Robust filtering for descriptor large scale systems via LMI

Robust filtering for descriptor large scale systems via LMI M. Mansouri ∗ L. Boutat-Baddas ∗∗ M. Darouach ∗∗ H. Messaoud ∗ ∗

Ecole Nationale d’Ing´enieurs de Monastir, Avenue Ibn El Jazzar, 5019 Monastir, Tunisie (e-mail: [email protected], [email protected]). ∗∗ CRAN Nancy Universit´e, IUT de Longwy, 186 Rue de Lorraine, 54400 Cosnes et Romain, France (e-mail: [email protected], [email protected]) Abstract: In this paper, we consider the problem of unbiased optimal H∞ filtering for linear descriptor large scale systems. Necessary and sufficient conditions for the existence of such filters are provided in terms of linear matrix inequalities (LMIs). A numerical example is presented to demonstrate the effectiveness of the developed method. 1. INTRODUCTION There were considerable interests in the research of largescale systems in past years [Ald-1999], [Bas-1992]. A large-scale dynamical system can be usually characterized by a large number of variables representing the system, a strong and a complex interaction between subsystem variables. Decomposition techniques allow certain type of problems to be solved in a decentralized form. However, most of studies developed about decentralized observer based control assume that interconnexions between subsystems are available [Gav and Sil-1989], [Gon and al-1996] which is not usually the case. In this paper we propose a new approach based on a decentralized H∞ filter which is calculated despite of the unavailability of the interconnexions. The paper extend recent studies developed in [Dar and Zas-2009] to decentralized case. this approach presented in [Dar and Zas-2009] is based on the recently developed bounded real lemma developed in [Zhang and al-2008] and gives only strict (LMIs) which have the advantage to have a reliable computation. The paper is organized as follows, the problem is stated in the second section, the third section illustrates the decentralized H∞ filter and provides necessary and sufficient conditions of the existence of such filter in term of a strict (LMI). An example is given in section 4 to illustrates our approach and section 5 conclude the paper. 2. PROBLEM STATEMENT Let us Consider a linear large scale interconnected system composed of N subsystems described by the following discrete time system: EX(k + 1) = AX(k) + Bu(k) + Dw(k)

output respectively. w (k) ∈
zN (k)

Matrices E, A, B ,H, D and R are constant real valued with appropriate dimension partitioned according to the partition of X(k) asfollows:     E= 

E1 0 .. . 0



B1  B2 B= .  .. BN



R1  R2 R= .  .. RN

0 ... 0 E2 . . . 0 .. .. ... . . . . . . . . EN





  , A =   

D1   D2 , D =  .   .. DN



A11 A12 . . . A1N A21 A22 . . . A2N .. .. .. . . ... . A N 1 A N 2 . . . AN N



H1   0 , H =  .   .. 0

0 ... 0 H2 . . . 0 .. .. ... . . . . . . . . HN

 , 

  , 

  . 

Remark 1. The above matrices partition is not a constraint because we can always find this form by similarity transformations (see [Bas and Dar-1992], [Daly-1976], [Fou and al-1987] and [Pet and al-1991] for more details). Now, the large scale system ((1) − (2)) can be rewritten as an N interconnected sub-systems in the following form:

(1)

Ei xi (k + 1) = Aii xi (k) + Bi u(k)

Z(k) = HX(k) + Rw(k) (2) where vectors X(k) ∈
+ Fi vi (k) + Di w(k) zi (k) = Hi xi (k) + Ri w(k), i ∈ {1 . . . N }

(3) (4)

£ ¤ i) where: Fi = Ai1 . . . Ai(i−1) Ai(i+1) . . . AiN and vi (k) = n ×n i i [ x1 (k) . . . xi−1 (k) xi+1 (k) . . . xN (k) ], Aii ∈ < , Aij ∈ ii)
(5)

x ˆi (k) = ξi (k) + (Ji2 − Ki Ji4 )zi (k) (6) where x ˆi is the estimate of xi , ξi is the state vector of the filter. System ((5) − (6)) is a filter for system ((3) − (4)) if lim xi (k) − x bi (k) = 0, when w (k) = 0 and when k→∞

w (k) = 0 and ke (k)k2 < γ kw (k)k2 . The problem of the decentralized filter design is to determine matrices Ni , Mi , Γi and Ki such that the estimation error lim xi (k) − x bi (k) = 0. k→∞

Now, under Assumption 1 there exists a non singular · matrix Ji partitioned as : Ji =

Ji1 Ji2 Ji3 Ji4

¸ such as

Ji1 Ei + Ji2 Hi = Ini

(7)

Ji3 E + Ji4 Hi = 0 (8) Then under Assumption 1 and by using the definition of matrix Ji the estimation error ei (k) = xi (k) − x bi (k) is: ei (k) = (Ji1 − Ki Ji3 )Ei xi (k) − ξi (k) − (Ji2 − Ki Ji4 )Ri w(k) and its dynamic is given by:

(9)

ei (k + 1) = Ni ei (k) + (Ψi Aii − Ni Ψi Ei − Γi Hi )xi (k) + (Ψi Bi − Mi )u(k) + Ψi Fi vi (k) + (Ψi Di + Ni (Ji2 − Ki Ji4 )Ri )w(k) − Γi Ri w(k) − (Ji2 − Ki Ji4 )Ri w(k + 1) (10) where Ψi = (Ji1 − Ki Ji3 ). The observer design is reduced to finding matrices Ni , Mi , Γi and Ki such as the filter asymptotically stable when w (k) = 0. From equation 10 bellow, we can deduce the following proposition which gives the condition for the existence and stability of the observer given by (5)-(6). Proposition 1. System ((3)−(4)) is an asymptotic observer i.e: lim x(k) − x b(k) = 0, when w (k) = 0 if : k→∞

Ni is a stability matrix. The generalized Sylvester equation holds: Ψi Aii − Ni Ψi Ei − Γi Hi = 0, under constraint Ψi Fi = 0. Ψi Bi − Mi = 0

The condition iii) is equivalent to following compact form: Yi Σi = Ωi " where: Yi = [ Ni Ki Li ], Σi =

I 0 Ji3 Aii Ji3 Fi Hi 0

(11)

# ,

Ωi = [ Ji1 Aii Ji1 Fi ] and Li = −(Ni (Ji2 − Ki Ji4 ) + Γi ), Ψi Bi = Mi . The necessary and sufficient conditions for the existence of the solution to (11) is given by the following lemma: Lemma 1. Under Assumption 1, there exists a solution to (11) if and only if · ¸ Ei Fi rank = ni + rankFi (12) Hi 0 Proof. First, the necessary and sufficient conditions for the existence of the solution to (11) are · ¸ Σ rank i = ni + rankDi (13) Ωi or equivalently

Ωi Σ+ i Σi = Ωi From Assumption 2 and the definition of matrix Ji we have ¸ · ¸· ¸ · J J Ei Di Ei Di = rank i1 i2 rank Ji3 Ji4 Hi 0 Hi 0 · ¸ I Ji1 Di = rank = ni + rankJi3 Di 0 Ji3 Di Then, condition (11) is equivalent to rank(Di ) = rank(Ji3 Di ) On the other hand (11) can be written as   " # I 0 I 0 Ji3 Aii Ji3 Di  = rank Ji3 Aii Ji3 Di rank  Hi 0  Hi 0 Ji1 Aii Ji1 Di which is equivalent to · ¸ · ¸· ¸ Ji1 Di Ji1 Ji2 Di rank(Ji3 Di ) = rank = rank Ji3 Di Ji3 Ji4 0 = rank(Di ) which is (11), this proves the Lemma. Under Assumption 1 and condition (12) the general solution to (11) is given by: + [ Ni Ki Li ] = Ωi Σ+ i − Zi (I − Σi Σi ) where Zi are arbitrary matrices of appropriate dimensions, specifying all filter matrices and that will be determined in the sequel using LMI. Σ+ is any generalized inverse satisfying ΣΣ+ Σ = Σ. Then matrices Ni , Ki , Li , Γi and Mi are given by:

Ni = Ai1 − Zi Bi1

(14)

Ki = Ai2 − Zi Bi2

(15)

Li = Ai3 − Zi Bi3

(16)

Γi = Li + (Ni (Ji2 − Ki Ji4 )

(17)

Mi = (Ji1 − Ki Ji3 )Bi

(18)

where: Ai1 = Ωi Σ+ i Bi1 = (I −

" # I 0 0

Σi Σ+ i )

Ai2 = Ωi Σ+ i

" # 0 I 0

(19) " # I 0 0

(20)

(21)

" # 0 + Bi2 = (I − Σi Σi ) I 0 " # 0 0 Ai3 = Ωi Σ+ i I " # 0 + Bi3 = (I − Σi Σi ) 0 I

(22)

(34)

(23)

Zi = Z¯i (I − βi2 βi+2 ) In this case we have:

(35)

(24)

Ni = Ai1 − Z¯i B¯i1 Ui = αi − Z¯i β¯i Vi = αi2

(37)

Vi = αi2 − Zi βi2

(25)

(26) "

#

0 Ji3 Di αi1 = Ji1 Di − Ωi Σ+ i Ri " # 0 + βi1 = (I − Σi Σi ) Ji3 Di Ri " # 0 + αi2 = −Ji2 Ri + Ωi Σi Ji4 Ri 0 " # 0 βi2 = (I − Σi Σ+ i ) Ji4 Ri 0

(27) (28)

(29)

(30)

(31)

In general, there exist many methods to represent the equation (25)[Dar and al-2001],[Bas-1992]... Here we propose another method which consists in rewriting the error system into a descriptor form from which we get the gain matrix Zi parameterizing the filter matrices. Hence, equation (25) is equivalent to: Ei χi (k + 1) = Ai χi (k) + Bi w(k) ei (k) = Ci χi (k)

1

where B¯i1 = (I −

where Ui = αi1 − Zi βi1

System (32) − (33) described by (Ei , Ai , Bi , Ci ) is restrictly equivalent to a singular system described by (E¯i , A¯i , B¯i , C¯i ) such as [Dai-1989]: · ¸ · ¸ I 0 Ni 0 E¯i = Θi Ei Λi = , A¯i = Θi Ai Λi = , 0 0 0 I · ¸ Ni Vi + Ui B¯i = Θi Bi = , C¯i = Ci Λi = [I Vi ], where Θi −I and Λ·i are nonsingular ¸ matrices· defined ¸ as following: I Ni Vi + Ui I Vi Θi = and Λi = . 0 −I 0 I However, matrix B¯i contains nonlinearity due to the term Ni Vi , to alleviate this nonlinearity let define the matrix parameter Zi as :

The dynamic of the estimation error (10) becomes: ei (k + 1) = Ni ei (k) + Ui w(k) + Vi w(k + 1)

· ¸ · ¸ · ¸ I −Vi ei (k) Ni Ui where Ei = , χi (k) = , Ai = , 0 0 w(k) 0 −I · ¸ 0 Bi = and Ci = [I 0]. I

(32) (33)

βi2 βi+2 )Bi1 ,

1

(36)

β¯i1 = (I − βi2 βi+2 )βi1 .

Then matrices A¯i , B¯i and C¯i become: · ¸ · ¸ I ¯ ¯ I ¯ ¯ Zi Bei2 and C¯i = Zi Aei2 , B¯i = B¯ei1 − A¯i = A¯ei1 − 0 0 ¸ · ¸ · Ai1 αi2 + αi1 Ai1 0 , B¯ei1 = , [ I αi2 ] where A¯ei1 = 0 I −I £ ¤ A¯ei2 = B¯i1 0 , and B¯ei2 = B¯i1 αi2 + βi1 . Now we are going to give solution to H∞ filter for system (3) − (4), but before, let us recall the following lemma: Lemma 2. [Zhang and al-2008] Consider the descriptor system given by (32) − (33). The pair (Ei , Ai ) is admissible and the transfer matrix Twei (z) = Ci (zEi − Ai )−1 Bi is H∞ norm-bounded by a positive real number γi (i.e. kTwei k∞ < γi , if and only if there exist a positive definite matrix Pi and a symmetric matrix Si such that · ¸ Φi11 Φi12 <0 (38) ΦTi12 Φi22 where Φi11 = ATi (Pi − Ei⊥T Si Ei⊥ )Ai − EiT Pi Ei + CiT Ci (39) Φi12 = ATi (Pi − Ei⊥T Si Ei⊥ )Bi = −γi2 I

BiT (Pi − ⊥

(40)

Ei⊥T Si Ei⊥ )Bi

Φi22 + (41) and where the matrix X denotes a matrix such that X ⊥ X ⊥T > 0 and X ⊥ X = 0. Now, the solution to the H∞ optimal filtering problem for system (3-4) is given by the following theorem.

Theorem 1. Under assumption 1, there exists a parameter matrix Z¯i such as the filter (5)-(6) is unbiased and solves the γi -suboptimal H∞ filtering problem for the descriptor system (3)-(4), if and only if there exists a positive definite matrix Pi , a symmetric matrix Si and a matrix Ti such that the following strict linear matrix inequality holds:  ⊥T ⊥ T T ⊥T E¯i Si E¯i − E¯i Pi E¯i + C¯i C¯i E¯i Si TaTi ⊥  Si E¯i −Si − γi2 I TbTi  < 0 Tai Tbi −Pi (42) ¯ ¯ ¯ ¯ where Ta·i = ¸ Pi Aei1 − Ti Aei·2 , Tbi ¸= Pi Bei1 − Ti Bei2 , £ ¤ I ¯ Ai1 0 Ti = Pi Zi and A¯ei1 = , A¯ei2 = B¯i1 0 , 0 0 I · ¸ Ai1 αi2 + αi1 B¯ei1 = and B¯ei2 = B¯i1 αi2 + βi1 −I 

In the above inequality, all matrices are constant, so using the LMI Control Toolbox, parameters Ti is determined and Z¯i and consequently Zi are deduced. Proof. From system (32-33) and using lemma 1 and the Schur lemma [Boy and al-1994], kTwei k∞ < γi if · and only ¸ Pi1 Pi2 if there exist a positive definite matrix Pi = PiT2 Pi3 and a symmetric matrix Si such that: · ¸ Φi11 Φi12 Φi = <0 (43) ΦTi12 Φi22 where

12

T ⊥T ⊥ Φi22 = −γi2 I + B¯i (Pi − E¯i Si E¯i )B¯i

(46)

⊥ ⊥ ⊥ Now, let E¯i = [ 0 I ], we have E¯i = [ 0 I ] = E¯i A¯i and ⊥ E¯i B¯i = [ 0 I ] = −I, so, using Schur lemma, inequality (43) can be written:



 ⊥T ⊥ T T ⊥T T E¯i Si E¯i − E¯i Pi E¯i + C¯i C¯i E¯i Si A¯i Pi ⊥ T  Si E¯i −Si − γi2 I B¯i Pi  < 0 Pi A¯i Pi B¯i −Pi · ¸ I ¯ Let Ti = Pi Zi , using the expression of A¯i and B¯i , we 0 have the following LMI: ⊥T ⊥ E¯i Si E¯i

T − E¯i Pi E¯i ⊥ Si E¯i

T C¯i C¯i

⊥T E¯i Si −Si − γi2 I Tbi



TaTi TbTi  < 0 Tai −Pi ¯ ¯ ¯ ¯ where Tai = ·Pi A ¸ ei1 − Ti Aei2 , Tbi = P·i B¸ei1 − Ti Bei2 and I ¯ ¯ I ¯ ¯ A¯i = A¯ei1 − Zi Aei2 , B¯i = B¯ei1 − Zi Bei2 0 0 

i) Step 1: Compute Σi and Ωi . ii) Step 2: Derive all matrices needed for the LMI resolution Ai1 , Bi1 , αi1 , αi2 , βi1 , βi2 respectively from: (19) , (20), (28), (29), (30), (31). iii) Step 3: Solve the LMI of theorem 1, to obtain the gain Z¯i which leeds to Zi using (34). iv) Step 4: Compute the filter matrices Ni , Ki and Mi respectively from (14), (15) and (18). v) Step 5: Finally, the filter given by (5 − 6) is completely determined by computing matrix Li , and consequently Γi respectively from (16) and (17). 4. NUMERICAL EXAMPLE Consider the following interconnected descriptor system composed of three subsystems with matrices E, A, B and H given by: 

1 0 0  0  E =0 0  0 0 0



T ⊥T ⊥ T T Φi11 = A¯i (Pi − E¯i Si E¯i )A¯i − E¯i Pi E¯i + C¯i C¯i (44) T ⊥T ⊥ Φi = A¯i (Pi − E¯i Si E¯i )B¯i (45)



The different steps of the decentralized H∞ filter design are summarized in what follow:

+

We obtain a strict LMI where all matrices are constant, so, using the LMI Control Toolbox of Matlab, parameter Ti is given, the admissible solution Z¯i = [ I 0 ] Pi−1 Ti and the desired parameter matrix Zi are deduced and therefore the filter is completely determined.

1 0 1  0  B =0 0  0 0 0

0 0 0 0 1 0 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 1 0 0

0 0 0 0 0 0 0 1 0





0 1 1 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 1 0 0 0 0 0

0 0 0 1 0 0 0 0 0

0 0 0 0 1 0 0 0 0

0 1 1 0 1 1 0 0 0     1 0 0     0 ,D =  1 ,H =  0    0 0 0  1 0 1  1 0 0 1 0 0



0 1 0  0  0 0    0 0   0 ,A =  1   0 0   −1 0   1 0 0 0

 



−1 −2 1 0 0 2 0 0 0

0 0 −1 0 1 0 0 0 1

1 2 0 2 0 0 0 1 0

0 0 1 0 1 1 −1 0 2

1 0 1 0 0 0 0 0 0

0 0 0 1 0 0 0 0 0

0 0 0 1 0 1 0 0 0

0 0 0 0 0 0 1 0 0

0 1 0 0 0 0 0 0 0

0 0 0 0 1 1 0 0 0

0 0 0 0 0 0 0 1 0

0 1 0 1 0 −1 0 0 0

0 0 1 1 0 0 1 −1 0



0 1 0 0 −2 0 0 −2 1

 

0 1 0 1 1 0    1 0    0 ,R =  0   1 0   1 0  1 0 1 0

System decentralization: According to the decomposition considered in section 1 we have : subsystem 1 : " E1 =

" A13 =

1 0 0 0 1 0 0 0 0 0 0 2 0 1 1 1 0 1

#

" , A11 =

#

" , B1 =

" # D1 =

1 1 0

" #

, R1 =

1 −1 0 0 −2 0 0 1 1

1 0 0 0 0 0 0 0 1 0 0 0

# , A12 =

#

" J13 =

" , H1 =

1 0 0 2 0 1 0 1 0

1 1 0 1 0 1 0 1 0

1 1 1

For this sub-system we have " # J11 =

"

2 1 1 0 0 1 −1 −1 1

1 1 1 0 −1 1 0 −1 0

"

, J12 =

#

" , J14 =

−1 0 0 0 0 1 0 1 1

−1 0 0 0 0 1 0 0 1

# ,

# .



2 1  1   0   0  1   0   0 −2

# ,

# ,

In this case from the results of section 3 we obtain " # N1 = 10−15

" −15

M1 = 10

" Γ1 = 10−14

0.00004 0.84 0.84 −0.000065 −0.35 −0.35 −0.000183 −0.97 −0.97

" A23 =

1 0 0 0 1 0 0 0 0

# , A22 =

1 0 0 0 −2 0 0 0 1

" J21 =

" J23 =

1 1 0

#

" " #

"

N2 = 10−15

" M2 = 10−15

" Γ2 = 10−15

#

" A32 =

0 −1 0 1 0 0 0 2 0

D3 =

" J31 =

1 1 0

1 1 0

0 −0.5 −1

0.2 0 −0.2

" , A21 =

−1 0 1 0 0 0 0 1 0

#

" , H2 =

0.5 0 −0.5

#

0 0 0 1 0 1 0 2 0

1 0 −1

,

1 0 1 0 1 0 0 1 1

#

5 0 −5

,

#

0.5 0 −0.5

1 0 0 −1 −2 0 0 1 −2 0 0 0 1 0 0 0 0 0 0 0 1

0 −1 −2

,

e11=x11−x11obs 0

10

20

30

40

50

60

70

80

90

100

0

10

20

30

40

50

60

70

e12=x12−x12obs 80 90 100

0

10

20

30

40

50

60

70

e13=x13−x13obs 80 90 100

0

10

20

30

40

50

60

70

e21=x21−x21obs 80 90 100

0

10

20

30

40

50

60

70

e22=x22 −x22obs 80 90 100

0

10

20

30

40

50

60

70

e23=x23−x23obs 80 90 100

0

10

20

30

40

50

60

70

e31=x31−x31obs 80 90 100

0

10

20

30

40

50

60

70

e32=x32−x32obs 80 90 100

0

10

20

30

40

50

60

70

80

e33=x33−x33obs 90

100

Fig. 1. The estimation error between decentralized system and decentralized H∞ filter

# .

1 0 −1

#

0.1 0 −0.1

,

#

"

, K2 =

0.08 0.15 −0.13 −0.21 −0.20 0.19 0.17 0.22 −0.10

" #

.

0.1 0.05 0

.

0 1 0 1 0 −1 0 1 1

0 −0.44 0.44 0 0 0 0.22 0 0 0.28 −0.66 0

"

#

#

0.0006 0.003 0.06 −0.00033 −0.046 −0.39 −0.00109 −0.036 −0.74

, B3 =

1 −1 0 −1 1 0 0 −1 1

"

, J24 =

#

#

0.5 0 −0.5

"

, A33 =

, R3 =

1.999 −0.999 2 0 1 −1 −0.999 1.999 −2

.

#

"

" #

,

#

0 1 0 0 0 0 1 0 0 0 0 0

, J22 =

subsystem 3 " # E3 =

1 0 1

, R2 =

1 −1 0 0 0 1 0 −1 0

2 0 1 0 1 0 0 1 −1

, B2 =

1 −1 0 −1 2 1 0 −2 0

1 0 0 0 1 0 0 0 0

, K1 =

"

" # D2 =

"

−0.058 0.103 0.0055 −0.023 0.0041 0.1151 −0.0083 −0.029 0.169

subsystem 2 : " E2 =

#

−0.66 0 0 0 −0.59 0 0 0 −0.66 0 0 0

observed state xikobs by decentralized H∞ filter. Figure 2 shows the estimation error ei between global system state and the observed state using decentralized H∞ filter. It can be seen that the presented method permits the convergence of the decentralized filter to the global system.

#

1 0 0 −1 1 −1 0 0 2,

# ,

2 0 −2

#

" , A31 =

" , H3 =

0 −0.2 −0.4 1 0.5 0

.

#

0.2 0 −0.2

−1 0 0 1 0 0 0 0 1

1 0 0 0 1 0 0 0 1

#

,

#

0.5 0 −0.5

,

0.5 0 −0.5 2 0 −2

e1=x1−x11obs 0

10

20

30

40

50

60

70

80

90

100

0

10

20

30

40

50

60

70

e2=x2−x12obs 80 90 100

0

10

20

30

40

50

60

70

80

e3=x3−x13obs 90 100

0

10

20

30

40

50

60

70

80

e4=x4−x21obs 90 100

0

10

20

30

40

50

60

70

80

e5=x5 −x22obs 90 100

0

10

20

30

40

50

60

70

80

e6=x6−x23obs 90 100

0

10

20

30

40

50

60

70

80

e7=x7−x31obs 90 100

0

10

20

30

40

50

60

70

80

e8=x8−x32obs 90 100

0

10

20

30

40

50

60

70

80

e9=x9−x33obs 90

100

Fig. 2. The estimation error between global system and decentralized H∞ filter

.

"

, J32 =

0 1 0 1 0 0 0 1 1

#

5. CONCLUSION ,

A new decentralized robust filter for linear large scale descriptor systems is proposed in this paper. The robustness of the present filter is very interesting to be exploited in J33 = , J34 = . practical application because it does’nt need any knowl" # " # edge of the statistical properties of the noise. The filter 0.32 0.32 −0.0004 0 0 0 −0.133 N3 = 10−5 −0.32 −0.32 0.0044 , M3 = 10−5 0 0 0 0.13 , design procedure is composed by two capital steps: first, the resolution of decentralized Sylvester equation under −0.75 −0.75 0.0006 0 0 0 −0.48 " # " # constraint using the generalized inverse, second, the ex−0.12 −0.16 0.0063 1 0 1 traction of a singular system, to get the optimal matrix 0.12 0.16 −0.0063 , K3 = −1 0 −1 . Γ3 = 10−5 gain using LMIs and to parameterize the matrices filter. −0.113 −0.145 −0.021 0 1 1 Further works will concern the observer-based control of Figure 1 shows the estimation error eik between k-th large scale descriptor system. component xik of the state of the i-th subsystem and the "

1 0 0 0 0 1 0 −1 0

#

"

−1 0 0 0 0 0 0 1 0

#

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