Since the projection method leads to discontinuous trajectories in the estimated states, a nonstandard Lyapunov - Krasovski functional is applied to derive the upper bound for estimation
Trang 10 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 -1.5
-1 -0.5 0 0.5 1 1.5 2 2.5x 10 -3
Time [s]
x3 DNN Observer without projection
x3 Projectional DNN Observer
x3
-2 -1 0 1 2 3 4 5 6
7x 10 -3
Time [s]
x3
x3 Projectional DNN Observer
x3 DNN Observer without projection
Trang 20 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -5
-4 -3 -2 -1 0 1 2 3 4
5x 10 -4
Time [s]
x4 DNN Observer without projection
x4 Projectional DNN Observer
x4
x4 min
x4 max
-5 -4 -3 -2 -1 0 1 2 3 4
5x 10 -4
Time [s]
x4
x4 Projectional DNN Observer
x4 DNN Observer without projection
As it can be seen, the projectional DNNO has significantly better quality in state estimation, especially in the beginning of the process, when negative values and over-estimation have been obtained by a non-projectional DNNO
Trang 36 Conclusion and future work
The complete convergence analysis for this class of adaptive observer is presented Also the boundedness property of the adaptive weights in DNN was proven Since the projection method leads to discontinuous trajectories in the estimated states, a nonstandard Lyapunov
- Krasovski functional is applied to derive the upper bound for estimation error (in "average sense"), which depends on the noise power (output and dynamics disturbances) and on an unmodelled dynamic It is shown that the asymptotic stability is attained when both of these uncertainties are absent The illustrative example confirms the advantages, which the suggested observers have being compared with traditional ones
Appendix (proof of Theorem 2)
Evidently that
( ) (t′ −δt−h)≤Lδt′− t−h( )t )
δ
η η η η η
η η η η η η
ϒ
− Λ
≤ Λ
− Λ
≤
⎟
=
2 1 2
1
2 1 1 2 1
/ )
(
t / , t / t
ξ ξ
ξ( )≤ Λ−1 /1 2ϒ
( )
2 1 2 1 1 0
2
f~
t x f~
f~
/ f~
) ( f~
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
Λ +
− Λ
≤
where δ( )t′ :=xˆ( ) ( )t′ −x t′ is the state estimation error at time t
Consider the next "nonstandard" Lyapunov-Krasovskii ("energetic") function
( ) δ(τ) p k( )τ tr{W ~ T( ) ( )τ W ~ τ}dτ
t t h t
external output disturbances we won't demonstrate that the time-derivative of this energetic function is strictly negative Instead, we will use it to obtain an upper bound for the averaged state estimation error Taking time derivative of Lyapunov-Krasovski function and
Trang 4( )[ ( ) ( ) ]
( ) ( ) { ( ) ( ) } ( ( ) ) { ( ) ( ) }
( ) { ( ) ( ) } ( ( ) ) { ( ) ( ) }
( ) { ( ) ( ) } ( ( ) ) { ( ) ( ) }
( ) { ( ) ( ) } ( ( ) ) { ( ) ( ))}
) ( +
W ˆ + Wˆ + x
+ +
+ +
) t
) ( x
-) xˆ
xˆ
t h t
W ~ ) t h t
W ~ t h t k t
W ~ t
W ~ t k
) t h t
W ~ ) t h t
W ~ t h t k t
W ~ t
W ~ t k
t h t -d ) ( ) ( f~
) ( u ) ( x ( ) ( x ) ( Ax
-))
t
h
-(
d )) ( C -( ( K ) ( u ) ( xˆ )(
( W ) ( xˆ ) ( W ) ( xˆ A ))
t
h
-(
xˆ
t h t
W ~ ) t h t
W ~ t h t k t
W ~ t
W ~ t k
) t h t
W ~ ) t h t
W ~ t h t k t
W ~ t
W ~ t k
) h t p t
d )) t xˆ C -( + ) ( Cx ( K + ) ( u ) ( )(
( W + ) ( ) ( W + ) ( xˆ A t h t +
))
t
h
-(
xˆ
) ( V dt d
T T
T T
p p
t t h t
t t h t
T T
T
t X
−
−
−
− +
−
−
−
− +
− +
≤
−
−
−
−
+
−
−
−
−
+
−
−
−
=
≤
−
=
−
∫
−
∫
⎪⎭
⎪
⎪⎩
2 2
2 2 2 2
1 1
1 1 1 1
2 2
2 1
2 1
2 2
2 2 2 2
1 1
1 1 1 1
2
2
2 1
tr tr
tr tr
tr tr
tr tr
δ τ τ ξ τ τ τ ϕ σ
τ τ δ τ η τ τ ϕ τ τ σ τ τ
δ
τ τ
η τ τ
τ ϕ τ τ
σ τ τ τ
π
τ τ
Taking into account that
(Pa, b)
P b P a P b
Defining:
( ) ( ) ( ) ( )xˆ (t) x(t) :
(t)
~
x(t) σ (t) xˆ σ (t):
σ~
, i i
W ˆ (t) i W (t):
i
W ~
KC, A :
A ~
ϕ ϕ
−
=
=
−
=
−
=
2 1
we derive
( ) ( ) ( ) { ( ) ( ) } ( ( ) ) { ( ) ( ) }
( )t tr{W ~ T( ) ( )t W ~ t} k (t h( )t)tr{W ~ T (t h( )t ) W ~ (t h( )t )}
k
) t h (t
W ~ ) t h (t T
W ~ tr t h t k t
W ~ t T
W ~ tr t k
t β t α V
−
−
−
−
+
−
−
−
−
+ +
≤
2 2
2 2 2 2
1 1
1 1 1 1
where:
( )
]
( ) ( )
] τ)
τ τ ξ τ η τ τ ϕ
τ τ ϕ τ τ σ τ σ τ τ δ τ
δ β
τ τ τ ξ τ η τ τ ϕ
τ τ ϕ τ τ σ τ σ τ τ δ τ
α
d ) ( f~
) ( ) ( K ) ( u ) (
~
W ˆ
) ( u ) ( xˆ )(
(
W ~ ) (
~ Wˆ ) ( xˆ ) (
W ~ ) (
A ~ t t h t , h t P :
t
P d ) ( f~
) ( ) ( K ) ( u ) (
~
W ˆ
) ( u ) ( xˆ )(
(
W ~ ) (
~
W ˆ ) ( xˆ ) (
W ~ ) ( A
~ t t h t : t
−
− + +
⎜
⎜
⎝
⎛
+ +
+
∫
−
=
−
=
−
− +
+ +
+ +
∫
−
=
=
2
2 1
1 2
2 2
2 1
1
Trang 5The term β( )t is expanded as
( ) ( ( ) )
( ) ( ) ( )
( )
( ) ( ) ( ) ( ( ) ) ( ) ( ) ( )
( )
( ) ( ) ( )
( )
( ) ( )
( )
( ) ( ( ) ( ) ) ( ( ) ) ( ) ( ) ⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∫
−
=
−
−
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
∫
−
=
− +
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜
⎝
⎛
∫
−
=
− +
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∫
−
=
− +
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∫
−
=
− +
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∫
−
=
−
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∫
−
=
−
=
τ τ τ
δ τ τ ξ τ η τ
δ
τ τ τ ϕ τ
δ
τ τ τ ϕ τ τ
δ
τ τ σ τ
δ τ τ σ τ τ
δ
τ τ δ τ
δ β
d f~
t t h t , t h t P d K
t t h t , t h t P
d ) ( u ) (
~
W ˆ t t h t , t h t P
d ) ( u ) ( xˆ )(
(
W ~ t t h t , t h t P
d
~
W ˆ t t h t , t h t P d xˆ
W ~ t t h t , t h t P
d
A ~ t t h t , t h t P t
2 2
2 2 2
2 2
1 2
2
2
( )
]
⎪
⎟⎟
⎞
⎜⎜
⎛
+ + +
∫
−
= +
⎪⎩
⎪
⎟⎟
⎞
⎜⎜
⎛
+ +
+
∫
−
=
≤
−
− + +
+ + +
∫
−
=
=
τ τ ξ τ τ η τ τ ϕ τ
τ τ τ ϕ τ τ
σ τ
σ τ τ
δ τ
τ τ τ ξ τ η τ τ ϕ
τ τ ϕ τ τ σ τ σ τ τ δ τ
α
d p p f~
p K p ) ( u ) (
~
W ˆ t t h t
d p ) ( u ) ( xˆ )(
(
W ~ p ) (
~ Wˆ p ) ( xˆ ) (
W ~ p
A ~ t t h t
P d ) ( f~
) ( ) ( K ) ( u ) (
~
W ˆ
) ( u ) ( xˆ )(
(
W ~ ) (
~ Wˆ ) ( xˆ ) (
W ~ ) ( A
~ t t h t : t
2 2 2 2 2
2 2
2 1
2 1
2 8
2 2
2 1 1
B
T
Y YΛ T XΛΛ T YX T
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ( ) ( ) ) ( ) ( ) ( ) ( ) ( ) ( )t C T ( )t ( )t C T C I ( )t e
T C
t t t C T C t T C t e T C
t t C T C t e T C
t t Cx t xˆ C t y t yˆ t e
δ ϖ ϖδ
η
ϖδ ϖδ δ η
η δ
η
⎟
= + +
−
− +
= +
−
−
=
−
−
−
=
−
=
−
Giving
( )t =Nϖ⎜−C T e( )t +C Tη( )t +ϖδ( )t ⎟
δ
Trang 6where:
1
−
⎟
:
obtained:
( ) ( )
( )+
−
⎥⎦
⎤ +
⎟
+
⎟
⎟
⎠
⎞ +
⎟
⎢⎣
−
≤
t h t δ Q Λ Λ I L μ σ L μ μ u L Λ σ
L
Λ
P Λ Λ T
W ˆ Λ
W ˆ T
W ˆ Λ
W ˆ Λ P
A ~ P T
A ~ T t h t δ t h ) ( V dt d
0
1 7
1 3
2 3 2 1
2 8 5
1 10
1 9 2
1 8 2 1
1 5 1
1 1
ϖ ϕ ϕ
( )
( )
( ) ( ) ( ) ( ( )) ( ) ( )( ) ( )σ T( )xˆ( )τ W ~ T( )τ PN (CΛ C Λ )N W ~ ( ) ( )τ σ( )xˆ τ dτ
t t h t τ
dτ τ xˆ σ τ
W ~ P CN t h t T e t t h t τ t h t δ Q T t h t δ η ξ ξ
Λ
P
Diam(x) f~
Λ f~
f~
f~
Λ P η
/ η Λ P K
f~
Λ t x f~
f~
f~
Λ Λ ξ
/ ξ Λ η
/ η Λ K Λ h(t)
δ L L Λ δ L L μ δ L μ δ L σ L μ δ L σ L Λ δ L A
~ Λ t
h
⎥⎦
⎤
⎢⎣
∫
−
= +
⎟
∫
−
= +
⎥⎦
⎤
−
−
− ϒ + ϒ
−
⎟
⎞ +
⎥
⎤
⎢
⎡ +
− + ϒ
−
⎢
⎢
⎢
⎣
⎡
+
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡ +
− +
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎟
⎞
⎜
⎛
ϒ
− + ϒ
−
+
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
+
ϒ + + +
+
1 3 2 1
1 2
0 2
1
2 1
1 0 1 2
1 2
2 1 1 0 1 10
2 2 1 2
1 9
3
2 2 8 3
2 2 3 3
2 1 3
2 2 3
2 5 4
2 2 1 3
ϖ ϖ ϖ
ϖ
ϕ ϕ
( )
( )t tr{W ~ 2 T( ) ( )t W ~ 2 t} k 2(t h( )t )tr{W ~ 2 T (t h( )t ) W ~ 2 (t h( )t )}
2 k
)dτ u(
) ( xˆ )(
( 2
W ~ )P ( T 2
W ~ T ) ( xˆ )(
( T u t t h t τ
)dτ u(
) ( xˆ )(
( 2
W ~ P N 7 Λ C 6 Λ T C PN τ T 2
W ~ τ xˆ T ) ( T u t
t h
t
τ
dτ ) u(
) ( xˆ )(
( 2
W ~ P CN t h t T e 2 t t h t τ
) t h (t 1
W ~ ) t h (t T 1
W ~ tr t h t 1 k
t 1
W ~ t T 1
W ~ tr t 1 k dτ τ xˆ σ τ
W ~ )P ( T 1
W ~ τ xˆ T σ t t h t τ
−
−
−
−
+
∫
−
= +
⎟
∫
−
=
+
⎟
∫
−
=
+
−
−
−
− +
∫
−
= +
τ τ ϕ τ τ τ
ϕ τ
τ τ ϕ τ ϖ ϖ ϖ
ϕ τ
τ τ ϕ τ ϖ τ
Trang 7Considering
0
1 7
1 3
2 3 2 1
2 8 5
3 2 1
1 10
1 9 2
1 8 2 1
1 5 1
1 1 1
0 3 2 1 1
Q
I L L
u L L
, , , Q
T
W ˆ
W ˆ T
W ˆ
W ˆ R
, , , Q P PR K
A ~ P K T
A ~
+
⎟
+
⎥⎦
⎤
⎢⎣
=
− Λ +
− Λ +
− Λ +
− Λ +
− Λ
=
− + +
ϖ
ϕ μ σ μ μ ϕ σ
μ μ μ δ
μ μ μ δ
implies:
( ) ( ) ( ) ] ( ) ( ) }
( ) { 1 ( ) ( )1 } 1( ( ) ) { 1 ( ) 1 ( ) } 0 1
1
1 2
3 2
1
=
−
−
−
− +
+
⎩
∫
−
=
) h t
W ~ ) h t T
W ~ tr t h t k t
W ~ t T
W ~ tr t k
d xˆ T xˆ
W ~
xˆ
W ~ P N C T N
t h t e T N P T
W ~ tr t
t h
t
τ τ σ τ σ τ
τ σ τ ϖ ϖ
ϖ ϖ
τ τ
that can be obtained selecting
( ) ( )
( ) ( ) ( ) ] ( ) ( )
( )
⎭
⎬
⎫
−
⎩
⎨
⎧
⎢⎣
−
=
t
W ~ dt
(t) dk
τ xˆ T σ τ xˆ σ τ
W ~
+ τ xˆ σ τ
W ~ P N C Λ T +C Λ +N t t-h e T C N P (t) k
-t W dt d
1 1 1
1 2
3 2
1 2 1
1
ϖ ϖ
ϖ ϖ
Analogously, for the second adaptive law
( ) ( ) ( ( ) ) ( ) ( )
( ) ] ( ( ) ) }
( ) { 2( ) ( )2 } 2( ( ) ) { 2 ( ) 2 ( ) } 0 2
2
1 7
6 2
2
=
−
−
−
− +
+
⎩
⎧
⎢⎣
∫
−
=
) h t
W ~ ) h t T
W ~ t h t k t
W ~ t T
W ~ t k
d xˆ T ) ( T ) ( u ) ( xˆ )(
(
W ~
) ( u ) ( xˆ (
W ~ P N C
T C N t h t e T C N P T
W ~ t
t
h
t
tr tr
tr
τ τ ϕ τ τ τ ϕ τ
τ τ ϕ τ ϖ ϖ ϖ
ϖ τ τ
leading to
( ) ( )
( ) ] ( ) ( )
( )
⎭
⎬
⎫
−
⎩
⎨
⎧
⎢⎣
−
−
=
t
W ~ dt
) ( dk
xˆ T ) ( T ) ( u ) ( xˆ )(
(
W ~
+ ) ( u ) ( xˆ (
W ~ P N + C T C N + t h -e T C N P )
(
k
t W dt d
2 2 2
1 7
6 2
1 2
2
2
τ ϕ τ τ τ ϕ τ
τ τ ϕ τ ϖ ϖ ϖ
ϖ
Trang 8Finally:
( )
( )
( ) − ( )⎥⎦⎤
−
− ϒ + ϒ
− Λ +
⎥
⎤
⎢
⎡ Λ +
− Λ + ϒ
− Λ +
⎢
⎢
⎢
⎣
⎡
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
Λ +
− Λ Λ +
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎟
⎞
⎜
⎛
ϒ
− Λ + ϒ
− Λ Λ +
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
Λ +
ϒ + + +
Λ + Λ
≤
t h t Q T t h t P
) x ( Diam f~
f~
f~
f~
P / P K
f~
t x f~
f~
f~
/ /
K ) h
L L L L L
L L L L L
A ~ t h ) V
dt
d
δ δ η ξ ξ
η η
ξ ξ η η
δ ϕ δ ϕ μ δ μ δ σ μ δ σ δ
0 2
1
2 1
1 0 1 2
1 2
2 1 1 0 1 10
2 2 1 2
1 9
3
2 2 8 3
2 2 3 3
2 1 3
2 2 3
2 5 4
2 2 1 3
or in the short form:
( ) ( )⎜ + − ( − ( ) ) ( − ( ) )⎟
) ( V dt
d
δ
2 where
( ) η ξ ξ η
η
ξ ξ η η
δ ϕ δ ϕ μ δ μ δ σ μ δ σ δ
ϒ + ϒ
− Λ +
⎥
⎤
⎢
⎡ Λ +
− Λ + ϒ
− Λ +
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
Λ +
− Λ Λ +
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎟
⎞
⎜
⎛
ϒ
− Λ + ϒ
− Λ Λ
=
ϒ Λ +
ϒ + + +
Λ + Λ
=
2 1 2
1 1 0 1 2
1 2
2 1 1 0 1 10
2 2 1 2
1 9
3
2 2 8 3
2 2 3 3
2 1 3
2 2 3
2 5 4
2 2 1
P ) x ( Diam f~
f~
f~
f~
P / P K
f~
t x f~
f~
f~
/ /
K : b
L L L L L
L L L L L
A ~ : a
So,
( ) (t h t )Q (t h( )t ) (ah( )t b) dV dt ( ) h( )t
δ
And integrating, we obtain
τ τ τ τ δ τ τ δ
) ( dV b ) ( ah
T d ) ( h Q ) ( h T
T
⎥⎦
⎤
⎢⎣
=∫
≤
−
−
=∫
1 2
0 0
0
And hence,
( ) h ( )
V ) ( h
V t
h t
V ) ( h
V d T
d ) ( h ) ( h V
T + ) ( h
V d
T -) ( h dV T
0
0 0
0 0
2 0 0
0
≤ +
−
=
⎟⎟
⎞
⎜⎜
⎛
=∫
−
≤
=∫
⎟⎟
⎞
⎜⎜
⎛
=∫
=
=∫
−
τ
τ τ
τ τ
ττ τ τ
τ τ
ττ τ
This implies
( )
) ( h
V bT d t h Q
T
a d ) ( h )
( h
T T
0
0 2
0
∫
=
∫
−
−
τ
τ τ τ
τ
Dividing by T and taking the upper limit we finally get (30)
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Integral Sliding Modes with Block Control of
Multimachine Electric Power Systems
Héctor Huerta, Alexander Loukianov and José M Cañedo
Centro de Investigación y de Estudios Avanzados del Instituto Politécnico Nacional,
Unidad Guadalajara Jalisco, México
1 Introduction
Over last 15 years the problem of rotor angle stability of electric power systems (EPS) has received a great attention A fundamental problem in the design of feedback controllers for EPS is that of robust stabilizing both rotor angle and voltage magnitude, and achieving a specified transient behavior Robustness implies operation with adequate stability margins and admissible performance level in spite of plant parameters variations and in the presence
of external disturbances
The EPS have nonlinearities and are subject to variations as a result of a change in the systems loading and/or configuration Then, the EPS are modeled as complex large-scale nonlinear systems and the generators may be interconnected over several kilometers in very large power systems Thus, the controller design is a challenging problem A complete centralized control scheme could be difficult to implement in EPS, due to the reliability and distortion in information transfer On the other hand, accurate prediction of system responses and system robustness to disturbances under different operation conditions are guarantee by robust decentralized control schemes The decentralized controllers are locally implemented, so do not need system information communication among subsystems In each subsystem, the effects of the other subsystems are considered as a disturbance To design decentralized control schemes for EPS, a controller is designed for each generator connected to the system
The control schemes of power systems are commonly based on reduced order linearized model and classical control algorithms that ensure asymptotic stability of the equilibrium point under small perturbations (Anderson & Fouad, 1994, DeMello & Concordia, 1969) Improvements on linear techniques have been analyzed in (Wang et al., 1998, Djukanovic et at., 1998a, Djukanovic et al., 1998b) Nevertheless, these controllers have been designed by using linear models To analyze the EPS entire operation region, nonlinear control design techniques are more appropriate Various nonlinear techniques have been implemented, e.g., control based on direct Lyapunov method (Machowsky et al., 1999), feedback linearization (FL) technique (Akhkrif, et al, 1999, Wu & Malik, 2006, ) including backstepping (Jung et al., 2005 King et al., 1994), intelligent neural networks (Venayagamoorthy et al., 2003, Mohagheghi et al., 2007), fuzzy logic (Yousef & Mohamed, 2004) and normal form analysis (Kshatriya, et al., 2005, Liu et al., 2006)