⎡ ⎤

= =⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

(7)

where F1 ∈ ℜm×r, F2 ∈ ℜm×(n−r), L1 ∈ ℜp×r, L2 ∈ ℜp×(n−r), D1 ∈ ℜr×q, D2 ∈ ℜ(n−r)×q, A11 ∈ ℜr×r, A21 ∈ ℜ(n−r)×r, A12 ∈ ℜr×(n−r) and A22 ∈ ℜ(n−r)×(n−r) are submatrices.

Incorporating (1d) and (7) into (3)-(6), the follow-ing two set of equations are obtained

1 1

1 11 2 21 1

M F KLQ L A L A ELH LB

= −⎧⎪ = + −⎨⎪ =⎩

(8)

and

⎪⎩

⎪⎨

⎧

=+=−−

=

.0Hurwitz is ,0

2211

2221212

22

DLDLEALALEL

KLF (9)

In order to avoid a trivial (i.e. zero) solution of (9), it is assumed that F2 ≠ 0 and rank(F2) = m (it is clear that when F2 = 0, a static observer of the form 1ˆ( ) ( )z t F y t= is obtained). We now have the following new theorem for the existence of a stable pth-order (m < p < n − r) linear functional observer (2). Theorem 1. There exists a stable pth-order linear func-tional observer (2) for the system (1) provided that there exists any arbitrary full row-rank matrix R of dimension (p − m) × (n − r) such that the following two conditions are satisfied

Condition 1:

2 22 2 2

12 122 2

212 1

2

000

0

F A F DA DRA RD

rank rank FA DRF

R

⎡ ⎤⎢ ⎥

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥ = ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥ ⎣ ⎦

⎢ ⎥⎣ ⎦

(10)

Condition 2:2 2 2

22 2

12 1

F F Fs A D

rank R R R

A D

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞− −⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎢ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎥⎢ ⎥⎣ ⎦

12 1

2 0 , Re( ) 0 .0

A Drank F s s

R

⎡ ⎤⎢ ⎥= ∀ ∈ ≥⎢ ⎥⎢ ⎥⎣ ⎦

C (11)

Proof. Substituting K = [Im 0] into (9) gives

22

FL

R⎡ ⎤

= ⎢ ⎥⎣ ⎦

(12)

2 1 12 2 22 0EL L A L A− − = (13)

and 02211 =+ DLDL , (14)

where L2 is a full row-rank matrix. Define the following full row-rank matrix

1 1 2 2 2[ ] [ ( )]H E L I L L+ += − (15)

where 12 2 2 2( )T TL L L L+ −= is the generalized inverse of L2

in (12). Post-multiply (13) by [H1 E1] to give

1 12 1 2 22 1E L A H L A H= + (16)

and

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

1 12 1 2 22 1 .L A E L A E= − (17)

Equations (17) and (14) can now be expressed as

1L Ω = Φ (18a)

where

12 1 1 2 22 1 2 2[ ] , [ ] .A E D L A E L DΩ = Φ = − (18b)

From (18), a solution for L1 ∈ ℜp×r exists iff [7]

( )rank rankΩ⎡ ⎤

= Ω⎢ ⎥Φ⎣ ⎦

or equivalently

2 22 1 2 212 1 1

12 1 1[ ] .

L A E L Drank rank A E D

A E D⎡ ⎤

=⎢ ⎥⎣ ⎦

(19)

We now show that (19) is ensured upon the satisfaction of condition 1 of Theorem 1.

Using (12), the left-hand side of (10) can be ex-pressed as

2 22 2 2

12 1

2 0

L A L Drank A D

L

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

2 22 2 21 1

12 1

2

00 0

0 q

L A L DH E

rank A DI

L

⎧ ⎫⎡ ⎤⎡ ⎤⎪ ⎪⎢ ⎥= ⎢ ⎥⎨ ⎬⎢ ⎥ ⎢ ⎥⎪ ⎪⎣ ⎦⎢ ⎥⎣ ⎦⎩ ⎭

2 22 1 2 22 1 2 2

12 1 12 1 1

0 0p

L A H L A E L Drank A H A E D

I

⎡ ⎤⎢ ⎥

= ⎢ ⎥⎢ ⎥⎣ ⎦

2 22 1 2 2

12 1 1.

L A E L Dp rank

A E D⎡ ⎤

= + ⎢ ⎥⎣ ⎦

(20)

Similarly, the right-hand side of (10) can be expressed as

1 112 1 12 1

2 2

00 00 0 q

H EA D A Drank rank

IL L

⎧ ⎫⎡ ⎤⎡ ⎤ ⎡ ⎤⎪ ⎪= ⎢ ⎥⎨ ⎬⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦⎪ ⎪⎣ ⎦⎩ ⎭

12 1 12 1 112 1 1[ ] .

0 0p

A H A E Drank p rank A E D

I⎡ ⎤

= = +⎢ ⎥⎢ ⎥⎣ ⎦

(21)

Thus upon the satisfaction of condition 1 of Theorem 1, a solution to equation (18) always exists [7] and is given by

1 ( )L Z I+ += ΦΩ + − ΩΩ (22)

where Z is an arbitrary matrix of appropriate dimension and Ω+ is the generalized inverse of Ω in (18b).

Substituting (22) into (16) gives

E N ZG= − (23a)

where

2 22 1 12 1N L A H A H+= + ΦΩ (23b)

12 1( ) .G I A H+= ΩΩ − (23c)

From (23), matrix E is stable provided that the pair (G, N) is detectable. This implies that the following condi-tion must be satisfied

, Re( ) 0 .psI Nrank p s s

G−⎡ ⎤

= ∀ ∈ ≥⎢ ⎥⎣ ⎦

C (24)

We now show that (24) holds provided that condition 2 of Theorem 1 is satisfied.

Using (12), the left-hand side of (11) can be ex-pressed as:

2 2 22 2 2

12 1

sL L A L Drank

A D− −⎡ ⎤

⎢ ⎥⎣ ⎦

1 12 2 22 2 2

12 1

00 0 q

H EsL L A L Drank

IA D

⎧ ⎫− − ⎡ ⎤⎡ ⎤⎪ ⎪= ⎢ ⎥⎨ ⎬⎢ ⎥⎢ ⎥⎣ ⎦⎪ ⎪⎣ ⎦⎩ ⎭

2 22 1

12 1

sI L A Hrank

A H− Φ⎡ ⎤

= ⎢ ⎥Ω⎣ ⎦

2 22 1

12 100

IsI L A H

rank IA H

+

+

+

⎧ ⎫⎡ ⎤−ΦΩ⎪ ⎪− Φ⎢ ⎥ ⎡ ⎤⎪ ⎪= ΩΩ −⎨ ⎬⎢ ⎥ ⎢ ⎥Ω⎣ ⎦⎪ ⎪⎢ ⎥ΩΩ⎪ ⎪⎣ ⎦⎩ ⎭

12 112 1

00

0sI N

Irank G

A H IA H

++

⎧ ⎫−⎡ ⎤⎡ ⎤⎪ ⎪⎢ ⎥= ⎨ ⎬⎢ ⎥⎢ ⎥ −Ω⎣ ⎦⎪ ⎪⎢ ⎥ΩΩ Ω⎣ ⎦⎩ ⎭

00

0

sI Nrank G

−⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥Ω⎣ ⎦

[ ]12 1 1 .psI Nrank rank A E D

G−⎡ ⎤

= +⎢ ⎥⎣ ⎦

(25)

From (21), condition 2 of Theorem 1 is thus proven.

H. Trinh et al.: Design of Reduced-Order Functional Observers for Linear Systems with Unknown Inputs 517

This completes the proof of Theorem 1.

By choosing matrix K = Im, we can easily derive new conditions for the existence and stability of linear functional observers for system (1) with an order p = m. The result is given in the following corollary. Corollary 1: There exists a stable mth-order linear functional observer of the form

( ) ( ) ( ) ( )w t Ew t Hu t Qy t= + +

ˆ( ) ( ) ( )z t w t My t= +

for the system (1) if the following two conditions are satisfied

Condition 1:

2 22 2 212 1

12 12

20

0

F A F DA D

rank A D rankF

F

⎡ ⎤⎡ ⎤⎢ ⎥ = ⎢ ⎥⎢ ⎥⎣ ⎦⎢ ⎥⎣ ⎦

(26)

Condition 2: 2 2 22 2 2 12 1

12 1 2 0sF F A F D A D

rank rankA D F− −⎡ ⎤ ⎡ ⎤

=⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

, Re( ) 0 .s s∀ ∈ ≥C (27)

Proof. By noting K = Im (and hence equation (12) is now reduced to L2 = F2), the rest of the proof of Corol-lary 1 is directly followed from the proof of the Theo-rem 1.

Remark 1. Subject to the satisfaction of conditions 1&2 of the Corollary 1, we can propose the following simple design procedure for the synthesis of a mth-order linear functional observer for system (1).

Design Procedure 1. p = m Step 1: Let K = Im and L2 = F2. Step 2: Obtain matrices N and G from (23b) and

(23c), respectively. Step 3: Use (23a) to derive Z and a stable matrix E. Step 4: Use (22) to obtain L1. Step 5: Matrices H, M and Q are obtained from (8).

Remark 2. For the case where the conditions of Corol-lary 1 are not satisfied, it is necessary to increase the order of the observer and apply Theorem 1. Ideally, in the application of Theorem 1, it is preferable that matrix R is of a small dimension in order to yield a low-order linear functional observer for system (1). How to search for a matrix R of minimal dimension is difficult and remains an open problem for further research.

In order to simplify the complexity of the problem, a simple way for choosing matrix R is now proposed. Examination of condition 1 of Theorem 1 shows it can be satisfied as long as the matrix [R 0] is chosen to be

linearly independent of 12 1

2 0A DF

⎡ ⎤⎢ ⎥⎣ ⎦

and the number of

rows in R is at most 12 1

2.

0A D

n q r rankF

⎛ ⎞⎡ ⎤+ − −⎜ ⎟⎢ ⎥⎜ ⎟⎣ ⎦⎝ ⎠

Furthermore, when the number of independent rows

in R is 12 1

2 0A D

n q r rankF

⎛ ⎞⎡ ⎤+ − −⎜ ⎟⎢ ⎥⎜ ⎟⎣ ⎦⎝ ⎠

, the matrix

12 1

2 00

A DFR

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

is a full column-rank matrix (i.e.

)(002

112

qrnRF

DArank +−=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡) and the Condition 1 of

Theorem 1 is always satisfied. Accordingly, the ob-server design problem is reduced to checking only the detectability condition (11). In this case, the detectabil-ity condition (11) is equivalent to the following condi-tion

2 2 22 2 2 2

12 1

n rsI L A L Drank

A D− − Γ −Γ⎡ ⎤

⎢ ⎥⎣ ⎦

( ) , Re( ) 0n r q s s= − + ∀ ∈ ≥C (28)

where Γ2 is defined as follows

1 2 12 1

3 4 2.

0A DL

+Γ Γ⎡ ⎤ ⎡ ⎤=⎢ ⎥ ⎢ ⎥Γ Γ⎣ ⎦ ⎣ ⎦

(29)

The above equivalent condition (28) can be easily

proven. Since matrix 12 1

2 0A DL

⎡ ⎤⎢ ⎥⎣ ⎦

is of full-column-

rank, therefore there always exists a matrix 1 2

3 4

Γ Γ⎡ ⎤⎢ ⎥Γ Γ⎣ ⎦

such that

1 2

3 4

Γ Γ⎡ ⎤⎢ ⎥Γ Γ⎣ ⎦

12 1

2 0 n r qA D

IL − +

⎡ ⎤=⎢ ⎥

⎣ ⎦ (30)

and the detectability condition (11) can be expressed as

2 2 22 2 2

12 1

sL L A L Drank

A D− −⎡ ⎤

⎢ ⎥⎣ ⎦

2 1 2 2 22 2 2

12 10 r

s sL L A L Drank

I A DΓ Γ − −⎡ ⎤ ⎡ ⎤

= ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

2 2 22 2 2 2

12 1.

sI L A L Drank

A D− Γ −Γ⎡ ⎤

= ⎢ ⎥⎣ ⎦

(31)

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

Remark 3. The Condition 1 of Theorem 1 is always guaranteed for any matrix R with its number of linearly independent rows equals or above

12 1

2 0A D

n q r rankF

⎛ ⎞⎡ ⎤+ − −⎜ ⎟⎢ ⎥⎜ ⎟⎣ ⎦⎝ ⎠

, therefore it is expected

that the detectability condition (11) is met by succes-sively increasing the order of the observer. Indeed, when L2 ∈ ℜ(n−r)×(n−r), i.e. p = (n−r), the condition (11) is equivalent to the following condition

22 2

12 1( )n rsI A D

rank n r qA D

− − −⎡ ⎤= − +⎢ ⎥

⎣ ⎦

, Re( ) 0 .s s∀ ∈ ≥C (32)

Condition (32) can be easily proven by using the fact that L2 ∈ ℜ(n−r)×(n−r) is a square and full-rank matrix, therefore there always exists a matrix Γ2 such that Γ2L2 = In−r . The detectability condition (11) of Theorem 1 is thus reduced to

2 2 22 2 2

12 1

sL L A L Drank

A D− −⎡ ⎤

⎢ ⎥⎣ ⎦

2 2 2 22 2 2

12 1

00 r

sL L A L Drank

I A DΓ − −⎡ ⎤ ⎡ ⎤

= ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

22 2

12 1.n rsI A D

rankA D

− − −⎡ ⎤= ⎢ ⎥

⎣ ⎦ (33)

which is equivalent to the well-known condition re-ported in the literature [1]-[5].

Therefore, based on the above development, the following design procedure is proposed. Design Procedure 2. m + rank(R) ≤ p < (n − r)

Set i = 0 . Step 1: Set the order of the observer (2) as p = m

+ rank(R) + i. Step 2: Choose matrix R according to Remark 2.

Hence use (12) to obtain L2. Step 3: Check the detectability condition (11). If

satisfies, then go to Steps 2-5 of the design Procedure 1. Otherwise, set i → (i + 1) and go to Step 1.

III. NUMERICAL EXAMPLES

This section presents two numerical examples to il-lustrate the attractiveness and simplicity of the proposed design procedures of this paper. Example 1 [8]. Consider an example of [8] with the following matrices

1 0 0 1 0 0 02 0 1 1 1 0 00 3 0 0 1 1 0

,0 0 0 3 0 1 10 0 0 0 1 0 11 0 0 0 0 1 00 1 0 0 1 0 2

A

−⎡ ⎤⎢ ⎥−⎢ ⎥⎢ ⎥⎢ ⎥

= −⎢ ⎥⎢ ⎥−⎢ ⎥⎢ ⎥−⎢ ⎥

−⎢ ⎥⎣ ⎦

1 0 0 0 0 0 01 1 0 0 0 0 01 0 1 0 0 0 0

C⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥−⎣ ⎦

and 3 2 2 1 2 1 0

.2 0 1 1 1 0 0

F− −⎡ ⎤

= ⎢ ⎥−⎣ ⎦

Tsui [8] derived a third-order linear functional observer for the case where there are no unknown inputs, i.e. D = 0. To illustrate the attractiveness and usefulness of the results of this paper, let us now design a third-order lin-ear functional observer for this system and with an added unknown input signal, where D = [1 0 1 0 1 1 1]T.

Using the following orthogonal transformation T = [C+ null(C)], matrix C is first transformed into the ca-nonical form (1d), and the following submatrices are obtained

1 23 2 2 1 2 1 0

, ,1 0 1 1 1 0 0

F F− −⎡ ⎤ ⎡ ⎤

= =⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦

11 12

1 0 0 1 0 0 02 0 1 , 0 1 0 0 ,2 3 0 1 1 1 0

A A−⎡ ⎤ ⎡ ⎤

⎢ ⎥ ⎢ ⎥= =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥− −⎣ ⎦ ⎣ ⎦

21 22

0 0 0 3 0 1 10 0 0 0 1 0 1

, ,1 0 0 0 0 1 01 1 0 0 1 0 2

A A

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

1 2

01

11 and .

10

1

D D

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

It is easy to verify that the Condition 1 of Corollary 1 is not satisfied, therefore the Proposed design Proce-dure 1 does not apply in this case. As a result, Theorem 1 and the design Procedure 2 are now used. We start the design Procedure 2 with a third-order observer, i.e. p = 3.

Let us choose, arbitrarily, matrix R as R = [0 0 0

H. Trinh et al.: Design of Reduced-Order Functional Observers for Linear Systems with Unknown Inputs 519

−1]. Accordingly, matrix 12 1

2 00

A DFF

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

has full col-

umn-rank and the Condition 1 of Theorem 1 is satisfied.

Matrix ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−=⎥

⎦

⎤⎢⎣

⎡=

100000110121

22 R

FL

Using (23b) and (23c), matrices N and G are ob-

tained as 1.6667 0.5 1

0 1.5 00.3333 0.5 2

N−⎡ ⎤

⎢ ⎥= −⎢ ⎥⎢ ⎥− −⎣ ⎦

and

0.3333 0.5 00.3333 0.5 0 .0.6667 1 0

G−⎡ ⎤

⎢ ⎥= −⎢ ⎥⎢ ⎥−⎣ ⎦

The pair (G, N) is observable and matrix E can be assigned with any prescribed stability. Let its eigenval-ues be, say, eig(E) = −3, −4, −5 Using pole placement technique, matrix Z is easily obtained as

82.1667 82.1667 164.333352.5 52.5 105

5.8333 5.8333 11.6667Z

− −⎡ ⎤⎢ ⎥= − −⎢ ⎥⎢ ⎥−⎣ ⎦

. Using (23a),

matrix 166 247 1105 156 012 18 2

E−⎡ ⎤

⎢ ⎥= −⎢ ⎥⎢ ⎥− −⎣ ⎦

.

Using (22), matrix 1

82 79 16652 51 10621 5 12

L− −⎡ ⎤

⎢ ⎥= − −⎢ ⎥⎢ ⎥−⎣ ⎦

. Using

(8), matrices M and Q are obtained as

85 81 16453 51 105

M−⎡ ⎤

= ⎢ ⎥−⎣ ⎦ and

201 24 1307132 21 84321 5 103

Q− −⎡ ⎤

⎢ ⎥= − −⎢ ⎥⎢ ⎥−⎣ ⎦

.

The observer design is thus completed and a third-order linear functional observer is obtained. It is easy to confirm the satisfaction of equations (3)-(5) by substituting the above derived matrices into (3)-(5).

This example thus illustrates the usefulness of the results presented in this paper. It shows that a third-order linear functional observer can indeed be de-signed for the system [8] and with the added unknown input signal. This was not considered in [8]. Further-more, the design procedures presented in this paper are straightforward and involve little computation.

To further illustrate the simplicity and the steps in-volve in the design of the proposed observer (2), we also consider the following example which was reported in [1-3].

Example 2 [1-3]: 1 1 0 11 0 0 , 0

0 1 1 0A D

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

, and

1 0 00 0 1

C⎡ ⎤

= ⎢ ⎥⎣ ⎦

.

For illustrative purpose, let us estimate state x2(t), i.e. F = [0 1 0].

Matrix C is first transformed into the canonical form (1d) by the following orthogonal transformation

1 0 0[ ( )] 0 0 1

0 1 0T C null C+

⎡ ⎤⎢ ⎥= = −⎢ ⎥⎢ ⎥⎣ ⎦

. Accordingly, using (7),

the following sub-matrices are obtained

F1 = [0 0], F2 = −1, A11 = 1 0

0 1−⎡ ⎤

⎢ ⎥−⎣ ⎦, A12 =

11−⎡ ⎤

⎢ ⎥⎣ ⎦

,

A21 = [1 0], A22 = 0, D1 = 1

0−⎡ ⎤

⎢ ⎥⎣ ⎦

and D2 = 0.

It is easy to check whether Conditions 1 and 2 of Corol-lary 1 in this paper are satisfied. Therefore the design Procedure 1 can now be used to design a stable first-order functional observer.

Step 1: K = 1 and L2 = −1 Step 2: Using (23b) and (23c) matrices N and G

are obtained as N = 0 and 01

G⎡ ⎤

= ⎢ ⎥⎣ ⎦

.

Step 3: Since the pair (G, N) is observable, matrix E can be assigned with any prescribed sta-bility. Let its eigenvalue be, say, eig(E) = −3 Accordingly, matrix Z is obtained as Z = [0 3].

Step 4: Using (22) matrix L1 is obtained as L1 = [0

3]. Step 5: Using (8) matrices M and Q are obtained as

M = [0 −3] and Q = [−1 6].

It is easy to confirm the satisfaction of equations (3)-(5) by substituting the above derived matrices into (3)-(5). Finally, the following first-order observer is obtained

( ) 3 ( ) [ 1 6] ( )w t w t y t= − + −

2ˆ ( ) ( ) [0 3] ( ) .x t w t y t= + −

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

IV. CONCLUSION

This paper has presented new conditions for the ex-istence of reduced-order linear functional state observers for linear systems with unknown inputs. Systematic procedures for the synthesis of reduced-order functional observers have been given. The attractive feature of the proposed observer is the simplicity with which the de-sign process can be accomplished. Numerical examples have been given to illustrate the attractiveness and sim-plicity of the new design procedures.

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