AN OBSERVER–BASED SCHEME FOR DECENTRALIZEDSTABILIZATION OF LARGE-SCALE SYSTEMS WITH APPLICATION TO
POWER SYSTEMS
Diego Langarica Córdoba and Romeo Ortega
ABSTRACT
An observer-based methodology for decentralized stabilization of large-scale linear time-invariant systems is presented.Each local controller is provided with available local measurements, it implements a deterministic observer to reconstruct the stateof the other subsystems and uses—in a certainty-equivalent way—these estimates in the control law. The observers are designedfollowing the principles of immersion and invariance. The class of systems to which the design is applicable is identified via alinear matrix inequality, from which the observer gains are obtained. It is shown that the use of immersion and invarianceobservers, instead of standard Luenberger’s observers, enlarges the class of stabilizable systems. The applicability of the proposedmethod is illustrated with a transient stabilization controller for a two-machine power system.
Key Words: Decentralized control, observers, linear matrix inequality, power systems.
I. INTRODUCTION
Large-scale interconnected dynamical systems havebeen studied for many years and continue to be a field withstrong practical significance. The increasing use of systemssuch as multi-agents in robotics [1–3], unmanned aerial andautonomous underwater vehicles [4,5] and widely geographi-cally distributed power systems [6], raises the question in howto coordinate their operation under constrained structuralcharacteristics, including non-classical information availabil-ity, large dimensionality and uncertainty.
In the early 70s, Wang and Davison [7] started formalresearch on decentralized control. They proved that a systemcan be stabilized with a decentralized controller (static ordynamic) if and only if it does not have unstable decentralizedfixed modes, see also [8]. The typical scenario of decentral-ized stabilization of linear time–invariant (LTI) systemsassumes that the input matrix is full and aims at finding localoutput feedbacks such the closed-loop is stable. Surveys andmore recent developments on this problem may be found in[9–15].
Motivated by some current applications, in particularthe transient stabilization problem of multimachine power
systems [16,17], we consider a different scenario. Namely,that the input matrix is block diagonal, the local states aremeasurable and we dispose of a full-state feedback controllerthat stabilizes the system. The approach adopted in the paperto solve this decentralized stabilization problem is to imple-ment local deterministic observers to reconstruct the state ofthe other subsystems and use—in a certainty-equivalentway—these estimates in the control law.
The observers are designed using the immersion andinvariance technique (I&I) proposed in [18]. As is well-known, the structure of I&I estimators differs considerablyfrom the structure of the classical Luenberger’s observer[20]. The current decentralized scenario yields some decen-tralized controller designs that—to the best of our knowl-edge—have not been reported in the literature. Moreover,the class of systems to which the design is applicablestrictly contains the class that uses standard (reduced or fullorder) Luenberger’s observers. The class is identified via alinear matrix inequality (LMI) that, if feasible, yields theobserver gains.
The remainder of this work is organized as follows.In Section II the decentralized stabilization problem is for-mulated and the main result is given. For ease of presenta-tion, the proof of the main result is given first for a systemcomposed of two subsystems in Section III. Its generaliza-tion to N > 2 subsystems is discussed later. The proposedmethod is applied to design a transient stabilization control-ler for a two-machine power system in Section IV, wheresome simulation results are also given. Finally, Section Vprovides some final conclusions and directions for futurework.
Manuscript received May 24, 2013; revised November 4, 2013; accepted January19, 2014.
The authors are with the Laboratoire des Signaux et Systémes, Supelec, Plateau duMoulon, 91192 Gif-sur-Yvette, France.
Diego Langarica Córdoba is the corresponding author (e-mail:diego.langarica(ortega)@lss.supelec.fr)
This work was financially supported by CONACYT (México).
Asian Journal of Control, Vol. 17, No. 1, pp. 1–9, January 2015Published online in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/asjc.872
with Bi ∈ Rn×m. To simplify the notation, and without loss ofgenerality, we have assumed that the dimensions of all thesubsystems states and inputs are the same.
For future reference we partition the system (1) into Ninterconnected subsystems
Σ i i ii i i i ij j
j j i
N
x A x B u A x: ,,
� = + += ≠∑1
(2)
with i N N∈ = …: { , , , }1 2 . We assume that each subsystem Σi
has available for measurement the corresponding state xi. (*)
Assumption II.1. The pair (A, B) is controllable and weknow a matrix F ∈ RmN×nN such that A + BF is a Hurwitzmatrix.
Problem Formulation. Consider the system (2) satisfyingAssumption II.1. Design, if possible, an observer-baseddecentralized certainty-equivalent controller of the form
�ξ ξ ξξ
j ij i j N i
j ij i j i
i ii i
i i i
i i
W x u
x R x u
u F x
Σ Σ Σ
Σ Σ
= …=
= +
( , , , , )
( , , )ˆ
FF x i j N j iij j
j j i
N
iˆ
,
Σ= ≠∑ ∈ ≠1
, , , ,
(3)
where Wij: Rn × Rn × . . . × Rn × Rm → Rn and Rij:Rn × Rn × Rm → Rn such that all trajectories are bounded and
tt
→∞=lim ( ) ,x 0
for all initial conditions ( ( ), ( ))xi jn n
i0 0ξ Σ ∈ ×R R . (†)
Remark. The proposed scheme is decentralized in the sensethat there is no (online) information exchange between sub-systems. Notice, however, that the matrix F is assumed to beknown and is designed from knowledge of the matrices Aand B.
The proposition below contains the main result of thepaper. Namely, the characterization—via an LMI—of a classof systems for which the aforementioned problem has a solu-tion.
Proposition II.1. Define the matrices
ΛΩ∈∈
− × −
− × −
RR
nN N nN N
nN N nN N
( ) ( )
( ) ( )
1 1
1 1(4)
which are determined by elements of A, B and F. For instance,in the case of two interconnected subsystems (N = 2), they aredefined as
Λ
Ω
=−
−⎡⎣⎢
⎤⎦⎥
= ⎡⎣⎢
⎤⎦⎥
A B F
B F A
A
A
22 2 21
1 12 11
12
21
0
0
(5)
where A A B Fii ii i ii:= + .The decentralized stabilization problem stated above
has a solution if there exists block diagonal matricesP ∈ RnN(N−1)×nN(N−1), U ∈ RnN(N−1)×nN(N−1) solutions of the LMI
P
P P U U
>+ − − <
0
0Λ Λ Ω ΩΤ Τ Τ .(6)
Furthermore, the mappings Wij, Rij are linear, that is theobserver takes the form
�ξ ξξj ijx
i jk k
k k i
N
iju
ii iT x T T uΣ Σ= + += ≠∑1,
ˆ , , , ,x H x H H u i j N j ij ijx
i ij j iju
ii iΣ Σ= + + ∈ ≠ξξ
* Unless indicated otherwise the subindex (·)i ranges in the set N . When clearfrom the context, this clarification is omitted.
† The signal x j iΣ is the estimate of the state xj generated with measurementsof the subsystem Σi, whose state is the (N − 1)n-dimensional vector
( , , )ξ ξ1ΣΤ
ΣΤ Τ
i iN… excluding ξi iΣΤ .
2 Asian Journal of Control, Vol. 17, No. 1, pp. 1–9, January 2015
For simplicity, we present first the proof for the case oftwo subsystems. The proof for the general case is given inSubsection 3.2.
3.1 Two-subsystem case
The two-interconnected subsystems take the form
ΣΣ
1 1 11 1 1 1 12 2
2 2 22 2 2 2 21 1
:
: .
��x A x B u A x
x A x B u A x
= + += + +
(7)
The certainty–equivalent decentralized control law is
u F x F x
u F x F x1 11 1 12 2
2 21 1 22 2
1
2
= += +
ˆ ,
ˆ ,Σ
Σ(8)
where the state feedback gain matrix of Assumption II.1 ispartitioned as
FF F
F F= ⎡⎣⎢
⎤⎦⎥
11 12
21 22
,
x2 1Σ and x1 2Σ represent the estimation of x2 generated by Σ1
and the estimation of x1 generated by Σ2, respectively. Theclosed-loop system (7)–(8) takes the form
��x A x A x B F x
x A x A x B F x1 11 1 12 2 1 12 2
2 22 2 21 1 2 21 1
1
2
= + += + +
ˆ
ˆ ,Σ
Σ
(9)
where we have defined A A B Fii ii i ii:= + .The main thrust of the proof is the construction of a
decentralized I&I observer for each subsystem Σi—that, werecall, disposes of its corresponding state xi. To this end, wefollow the methodology of the I&I technique in [18] anddefine the observed states as the sum of an integral and aproportional term. In this way, the estimate of x2 generated bysubsystem Σ1 is defined as
ˆ ,x K x2 2 2 11 1 1Σ Σ Σ= +ξ
where ξ2 1Σ is the integral part and K n n2 1Σ ∈
×R is a matrix tobe defined. Define the first estimation error as
�x x x2 2 2: .1= −ˆ Σ
Taking its time derivative yields
�� �x K A x B u A x
A x A x B F x2 2 2 11 1 1 1 12 2
22 2 21 1 2 21 1
1 1
2
( )
( )
= + + +− + +ξ Σ Σ
Σˆ ..
Replacing the unmeasurable signal x2 above by
x K x x2 2 2 1 21 1= + −ξ Σ Σ � ,
and splitting into measurable and unmeasurable signalssuggests the selection
�ξ ξ2 2 12 22 2 2 1
2 21 2 11 21 1 2
1 1 1 1
1
Σ Σ Σ Σ
Σ
= − − ++ − + −
( )( )
( )
K A A K x
B F K A A x K ΣΣ1 1 1B u .(10)
This yields the first observer error equation dynamics
�� � �x A K A x B F x2 22 2 12 2 2 21 11= −( ) −Σ ,
where we defined the observation error of the secondsubsystem as
�x x x1 1 1: .2= −ˆ Σ
Proceeding analogously with the derivative of �x1 anddefining
�ξ ξ1 1 21 11 1 1 2
1 12 1 22 12 2 1
2 2 2 2
2
Σ Σ Σ Σ
Σ
= − − ++ − + −
( )( )
( )
K A A K x
B F K A A x K ΣΣ2 2 2B u ,(11)
we obtain the second observer error equation dynamics
�� � �x A K A x B F x1 11 1 21 1 1 12 22= −( ) −Σ .
The two error equations can be compactly written as
����
xx
x= ⎡
⎣⎢⎤⎦⎥
A 2
1
(12)
where
A : .=− −− −
⎡⎣⎢
⎤⎦⎥
A K A B F
B F A K A22 2 12 2 21
1 12 11 1 21
1
2
Σ
Σ
Now, the closed–loop system (9) can be expressed interms of the observation errors as
� �x x x= + + ⎡⎣⎢
⎤⎦⎥
( ) .A BFB F
B F1 12
2 21
0
0(13)
Given Assumption II.1 and the cascaded structure of(12) and (13) it is clear that the control objective is attained ifthe matrices K2 1Σ , K1 2Σ are selected such that the matrix Ais Hurwitz.
The matrix A can be written as
A = −Λ ΩK , (14)
where the matrices
3D. Langarica Córdoba and R. Ortega: Observer–Based Scheme for Decentralized Stabilization
To establish the proof we invoke a standard LMI argu-ment given in [19] that is detailed here for the sake of com-pleteness.
The matrix A is Hurwitz if and only if there exists ablock diagonal symmetric matrix P such that
P
P K K P
>− + − <
0
0( ) ( ) .Λ Ω Λ Ω Τ (16)
Defining the block diagonal matrix U := PK theinequalities (16) become the LMI
P
P P U U
>+ − − <
0
0Λ Λ Ω ΩΤ Τ Τ .(17)
that should be solved for P and U. The gain matrix K can berecovered using K = P−1U.
The proof is completed identifying from the observerequations above the matrices of the proposition as
T B F K A A K A K A Kx12 2 21 2 11 21 2 12 2 22 21 1 1 1= − + − +Σ Σ Σ Σ
T K A A12 2 12 221
ξ = − −( )Σ
T K Bu12 2 11= − Σ
H Kx12 2 1= Σ
H In n12ξ = ×
H un n12 0= ×
and
T B F K A A K A K A Kx21 1 12 1 22 12 1 21 1 11 12 2 2 2= − + − +Σ Σ Σ Σ
T K A A21 1 21 112
ξ = − −( )Σ
T K Bu21 1 22= − Σ
H Kx21 1 2= Σ
H In n21ξ = ×
H un n21 0= × .
In order to illustrate the ideas given by the proposedmethod above, the decentralized scheme diagram for twointerconnected subsystems is presented in Fig. 1.
Remark. Some simple calculations show that the standardreduced order Luenberger’s observer in [20]
ˆ ˆ�x A x A x B F x2 21 1 22 2 2 21 11 1Σ Σ= + +
ˆ ˆ ,�x A x A x B F x1 12 2 11 1 1 12 22 2Σ Σ= + +
yields the error dynamics
�� �x x=−
−⎡⎣⎢
⎤⎦⎥
A B F
B F A22 2 21
1 12 11
,
that coincides with (12) when K K2 11 2 0Σ Σ= = . Hence, the setof stabilizable systems using this observer is strictly smallerthan the one corresponding to the I&I observer. Similarderivations in [20–22] have been made to overcome the lackof the correction term in the Luenberger’s reduced orderobserver above. In addition, it can be shown that the errordynamics of a full order Luenberger’s observer is of the form
�� �χ χ=−
−
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
L A
A B F
B F A
A L
1 12
22 2 21
1 12 11
21 2
0 0
0 0
0 0
0 0
,
where �χ∈R4n is the observation error and Li ∈ Rn×n are freematrices. From the structure of the matrix above it can be seenthat a necessary condition for stability is that the matrix of thereduced order observer is Hurwitz. Hence, the use of a fullorder observer does not enlarge the class of stabilizablesystems.
Remark. In the derivations above, the full-state feedbackstabilizing matrix F was supposed fixed to obtain an LMI inthe observer gains K j iΣ . Posing the problem of findingboth F and K j iΣ , for the given data A and B, results in a
Fig. 1. Decentralized stabilization scheme.
4 Asian Journal of Control, Vol. 17, No. 1, pp. 1–9, January 2015
nonlinear matrix inequality. Indeed, using the standardreparametrization of the Lyapunov equation
Q Q A BF A BF Q> + + + <0 0, ( ) ( )Τ
given in [19]
S Q T FQ: , : ,= =− −1 1
shows that stability of A + BF is equivalent to solvability ofthe LMI
S > 0
AS BT SA T B+ + + <Τ Τ Τ 0.
On the other hand, the LMI (17) takes the form
P
P TS TS P U U
>+ + + − − <
0
00 1 0 1( ) ( ) ,Λ Λ Λ Λ Ω ΩΤ Τ Τ
where
Λ Λ Λ= +0 1TS,
with Λ and Ω given in (15). Taking together‡ the two sets ofinequalities yields a nonlinear equation in the unknowns S, T,P, U. Although numerical techniques are available to solvenonlinear matrix inequalities, for ease of presentation, wehave preferred to avoid this complexity.
3.2 N-subsystem case
We now present the proof for the case of N subsystems.Consider (2) connected in closed-loop with ui given in (3).The overall state feedback gain matrix is assembled as
F
F F F
F F F
F F F
N
N
N N NN
=
……
…
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
11 12 1
21 22 2
1 2
� � � �,
consequently, the closed-loop system takes the form
Σ Σi i ii i ij j i ij j
j j i
N
x A x A x B F x i: ( ).� = + += ≠∑ ˆ
,1
The state estimation of the jth subsystem provided tothe ith subsystem controller is
ˆ , , , ,x K x i j N j ij j j ii i iΣ Σ Σ= + ∈ ≠ξ (18)
where ξ j iΣ is the integral part and K xj iiΣ is the proportionalpart, K j
n niΣ ∈
×R . We define the estimation error with thegeneral form
�x x xj j ji iΣ Σ= −ˆ . (19)
Its time derivative results in
�� �x K A x B u A x
A x A
j j j ii i i i ik k
k k i
N
jj j jk
i i iΣ Σ Σ= + + +⎛⎝⎜
⎞⎠⎟
− +
= ≠∑ξ1,
xx B F xk j jk k
k k j
N
j+( )⎛⎝⎜
⎞⎠⎟= ≠
∑ ˆ ,,
Σ1
and the unmeasurable signal xj above is replaced by
x K x xj j j i ji i i= + −ξ Σ Σ Σ� .
Separating into measurable and unmeasurable signalssuggests the selection
�ξ j i j ii ji j ji i
j ik jk j jk k
k k
K A A B F x
K A A B F x
i
i i
Σ Σ
Σ Σ
= − − −
− − −=
( )
( )1
ˆ, ≠≠∑
− ∈ ≠i
N
j i iK B u i j k N j iiΣ , , , , ,
(20)
which renders the corresponding observer error dynamics
�� �
�
x K A A B F x
B F x
j j ik jk j jk k
k k i
N
j jk k
k k
i i i
j
Σ Σ Σ
Σ
= − − −
−
= ≠
= ≠
∑ ( ),
,
1
1 jj
N
i j k N j i∑ ∈ ≠, , , , .
(21)
Notice the auxiliary subscript k follows k ≠ i and k ≠ j inthe first and the second sum of (21), respectively. From theclosed-loop system (2)–(3) and the estimation errors (19), weobtain
� �x x x= + +( ) ,A BF I
where � � � �x x x x= …[ ] ∈ −ΣΤ
ΣΤ
ΣΤ Τ
1 21, , , ( )
nnN NR , is the overall
observation error vector formed by
�
���
�
�
���
�
x xΣ
Σ
Σ
Σ
Σ
Σ
Σ
Σ
1
2
3
1
3
1
1
1
2
2
2
2
=
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
=
⎡
⎣
⎢x
x
x
x
x
xN N
, ⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
,
�
���
�
xΣ
Σ
Σ
Σ
n
N
N
N
x
x
xN
n N=
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
∈
−
−
1
2
1
1R ( ).
‡ We have split Λ into two to underscore that, since it depends linearly on theunknown F, it depends on TS. Notice that Λ0 and Λ1 depend only on A and B.
5D. Langarica Córdoba and R. Ortega: Observer–Based Scheme for Decentralized Stabilization
The interconnection matrix between the closed-loopsystem and the observer error dynamics is defined as
I
I
I
I
=
……
…
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
∈ × −
11
22 1
0 0
0 0
0 0
� � � �
NN
nN nN NR ( ),
formed by submatrices
I11 1 12 1 13 1 1= …[ , , , ]B F B F B F N
I22 2 21 2 23 2 2= …[ , , , ]B F B F B F N
INN N N N N N N NB F B F B F= … −[ , , , ].( )1 2 1
The resulting observer error dynamics in (21) can beunited as
�� �x x=A ,
where
A
A A AA A A
A A A
=
……
…
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
∈ − ×
11 12 1
21 22 2
1 2
1
N
N
N N NN
nN N
� � � �R ( ) nnN N( ),−1
with sub-matrices Aii defined by the first sum in (21) and theoff-diagonal sub-matrices Aij defined by the second sum. Forconvenience, the matrix A can be reformulated similar to(14). To this end, the resulting matrices K j iΣ are placed in adiagonal formation leading to the overall observation gainmatrix
K K K K K K K K
K KN N
N
N
N
= … …… −
diag{ , , , , , , , , ,
, ,2 3 1 3 1
2 1
1 1 1 2 2 2Σ Σ Σ Σ Σ Σ Σ
Σ ΣNNnN N nN N} .( ) ( )∈ − × −R 1 1
According to Proposition II.1, it is possible to find afeasible matrix K by solving the LMI in (17). Moreover, theproof is completed identifying from the observer equationsabove the matrices of the proposition as
T K A A B F T Kijx
j ii ji j ji jk k
k k i
N
i i= − − − += ≠∑( )
,
Σ Σξ
1
T K A A B Fjk j ik jk j jki
ξ = − − −( )Σ
T K Biju
j ii= − Σ
H Kijx
j i= Σ
H Iij n nξ = ×
Hiju
n n= ×0 .
Finally, in the above description, each subsystem con-troller ui is provided with (N − 1) local estimations and thetotal number of subsystem observers used by the wholescheme is (N − 1)N.
IV. SIMULATION RESULTS
The theory presented in this work is verified throughnumerical simulation results. These results are obtained viaMatlab as well as Yalmip-SeDuMi package [25] to computethe corresponding LMI’s. Moreover, the controller robustnessis tested introducing white noise in sensors.
The control law (8) is used to stabilize a nonlinearpower system represented by two machines interconnectedvia a lossy transmission line as depicted in Figure (5). Thedynamics of this system are [16,17,23]:
�
�
�
δ ωω ω δ δ α
1 1
1 1 1 1 1 12
1 2 12 1 2 12
1 1 1 1
== − + − − − +
= − +
D P G E E E Y
E a E b
sin( )
EE E
D P G E E E
F2 1 2 121
1 1
2 2
2 2 2 2 2 22
2
1cos( ) ( )δ δ α
τν
δ ωω ω
− + + ∗ +
== − + − −
�
� 11 21 2 1 21
2 2 2 2 1 2 1 212
2
1
Y
E a E b E EF
sin( )
cos( ) (
δ δ α
δ δ ατ
ν
− +
= − + − + + ∗ +�22),
(22)
where δ π πi ∈ = −S : ( , )2 2 , ωi ∈ R and Ei ∈ R+ are thestates, νi ∈ R are the control inputs, Di, Pi, Gi, ai, bi, τi and EFi
∗are positive constants depending on the physical parametersof the i-th machine, and Yij and αij, with i, j N∈ and j ≠ i, areconstants depending on the topology, the physical propertiesof the network and the loads (admittance matrix). Moreover,we assume all parameters are known exactly. The equilibriumto be stabilized is
ε : , , ,= ( )∈ × × +d 0 E S n n nR R (23)
with d := ( )col δ iand E := ( )col Ei . d is such that (22), with
ωi = 0 and δ δi i= , has a unique solution in E = col(Ei), withEi > 0 for all i.
The system parameters are grouped in Table I. Theinitial conditions for the nonlinear model in (22) are gatheredin Table II. They have been selected to be around ±20% of theequilibrium points. The initial condition for the state of theobservers (10) and (11) are settled to zero.
The matrices A and B resulting from the linearization of(22) and the controller gain matrices F11, F12, F21, F22 aregiven below. The spectrum of the closed-loop systems isλ(A + BF) = {−0.31 ± 12.76i, −0.66 ± 1.45i, −0.83, −0.71}.The LMI (17) is feasible and the resulting matrix K places all
6 Asian Journal of Control, Vol. 17, No. 1, pp. 1–9, January 2015
eigenvalues of A at the left hand side of the complex plane.However, it is needed to assign the spectrum of A at the lefthand side of the spectrum of A + BF to provide the observerwith a faster dynamics than the closed-loop dynamics. Thespectrum of A can be placed in a desired disk region in thecomplex plane if the LMI [24]
P P
L P M P U M P U
= >⊗ + ⊗ −( ) + ⊗ − <
Τ
Τ Τ Τ ΤΛ Ω Λ Ω0
0( )(24)
holds, where ⊗ defines de Kronecker product and
Lr q
q rM=
−−
⎡⎣⎢
⎤⎦⎥
= ⎡⎣⎢
⎤⎦⎥
, ,0 1
0 0
characterize a disk region D with center −q and radius r in thecomplex plane. Under this consideration, the observer gainmatrices K2 1Σ and K1 2Σ given below, are obtained from thesolution of (24) where all the eigenvalues of A lie within thedisc D(−15, 5).
In Figs 2 to 4, the evolution of the state variablesof machines 1 and 2 are depicted together with thecorresponding estimations, the latter are expressed in the
A methodology to stabilize large-scale interconnectedLTI systems via decentralized observer-based control hasbeen presented. The restriction of limited exchange ofinformation between subsystem has been overcome usingI&I observers that, as is well-known [18], have a novelstructure consisting of an integral part and a proportionalpart. A similar construction has been proposed in [21]and [22] to include a correction term in standardLuenberger’s reduced order observer. The proposed proce-dure, however, seems more involved and less systematicthan the I&I observer given here. Finally, the proof of exist-ence of the observer gains that stabilize the system reducesto a simple LMI test. Current research is underway toextend the result by developing a nonlinear observer insteadof a linear one.
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Fig. 4. Time histories of E1, E2 and their estimations E1 2Σ andE2 1Σ .
Fig. 5. Two generator system with PLi and QLi the active andreactive power of the loads.
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Diego Langarica Córdoba was born inVeracruz, Mexico. He received the B.Eng.degree in Electronic Engineering fromthe Technological Institute of Veracruz,Mexico in 2008, the M.Sc. degree fromCENIDET, Morelos, Mexico in 2010. Now,
he is pursuing the Ph.D. degree in Engineering at theLaboratoire des Signaux et Systemes of the Universite deParis Sud XI-Supelec, France.
Romeo Ortega was born in Mexico. Heobtained his B.Sc. in Electrical andMechanical Engineering from the NationalUniversity of Mexico, Master of Engineer-ing from Polytechnical Institute of Lenin-grad, USSR, and the Docteur D’Etat fromthe Politechnical Institute of Grenoble,France in 1974, 1978 and 1984, respec-
tively. He then joined the National University of Mexico,where he worked until 1989. He was Visiting Professor at theUniversity of Illinois in 1987–88 and at the McGill Universityin 1991–1992, and Fellow of the Japan Society for Promotionof Science in 1990–1991. He has been a member of theFrench National Researcher Council (CNRS) since June1992. Currently he is in the Laboratoire de Signaux etSystemes (SUPELEC) in Paris. His research interests are inthe fields of nonlinear and adaptive control, with specialemphasis on applications.
9D. Langarica Córdoba and R. Ortega: Observer–Based Scheme for Decentralized Stabilization
