Post on 13-May-2023
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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