minimal single linear functional observers for linear systems

6
Automatica 47 (2011) 164–169 Contents lists available at ScienceDirect Automatica journal homepage: www.elsevier.com/locate/automatica Brief paper Minimal single linear functional observers for linear systems Frédéric Rotella a,, Irène Zambettakis b a Ecole Nationale d’Ingénieurs de Tarbes, Laboratoire de Génie de production, 47 avenue d’Azereix, 65016 Tarbes CEDEX, France b Institut Universitaire de Technologie de Tarbes, Université Paul Sabatier de Toulouse, 1 rue Lautréamont, 65016 Tarbes CEDEX, France article info Article history: Available online 4 December 2010 Keywords: Linear system Single functional observer Luenberger observer abstract A constructive procedure to design a single linear functional observer for a time-invariant linear system is given. The proposed procedure is simple and is not based on the solution of a Sylvester equation or on the use of canonical state space forms. Both stable observers or fixed poles observers problems are considered for minimality. © 2010 Elsevier Ltd. All rights reserved. 1. Introduction Since Luenberger’s works (Luenberger, 1963, 1964, 1966)a significant amount of research has been devoted to the problem of observing a linear functional v(t ) = Lx(t ), (1) where L is a constant full row rank (l×n) matrix, and, for every time t in R + , x(t ) is the n-dimensional state vector of the state space system ˙ x(t ) = Ax(t ) + Bu(t ), y(t ) = Cx(t ), (2) where u(t ) is the p-dimensional control, and y(t ) is the m- dimensional measure. A(n × n), B(n × p) and C (m× n) are constant matrices. For a survey of the main results see for instance (Aldeen & Trinh, 1999; O’Reilly, 1983; Trinh & Fernando, 2007; Tsui, 1998) and the references therein. The observation of v(t ) can be carried out with the design of the Luenberger observer ˙ z (t ) = Fz (t ) + Gu(t ) + Hy(t ), w(t ) = Pz (t ) + Vy(t ), (3) where z (t ) is the q-dimensional state vector. Constant matrices F , G, H, P and V are determined such that This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Delin Chu under the direction of Editor Ian R. Petersen. Corresponding author. Tel.: +33 056244 2763; fax: +33 056 244 2716. E-mail addresses: [email protected] (F. Rotella), [email protected] (I. Zambettakis). lim t →∞ (v(t ) w(t )) = 0. This asymptotic tracking is ensured if F is a Hurwitz matrix. Namely, if all the eigenvalues of F are such that their real part is negative. We know from Fortmann and Williamson (1972) that the linear functional observer (3) exists if and only if there exists a (q × n) matrix T such that: G = TB, TA FT = HC , (4) L = PT + VC , (5) F is a Hurwitz matrix. (6) Notice that a rigorous proof of this result has been established in Fuhrmann and Helmke (2001) for the case V = 0. According to the value of q, we distinguish several observers: q = n: the Kalman observer; q = n m: the reduced-order observer or Cumming–Gopinath observer (Cumming, 1969; Gopinath, 1971); l < q < n m and q is such that no observer of v(t ) with an order less than q exists: the minimal-order observer or minimal observer; l = q: the minimum-order observer or minimum observer. With (Roman & Bullock, 1975; Sirisena, 1979), we know that l is a lower bound for the order of the observer (3). Until now the direct design of a minimal observer of a given linear functional is an open question. Since (Fortmann & Williamson, 1972), design schemes have been proposed to reduce the order of the observer (3) with respect to the reduced-order observer. Mainly, these designs are based on the determination of the matrices T and F such that the Sylvester equation (4) is fulfilled (Trinh, Nahavandi, & Tran, 2008; Tsui, 2004). Unfortunately, the problem rests in satisfying conditions (4)–(6) with the dimension 0005-1098/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2010.10.027

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Page 1: Minimal single linear functional observers for linear systems

Automatica 47 (2011) 164–169

Contents lists available at ScienceDirect

Automatica

journal homepage: www.elsevier.com/locate/automatica

Brief paper

Minimal single linear functional observers for linear systems✩

Frédéric Rotella a,∗, Irène Zambettakis b

a Ecole Nationale d’Ingénieurs de Tarbes, Laboratoire de Génie de production, 47 avenue d’Azereix, 65016 Tarbes CEDEX, Franceb Institut Universitaire de Technologie de Tarbes, Université Paul Sabatier de Toulouse, 1 rue Lautréamont, 65016 Tarbes CEDEX, France

a r t i c l e i n f o

Article history:Available online 4 December 2010

Keywords:Linear systemSingle functional observerLuenberger observer

a b s t r a c t

A constructive procedure to design a single linear functional observer for a time-invariant linear system isgiven. The proposed procedure is simple and is not based on the solution of a Sylvester equation or on theuse of canonical state space forms. Both stable observers or fixed poles observers problems are consideredfor minimality.

© 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Since Luenberger’s works (Luenberger, 1963, 1964, 1966) asignificant amount of research has been devoted to the problemof observing a linear functional

v(t) = Lx(t), (1)

where L is a constant full row rank (l×n)matrix, and, for every timet in R+, x(t) is the n-dimensional state vector of the state spacesystem

x(t) = Ax(t) + Bu(t),y(t) = Cx(t),

(2)

where u(t) is the p-dimensional control, and y(t) is the m-dimensionalmeasure. A(n×n), B(n×p) and C(m×n) are constantmatrices. For a survey of the main results see for instance (Aldeen& Trinh, 1999; O’Reilly, 1983; Trinh & Fernando, 2007; Tsui, 1998)and the references therein.

The observation of v(t) can be carried outwith the design of theLuenberger observer

z(t) = Fz(t) + Gu(t) + Hy(t),w(t) = Pz(t) + Vy(t),

(3)

where z(t) is the q-dimensional state vector. Constant matricesF ,G,H, P and V are determined such that

✩ This paper was not presented at any IFAC meeting. This paper wasrecommended for publication in revised form by Associate Editor Delin Chu underthe direction of Editor Ian R. Petersen.∗ Corresponding author. Tel.: +33 056244 2763; fax: +33 056 244 2716.

E-mail addresses: [email protected] (F. Rotella), [email protected](I. Zambettakis).

0005-1098/$ – see front matter© 2010 Elsevier Ltd. All rights reserved.doi:10.1016/j.automatica.2010.10.027

limt→∞

(v(t) − w(t)) = 0.

This asymptotic tracking is ensured if F is a Hurwitz matrix.Namely, if all the eigenvalues of F are such that their real part isnegative.

We know from Fortmann andWilliamson (1972) that the linearfunctional observer (3) exists if and only if there exists a (q × n)matrix T such that:

G = TB,TA − FT = HC, (4)L = PT + VC, (5)F is a Hurwitz matrix. (6)

Notice that a rigorous proof of this result has been establishedin Fuhrmann and Helmke (2001) for the case V = 0. According tothe value of q, we distinguish several observers:

• q = n: the Kalman observer;• q = n−m: the reduced-order observer or Cumming–Gopinath

observer (Cumming, 1969; Gopinath, 1971);• l < q < n − m and q is such that no observer of v(t) with an

order less than q exists: theminimal-order observer orminimalobserver;

• l = q: the minimum-order observer or minimum observer.With (Roman & Bullock, 1975; Sirisena, 1979), we know thatl is a lower bound for the order of the observer (3).

Until now the direct design of a minimal observer of a given linearfunctional is an open question. Since (Fortmann & Williamson,1972), design schemes have been proposed to reduce the orderof the observer (3) with respect to the reduced-order observer.Mainly, these designs are based on the determination of thematrices T and F such that the Sylvester equation (4) is fulfilled(Trinh, Nahavandi, & Tran, 2008; Tsui, 2004). Unfortunately, theproblem rests in satisfying conditions (4)–(6) with the dimension

Page 2: Minimal single linear functional observers for linear systems

F. Rotella, I. Zambettakis / Automatica 47 (2011) 164–169 165

of F minimal. Moreover the distinction between the fixed-poleobserver problem where the poles are specified at the outset andthe stable observer problem where the poles are permitted tolie anywhere in the left half-plane is not well defined. Indeed, inseveral cases, necessary and sufficient existence conditions for acandidate observer to be minimal are obtained for the fixed-poleobserver problem only.

Apart from results on the multifunctional case (l > 1),for simplification purpose several authors have considered theobservation of a single linear functional (l = 1). Luenbergerhas shown in Luenberger (1966) that any specified single linearfunctional of the state vector may be obtained by means of anobserver of order (ν − 1), with arbitrary dynamics, where ν is theobservability index of the system. Namely, ν is the smallest integerfor which the matrix

O(A,C),ν =

CCA...

CAν−1

has rankn. InMurdoch (1973), an effective design procedure is pro-posed to design an observer of order (ν−1), with arbitrary dynam-ics. Some constraints are imposed on the choice of the observerpoles or on structures of some matrices. In addition, a method toapply this result to the multifunctional case has been proposedin Murdoch (1974). Some years later, based on simplificationsbrought about by considering (Roman & Bullock, 1975)C =

0m×(n−m) Im

,

a necessary and sufficient condition is proposed in Gupta, Fair-mann, and Hinamoto (1981) for the design of a single functionalobserver in a fixed-pole problem. Another condition for the ex-istence of a q-order observer can be found in Kondo and Sunaga(1986) which can be related to the Fortmann–Williamson condi-tion (Fortmann &Williamson, 1972). In the particular case of a sin-gle functional, the minimum order observer is a one-order filterand may not exist. Some conditions for the existence of a secondorder observer are given in Trinh and Zhang (2005). Based on theDuan procedure to solve a Sylvester equation (Duan, 1993), the de-sign of a general order observer for a single functional is proposedin Trinh, Tran, and Nahavandi (2006). To apply this method, whichis extended to the multifunctional observer in Trinh et al. (2008),the poles of the observer have to bedistinct andmust be fixed at theoutset. Nevertheless, the proposed procedures are based on the so-lution of the Sylvester equation and somenumerical algorithms ex-ist to increase the numerical robustness in the design (Datta, 2004;Datta & Sarkissian, 2000).

The present paper describes a direct and iterative procedure toget a minimal order for a single linear functional observer. On theone hand the word direct means that the design method is notbased on the solution of the Sylvester equation (4). It has beenunderlined in Tsui (1998), that the calculus of the matrix T is nota necessary step. This point is a specific feature of the procedurewe propose. On the other hand the word iterative indicates thatwe test an increasing sequence for the orders of the observers toobtainminimality.Moreover, the procedure points out if we face tothe stable observer case or if some poles can be fixed at the outset.

The paper is organised as follows. In the first place, a necessaryand sufficient condition is outlined for the existence of a singlefunctional observer. From these conditions, in the second section adesignmethod for the observer is proposed. An example illustratesthe procedure and points out that the minimal order depends onconstraints on the poles of the observer.

Notice that, in all the following, we suppose

rank[CL

]= m + 1.

This condition rules out the design of an obvious nondynamicobserver and can be verified easily.

2. Minimal observers existence condition

Let us define q as the smallest integer such that

rank Σq = rank[ΣqLAq

], (7)

with

Σq =

CLCALA...

CAq−1

LAq−1

CAq

.

After q derivatives of v(t) = Lx(t) we obtain

v(q)(t) = LAqx(t) +

q−1−i=0

LAiBu(q−1−i)(t). (8)

From (7) there exist Γ(i), for i = 0 to q, andΛi, for i = 0 to q−1,such that

LAq=

q−i=0

Γ(i)CAi+

q−1−i=0

ΛiLAi. (9)

Thus (8) can be written

v(q)(t) =

q−i=0

Γ(i)CAix(t) +

q−1−i=0

ΛiLAix(t) +

q−1−i=0

LAiBu(q−1−i)(t).

To eliminate the state x(t) we have the equalities

Lx(t) = v(t),LAx(t) = v(t) − LBu(t),...

LA(q−1)x(t) = v(q−1)−

q−2−i=0

LAiBu(q−2−i)(t),

Cx(t) = y(t),CAx(t) = y(t) − CBu(t),CA2x(t) = y(t) − CABu(t) − CBu(t),...

CA(q)x(t) = y(q)−

q−1−i=0

CAiBu(q−1−i)(t).

It yields

v(q)(t) = Γ(0)y(t) +

q−i=1

Γ(i)

y(i)

i−1−j=0

CAjBu(i−1−j)(t)

+ Λ0v(t) +

q−1−i=1

Λi

v(i)

i−1−j=0

LAjBu(i−1−j)(t)

+

q−1−i=0

LAiBu(q−1−i)(t),

=

q−i=0

Γ(i)y(i)+

q−1−i=0

Λiv(i)

+

q−1−i=0

Φiu(i)(t), (10)

Page 3: Minimal single linear functional observers for linear systems

166 F. Rotella, I. Zambettakis / Automatica 47 (2011) 164–169

where for i = 0 to q − 2

Φi =

LAq−1−i

q−j=i+1

Γ(j)CAj−i−1−

q−1−j=i+1

ΛjLAj−i−1

B,

and

Φq−1 =L − Γ(q)C

B.

The input–output differential equation (10) can be realized asthe q-order state space observable system

z(t) =

0 Λ0

1. . . Λ1. . . 0

...1 Λq−1

z(t) +

Φ0Φ1...

Φq−1

u(t)

+

Γ(0) + Λ0Γ(q)Γ(1) + Λ1Γ(q)

...Γ(q−1) + Λq−1Γ(q)

y(t),

v(t) =0 · · · 0 1

z(t) + Γ(q)y(t).

(11)

Theorem 1. Let us define q as the smallest integer such that thereexist Γi, for i = 0 to q, and Λi, for i = 0 to q − 1, such that

LAq=

q−i=0

Γ(i)CAi+

q−1−i=0

ΛiLAi,

and the matrix

F =

0 Λ0

1. . . Λ1. . . 0

...1 Λq−1

is a Hurwitz matrix. Then the q-order Luenberger observer (11) is aminimal observer of the single linear functional (1) for the system (2).

Proof. As G = TB, the form of the Φi leads to the matrix

T =

LAq−1

− Γ(1)C − · · · − Γ(q)CAq−1

−Λ1L − · · · − Λq−1LAq−1

...LA − Γ(q−1)C − Γ(q)CA − Λq−1LL − Γ(q)C

.

Somecalculations point out that (5) and (4) are verified. When Fis a Hurwitz matrix, the necessary and sufficient conditions forthe existence of a single functional observer of the single linearfunctional (1) for the system (2) are fulfilled. To prove minimalitylet us consider that there exists a p-observer solving the sameproblem with p < q. Then there exist matrices such that

v(p)(t) =

p−i=0

Riy(i)+

p−1−i=0

Siv(i)+

p−1−i=0

Tiu(i)(t).

Taking into account

v(t) = Lx(t),v(t) = LAx(t) + LBu(t),...

v(p)= LA(p)x(t) +

p−1−i=0

LAiBu(p−1−i)(t),

y(t) = Cx(t),y(t) = CAx(t) + CBu(t),...

y(p)= CApx(t) +

p−1−i=0

CAiBu(p−1−i)(t),

and supposing, for simplicity sake, that u(t) vanishes for every t ,we get

LApx(t) =

p−i=0

RiCAix(t) +

p−1−i=0

SiLAix(t).

This relation must be fulfilled for all solutions of x(t) = Ax(t).Consequently LAp is linearly dependent on the rows CAi and LAi.Let us suppose that the matrix0 S0

1. . . S1. . . 0

...1 Sp−1

is a Hurwitz matrix. This point is a contradiction, because q is thesmallest integer such that the writing (9) exists. �

For completeness, let us describe the existence condition ofa minimum order observer for a single linear functional. In thiscase, it can readily be seen that the Theorem 1 gives: a one-orderobserver for Lx(t) exists if and only if we can write

LA = Γ(0)C + Λ0L + Γ(1)CA, (12)

whereΛ0 is strictly negative. It can been shown that this conditionis equivalent to the condition established in Darouach (2000).

3. Design procedure and pole placement

3.1. Design procedure

We develop in this section the procedure to design a q-ordersingle functional observer when the conditions for the existenceof a p-order single functional observer with 1 ≤ p < q are notfulfilled. In order to examine if some of the poles of the observercan be fixed at the outset we introduce the following partitions

C = C0 =

[C1C∗

1

]=

[C2C∗

2

]= · · · =

[CqC∗

q

], (13)

where q is defined by (9) and for i = 1 to q, the rows of Ci and C∗

iare such that

• CiAi is linearly independent of C, L, CA, LA, . . . , CAi−1;• C∗

i Ai is linearly dependent of C, L, CA, LA, . . . , CAi−1.

It is obvious that the previous partitions of C possibly necessitatesa permutation in the measure variables. Let us denote πi, i = 1 toq, the number of rows in C∗

i . We have 0 ≤ π1 ≤ π2 ≤ · · · ≤ πq.Associated with the partitions (13) we have the partitions, for

i = 1 to q, Γ(i) =Γi Γ ∗

i, where Γ ∗

i is a (l × πi) matrix. From(9), it yields

Page 4: Minimal single linear functional observers for linear systems

F. Rotella, I. Zambettakis / Automatica 47 (2011) 164–169 167

LAq=

q−i=0

ΓiCiAi+

q−i=1

Γ ∗

i C∗

i Ai+

q−1−i=0

ΛiLAi. (14)

The matrix

Σ∗

q =

CL

C1ALA...

Cq−1Aq−1

LAq−1

CqAq

,

is a full-row rank matrix with

rank Σ∗

q = (q + 1)m + q −

q−i=1

πi.

Thus, the matrices Πi and ∆i defined by

LAq=

q−i=0

ΠiCiAi+

q−1−i=0

∆iLAi, (15)

and the matrices Γi,j and Λi,j defined, for i = 1 to q, by

C∗

i Ai=

i−j=0

Γi,jCjAj+

i−1−j=0

Λi,jLAj, (16)

are unique. Taking into account (16) in (14) yields

LAq=

Γ0 +

q−i=1

Γ ∗

i Γi,0

C0 +

q−i=1

Γi +

q−j=i

Γ ∗

j Γj,i

CiAi

+

q−1−i=0

Λi +

q−j=i+1

Γ ∗

j Λj,i

LAi.

From the unique property in (15) we deduce the coefficients forthe minimal observer

Γ0 = Π0 −

q−i=1

Γ ∗

i Γi,0,

for i = 1 to q, Γi = Πi −

q−j=i

Γ ∗

j Γj,i,

for i = 0 to q − 1, Λi = ∆i −

q−j=i+1

Γ ∗

j Λj,i,

where theΓ ∗

i are designparameters. Specifically, the characteristicpolynomial of the matrix

F =

0 Λ0

1. . . Λ1. . . 0

...1 Λq−1

is given by

pF (λ) = λq−

q−1−i=0

∆i −

q−j=i+1

Γ ∗

j Λj,i

λi.

The total number of design parameters to obtain stable polesfor the observer or to solve the fixed-pole observer problem isσ =

∑qi=1 πi. When σ ≥ q, all the poles can be fixed at the

outset. On the opposite side, when σ < q, σ indicates the numberof poles which can be fixed at the outset. In this case, when the

stable observer problem cannot be solved, the solution consistsin increasing the order of the observer. This step is performedby taking the derivative of v(q). Namely, the procedure we havedetailed in this section is applied on

v(q+1)(t) = LAq+1x(t) +

q−i=0

LAiBu(q−i)(t).

We can remark that the decomposition (9) yields

LAq+1= LAqA =

q−

i=0

Γ(i)CAi+

q−1−i=0

ΛiLAi

A,

=

q+1−i=1

Γ(i−1)CAi+

q−i=1

Λi−1LAi.

Thus this decomposition is immediate and the previous methodcan be used to design a minimal observer with some constraintson the poles.

3.2. Illustrative example

With (Trinh et al., 2006), let us consider the system (2) and thesingle functional (1) defined by

A =

−1 0 0 1 −20 −5 3 4 01 1 −8 3 0

−4 0 2 −6 00 0 0 1 −1

, B =

00001

,

C =1 0 0 0 0

,

L =1 14 42 79 2

.

(17)

The following steps illustrate the design procedure of minimalobservers.

(1) Test for the minimum observer. As

CA =−1 0 0 1 −2

,

LA =−275 −28 −136 −289 −4

,

we obtain rank (Σ1) = 3 and

rank[Σ1LA

]= 4.

Thus a first-order minimum observer cannot be designed.(2) Tests for a second-order observer. As

CA2=

−3 0 2 −9 4

,

LA2=

1295 4 426 935 554

,

we get rank (Σ2) = 5 and

rank[Σ2

LA2

]= 5.

As

LA2Σ−12 =

−1343.7 −16.6 −301 −8.4 −12.1

,

we deduce Λ0 = −16.6 and Λ1 = −8.4. It yields

F =

[0 −16.61 −8.4

].

The eigenvalues of F are {−3.12, −5.22}. Thus a minimalsecond-order observer can be designed.

Page 5: Minimal single linear functional observers for linear systems

168 F. Rotella, I. Zambettakis / Automatica 47 (2011) 164–169

(3) Design of the minimal second-order observer. From LA2Σ−12

we get Γ(0) = −1343.7, Γ(1) = −301 and Γ(2) = −12.1. From(11) we obtain

G =

[22.29 90 218 389 −11.4213.14 14 42 79 2

]B =

[−11.42

2

],

H =

[−1142.5−198.7

], P =

0 1

, V = −12.1.

The design of this observer is finished and the procedurecan be stopped. Nevertheless the poles are fixed. If the poles{−3.12, −5.22} are not acceptable we have to augment theorder of the observer. In the next step we tackle this point.

(4) Design of a minimal third-order observer with partially fixedpoles. In order to illustrate the design procedure we detailsome calculations here. From

CA3=

Γ20 Λ20 Γ21 Λ21 Γ22

Σ2,

LA2=

Π0 ∆0 Π1 ∆1 Π2

Σ2,

we deduce, on the one hand

LA3= LA2A = Π0CA + ∆0LA + Π1CA2

+ 1LA2+ Π2CA3,

= Π0CA + ∆0LA + Π1CA2

+ ∆1Π0 ∆0 Π1 ∆1 Π2

Σ2

+ Π2Γ20 Λ20 Γ21 Λ21 Γ22

Σ2,

= [∆1Π0 + Π2Γ20 ∆1∆0 + Π2Λ20 Π0 + ∆1Π1 + Π2Γ20

∆0 + ∆1∆1 + Π2Λ21 Π1 + ∆1Π2 + Π2Γ22]Σ2,

and on the other handLA3

= Γ0C + Λ0L + Γ(1)CA + Λ1LA + Γ(2)CA2+ Λ2LA2

+ Γ(3)CA3,

= [Γ0 + Λ2Π0 + Γ(3)Γ20 Λ0 + Λ2∆0 + Γ(3)Λ20

Γ(1) + Λ2Π1 + Γ(3)Γ21 Λ1 + Λ2∆1 + Γ(3)Λ21

Γ(2) + Λ2Π2 + Γ(3)Γ22]Σ2,

where Λ2 and Γ(3) are two design parameters. It yields

Γ0 = ∆1Π0 + Π2Γ20 − Λ2Π0 − Γ(3)Γ20,

Γ(1) = Π0 + ∆1Π1 + Π2Γ21 − Λ2Π1 − Γ(3)Γ21,

Γ(2) = Π1 + ∆1Π2 + Π2Γ22 − Λ2Π2 − Γ(3)Γ22,

and

Λ0 = ∆1∆0 + Π2Λ20 − Λ2∆0 − Γ(3)Λ20,

Λ1 = ∆0 + ∆1∆1 + Π2Λ21 − Λ2∆1 − Γ(3)Λ21.

WhenΛ2 andΓ(3) are chosen, these five parameters are knownand we can implement the observer (11) for q = 3. The polesof the matrix F are the roots of the characteristic polynomialpF (λ) = λ3

− Λ2λ2

− Λ1λ − Λ0 which depends on theparameters Λ2 and Γ(3). In our example

CA3=

41 2 −34 61 2

,

LA3=

−4609 406 −1526 −2467 −3144

.

With CA3Σ−12 and LA3Σ−1

2 we obtain the values

Γ20 = 55.14, Γ21 = −26, Γ22 = −12.57,Λ20 = 0.71, Λ21 = 0.29,∆1Π0 + Π2Γ20 = 10 656,Π0 + ∆1Π1 + Π2Γ21 = 1509,Π1 + ∆1Π2 + Π2Γ22 = −46,∆1∆0 + Π2Λ20 = 131,∆0 + ∆1∆1 + Π2Λ21 = 51.

As ∆1 = −8.4 and ∆0 = −16.6, it can be read

Λ0 = 131 + 16.6Λ2 − 0.71Γ(3),

Λ1 = 51 + 8.4Λ2 − 0.29Γ(3),

Fig. 1. Simulation results for the implementation of the reduced order observersfor (17).

and the polynomial

pF (λ) = λ3− 51λ − 131 − Λ2(λ

2+ 8.4λ + 16.6)

+ Γ(3)(0.29λ + 0.71).

In order to compare it with (Trinh et al., 2006) where the rootsof pF (λ) are {−3, −4, −5} we get Λ2 = −12, Λ1 = −47 andΛ0 = −60. These equalities are consistent and yield Γ(3) =

−11. Despite this result it is not obvious to give three poles atthe outset such that the constraints are satisfied. In order tocompare with (Trinh et al., 2006) we can fix Λ2 = −12. Weare then led to

pF (λ) = λ3+ 12λ2

+ 50.14λ + 67.86 + Γ(3)(0.29λ + 0.71).

The use of the root locus method yields to the poles{−3.3, −3.3, −5.4} for Γ(3) = −12.56. For these values weget the Luenberger observer defined by

G =

83.8 319.8 774.4 1381.3 −49.974.1 140 368 671.6 −5.113.6 14 42 79 2

B,

=

−49.92−5.12

2

, F =

0 0 −58.881 0 −46.550 1 −12

,

H =

−4036.1−1844.8−198.9

, P =

0 0 1

, V = −12.56.

In Fig. 1 the simulation results are displayed showing theperformances of the second-order and the third-order designedobservers.

4. Conclusion

We have proposed a new and direct design procedure of aminimal Luenberger observer for a single linear functional. Ouralgorithm is based on linear algebraic operations in a state spacesetting. With respect to other procedures the design proceduredoes not require the solution of a Sylvester equation. Moreover,the proposed solution exhibits design parameters for the candidateobserver to achieve asymptotic stability or pole placement whensome poles are fixed at the outset. Let us mention that we donot suppose any canonical form either for the system or for theobserver. The proposed constructive procedure is simpler thanthe Trinh procedure (Trinh et al., 2006) which is based on theresolution of the Sylvester equation with the Duan method.

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F. Rotella, I. Zambettakis / Automatica 47 (2011) 164–169 169

The proposed design principle can be extended twofold. On theone hand to time-varying linear systems. On the other hand for themultifunctional case. Such developments are under investigationand will be the subjects of future works. Nevertheless it can beenshown that in the multifunctional case the existence condition ofa minimum observer can be given by an obvious extension of theTheorem 1.

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Frédéric Rotella received his Ph.D. in 1983 and hisDoctorat d’Etat in 1987 from the University of Science andTechnology of Lille-Flandres-Artois, France. During thisperiod, he served at the Ecole Centrale de Lille, France,as assistant professor in automatic control. In 1994, hejoined the Ecole Nationale d’Ingénieurs de Tarbes, France,as professor of automatic control where he is in chargeof the Department of Electrical Engineering. His personalfields of interest are about control of nonlinear systemsand linear time-varying systems. Professor Rotella hascoauthored seven French textbooks in automatic control

(e.g. Automatique élémentaire, Hermés-Lavoisier, 2008) or in applied mathematics(e.g. Théorie et Pratique du Calcul Matriciel, Technip, 1995).

Irène Zambettakis received her Ph.D. in 1983 and herDoctorat d’Etat in 1987 from the University of Science andTechnology of Lille-Flandres-Artois, France. During thisperiod, she served at the Ecole Centrale de Lille, France,first as lecturer and then as assistant professor from1988 to 1994, in automatic control, image processing andmathematics. In 1994, she joined the Institut Universitairede Technologie de Tarbes (University Paul Sabatier,Toulouse, France) where she became professor in 2000in automatic control and signal processing. Her researchinterests include distributed parameter systems and linear

and nonlinear systems control. Professor Zambettakis has coauthored six Frenchtextbooks in automatic control. The last one, coauthored with Professor Rotella, isentitled Automatique élémentaire (Hermés-Lavoisier, 2008).