where F
i
A R
p
; G
i
A R
pp
; c
i
ðX; tÞ A R
p
are nonlinear functions, c
i
ðX; tÞ
represents the overall influence of the interconnections between
subsystems and bounded external disturbances; we abused the
notation a little and still use X
i
¼ðy
T
i
; ψ
T
i
Þ
T
to denote the state
vector in the new coordinate system, with
ψ
i
A R
n
i
r
i
being the
possible inner state; W
i
is a specified continuous mapping; and
y
ðr
i
Þ
i
9½y
ðr
i1
Þ
i1
; …; y
ðr
ip
Þ
ip
T
: Moreover, we define
where y
i0
is the system output; y
ir
is the available bounded
reference signal; e
i
is the output tracking error; and
r
i
¼ r
i1
þ… þ r
ip
r n
i
; with j ¼ 1; …; p: Only the output vector
y
i0
A R
p
of the subsystem is assumed to be measurable.
Remark 1. The decentralized CAFC algorithm investigated in this
paper is restricted to the class of large-scale systems in the form
(2), whose interaction does not appear in the term involving the
control input.
Assumption 1. [15,27,32]: Subsystem i is a feedback linearizable
system based upon static-state feedback and it is an exponential
minimum phase system when r
i
o n
i
.
Assumption 2. [15,17,27]: The input gain matrix G
i
ðX
i
Þ is inver-
tible and bounded for all X
i
A U
ic
R
n
i
where U
ic
is the controll-
ability region of X
i
.
2.2. Control objective
Let
^
X
n
be an estimate of X
n
: Under the above assumptions, we
design the following decentralized CAFC via output feedback
u
i
¼ α
i
u
iI
þð1α
i
Þu
iD
þu
iC
ð3Þ
and its update laws for the large-scale MIMO nonlinear system (2)
where
αA ½ 0; 1 is a weighting coefficient, u
iI
; u
iD
and u
iC
are
primary IAFC, DAFC and an auxiliary compensator, respectively,
such that the resultant closed-loop large-scale system is asympto-
tically stable and all the signals are UUB. The weighting factor
sums together the control efforts from the IAFC and the DAFC,
which is a flexible design methodology balancing the knowledge
of the plant and controller. If fuzzy descriptions are superior to
control rules, we choose
α
i
to be larger than 0.5; otherwise, α
i
is
less than 0.5. In general, we take
α
i
¼ 0:5:
2.3. Observer-based fuzzy system
A fuzzy logic (or inference) system for decentralized control is a
mapping from an input vector X
i
¼ðx
i1
; …; x
i;n
i
Þ
T
A U
i
R
n
i
to an
output scalar V
i
A R where U
i
is the universe of discourse of the
input that belongs to the ith subsystem. The fuzzy system, in
general, is characterized by a set of “If-Then” rules. We define
fuzzy sets to construct the fuzzy rule base for the observer-based
CAFC which are constituted of the following fuzzy information.
R
ðl
1
;…;l
n
i
Þ
if
: If
^
x
i1
is F
l
1
i1f
and …; and
^
x
in
i
is F
l
n
i
in
i
f
; then
^
F
i
ð
^
X
i
j
Θ
if
Þ is F
ðl
1
;…;l
n
i
Þ
if
ð4Þ
R
ðl
1
;…;l
n
i
Þ
ig
: If
^
x
i1
is F
l
1
i1g
and … and
^
x
in
i
is F
l
n
i
in
i
g
; then
^
G
i
ð
^
X
i
j
Θ
ig
Þ is G
ðl
1
;…;l
n
i
Þ
ig
ð5Þ
R
ðl
1
;…;l
n
i
Þ
iu
: If
^
x
i1
is F
l
1
i1u
and…; and
^
x
in
i
is F
l
n
i
in
i
u
; then
^
u
id
ð
^
X
i
j
Θ
iu
Þ is F
ðl
1
;…;l
n
i
Þ
iu
ð6Þ
Where F
l
1
i1f
; … ; F
l
n
i
in
i
f
; F
l
1
i1g
; …; F
l
n
i
in
i
g
; F
l
1
i1u
; …; F
l
n
i
in
i
u
; F
ðl
1
;…;l
n
i
Þ
if
; G
ðl
1
;…;l
n
i
Þ
ig
;
F
ðl
1
;…;l
n
i
Þ
iu
are appropriately chosen fuzzy sets;
^
F
i
;
^
G
i
;
^
u
id
are the
estimates of F
i
; G
i
; u
n
i
; respectively; u
n
i
is the desired control;
l
τ
¼ 1; …; m
τ
; τ ¼ 1 ; …; n
i
; m
τ
is denoted as the overall number of
fuzzy rules; (4) and (5) are fuzzy descriptions for IAFC; and (6) is
control rules for DAFC. Based on a singleton fuzzifier, a product
inference engine and a center-average defuzzifier, we construct
^
V
ik
ð
^
X
i
jΘ
if
Þ¼Φ
T
ik
ð
^
X
i
ÞΘ
ik
; k ¼ 1; …; p; ð7Þ
as the final output of the observer-based fuzzy logic system where
Θ
ik
¼½θ
1
ik
; …; θ
m
ik
T
is an adjustable parameter vector and
Φ
T
ik
ð
^
X
i
Þ¼½ξ
1
ik
ð
^
X
i
Þ; …; ξ
m
ik
ð
^
X
i
Þ is a fuzzy basis function vector with
m ¼ ∏
n
i
τ
¼ 1
m
τ
whose scalar entity is in the following form
ξ
ðl
1
;…;l
n
i
Þ
ik
ð
^
X
i
Þ¼
∏
n
i
τ
¼ 1
μ
V
l
τ
i
ð
^
x
i
τ
Þ
∑
m
1
l
1
¼ 1
…∑
m
n
i
l
n
i
¼ 1
∏
n
i
τ
¼ 1
μ
V
l
τ
i
ð
^
x
i
τ
Þ
ð8Þ
where
μ
V
l
τ
i
ð
^
x
i
τ
Þ; τ ¼ 1; …; n
i
; are the membership function of the
fuzzy set V
l
τ
i
to be specified. In accordance with the universal
approximation theorem [15], the fuzzy logic system in the form of
(7) can approximate to an arbitrary degree of accuracy any
continuous function whose universe of discourse is defined based
on a compact set, so long as the number of fuzzy rules is large
enough.
3. Observer-based decentralized CAFC
3.1. Decentralized controller design
Assume F
i
ðX
i
Þ and G
i
ðX
i
Þ are known, then there exists an ideal
controller for the ith nominal subsystem, i.e. c
i
ðX; tÞ¼0
u
n
i
¼ G
1
i
ðX
i
Þ½y
ðr
i
Þ
ir
þK
T
ic
e
i
F
i
ðX
i
Þ ð9Þ
where the feedback gain matrix K
ic
is chosen such that A
i
B
i
K
T
ic
is
Hurwitz, because the matrix pair ðA
i
; B
i
Þ is controllable. Since F
i
ðX
i
Þ
and G
i
ðX
i
Þ are unknown, (9) cannot be implemented directly.
Instead, we construct fuzzy systems
^
F
i
ðX
i
jΘ
n
if
Þ¼½Φ
T
ijf
ðX
i
ÞΘ
n
ijf
p1
;
^
G
i
ðX
i
jΘ
n
ig
Þ¼½Φ
T
ijkg
ðX
i
ÞΘ
n
ijkg
pp
and
^
u
iD
ðX
i
jΘ
n
iu
Þ¼½Φ
T
iju
ðX
i
ÞΘ
n
iju
p1
F
i
ðX
i
Þ9½f
i1
ðX
i
Þ; …; f
ip
ðX
i
Þ
T
; G
i
ðX
i
Þ9½G
i1
ðX
i
Þ; …; G
ip
ðX
i
Þ
T
;
G
ik
ðX
i
Þ9½g
ik1
ðX
i
Þ; …; g
ikp
ðX
i
Þ
T
; c
i
ðX; tÞ 9 ½c
i1
ðX; tÞ; …; c
ip
ðX; tÞ
T
;
y
i
9½y
i1
; …; y
ðr
i1
1Þ
i1
; …; y
ip
; …; y
ðr
ip
1Þ
ip
; y
ir
9½y
i1r
; …; y
ðr
i1
1Þ
i1r
; …; y
ipr
; …; y
ðr
ip
1Þ
ipr
T
;
y
i0
9½y
i1
; …; y
ip
T
; y
ir0
9½y
i1r
; …; y
ipr
T
; u
i
9ðu
i1
; …; u
ip
Þ
T
; e
i0
9y
ir0
y
i0
;
y
ðr
i
Þ
ir
9½y
ðr
i1
Þ
i1r
; …; y
ðr
ip
Þ
ipr
T
; e
i
9y
ir
y
i
9½e
i1
; … ; e
ðr
i1
1Þ
i1
; …; e
ip
; …; e
ðr
ip
1Þ
ip
T
;
e
i0
9ðe
i1
; …; e
ip
Þ
T
; e
ij
9½e
ij
; …; e
ðr
ij
1Þ
ij
T
; e
ðr
i
Þ
i
9½e
ðr
i1
Þ
i1
; …; e
ðr
ip
Þ
ip
; r
i
9ðr
i1
; …; r
ip
Þ
T
;
Y.-S. Huang et al. / ISA Transactions 53 (2014) 1569–1581 1571