Design of observers with unknown inputs using eigenstructure assignment

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International Journal of Systems Science

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Design of observers with unknown inputs using

eigenstructure assignment

Sheng-Fuu Lin a & An-Ping Wang a a

Department of Electrical and Control Engineering , National Chiao Tung University , 1001 Ta Hsueh Road, Hsinchu, Taiwan, 30010, Republic of China Published online: 26 Nov 2010.

To cite this article: Sheng-Fuu Lin & An-Ping Wang (2000) Design of observers with unknown inputs using

eigenstructure assignment, International Journal of Systems Science, 31:6, 705-711

To link to this article: http://dx.doi.org/10.1080/00207720050030752

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Design of observers with unknown inputs using eigenstructure

assignment

S

HENG

-

F

UU

L

IN

_{{ and}

A

N

-

P

ING

W

ANG

_{

Obsevers with unknown inputs using eigenstructure assignment are established in this paper. Orders of observers may vary from minimum order to full order. Complete and parametric solutions for observer matrices and generalized eigenvectors are obtained. Owing to the completeness and parametric forms of the solution, more properties of the observer may be obtained; hence, the solution is quite suitable for advanced applica-tions.

1. Introduction

Over the past few decades, many researchers have devel-oped the reduced and/or full order unknown input observers by di€ erent approaches (Meditch and Hostetter 1974, Wang et al. 1975, Kudva et al. 1980,

Kobayashi and Nakamizo 1982, Miller and

Mukundan 1982, Fairman et al. 1984, Hou and Muller 1992, 1994, Syrmos 1993, Yang and Wilde 1988, Darouach et al. 1994, Chang and Hsu 1994, Schreier

et al. 1995), because some of the inputs to the system

are inaccessible. It is well known that the observer design problem where all inputs are measurable is dual to the state feedback design problem. However, the way to apply the existent results of the state feedback design to the unknown input observer design is not trivial. Eigenstructure assignment is one of the existing methods for state feedback design. The method designs the feed-back gain by ® nding eigenvectors with assigned eigen-values of the resulting system (Moore 1976, Fahmy and O’Reilly 1982, Andry et al. 1984) and the solution is often represented in parametric forms. More control objectives such as robustness, sensitivity, minimum gain design etc can be achieved by choosing a set of suitable parameters from optimizing certain objective functions (Roppenecker 1983, Andry et al. 1984, Owens and O’Reilly 1987, Burrows and Patton 1991, Liu and Patton 1996).

In this paper, unknown input observers with orders varying from minimum order to full order using an eigenstructure assignment method are designed. Complete solutions of the observer matrices are repre-sented in parametric forms. The solution is suitable for other optimization processes. In most eigenstructure assignment approaches based on state feedback design, the system is assumed to be controllable. Therefore, these approaches did not develop solutions of general-ized eigenvectors with any uncontrollable eigenvalue. Transmission zeros in the unknown input observer design play the same role as uncontrollable eigenvalues in the feedback design. If a system contains a transmis-sion zero, the transmistransmis-sion zero must be an eigenvalue of any possible unknown input observers. Here, solution for transmission zeros are found.

This paper is organized as follows. In section 2, an unknown input observer is introduced. Some preli-minary results are presented in section 3. In section 4 the problem is formulated and in section 5 solutions for unknown input observers are established. Some illustra-tive examples are provided in section 6, and section 7 concludes the paper.

2. Observer design

Consider the following linear system _xˆ Ax

‡

Bu

‡

Dv; y ˆ Cx;

)

…

1

_†

where x 2 Rn is the state variable, u 2 Rp is the input variable, v 2 Rmis the unknown disturbance, y 2 Rq is the measurable output and A, B, D, C are matrices with

International Journal of Systems Science ISSN 0020± 7721 print/ISSN 1464± 5319 online#2000 Taylor & Francis Ltd http://www.tandf.co.uk/journals

Received 9 October 1998. Accepted 15 June 1999.

{ Department of Electrical and Control Engineering, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu, Taiwan 30010, Republic of China.

appropriate dimensions. The main purpose of this paper is to design an rth order observer of the following type

_zˆ Nz

‡

L y

_‡

Gu;

^

x ˆ Pz

‡

Qy;

)

…

2

_†

without any knowledge of disturbance v, where r is an integer in the range n ¡ q µ r µ n,x is the estimation of

^

x, z 2 Rris the state variable of the observer, and N, L ,

G, P, Q are matrices with appropriate dimensions. If

there exists a matrix T 2 Rr£n satisfying the following conditions: NT ¡ TA

‡

L C ˆ 0;

…

3

_†

T D ˆ 0;

…

4

_†

G ¡ TB ˆ 0;

…

5

_†

PT

_‡

QC ˆ I;

_…

6

_†

then we have

…

_z¡ T _x

†

ˆ N

_…

z ¡ Tx

†

;

…

x ¡ x

^

_†

ˆ P

_…

z ¡ Tx

†

:

If N is stable, then

^

x ! x. Hence, the problem of the rth

order unknown input observer is to ® nd matrices N, L ,

G, P, Q and T satisfying (3), (4), (5) and (6).

From right-multiplying both sides of (6), it follows that PT D

_‡

QCD ˆ QCD ˆ D ; hence, if

rank CD < rank D, the matrix Q does not exist. This means that the observer (2) cannot be found. From this fact, without loss of generality, assumptions that

C is of full row rank, D is of full column rank and q

_¶

m are made.

3. Some preliminary results

Before solving the problem, we will ® rst discuss the possible eigenvalues of matrix N.

De® nition 1: A complex number ¶i is a transmission zero of system (1) if and only if

rank

…

A ¡ ¶iI

†

C

D

0 < n

‡

m u

De® nition 2: If there exists a set of linear independent vectors vkij wk ij " # ; jˆ 1;. . . ;³-i; kˆ 1;. . . ; -»ij; satisfying that

…

A ¡ ¶iI

†

C D 0 vk ij wk ij ˆ I 0 0 0 vk¡1 ij wk¡1 ij ; v0ij ˆ 0;

…

7

_†

then we say that vk

ij is the right generalized transmission vector of order k with transmission zero ¶i.

Furthermore, if rank

…

A ¡ ¶iI

†

C D 0 ˆ n

‡

m ¡ -³i

and any non-zero linear combination of

v-»i1 i1 0 ;. . . ; v-»i³i -i -³i 0 " #

is not in the column space of

…

A ¡ ¶iI

†

C

D

0 ; then we say that vk

ij, j ˆ 1;. . . ;³-i; k ˆ 1;. . . ; -»ij, form a complete set of right transmission vectors with

transmis-sion zero ¶i. u

The following lemma discusses the role of transmis-sion zeo in an observer with unknown inputs.

Lemma 1: If observer (2) exists, the eigenvalues of N contains the transmission zeros of (1) counting multiplicity and the rest of eigenvalues can be assigned arbitrarily. Proof : Let L TZbe a Jordan form matrix with the

trans-mission zeros as its eigenvalues counting multiplicity and

F 2 Cn£s_{be a matrix whose columns are all generalized}

transmission vectors arranged in the corresponding order. Since CF ˆ 0, T F is of full column rank; other-wise, from (6), the observer does not exist. We can, therefore, de® ne a matrix K 2 Cn£…r¡s† _{such that}

‰

F K

_Š

and T

_‰

F K

_Š

are of full column rank. Three matrices C2, L1 and L2 can then be found such that

C

_‰

F K

_Š

ˆ

‰

0 C2

Š

and L ˆ

‰

TF T K

Š

L_L1

2 :

…

8

†

By de® nition, a matrix W exists such that

AF

‡

DW ˆ FL TZ; hence, T AF ˆ TFL TZ and A1 and A2 can be found such that

T A

‰

F K

Š

ˆ

‰

TF TK

Š

L ₀TZ A_A1

2 :

…

9

†

From (3), we have NT F ˆ T AF ˆ T FL TZ; hence, N1

and N2 can be found such that

N

‰

TF T K

Š

ˆ

‰

TF TK

Š

L ₀TZ N_N1

2 :

…

10

†

It follows that the eigenvalues of N are those of L TZand N2. From right-multiplying by

‰

F K

Š

and

left-multi-plying by T

_‰

F K

_Š

¡1_{to (3), we have} L TZ¡ L TZ 0 N1¡ A1

‡

L1C2 N2¡ A2

‡

L2C2 ˆ 0:

…

11

_†

Noticing that if the matrices pair

_…

A2; C2

†

is

unobser-vable, then its unobservable eigenvalue is a transmission

706 S.-F. L in and A.-P. W ang

zero of the original system. Since L TZcontains all

trans-mission zeros,

_…

A2; C2

†

is an observable pair and the

eigenvalues of N2 can be assigned arbitrarily. u

For a transmission zero ¶i, assume that elements of

set f-»i1; . . . ; -»i -³ig are arranged as follows:

-»i1µ -»i2µ ¢ ¢ ¢ µ -»i -³i. Denote ¿i as the number of

distinct elements in the set

{

-»i1;. . . ; -»i -³i

}

. In addition,

the notations ¼i1;¼i2; . . . ;¼i¿i, satisfying ¼il<

¼i2 <¢ ¢ ¢ < ¼i¿i, represent all distinct elements of the

set

{

-»i1;. . . ; -»i -³i

}

. Assume that there are ²il elements

with value ¼il, l ˆ 1;. . . ;¿i, within the set

f-»i1;. . . ; -»i -³ig. Let Vk ilˆ

‰

vki_…²_i0‡²_i1‡¢¢¢‡²_i…l¡1†‡1_† vki_…²_i0‡²_i1‡¢¢¢‡²_i…l¡1†‡2_† ¢ ¢ ¢ vki_…²i0‡²i1‡¢¢¢‡²i…l¡1†‡²il†

Š

; Wilkˆ

‰

wki_…²i0‡²i1‡¢¢¢‡²i…l¡1†‡1† w k i_…²i0‡²i1‡¢¢¢‡²i…l¡1†‡2† ¢ ¢ ¢ wki_…²i0‡²i1‡¢¢¢‡²i…l¡1‡²il†

Š

; k ˆ 1;. . . ;¼il; lˆ 1;. . . ;¿iand ²i0ˆ 0;

where the column vectors of V kilare the right generalized

transmission vectors of grade k, in all chains with length ¼il. Then by (7), we have

…

A ¡ ¶iI

†

C D 0 V kil W k il ˆ I₀ 0₀ V k¡1il W k¡1 il ; k ˆ 1;. . . ;¼il; V 0ilˆ 0:

…

12

†

Let Uilˆ

‰

V ¼i1i1¢ ¢ ¢ V ¼ilil

Š

, which gathers all the last

gen-eralized transmission vectors in those chains with length less than or equal to ¼il.

4. Problem formulation

The key problem in this paper is now stated as follows.

Design an observer with unknown inputs using eigenstruc-ture assignment.

Given a symmetric set of complex numbers

{

¶i;. . . ;¶º

}

, which contain transmission zeros, and a

set of positive integers »i1;. . . ;»i³i, i ˆ 1;. . . ;º,

repre-senting the multiplicities and satisfying

Pº

iˆ1P³jˆ1i »ijˆ r, we want to ® nd the parametric

sol-utions of matrices N, L , G, P, Q and T over the ® eld of real number satisfying (3), (4), (5) and (6) where the matrix N has eigenvalues

{

¶i;. . . ;¶º

}

and left

general-ized eigenvectors h1ij; . . . ;h»ijij; j ˆ 1;. . . ;³i; i ˆ 1;. . . ;º

satisfying the relation:

hkijN ˆ ¶ihkij

‡

hk¡1ij ; for iˆ 1;. . . ;º; j ˆ 1;. . . ;³i; k ˆ 1;. . . ;»ij; and h0ijˆ 0:

…

13

†

Since hkij, i ˆ 1;. . . ;º; j ˆ 1;. . . ;³i; k ˆ 1;. . . ;»ij are

linearly independent, (3) and (4) are equivalent to the

following conditions:

‰

~_tk_ij ~_lk ij

Š

…

A ¡ ¶iI

†

C D 0 ˆ

‰

~tk¡1ij 0

Š

i ˆ 1;. . . ;º; j ˆ 1;. . . ;³i; kˆ 1;. . . ; »ij;

…

14

†

where ~tk ijˆ ¡hkijT and ~lkij ˆ hkijL . De® ne H as H ˆ H1 .. . Hº

2

6

₄

3

7

_{5; where Hiˆ} Hi1 .. . Hi³i

2

6

6

4

3

7

7

5 and Hij ˆ h1 ij .. . h»ij ij

2

6

6

4

3

7

7

5: The matrices ~T and ~L are de® ned in the same way. If

solutions of ~T and ~L are found from (14), then by

choosing a non-singular H, the solutions of T , L , N and G are:

T ˆ ¡H¡1_{T ; L ˆ H~} ¡1_{L ; N ˆ H~} ¡1_{L H and G ˆ TB;}

where L is a Jordan form matrix in lower case with eigenvalues

{

¶1;. . . ;¶·

}

and multiplicity

{

»i1;. . . ;»i³i;

i ˆ 1;. . . ;·

}

. Let X‡ _{represent the pseudo-inverse of}

the matrix X satisfying X‡_ˆ

_…

_XH_X

†

¡1_XH_{. If the}

matrix ~T or T satis® es the following relation:

rank T

C ˆ rank

~

T C ˆ n;

then solutions of P and Q are as follows:

‰

P Q

_Š

ˆ T_C

‡

‡

K I_rˆq¡ T_C T_C

‡

…

†

;

where K 2 Rn£…r‡q_{†represents free parameters.}

From the above discussion, the following lemma similar to that given by Moore (1976) and Klein and Moore (1977), characterizing all possible solutions of the observer, is given.

Lemma 2: L et

{

¶1;. . . ;¶·

}

be a symmetric set of com-plex numbers and let

{

»i1;. . . ;»i³i; i ˆ 1;. . . ;·

}

be a set

of positive integers satisfying P· iˆ1

P³i

jˆ1»ijˆ r. T here exist N, L , G, P, Q, T and h1ij;. . . ;h»ijij; j ˆ 1;. . . ;³i; i ˆ 1;. . . ;º, satisfying (3), (4), (5), (6) and (13) if and only if

(1) matrix H is non-singular,

(2) for each i 2 f1;. . . ;ºg there exists i0_{2 f1;. . . ;ºg} such that ¶iˆ

…

¶i0

†

¤, ³_iˆ ³_i0, »_ijˆ »_i0_j,

j ˆ 1;. . . ;³i, and vkijˆ

…

vkij

†

¤, i ˆ 1;. . . ;º; j ˆ 1;. . . ;³i; k ˆ 1;. . . ;»ij, where

…

vkij

†

¤ means the component-wise conjugate of the vector vk

ij.

(3) for each i 2 f1;. . . ;·g, there exists a set of vectors f~tkij; ~lkij; iˆ 1;. . . ;º; j ˆ 1;. . . ; ³i; k ˆ 1;. . . ;»ijg satisfying (14) and

rank T~

C ˆ n:

…

15

†

Proof : The su cient part has been stated above. Assume that the solution exists. Since matrix H repre-sents the left generalized eigenvectors, it is non-singular. If (15) is not satis® ed, P and Q do not exist. Hence, the

necessity part follows. u

5. Main results

Complete parameteric solutions of ~t1ij;. . . ;~t»ijij and

~

l1ij;. . . ;~l»ijij with an eigenvalue ¶i from (14) according

to whether the eigenvalue ¶i is a transmission zero or

not will be shown in this section.

5.1. Solutions for eigenvalues which are not

transmission zeros

If ¶i is not a transmission zero, then

rank

…

A ¡ ¶iI

†

C

D

0 ˆ n

‡

m:

There exists a non-singular transformation such that

P11i P21 i P12i P22 i

…

A ¡ ¶iI

†

C D 0 Q1i Q2 i ˆ I_…n‡m_†£…n‡m_† 0 ;

…

16

_†

where P11i 2 C…n‡m†£n, P12i 2 C…n‡m†£q, P21i 2 C…q¡m†£n, P22i 2 C…q¡m†£q, Q1i 2 Cn£…n‡m†, Q2i 2 Cm£…n‡m† and I_…n‡m_†£…n‡m_†is an

…

n

‡

m

†

£

…

n

‡

m

†

identity matrix.

Theorem 1: If ¶i is not a transmission zero, complete solutions of (14) can be represented as follows:

‰

~tk ij ~lkij

Š

ˆ

‰

~tk¡1ij Q1i f k¡1ij

Š

P 11 i P12i P21 i P22i " # ; k ˆ 1;. . . ;»ij; ~t0ijˆ 0;

…

17

†

where f k¡1ij , k ˆ 1;. . . ;»ij, are free vectors and if

¶iˆ

…

¶i0

†

¤ then the free parameters should be chosen as

f kijˆ

…

fki0_j

†

¤ for consideration of realness.

Proof : The constraint for consideration of realness can be seen easily. Here, we will show that (17) is equivalent to (14).

(Necessity) If the following variable transformation is introduced:

‰

~tkij ~lkij

Š

ˆ

‰

akij f k¡1ij

Š

P 11 i P12i P31 i P32i " # ; k ˆ 1;. . . ;»ij;

…

18

†

then from (14) and (16), we have akij ˆ ~tk¡1ij Q1i, k ˆ 1;. . . ;»ij. By applying this relation into (18), (17)

is obtained.

(Su ciency) From (17) and (16), we have

‰

~tkij ~lkij

Š

…

A ¡ ¶_C iI

†

D₀ ˆ

‰

~tk¡1 ij Q1i ~zkij

Š

I…n‡m†₀£…n‡m† Q 1 i Q2 i ¡1 ˆ

‰

~tk¡1 ij 0

Š

:

Therefore the vectors given by (17) satisfy (14). u 5.2. Solutions for eigenvalues which are transmission

zeros

If ¶i is a transmission zero, then

rank

…

A ¡ ¶iI

†

C

D

0 ˆ ri< n

‡

m:

The following non-singular transformation can be obtained : P11i P12i P21 i P22i

…

A ¡ ¶iI

†

C D 0 Q11i Q12i Q21 i Q22i ˆ Ir_i 0 0 0 ;

…

19

_†

where P11i 2 Cri£n, P12i 2 Cri£q, P21i 2 C…n‡q¡ri†£n, P22i 2 C…n‡q¡ri†£q, Q11i 2 Cn£ri, Q12i 2 Cn£…n‡m¡ri†, Q21i 2 Cm£ri, Q22i 2 Cm£…n‡m¡ri†; and Iri is an ri£ ri identity matrix.

Lemma 3: P21i Uilis of full column rank for l ˆ 1;. . . ;¿i.

Proof : Assume that a non-zero vector f cn be found such that P21 i Uil f ˆ 0. Since

‰

Ir_i 0

Š

Q 11 i Q12i Q21 i Q22i " #¡1

is of full row rank, a vector -v

-w

can be found such that

‰

Ir_i 0

Š

Q 11 i Q12i Q21 i Q22i ¡1 _-v -w ˆ

‰

P 11 i P12i

Š

U₀ilf :

Then it follows that

A ¡ ¶iI C D 0 -v -w ˆ Uilf 0 :

It is a contradiction to the de® nition of Uil. u

From lemma 3, for each Uil, l ˆ 1;. . . ;¿i, we can

obtain the following non-singular transformation:

S1il S2 il

P21i Uilˆ I²i1‡₀¢¢¢‡²il ;

…

20

†

708 S.-F. L in and A.-P. W ang

where

S1

il 2 R²i1‡¢¢¢‡²il†£…n‡q¡ri†; S2

il2 R…n‡q¡ri¡²i1¡¢¢¢¡²il†£…n‡q¡ri†:

Theorem 2: Assume that ¶i is a transmission zero.

(a) If »ijµ ¼i1, complete solutions of (14) can be repre-sented as follows:

‰

~tk ij ~lkij

Š

ˆ

‰

~tk¡1ij Q11i f k¡1ij

Š

P 11 _P12 P21 _P22 " # ~t0 ijˆ 0; k ˆ 1;. . . ;»ij;

…

21

†

where f k¡1

ij , k ˆ 1;. . . ;»ij are free vectors.

(b) If ¼i1< »ijµ ¼i¿i, there exists a number b such that

¼ib< »ij< ¼i_…b‡1_†, and if »ij> ¼i¿i, then b ˆ ¿i.

Under the above conditions, the complete solutions of (14) can be represented as follows:

‰

~tkij ~lkij

Š

ˆ

‰

t~k¡1ij Qk¡1i f k¡1ij

Š

I₀ ¡P 11 i UibS1ib S2 ib " # £ P 11 i P12i P21 i P22i " # ; ~t0ij ˆ 0; k ˆ 1;. . . ;

…

»ij¡ ¼ib

†

;

‰

~tkij ~lkij

Š

ˆ

‰

t~k¡1ij Q11i f k¡1ij

Š

I₀ ¡P 11 i UilS1il S2 il " # £ P 11 i P12i P21 i P22i " # ; k ˆ

_…

»ij¡ ¼i_…l‡1_†

†

‡

1;. . . ;

…

»ij¡ ¼il

†

; l ˆ

…

b ¡ 1

†

;. . . ;1;

‰

~_tk_ij ~lk ij

Š

ˆ

‰

~tk¡1ij Q11i f k¡1ij

Š

P 11 _P12 P21 _P22 " # ; k ˆ

_…

»ij¡ ¼i1

†

‡

1;. . . ;»ij;

…

22

†

where f k¡1ij , k ˆ 1;. . . ;»ij are free vectors.

(c) In (a) and (b) , if ¶iˆ

…

¶i0

†

¤then the free parameters

should be chosen as f k

ijˆ

…

f ki0_j

†

¤ for consideration of

realness.

Proof : The constraint for consideration of realness can be easily seen. Here, we will show that (21) is equivalent to (14), or (22) is equivalent to (14).

(Necessity) A variable transformation is adopted as follows:

‰

~tkij ~lkij

Š

ˆ

‰

akij ckij

Š

P 11 _P12 P21 _P22 " # ; k ˆ 1;. . . ;»ij;

…

23

†

where akij 2 C1£r and ckij2 C1£…m‡n¡r†; hence, (14) is

equivalent to the following relations:

akij ˆ ~tk¡1ij Q11i ;

…

24

†

0 ˆ ~tk¡1ij Q12i ; k ˆ 1;. . . ;»ij:

…

25

†

Since the column spaces of Q12i and V 1il, l ˆ 1;. . . ;¿iare

the same, (25) is equivalent to ~tk¡1ij V 1ilˆ 0, l ˆ 1;. . . ; ¿i; k ˆ 1;. . . ;»ij. From (12) and (14), it can be further

shown that ~ tk¡1ij V 1ilˆ ~_t…k¡¼il† ij V ¼ilil; ~_t0 ijV kil² 0; if k > ¼il; if k µ ¼il: (

…

26

_†

We discuss the problem in the following two cases:

(a) From (26), if »ijµ ¼i1, (14) is equivalent to (24). Let fk¡1ij ˆ ckij. By applying (24) to (23), (21) is obtained.

(b) If ¼i1< »ijµ ¼i¿i, ® nd a number b such that

¼ib< »ijµ ¼i_…b_‡1_†, and if »ij > ¼i¿i, let b ˆ ¿i.

From (26), (25) is equivalent to the following rela-tions: ~ tk¡1 ij Uibˆ 0; where k ˆ 2;. . . ;

…

»ij¡ ¼bi

†

‡

1; ~tk¡1ij Ui_…b¡1†ˆ 0; where k ˆ

…

»ij¡ ¼ib

†

‡

2;. . . ;

…

»ij¡ ¼i_…b¡1†

†

‡

1; .. . ~tk¡1ij Ui1 ˆ 0; where k ˆ

…

»ij¡ ¼i2

†

‡

2;. . . ;

…

»ij¡ ¼i1

†

‡

1:

From the above relations and (23), we have

…

ak¡1 ij P11i

‡

ck¡1ij P21i

†

Uibˆ 0; k ˆ 2;. . . ;

_…

»ij¡ ¼ib

†

ˆ 1;

…

ak¡1ij P11i

‡

ck¡1ij P21i

†

Uilˆ 0; k ˆ

…

»ij¡ ¼i_…l‡1_†

†

‡

2;. . . ;

…

»ij¡ ¼il

†

‡

1; l ˆ 1;. . . ;

…

b ¡ 1

†

:

From (20) and (24), the above relations are equivalent to

ck ijˆ ¡~tk¡1ij Q11i P11i U1bS1ib

‡

f k¡1ij Sib2; k ˆ 1;. . . ;

_…

»ij¡ ¼ib

†

; ckijˆ ¡~tk¡1ij Q11i P11i U1lS1il

‡

f k¡1ij Sil2; k ˆ

_…

»ij¡ ¼i_…i¡1†

†

‡

1;. . . ;

…

»ij¡ ¼il

†

; l ˆ 1;. . . ;

…

b ¡ 1

†

;

…

27

†

where f k

ij are free vectors. Note that ckij is free for k ˆ

_…

»ij¡ ¼il

†

; . . . ;»ij. Let

ck

ijˆ f k¡1ij ; k ˆ

…

»ij¡ ¼i1

†

‡

1;. . . ;»ij:

…

28

†

By applying (24), (27) and (28) to (23), (22) is obtained.

(Su ciency) Since (23) is a non-singular transfor-mation and (26) is an equivalent relation, (14) can be derived from (21) or (22) by reversing the procedure.

u

6. Examples

Example 1: Consider system (1) with the following matrices A ˆ ¡2 ¡2 0 0 0 1 0 ¡3 ¡4

2

6

6

4

3

7

7

5; D ˆ 1 0 0 1 0 0

2

6

6

4

3

7

7

5 and C ˆ 1 0 1_{0 1 0} :

This example has been studied by Miller and Mukundan (1982), Yang and Wilde (1988), Hou and Muller (1992). This system has one transmission zero ¡4 with ³-1ˆ 1

and »11ˆ 1. If a ® rst-order observer is designed, the

eigenvalue must be ¡4. We have

T ˆ h¡1_f0 11

‰

0 0 ¡ 1

Š

; L ˆ h¡1f 011

‰

0 3

Š

;

‰

P Q

Š

ˆ h f 0 11 1 0 0 0 1 ¡_fh0 11 0 0

2

6

6

6

6

6

4

3

7

7

7

7

7

5: Let a ˆ h¡1_f0

11, then the parametric solutions of the

observer are _zˆ ¡4z

‡‰

0 3a

_Š

y;

^

x ˆ 1 a 0 ¡1_a

2

6

6

6

6

4

3

7

7

7

7

5z

‡

1 0 0 1 0 0

2

6

4

3

7

_5y:

_…

29

_†

Putting the existing results in the previous research studies (Miller and Mukundan 1982. Yang and Wilde 1988, Hou and Muller 1992), only solutions when

a ˆ ¡1 are obtained. However, in (29), there is one

more free parameter a, and a is not a trivial parameter. It provides a greater degree of freedom to assign a suit-able observer according to the control requirement. u Example 2: Consider system (1) with the following matrices (Hou and Muller 1992)

A ˆ ¡1 ¡1 0 0 ¡1 1 0 _{0 ¡1}

2

6

6

4

3

7

7

5; D ˆ 1 0 0

2

6

6

4

3

7

7

5 and C ˆ 1 0 1_{0 0 1} :

This system has one transmission zero ¡1 with ³1ˆ 1

and »11 ˆ 1.

(a) First (minimum) order observer: the eigenvalue of the observer must be ¡1. Let f 011ˆ

‰

g1 g2

Š

. It

follows that T ˆ 0 1 hg1 ¡ 1 hg2 ; L ˆ 0 1 hg1 ;

‰

P Q

_Š

ˆ 0 1 _¡1 1 g1h 0 1 g1 f2 0 0 1

2

6

6

6

6

4

3

7

7

7

7

5:

Let a1ˆ h¡1g1and a2ˆ h¡1g2, then the solutions are

_zˆ ¡z

‡‰

0 a1

Š

y;

^

x ˆ 0 1 a1 0

2

6

6

6

4

3

7

7

7

5z

‡

1 ¡1 0 a2 a1 0 1

2

6

6

₄

3

7

7

5y:

…

30

†

In (30), there are two more free parameters a1 and a2

then the ® rst-order results given by Hou and Muller (1992) whose solutions are those with a1ˆ 1 and a2ˆ 0 in (30).

(b) Third (full) order observer: here, we assigned the eigenvalues to be the transmission zero ¶1ˆ ¡1

with ³1ˆ 1 and »11ˆ 3. By a simple computation,

it follows that

‰

~tk1j ~lk1j

Š

ˆ

‰

~tk¡11j f k¡11j

Š

0 0 0 1 ¡1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0

2

6

6

6

6

6

₄

3

7

7

7

7

7

5; ~ t0 ijˆ 0; k ˆ 1;2;

‰

~tk1j ~lk¡11j

Š

ˆ

‰

~tk¡11j f k¡11j

Š

0 ¡1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 ¡1 0 0 1 0 0 1 0 0

2

6

6

6

6

6

6

6

6

₄

3

7

7

7

7

7

7

7

7

5 ; k ˆ 3: Choose f011 ˆ ¡1, f111 ˆ 2, f 211 ˆ

‰

1 1

Š

,

710 S.-F. L in and A.-P. W ang

H ˆ V ¡1_ˆ 0 1 0 1 0 1 1 0 0

2

6

4

3

7

5 and K ˆ 0 ¡1 0 ¡1 0 1 0 0 0 1 ¡1 ¡1 ¡1 0 0

2

6

4

3

7

_5; then the resulting system is

_zˆ 0 0 1 0 ¡1 0 ¡1 1 ¡2

2

6

6

4

3

7

7

5z

‡

0 1 0 0 0 ¡2

2

6

6

4

3

7

7

5y;

^

x ˆ 0 ¡1 0 0 0 1 ¡1 ¡1 ¡1

2

6

6

4

3

7

7

5z

‡

1 0 0 1 0 0

2

6

6

4

3

7

7

5y: 7. Conclusions

In this paper, the eigenstructure assignment method for state feedback design has been further applied to the unknown input observer design with orders varied from minimum order to full order. Complete and para-metric solutions of the observer matrices and the gener-alized eigenvector are obtained. In the illustrative examples, we can see that results obtained by the pro-posed method have more meaningful free parameters than the previous results. It is because the proposed sol-utions are complete, as shown in the main theorems. The completeness and parametric form of the solutions makes then more suitable for advanced applications. Acknowledgements

This research was supported in part by the National Science Council of Taiwan, R.O.C., under contract NSC-87-2213-E-009-135.

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