on the functional observers for linear descriptor systems

Systems & Control Letters 61 (2012) 427–434

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Systems & Control Letters

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On the functional observers for linear descriptor systemsM. Darouach ∗

CRAN-CNRS (UMR 7039), Nancy University, IUT de Longwy, 186, Rue de Lorraine, 54400 COSNES et ROMAIN, France

a r t i c l e i n f o

Article history:Received 9 December 2009Received in revised form26 November 2011Accepted 12 January 2012Available online 14 February 2012

Keywords:Functional observerLinear systemsDescriptor systemsStabilityExistence conditions

a b s t r a c t

This paper is concerned with the design of functional observers for linear time-invariant descriptorsystems. Contrary to the functional observers considered for the standard systems in Darouach [8], wherethe order of these observers is equal to the dimension r of the functional to be estimated, in this paper theorder can be of dimension different of that of this functional. When this order is equal to the dimensionof the functional and the system is in a standard form, the presented design method becomes that ofDarouach [8]. It also generalizes the existing results for descriptor systems. The approach is based onthe new definition of partial impulse observability. Sufficient conditions for the existence and stabilityof these observers are given. Continuous time and discrete time systems are considered. Two numericalexamples are given to illustrate our approach.

© 2012 Elsevier B.V. All rights reserved.

1. Introduction

The problem of observing the state vector of deterministiclinear time-invariant multivariable systems has been the object ofnumerous studies ever since the original work of Luenberger [1,2]first appeared. This is because state estimation is of greatimportance for a wide range of applications ranging from controlapplications, like output feedback design, fault detection toapplications in signal and image processing and cryptography bysynchronization [3–5]. Generally, in practical cases, only a portionor a linear function of the state is required, this is the casefor example for the state feedback control. It can be seen thatthe conditions for the existence of the functional observers areweaker than the detectability condition which is required in fulland reduced order observers design. Theory and algorithms fordesigning an observer which can give an estimate of the entirestate vector or of a linear functional for standard systemshave beenreported in [1,2,6–14], and in the books [15,16].

On the other hand singular or descriptor systems have a greattheoretical and practical importance, since they describe a largeclass of systems encountered in chemical, mineral, electrical andeconomical systems [17]. In recent years a great deal of workshave been devoted to the analysis and design techniques forthese systems. The observers design for descriptor systems with

∗ Tel.: +33 382396221.E-mail address:[email protected].

0167-6911/$ – see front matter© 2012 Elsevier B.V. All rights reserved.doi:10.1016/j.sysconle.2012.01.006

or without unknown input has been treated in [18,24,19], andreferences therein.

In this paper, functional observers design for descriptor systemsis proposed, the order of these observers can be different fromthe dimension of the linear function to be estimated. In fact ifconditions for the existence of the observer of an order equalto that of the functional to be estimated are not satisfied, anobserver of another order may exist. The approach is based onthe new introduced definition of the partial impulse observabilitywith respect to a functional and on the parametrized solutions ofthe generalized Sylvester equations. Sufficient conditions for theexistence and stability of these observers are given. A systematicmethod for their design is presented. Continuous-time as well asdiscrete-time systems are considered.

2. Preliminary results

In this section we recall some basic results from linear algebrawhich are used in the sequel of the paper. We shall use thefollowing notations:

The symbol R(A) will be used to denote the row space of amatrix A, Im(A) = Ax, x ∈ Rn

, and R(A) = Im(AT ), Σ+ denotesany generalized inverse of the matrix Σ , i.e. verifies ΣΣ+Σ = Σ ,this generalized inversematrix is also denoted byΣ− orΣ (1) in theliterature (see [20] for example). The symbol E⊥ denotes amaximalrow rank matrix such that E⊥E = 0. When E is of full row rankmatrix, E⊥

= 0.Let Σ+ be any generalized inverse of Σ , then we have the

following lemma [20].

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428 M. Darouach / Systems & Control Letters 61 (2012) 427–434

Lemma 1. The general solution to ΣXΣ = Σ is given by X =

Σ++V−Σ+ΣVΣΣ+, where V is an arbitrarymatrix of appropriate

dimension.

The following two lemmas are proved in standard linear algebrareferences (see for example [21]).

Lemma 2. Let X represent an m × n matrix and Y an n × p matrixthen rank (XY ) = rank (Y ), if and only if rank

X

I − YY+

= n.

Lemma 3. For any matrices A ∈ Rm×n and L ∈ Rr×n, R(L) ⊂ R(A)if and only if there exists a matrix Φ ∈ Rr×m such that L = ΦA.

Now consider the following consistent linear equation systemof unknowns x and z.

Ax = b (1a)z = Lx (1b)

where x ∈ Rn, z ∈ Rr , b ∈ Rm, A is an (m × n) matrix and L is an(r × n) matrix.

Let the set of solutions of (1a) be defined by S = x ∈ Rn,such that (1a) is satisfied = ∅, thenwe have the following lemma(see the similar results presented in the framework of theestimable function in linear models in statistics [22]).

Lemma 4. The following assertions are equivalent(1) The functional z is uniquely determined from (1a).(2) R(L) ⊂ R(A).(3) rank

AL

= rank A.

Proof. Let x1 be any element of S. Assume that R(L) ⊂ R(A), thenfrom Lemma 2 there exists a matrix Φ such that L = ΦA, let x2be another element of S, we have Lx2 = ΦAx2 = Φb = ΦAx1 =

Lx1 = z, thus z is unique. Assume that (1) is satisfied, that thevalue of z is unique for every x ∈ S, then from [20] the value of xis given by x = A+b + (I − A+A)y, where y is an arbitrary vectorof appropriate dimension. Then z = Lx = L(A+b + (I − A+A)y)is unique for every value of y or equivalently L(I − A+A)y = 0 forevery y or equivalently L = LA+A which means that R(L) ⊂ R(A)the proof that (2a) is equivalent to (3) is direct.

3. Functional observers design

Consider the linear time-invariant multivariable system de-scribed by

Eσ x(t) = Ax(t) + Bu(t) (2a)y(t) = Cx(t) (2b)z(t) = Lx(t) (2c)

where σ denotes the derivative operator σ x(t) = dx(t)/dt forcontinuous time systems and the forward-shift operator σ x(t) =

x(t + 1) for discrete time systems, x ∈ Rn and y ∈ Rp are the semistate vector and the output vector of the system, u ∈ Rm is theknown input and z ∈ Rr is the vector to be estimated,where r 6 n.Matrix E ∈ Rn1×n, when n1 = n matrix E is singular, matrices A, B,C , and L are known constant and of appropriate dimensions.

Before presenting the observers design for system (2a) we shallgive the following results which can be used in the sequel ofthis paper. These results extends the notion of Y observability orimpulse observability (causal observability for the discrete timecase) (see Refs. [17,23]).

Definition 1. The descriptor system (2) with u(t) = 0, or thetriplet (C, E, A) is said to be partially impulse (causal for thediscrete time case) observable with respect to L if y(t) is impulsefree (causal) for t > 0, only if Lx(t) is impulse free (causal) for t > 0.

The following lemma gives the conditions for the partialimpulse observability (partial causal observability for the discretetime case).

Lemma 5. The following statements are equivalent:

(1) The triplet (C, E, A) is partially impulse (causal) observable withrespect to L.

(2) If there exist a vector v ∈ Rn and a vector w ∈ Rn such thatsE − A

C

v =

E0

w for all s ∈ C, then Lv = 0.

(3) If there exist a vector v ∈ Rn and a vector w ∈ Rn such thatECA

v =

00E

w, then Lv = 0;

(4) A−1Im(E) ∩ KerE ∩ KerC = KerL;

(5) rank

E A0 C0 E0 L

= rank

E A0 C0 E

;

(6) rank

LE

E⊥AC

= rank EE⊥AC

.

Proof. First let rank E = r1, then there always exist twononsingular matrices U and V such that E = UEV =

Ir1 00 0

,

A = UAV =

A11 A12A21 A22

, C = CV =

C1 C2

and L =

LV = [L1 L2]. Let S1 and S2 be two nonsingular matrices defined

by S1 =

U 0 0 00 I 0 00 0 U 00 0 0 I

and S2 =

V 00 V

, then condition (5)

becomes rank S1

E A0 C0 E0 L

S2 = rank S1

E A0 C0 E

S2, or equivalently

rankA22C2L2

= rank

A22C2

, or R(L2) ⊂ R

A22C2

. From Lemma 3

there exists a matrix Ω of appropriate dimension such that L2 =

Ω

A22C2

.

Now, the equivalence of (2)–(4) is direct. We shall show theequivalence between (3) and (5), without loss of generality, let E,A and C be in the above form, i.e (E = E, A = A, C = C) and letV−1v =

v1v2

and V−1w =

w1w2

, then condition (3) can be written

as: Ir1 0C1 C2A11 A12A21 A22

v1v2

=

0 00 0Ir1 00 0

w1w2

which leads to v1 = 0, w1 = A12v2 andA22C2

v2 = 0.

Then Lv = 0, or equivalently L2v2 = 0 since v1 = 0, if and only if

rankA22C2L2

= rank

A22C2

, which is equivalent to (5).

The equivalence between (5) and (6) can be obtained as follows:

Let M1 =

E⊥ 0 0 0EE+ 0 0 00 I 0 00 0 I 00 0 0 I

be a full column rank matrix and

let N1 =

I −E+A0 I

be a nonsingular matrix of appropriate

dimension, then we have rank

E A0 C0 E0 L

= rank M1

E A0 C0 E0 L

N1 =

M. Darouach / Systems & Control Letters 61 (2012) 427–434 429

rank E + rank

LE

E⊥AC

, on the other hand we have rankE A0 C0 E

=

rankM1

E A0 C0 E

N1 = rank E + rank

EE⊥AC

, by using condition (5)

we obtain the equivalence between (5) and (6).To show that (1) is equivalent to (5) without loss of generality

let system (2) with u(t) = 0 be in the formIr1 00 0

σ x1(t)σ x2(t)

=

A11 A12A21 A22

x1(t)x2(t)

(3a)

y(t) = [C1 C2]

x1(t)x2(t)

(3b)

z(t) = [L1 L2]x1(t)x2(t)

. (3c)

Then we have

σ x1(t) = A11x1(t) + A12x2(t) (4a)A22C2

x2(t) = −

A21C1

x1(t) +

0

y(t)

(4b)

z(t) = [L1 L2]x1(t)x2(t)

. (4c)

From (4b) and (4c) and by using Lemma 4, one can see thatthe necessary and sufficient condition for the determination of

L2x2 is that rankA22C2L2

= rank

A22C2

which is equivalent to

condition (5) which is also equivalent to R(L2) ⊂ R

A22C2

.

From Lemma 3 there exists a matrix Ω of appropriate dimensionsuch that L2 = Ω

A22C2

and from (4b) and (4c) we obtain z(t) =

L1 − Ω

A21C1

x1(t)+Ω

0I

y(t)which shows that z(t) is impulse

free (causal) when y(t) is. Conversely, assume that (5) is not

satisfied, then we have rankA22C2L2

= rank

A22C2

, from (4b) and

(4c) and Lemma 4 we have

z(t) =

−L2

A22C2

+ A21C1

+ L1

x1(t) + L2

A22C2

+ 0

y(t)

+ L2

I −

A22C2

+ A22C2

η

where η = δ(t), the δ-distribution for the continuous time caseand η = ν(t+1), with ν(t) an arbitrary signal, for the discrete timecase. From the expression of z(t) we can see that z(t) contains animpulsive (non causal) part, independently of y(t). This completesthe proof of the lemma.

Remark 1. • For L = I , the partial impulse observability(causality) becomes the standard impulse (causal) observability(see [17]).

• For E = I the conditions of Lemma 5 are always satisfied.• No assumption is made on the rank of the matrix L contrarily

to [8] where L is assumed to be of full row rank.

The following assumption is used in the sequel of the paper.

Assumption I. We assume that system (2) is partial impulse(causal) observable with respect to L.

3.1. Observers design

In this section we shall present a method for the observersdesign for system (2). Let us consider the following reduced orderobserver

σζ (t) = Nζ (t) + F−E⊥Bu(t)

y(t)

+ Hu(t), (5a)

z(t) = Pζ (t) + Q−E⊥Bu(t)

y(t)

, (5b)

where ζ (t) ∈ Rq is the state of the observer,z(t) ∈ Rr is theestimate of z(t). Matrices N , F , H , P and Q are constant and ofappropriate dimensions to be determined such that limt→∞(z(t)−z(t)) = 0.

The following theorem gives the conditions for system (5) to bean qth order observer for the functional z(t) in system (2).

Theorem 1. The qth-order observer (5) will estimate (asymptoti-cally) z(t) if there exists a matrix parameter T such that the followingconditions hold.

(1) N is a stability matrix,

(2) NTE − TA + FE⊥AC

= 0,

(3) [ P | Q ]

TE

E⊥AC

= L,

(4) H = TB.

Proof. Let ϵ(t) be the error between ζ (t) and TEx(t), i.e ϵ(t) =

ζ (t) − TEx(t), then its dynamic is given by

σϵ(t) = Nϵ(t) +

NTE − TA + F

E⊥AC

x(t)

+ (H − TB)u(t). (6)

On the other hand from (5b) the estimate of z(t) can be writtenas

z(t) = Pϵ(t) + [P | Q ]

TE

E⊥AC

x(t). (7)

If conditions (2)–(4) are satisfied, then (6) and (7) reduce to

σϵ(t) = Nϵ(t)

and

e(t) =z(t) − z(t) =z(t) − Lx(t) = Pϵ(t).

In addition if (1) is satisfied we have limt→∞ ϵ(t) = 0 andconsequently limt→∞ e(t) = 0 for any x(0),z(0), and u(t). Hencez(t) in (5) is an estimate of z(t). This completes the proof.

Remark 2. The functional observers design is independent of thechoice of the matrix E⊥, in fact let E⊥

1 = ME⊥ be another maximalfull row rank matrix such that E⊥

1 E = 0, where M is a nonsingularmatrix parameter of appropriate dimension, in this case Eqs. (2)and (3) of Theorem 1 become NTE − TA + F

E⊥AC

= 0 and

[P | Q ]

TE

E⊥AC

= L , where F = FM 00 I

and Q =

QM 00 I

.


From Theorem 1, the design of the observer (5) is reduced tofind the matrices T , N , P , Q , F and H such that conditions (1)–(4)are satisfied.

Now, define the following matrix Γ =

E

E⊥AC

and let

R ∈ Rq×n be a full row rank matrix such that rankRΓ

= rank Γ .

In this case, there exist always two matrices T and K such that

TE = R − KE⊥AC

,

which can also be written as

[T K ]Γ = R.

Since rankRΓ

= rank Γ , the general solution to this equation is

given by :

[T K ] = RΓ +− Y (I − Γ Γ +)

where Y is an arbitrary matrix of appropriate dimension. In thiscase, matrices T and K are given by

T = α − Yβ

and

K = α1 − Yβ1

with α = RΓ +

I0

, α1 = RΓ +

0I

, β = (I − Γ Γ +)

I0

and

β1 = (I − Γ Γ +)0I

.

On the other hand equations (2) and (3) of Theorem 1 can bewritten as

NR − K

E⊥AC

+ F

E⊥AC

− αA + YβA = 0

and

PR − K

E⊥AC

+ Q

E⊥AC

= L

or equivalentlyN K1 Y

Σ = Θ (8)

andP K2

Π = L (9)

respectively, with Σ =

R

E⊥AC

βA

, Θ = αA, K1 = F − NK ,

K2 = Q − PK and Π =

R

E⊥AC

. One can see that upon matrices

N , P , Y , K1 and K2 are determined, we can deduce the values ofmatrices F and Q .

The necessary and sufficient conditions for the existence of thesolution to (8) and (9) can then be given by the following lemma.

Lemma 6. The necessary and sufficient conditions for the existence ofthe solution to (8) and (9) are given by

rank

R

E⊥ACβAαA

= rank

RE⊥ACβA

(10)

and

rank

RE⊥ACL

= rank

RE⊥AC

. (11)

Proof. From the general solution of linear matrix equations [20],there exists a solution to (8) if and only if:

rankΣ

Θ

= rank[Σ], (12)

or equivalently

ΘΣ+Σ = Θ (13)

where Σ+ is any generalized inverse matrix of Σ .Condition (12) is exactly (10).Also there exists a solution to (9) if and only if:

rank

Π

L

= rank Π, (14)

or equivalently

LΠ+Π = L. (15)

Condition (14) is exactly (11). This completes the proof.

The following remark summarizes the constraints introduced onthe matrix R.

Remark 3. From Assumption I and the above results, one can seethat matrix R must be chosen such that:

(1)

rank

EE⊥ACR

= rank

EE⊥ACL

= rank

EE⊥AC

,

and(2)

rank

RE⊥ACL

= rank

RE⊥AC

.

From these conditions matrix R must be chosen such that system(2) is partially impulse (causal) observable with respect to R and

R(L) ⊂ R

RE⊥AC

. Conditions (1) and (2) are always satisfied for

R = L, this case corresponds to that of functional observers of orderq = r , the dimension of the functional to be estimated. One can seethat in this case the conditions of Lemma6 reduce to condition (10)with R = L.

Now from [20], under condition (10) and (11), the generalsolutions of Eqs. (8) and (9) are given byN K1 Y

= ΘΣ+

− Z(I − ΣΣ+) (16)

andP K2

= LΠ+

− Z1(I − ΠΠ+) (17)

where matrices Z and Z1 are arbitrary of appropriate dimensions.


From solution (16) and (17) we obtain

N = A1 − ZB1, (18)K1 = A2 − ZB2, (19)Y = A3 − ZB3, (20)P = A4 − Z1B4, (21)K2 = A5 − Z1B5, (22)

where A1 = ΘΣ+

I00

, B1 = (I − ΣΣ+)

I00

, A2 = ΘΣ+

0I0

,

B2 = (I − ΣΣ+)

0I0

, A3 = ΘΣ+

00I

, B3 = (I − ΣΣ+)

00I

,

A4 = LΠ+

I0

, B4 = (I − ΠΠ+)

I0

, A5 = LΠ+

0I

and B5 =

(I − ΠΠ+)0I

.

Under condition (10) and by using (18), the observer errordynamics can be written as

σϵ(t) = Nϵ(t) = (A1 − ZB1)ϵ(t).

The condition for N to be a stability matrix is given by thefollowing lemma.

Lemma 7. Under condition (10) there exists a matrix parameter Zsuch that N is Hurwitz if and only if

rank

λR − αAE⊥ACβA

= rank Σ, (23)

∀λ ∈ C, Re(λ) ≥ 0 for the continuous time case (|λ| > 1 for thediscrete time case).

Proof. From (18), there exists a matrix parameter Z such thatmatrix N is Hurwitz if and only if the pair (B1,A1) is detectableor equivalently rank

λIq − A1

B1

= q, ∀λ ∈ C, Re(λ) ≥ 0 (|λ| > 1 for

the discrete time case).Now, we have:

rank


= rank

λ[Iq 0 0]Σ − ΘΣ+Σ0 I 00 0 I

Σ

= rank

λI − A1 −A2 −A3B1 B2 B30 I 00 0 I

Σ

where we have used condition (12) or equivalently (13), i.eΘΣ+Σ = Θ .

Now, by using Lemma 2 and (10), we obtain that the followingmatrixλI − A1 −A2 −A3

B1 B2 B30 I 00 0 I

,

must be of full column row. This proves the lemma.

Lemma 8. The observer design is independent of the choice of thegeneralized inverse.

Proof. We shall prove this lemma for solution (16), the sameapproach can be applied to (17). From (16) we have

[N K1 Y ] = ΘΣ+− Z(I − ΣΣ+).

Let X be the generalized inversematrix given by Lemma 1, thenweobtain

[N K1 Y ] = Θ(Σ++ V − Σ+ΣVΣ+Σ)

− Z(I − Σ(Σ++ V − Σ+ΣVΣ+)).

By using (13) we obtain

[N K1 Y ] = ΘΣ+− Z(I − ΣΣ+),

where Z = Z(I − ΣV ) − ΘV is a parameter matrix to bedetermined, which is in the form of solution (16).

From the above results we have the following theorem.

Theorem 2. Under Assumption I, the qth-order observer (5) willestimate (asymptotically) z(t) if conditions (10), (11) and (23) aresatisfied.

3.2. Particular cases

In this section we shall consider four particular cases of ourresults.

3.2.1. Static observerThis case corresponds to q = 0 or R = 0, i.e the functional

observer is reduced to an algebraic one. Here Assumption Iand conditions (10) and (11) are satisfied since rank

C

E⊥A

=

rank L

CE⊥A

. From system (2) we obtain the following algebraic

system:

Ax = b, (24a)z = Lx, (24b)

where A =

E⊥AC

and b =

−E⊥Bu

y

. From Lemma 4, the solution

z is given by z = L(A+b − (I − A+A)w), where w is an arbitraryvector of appropriate dimension.

3.2.2. Case where q = rIn this case, the dimension of the observer is equal to that of

the functional z to be estimated. It corresponds to matrix P = I ,then (3) of Theorem 1 becomes TE = L − Q

E⊥AC

, by using the

definition of matrices R and K we obtain R = L, K = Q and K2 = 0.Therefore conditions 10 and 11 of Lemma 6 reduce to condition 10with R = L.

3.2.3. Full state estimationThis case corresponds to matrix L = I , then assumption I

becomes rankE A0 C0 E

= n+ rank E or rank

EE⊥AC

= rank Γ = n

which is exactly the impulsive observability of system (2) or ofthe triplet (E, A, C). Since matrix Γ is of full column rank we have

Γ +Γ = I . Condition (11) becomes rank RE⊥AC

= n and condition

(10) is always satisfied. Then the conditions for the existence of the

functional observer reduce to condition (23), i.e rank


=

rankΣ = n,∀λ ∈ C, Re(λ) ≥ 0 for the continuous time case (|λ| >1 for the discrete time case), we shall show that this conditionis equivalent to the detectability of system (2) or equivalently torank

λE − A

C

= n, ∀λ ∈ C, Re(λ) ≥ 0 for the continuous time


case (|λ| > 1 for the discrete time case). In fact, we have:

rankλE − A

C

= rank

I 0−E⊥ 00 I

λE − AC

= rank

λE − AE⊥AC

= rank

λE − AλE⊥AλCE⊥AC

= rank

λΓ −

A0

E⊥AC

= rank

RE⊥AC

Γ + 0

(Γ Γ +− I) 0

0 I

λΓ −

A0

E⊥AC

= rank

λR − αA

λE⊥A − E⊥AΓ +

A0

λC − CΓ +

A0

E⊥ACβA

= rank

λR − αA

E⊥AΓ +

A0

CΓ +

A0

E⊥ACβA

= rank


wherewehaveused the fact thatΓ +Γ = I , the last equality results

from the fact that R

E⊥AΓ +

A0

CΓ +

A0

⊂ R(βA).

3.2.4. Functional observers for standard systemsThis case corresponds to matrix E = I and assumption I is

always satisfied. On the other hand we have E⊥= 0, Γ =

IC

,

rank Γ = rankRΓ

= n, Γ +

= [I 0], α = R and β =

0

−C

.

Condition (11) becomes rankRCL

= rank

RC

, which means that

R(L) ⊂ RRC

, on the other hand conditions (10) and (23) become:

rank

RCCARA

= rank

RCCA

and

rank

λR − RACCARA

= rank

RCCA

.

For R = L, i.e q = r , the dimension of the functional observer isequal to that of the vector z to be estimated, this case correspondsto that considered in [8], thenmatrix P of the observer (5) is P = Ir ,from conditions (10) and (23) we obtain

rank

LCCALA

= rank

LCCA

and

rank

λL − LACCALA

= rank

LCCA

which are exactly those obtained in [8].

3.3. Design procedure

In this section we shall give, from the presented results, asystematic procedure to design the functional observer for lineardescriptor systems. In summary, this design can performed asfollows.

Let Γ =

EE⊥AC

and let R be a full row rank matrix such that

system (3) is partially impulse (causal) observable with respect

to R and R(L) ⊂ R(Π), with Π =

RE⊥AC

, compute the matrix

parameters α = RΓ +

I0

, α1 = RΓ +

0I

, β = (I − Γ Γ +)

I0

and

β1 = (I − Γ Γ +)0I

, then compute matrices Σ =

Π

βA

and Θ =

αA. Verify that condition (12) or rankΣ

Θ

= rank Σ is satisfied,

then compute matrices A1 = ΘΣ+

I00

, B1 = (I − ΣΣ+)

I00

,

A2 = ΘΣ+

0I0

, B2 = (I − ΣΣ+)

0I0

, A3 = ΘΣ+

00I

, and

B3 = (I−ΣΣ+)

00I

. Verify that condition (23) or the detectability

of the pair (B1,A1) is satisfied, then determine the parametermatrix Z such that N = A1 − ZB1 is Hurwitz. Compute matricesK1 = A2 − ZB2 and Y = A3 − ZB3. Deduce matrices P and K2from the following expression [P K2] = LΠ+

− Z1(I − ΠΠ+),where Z1 is an arbitrarymatrixwhich can be taken equal zero. Thendeduce the values of T and K from the expressions T = α−Zβ andK = α1 − Zβ1. In this case matrices F , Q and H of the observer canbe determined as follows F = K1 + NK , Q = K2 + PK and H = TB.

Remark 4. In the above design procedure, if we do not considerthe two first particular cases of Section 3.2, we can start theprocedure by taking R = L, if conditions (12) and (23) are notsatisfied, then increase the size ofmatrixR, such thatR(L) ⊂ R(R),by adding a new rowmatrix. The procedure can be reiterated up toq = n (this case corresponds to the impulse observable system),if the conditions (12) and (23) are not satisfied, then no functionalobserver of the form (5) for the considered descriptor system (2)can be designed from the presented approach.

4. Numerical examples

4.1. Example 1

To illustrate the main idea, consider the following continuoustime system described by : Ex = Ax + Bu, y = Cx and z = Lx

where E =

1 0 0 00 1 0 00 0 0 00 0 0 0

, A =

1 0 2 11 −1 1 01 1 0 00 1 −1 0

, B =

0 11 01 10 0

,

C =

1 0 0 00 1 1 0

, and L = [0 0 1 0]. For this system, we have

E⊥=

0 0 1 00 0 0 1

and E⊥A =

1 1 0 00 1 −1 0

, it is easy to see

that rankEC

= 3 = n = 4 and rank

EC

E⊥A

= 3 = n = 4,


Fig. 1. z(t) and its estimate for Example 1.

then the conditions generally adopted for the observers design arenot satisfied, see [24,19] for example. However our Assumption Iis satisfied since rank Γ = rank

Γ

L

= 3. Now, we can see that

rank

CE⊥A

= rank

LC

E⊥A

= 3, in this case a functional observer

of dimension q = 0 exists, this can be seen from the followingequations obtained from system (2):

Ax = b, (25a)z = Lx, (25b)

where A =

E⊥AC

and b =

−E⊥Bu

y

. From Lemma 4, the

general solution z is given by z = L(A+b − (I − A+A)w),where w is an arbitrary vector of appropriate dimension. For our

numerical example, we have: A =

1 1 0 00 1 −1 01 0 0 00 1 1 0

, b =

u1 + u20y1y2

and A+=

0.4 −0.2 0.6 −0.20.2 0.4 −0.2 0.40 −0.5 0 0.50 0 0 0

. Then z is given by z =

L

0.4(u1 + u2) + 0.6y1 − 0.2y20.2(u1 + u2) − 0.2y1 + 0.4y2

0.5y2−w4

= 0.5y2.

If we choose as a functional observer for this system an observerof dimension q = r = 1, the dimension of the functionalto be estimate, then we can take R = L, in this case we

obtain Σ =

LE⊥ACβA

and conditions (10) and (23) are satisfied,

i.e rank Σ = rank

L

E⊥AαACβA

= 4 and rank


=

rank Σ = 4, ∀λ ∈ C, Re(λ) ≥ 0. From the results ofSection 3.1 we obtain T = [0 −0.0747 −0.0174 0.2142], H =

[−0.0921 −0.0174], K = [0.0149 −0.4701 −0.0149 0.5298],P = 1, K2 = 0, Z = [1.3128 0 0 0 0 0 0 0 0], N =

−0.9715, K1 = [0.0174 −0.2142 −0.1096 0.4683], Y =

[0 0.07473 0.0174−0.2142−0.01494−0.0298 0.0149−0.0298],F = K1 + NK = [−0.0548 0.2226 −0.0194 −0.1516], andQ = K .

The estimate of z is given by the following observer

ζ = −0.9715ζ − 0.0950u1 − 0.0203u2 − −0.0950y1 − 0.0464y2z = ζ − 0.01494(u1 + u2) − 0.014947y1 + 0.5298y2.

Fig. 2. Estimation error for Example 1.

The simulation for the proposed functional observer was per-formed with z0 = 30. Figs. 1 and 2 show the true and estimatedtrajectory of the state z(t) and its estimation error. These simula-tion results demonstrate that our proposed design is very effective.

4.2. Example 2

This example is a rectangular systems, i.e the matrix E isrectangular, described by the following continuous time system:

Ex = Ax + Bu, y = Cx and z = Lx, where E =

1 0 0 00 1 0 00 0 0 0

,

A =

−3 0 2 11 −1 1 00 0 0 1

, B =

0 11 01 1

, C = [1 0 0 0] and L = [0 1 0 0].

For this system, we have E⊥= [0 0 1] and E⊥A = [0 0 0 1], it is

easy to see that rankEC

= 2 = n = 4 and rank

EC

E⊥A

= 3 =

n = 4, then the conditions generally adopted for the observersdesign are not satisfied. However ourAssumption I is satisfied sincerankΓ = rank

Γ

L

= 3. Now, we can see that rank

C

E⊥A

= 2 =

rank L

CE⊥A

= 3, in this case a functional observer of dimension

q = 0 does not exist. If we choose as a functional observer for thissystem an observer of dimension q = r = 1, the dimension ofthe functional to be estimate, then we can take R = L. For this

example, we have β =

0.5 0 00 0 00 0 10 0 0

−0.5 0 0

, α = [0 1 0], in this case we

obtain Σ =

LE⊥ACβA

and conditions (10) and (23) are satisfied, i.e

rank Σ = rank

L

E⊥AαACβA

= 4 and rank


= rank Σ = 4,

∀λ ∈ C, Re(λ) ≥ 0. From the results of Section 3.1 we obtainT = [−0.5 1 0.25], H = [1.25 − 0.25], K = [0 0.5], P = 1,K2 = 0, Z = [1 0 0 0 0 0 0 0], N = −1, K1 = [−0.25 2.5],Y = [0.5 0 −0.25 0 −0.5], F = K1 + NK = [−0.25 2], andQ = K .

The estimate of z is given by the following observerζ = −ζ + 1.5u1 + 2yz = ζ + 0.5y.


Fig. 3. z(t) and its estimate for Example 2.

Fig. 4. Estimation error for Example 2.

The simulation for the proposed functional observer was per-formed with z0 = 40. Figs. 3 and 4 show the true and esti-mated trajectory of the state z(t) and its estimation error. Thesesimulation results demonstrate that our proposed design is veryeffective.

5. Conclusion

In this paper, we have presented a simple method to designfunctional observers for linear descriptor systems. The order of

these observersmay be different of the dimension of the functionalto be estimated. The existence and stability conditions are given,and generalize those generally adopted for the standard anddescriptor systems.

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on the functional observers for linear descriptor systems

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