Nonlinear model reference adaptive control using Takagi-Sugeno fuzzy systems

Abstract

This paper develops a new direct model reference fuzzy adaptive control of SISO continuous-time nonlinear systems. The model following conditions are assured by using an adaptive Takagi-Sugeno (TS) fuzzy system as nonlinear state feedback controller. Both full state information and observer-based control schemes are investigated. It is shown, that the proposed control algorithm guarantees the stability of the nonlinear system with the tracking and state estimation errors converging to the neighborhood of the origin for all realizations of uncertainties and disturbances. Compared to previous works this approach is much simpler and less assumptions are required. Simulation results for controlling inverted pendulum system are given.

Full text

Page 1

Journal of Intelligent & Fuzzy Systems 17 (2006) 47–57
IOS Press
47
Nonlinear model reference adaptive control
using Takagi-Sugeno fuzzy systems
Noureddine Gol´ eaa,∗, Amar Gol´ eaband Mohamed Kadjoudjc
aEE Institute, Oum El-Bouaghi University, 04000 Algeria
bEE Institute, Biskra University, 07000 Biskra, Algeria
cEE Institute, Batna University, 05000 Batna, Algeria
Abstract. This paper develops a new direct model reference fuzzy adaptive control of SISO continuous-time nonlinear systems.
The model following conditions are assured by using an adaptive Takagi-Sugeno (TS) fuzzy system as nonlinear state feedback
controller. Both full state information and observer-based control schemes are investigated. It is shown, that the proposed
control algorithm guarantees the stability of the nonlinear system with the tracking and state estimation errors converging to the
neighborhood of the origin for all realizations of uncertainties and disturbances. Compared to previous works this approach is
much simpler and less assumptions are required. Simulation results for controlling inverted pendulum system are given.
Keywords: TS fuzzy systems, reference model, adaptive control, observer, stability
1. Introduction
Model reference adaptive control is well established
subject in linear systems theory [11]. However, most
practical plants are nonlinear. The application of lin-
ear control techniques to these plants rely on the as-
sumptionofsmall variationaboutthedesiredoperating
point. Whentherequiredoperatingrangeislarge,alin-
ear controlis likely to performpoorly. Consequently,a
nonlinear controller is required to satisfactory control
such plants. However, the main difficulty in designing
nonlinear controller is the lack of unified structure.
Recently,themodelreferenceadaptivecontrolisalso
extended to the nonlinear systems using direct and in-
direct approaches. The fuzzy model reference control
of SISO nonlinear systems is addressed in [2] using
neuro-fuzzy systems and good performance is shown
through Lyapunov stability approach. In [3,9] indirect
modelreferencecontrol,usingfuzzylinguisticmodels,
are proposed. In [5] a direct adaptive approach, based
onTakagi-Sugenosystem[14],is proposedandthesta-
bility is analyzed in the hyperstability framework [11].
∗Corresponding author. E-mail: nour golea@yahoo.fr.
The multivariable case was considered using the in-
direct approach and standard fuzzy systems in [10],
and in [7] using TS fuzzy systems. The direct case is
developed in [6]. More Recently, the observer-based
approach was considered in [7].
This work extends the model reference control to
the nonlinear systems case. It exploits the quasi-linear
structure of TS systems, i.e., the TS fuzzy being non-
linear superposition of linear local systems, and their
ability to exactly represent some nonlinear systems [8,
17], to achieve model reference adaptive control of
SISO continuous-time nonlinear plants. The idea be-
hind this approach,is that fuzzy membershipfunctions
playthe role of continuousschedulerof the local linear
control rules, such that at each operating point there
exists a controller that achieves the model-following
objective. Both full state information and observer-
based approaches are developed. The stability in both
cases is established using Lyapunov tools. It is shown
thatthisapproachis robustagainstexternaldisturbance
and the tracking and observation errors are bounded.
Compared to previously developed direct adaptive ap-
proaches[12,13,17],this scheme requires few assump-
tions, and it’s implementation is much simpler, i.e., no
supervisory or switching control is needed to ensure
1064-1246/06/$17.00  2006 – IOS Press and the authors. All rights reserved

Page 2

48
N. Gol´ ea et al. / Nonlinear model reference adaptive control using Takagi-Sugeno fuzzy systems
the stability, and the upper bound on high-frequency
derivative is not required. Simulation results, for un-
stable invertedpendulum,show the proposedapproach
capabilities.
2. Problem formulation
Consider the continuous-time nonlinear systems
class given by
˙ x = Ax + B [f(x) + g(x)u + d]
(1)
y = Cx
(2)
where xT=
u,y ∈ ? are the system input and output respec-
tively,f (x),g(x)aresmoothnonlinearunknownfunc-
tions, d is bounded external disturbance, and BT=
?0 ... 0 1?
A =
0
n×n
where Inis the n × n identity matrix.
The nonlinear system (1)–(2) satisfies the following
assumptions:
?x1... xn
?
∈ ?nis the state vector,
1×n, C =?1 0 ... 0?
1×n,
?0 In−1
?
P1. The high-frequency gain g (x) is strictly posi-
tive and bounded by 0 < g < g (x) < g, in
the relevant control space Ωx, for some known
constants g and g. The negative case can also
be handled.
P2. The nonlinear function f(x) can be decom-
posed as f (x)= a(x)x with a(x) =
?a1(x) ... an(x)?
Note that P1 is an usual assumption in the adaptive
control (see [6,13,17]), and P2 specifies the consid-
ered class of nonlinear systems. Note that many phys-
ical plants satisfy P2, e.g., manipulators and electrical
drives. As will be remarked later, this approach can
be easily extended to more broader class of nonlinear
systems.
The reference model is defined by
∈ ?nis unknown smooth
nonlinear vector function.
˙ xm= Amxm+ Bbmr
m=?xm1... xmn
Am=
−am
with am=?am1am2... amn
(3)
where xT
r is bounded reference input, bm> 0 and
?0 In−1
?∈ ?nis the state vector,
?
n×n
?∈ ?n.
SubstractingEq.(1)fromEq.(3)andusingP2yields
the following tracking error dynamic
˙ e = Ame + B [−(am+ a(x))x
+bmr − g(x)u − d]
where e = xm− x is the tracking error.
The control problem can be stated as: Design the
control input u such as the reference model tracking
is achieved, i.e., the states of the system Eq. (1) fol-
low those of the reference model Eq. (3), under the
condition that all involved signals remain bounded.
(4)
3. Fuzzy adaptive approach
3.1. State feedback design
Thefuzzyadaptivecontroller,tobedesigned,iscon-
stituted by two TS fuzzy systems defined by the fol-
lowing sets of if-then rules
R1
j: If x is X1
Then uf1= K1jx, j = 1...m1
j
(5)
R2
j: If x is X2
Then uf2= k2jr, j = 1...m2
where K1j ∈ ?1×nand k2j ∈ ? are the jth rule
consequence parameters, m1(resp. m2) is the number
of rules, x is the fuzzy systems input vector, and the
fuzzy sets X1
jj
the fuzzy systems input spaces.
The output of the fuzzy systems Eqs (5)–(6) are in-
ferred, respectively, as follows
j
(6)
?resp. X2
?
operate fuzzy partitions of
uf1= φ1(x)K1x
uf2= φ2(x)K2r
where
(7)
(8)
φ1(x) =
1
?m1
?m2
K11
...
K1m1
j=1µ1j(x)
1
j=1µ2j(x)


?µ11(x) ... µ1m1(x)?
?µ21(x) ... µ2m2(x)?


φ2(x) =
K1=




m1×n
,K2=

k21
...
k2m2



m2×1
and µ1j(x) (resp. µ2j(x)) is the grade of membership
of x in X1
Let’s define the control input as
j(resp. X2
j).

Page 3

N. Gol´ ea et al. / Nonlinear model reference adaptive control using Takagi-Sugeno fuzzy systems
49
u = uf1+ uf2
(9)
Now, introducing Eq. (9) in Eq. (4) yields
˙ e = Ame + B[−(am+ a(x)
+g(x)φ1(x)K1)x
(10)
+(bm− g(x)φ2(x)K2)r − d
According to the universal approximationresults [1,
4,16], there exist optimal parameters K∗
fined as
K∗
?supx∈Ωx|am+ a(x) + g(x)φ1(x)K∗
K∗
argminK2∈Ω2
where Ω1, Ω2and Ωxare constraint sets for K1, K2
and x, respectively. Meanwhile, define the minimum
approximation errors
1and K∗
2de-
1= argminK1∈Ω1
1|?
2=
?supx∈Ωx|bm− g(x)φ2(x)K∗
2|?
am+ a(x) + g(x)φ1(x)K∗
1= ε1(x)
(11)
bm− g(x)φ2(x)K∗
where ε1(x) =
ε2(x) ∈ ? are the minimum approximation errors
achieved by the fuzzy systems Eqs (7)–(8) with the
optimal parameters K∗
ProprietiesEqs(11)–(12)canbeviewedasnonlinear
version of the linear model-following control proper-
ties, where the fuzzy membership functions play the
role of scheduling parameters, i.e., for every sample
x(t), the quantities φ1(x) and φ2(x) quantify the con-
tributionofthelinearcontrolrulestothecontrollerout-
put. In some special cases, it is possible to achieveper-
fectapproximation[8,17],thenε1(x) = 0,ε2(x) = 0,
andperfectreferencemodelfollowingcanbeachieved.
The fuzzy systems Eqs (7)–(8) verify the following
assumptions:
2= ε2(x)
?ε11(x) ... ε1n(x)?
(12)
∈ ?nand
1and K∗
2.
C1. The membership functions µ2j(x) ∈ C1, and
???˙φ2(x)
{K1| ?K1? < κ1} and Ω2= {K2| |K2| <
κ2}, with κ1,κ2> 0, are given.
C3. The approximation errors are upper bounded
by |ε1(x)| ? ε1and |ε2(x)| ? ε2, for some
positive constants ε1and ε2.
??? < ϕ0, where ϕ0is a known constant.
C2. φ2(x)K∗
2
> 0, and upper bounds Ω1 =
AssumptionC1 indicatesthat membershipfunctions
µ2j(x), j = 1...m2shouldbechosenassmoothfunc-
tionswithboundedrateofchange,notethat˙φ2(x)may
not necessarilybe globallybounded,it will havea con-
stant bound within Ωxdue to the continuity of µ2j(x).
Assumption C2 is related to the assumption P1, i.e., to
the sign and bounds on the high-frequency gain g (x).
Assumption C3 follows from universal approximation
property of fuzzy systems.
Hence, using Eqs (11)–(12) in Eq. (10) becomes
˙ e = Ame + B[g(x)φ1(x)? K1x
(13)
+g(x)φ2(x)? K2r − ε1(x)x + ε2(x)r − d]
where? K1 = K∗
Then, using Eq. (12), Eq. (13) can be arranged as
1− K1and? K2 = K∗
2− K2are the
parameters estimation errors.
˙ e = Ame +
bm
φ2(x)K∗
2
B
?
+
φ1(x)? K1x + φ2(x)? K2r
φ2(x)K∗
2
?
(14)
1
B [α1e + ξ]
with
ξ = −α1xm+ α2r − φ2(x)K∗
α1= ε2(x)φ1(x)? K1+ ε1(x)φ2(x)K∗
2d
2
α2= ε2(x)φ2(x)K2
The fuzzy controller parameters are updated using
the following projection algorithm:
˙K1= γ1bmeTPB
?
φT
1(x)xT
−IK1tr?KT
?1 + ?K1?
1φT
?2
1(x)xT?
K1
(15)
κ1
?
˙K2= γ2bmeTPB
?
(16)
φT
2(x)r − IK2
K2KT
2φT
|K2|2
2(x)r
?
where

Page 4

50
N. Gol´ ea et al. / Nonlinear model reference adaptive control using Takagi-Sugeno fuzzy systems
IK1=


IK2=







0, if
??K1? < κ1or ?K1? = κ1and
?
eTPBtr?KT
eTPBtr?KT
1φT
1(x)xT?? 0
1φT
1, if
?K1? = κ1and
1(x)xT?> 0

0, if
?|K2| < κ2or |K2| = κ2
1, if |K2| = κ2and eTPBφ2(x)K2r > 0
where γ1,γ2 > 0 are design parameters, and P =
PT> 0 is the solution, for some Q = QT> 0, of the
Lyapunov equation
and eTPBφ2(x)K2r ? 0
AT
mP + PAm= −Q
The update laws Eqs (15)–(16) ensure that ?K1? ?
κ1and |K2| ? κ2 ∀t ? 0, if the parameters initial
values are set properly.
(17)
Proof: Toprovethat?K1? ? κ1,letVK=1
1
1K1
we have either ?K1? < κ1 or ?K1? = κ1 and
˙VK= γ1bmeTPBtr?KT
and
˙VK= γ1bmeTPBtr?KT
1 −
κ1
Since γ1bmeTPBtr?KT
Therefore, we get that ?K1? ? κ1, ∀t ? 0. Similar
analysis can be used to show that |K2| ? κ2, ∀t ? 0.
Now, consider the Lyapunov function
2?K1?2=
. If IK1= 0,
2tr?KT
?, then˙VK = tr
?
KT
1˙K1
?
1φT
1(x)xT?? 0,i.e.,wehave
always ?K1? ? κ1. If IK1= 1, we have ?K1? = κ1
1φT
?2
1(x)xT?> 0 and ?K1?
1(x)xT?×
tr?KT
(18)
?
?1 + ?K1?
1K1
?
?
1φT
= κ1, we get˙VK? 0 which implies that ?K1? ? κ1.
V =1
2φ2(x)K∗
+
2eTPe +
1
2γ1tr
?? KT
1? K1
?
1
2γ2? KT
2φ2(x)K∗
2? K2
(19)
The differentiation of Eq. (19) along Eq. (14) yields
˙V = −1
?
+eTPBα1e + eTPBξ
+1
γ1tr
Further, Eq. (20) can be arranged as
2eTQe +1
2
˙φ2(x)K∗
2eTPe
+bmeTPBφ1(x)? K1x + φ2(x)? K2r
˙? K1
?
(20)
?? KT
1
?
+1
γ2
? KT
2
˙? K2
˙V = −1
2φ2(x)K∗
+eTPBα1e + eTPBξ
+1
γ1tr
+1
γ2
Then, using the update laws Eqs (15)–(16),It can be
shownthatthirdandfourthtermsinEq.(21)arealways
? 0. Hence, Eq. (21) can be rewritten as
˙V ? −1
+eTPBα1e + eTPBξ
2eTQe +1
2
˙φ2(x)K∗
2eTPe
?? KT
2
1
?
γ1bmeTPBφT
2(x)r −˙K1
??
(21)
? KT
?
γ2bmeTPBφT
2(x)r −˙K2
?
2φ2(x)K∗
2eTQe +1
2
˙φ2(x)K∗
2eTPe
(22)
Further, using the fact that ?Bα1? ? α1(where α1
is th uper bound on α1) in Eq. (22) becomes
˙V ? −1
+α1eTPe +1
2φ2(x)K∗
2eTQe +1
2
˙φ2(x)K∗
2eTPe
2eTPBBTPe +1
2ξ2
(23)
which is upper bound by
˙V ? −1
+α1eTPe +1
2κ2eTQe +1
2ϕ0κ2eTPe
2eTPBBTPe +1
(24)
2ξ2
Finally, Eq. (24) can be arranged as
?
−1
If the matrices Q and P are chosen such as
?
forsome positivedefinite matrix Q1, Eq. (25)becomes
˙V ? −1
which can be upper bounded by
˙V ? −1
where λQ1is the smallest eigenvalue of Q1.
Then,˙V ? 0 outside the region bounded by
1
?κ2λQ1
˙V ? −1
2κ2eT
κ2PBBTP)e +1
Q − (ϕ0+ 2α1
κ2)P
(25)
2ξ2
Q −
ϕ0+ 2α1
κ2
?
P −
1
κ2PBBTP ? Q1 (26)
2κ2eTQ1e +1
2ξ2
(27)
2κ2λQ1|e|2+1
2|ξ|2
(28)
|e| ?
|ξ|
(29)

Page 5

N. Gol´ ea et al. / Nonlinear model reference adaptive control using Takagi-Sugeno fuzzy systems
51
The properties of the proposed adaptive fuzzy state
feedback approach are summarized by the following
theorem.
Theorem 1. The controlsystem composedby the non-
linear system Eqs (1)–(2) satisfying P1–P2, the refer-
ence model Eq. (3), the fuzzy controller Eq. (9) satis-
fying C1–C3, the update laws Eqs (15)–(16), and the
matrices P, Q satisfying Eqs (17), (26), guarantees the
following:
|x| ? |xm| + c1|ξ|
|u| ? κ1|xm| + κ2|r| + c2|ξ|
Proof: UsingEq.(29)andthefactthat|x| ? |xm|+|e|
yields
1
?κ2λQ1
Then Eq. (30) holds by setting c1=
(30)
(31)
|x| ? |xm| +
|ξ|
(32)
1
√κ2λQ1.
From Eq. (9) and the parameters bounds,the control
input is bound by
|u| ? κ1|x| + κ2|r|
where the fact that |φ1|,|φ2| ? 1 is used. Then, using
Eq. (32) in Eq. (33) becomes
|u| ? κ1|xm| + κ2|r| + κ1c1|ξ|
Therefore Eq. (31) is proved with c2= κ1c1.
Remarks:
(33)
(34)
1. It would be more simple to choose the same
membership functions in Eqs (7)–(8). But in
practice f(x) and g(x) maydependon different
statevariables,then,theproposedapproachpro-
vides a mean to design the fuzzy control input
with a reduced number of rules.
2. Compared to the direct approach in [6], this ap-
proachissimplersinceonlythefuzzystatefeed-
backisused,andnosupervisorytermisrequired
to ensure the variables boundedness.
3. comparedto[13]wheretheupperboundon ˙ g(x)
is required to achieve the control objective, this
scheme needs only the upper bound on˙φ2(x),
which is user designed parameter and therefore
is available to the evaluation.
4. The stability condition encountered in Eqs (17)
and (26) can be expressed as
P > 0
−AT
−PBBTP
mP − PAm− (ϕ0+ 2α1
> 0
κ2)P
κ2
(35)
Then, an LMI based stability analysis is carried
out to check wether the stability condition is
satisfied. In the case where Eq. (35) are not
satisfiedthereferencemodeldynamicshouldbe
modified (i.e., the eigenvalues of Am).
5. The proposed approach can be extended, with
little change, to the case where f(x) is of the
form f(x) = a0(x) + a(x)x with a0(x) is
bounded nonlinear function, that can be ac-
counted for by modifying the control input as
u = φ1(x)K1x + φ2(x)K2r
(36)
+φ3(x)K3
where the third term is added to compensate for
a0(x).
3.2. Observer-based design
The above design was based on full state informa-
tion. Since, this is seldom the case in many practical
situations, the problem will be solved using only the
system’s output as available information.
Let’s define the following observer
˙? e = Am? e − L? y
where ? e = xm− ? x is the estimated tracking error,
?l1l2... ln
Using the fact that ? y = C? e, where ? e = ? e − e is
further arranged as
˙? e = Am? e − LC? e
u = φ1(? x)K1? x + φ2(? x)K2r
Eqs (11)–(12), yields
(37)
? y = C? x
(38)
? y = y − ? y is the output estimation error, ? x, ? y are the
Hurwitz matrix.
estimated states and output, respectively, and LT=
?∈ ?nis selected such as [Am− LC] is
the state estimation error, the observer Eq. (37) can be
(39)
The control input is defined as
(40)
Then, introducing Eq. (40) in Eq. (4) and using
˙ e = Ame +
bm
φ2(x)K∗
2
B
?
+
φ1(? x)? K1? x + φ2(? x)? K2r
φ2(x)K∗
2
?
(41)
1
B [α3? e + α4? e + ζ]
with

Page 6

52 N. Goléa et al. / Nonlinear model reference adaptive control using Takagi-Sugeno fuzzy systems
ζ = −α4xm+ α5r − φ2(x)K∗
α3= φ2(x)K∗
2d
2(g(x)φ1(x)K∗
1− ε1(x))
α4= ε2(x)φ1(? x)? K1
−φ2(x)K∗
2
?
?
g(x)?φ2(x)K∗
g(x)?φ1(x)K∗
1− ε1(x)
?
α5= φ2(x)K∗
22+ ε2(x)
?
−ε2(x)φ2(? x)? K2
Now, substracting Eq. (41) from Eq. (39) gives
˙? e = [Am− LC]? e
−
φ2(x)K∗
2
bm
φ2(x)K∗
2
Using the fact that ? y = C? e and Eq. (42), the input-
? y = −G(s)
φ1(? x)? K1? x + φ2(? x)? K2r
−
φ2(x)K∗
2
where s is the Laplace operator and G(s)
C [sIn− Am+ LC]−1B.
Let’s define the polynomial H(s) = sn−1+
β2sn−2+ ... + βn−2s + βn−1, such that H−1(s) is
proper stable transfer function. Then, devising and
multiplying Eq. (43) by H(s) yields
and?φi(x) = φi(x) − φi(? x), i = 1,2.
1
B [α3? e + α4? e + ζ]
Bφ1(? x)? K1? x + φ2(? x)? K2r
(42)
−
??
output transfer function is given by
bm
φ2(x)K∗
2
?
?
(43)
G(s)
[α3? e + α4? e + ζ]
=
? y = −Go(s)
φ1f(? x)? K1? x + φ2f(? x)? K2r
−
φ2(x)K∗
2
where
bm
φ2(x)K∗
2
?
?
(44)
Go(s)
[α3f? e + α4f? e + ζf]
ζf= −α4fxm+ α5fr − φ2f(x)K∗
α3f= φ2f(x)K∗
2d
2(g(x)φ1(x)K∗
1− ε1(x))
α4f= ε2(x)φ1f(? x)? K1
−φ2f(x)K∗
2
?
g(x)?φ1(x)K∗
1− ε1(x)
?
α5f= φ2f(x)K∗
2
?
g(x)?φ2(x)K∗
2+ ε2(x)
?
−ε2(x)φ2f(? x)? K2
and Go(s) = H(s)G(s), φif(x) = H−1(s)φi(x),
φif(? x) = H−1(s)φi(? x),i = 1,2.
˙? e = [Am− LC]? e −
φ1f(? x)? K1? x + φ2f(? x)? K2r
−
φ2(x)K∗
2
? y = C? e
Hence, the augmented dynamic of Eqs (39), (45)–
(46) can be arranged as
Hence, Eq. (44) can be rewritten as
bm
φ2(x)K∗
2
Bo
?
?
(45)
1
Bo[α3f? e + α4f? e + ζf]
(46)
where BT
o=?0...01 β2... βn−1
?∈ ?n.
˙ z = Az−
?
−
? y = Cz
?01×nC?, α0=?α4fα3f
A =
0 Am− LC
The fuzzy controller parameters are updated by:
?
−IK1tr?KT
(1 + ?K1?)2
κ2
1
bm
φ2(x)K∗
2
B
φ1f(? x)? K1? x + φ2f(? x)? K2r
φ2(x)K∗
2
?
(47)
1
B?α?z+ ζf
?
(48)
with zT
=
?? eT? eT?, BT
?Am
=
?01×nBT
o
?, C =
?and
?
−LC
˙K1= −γ1bm? yφT
1f(? x)? xT
1φT
1f(? x) ? xT?
K1
(49)
?
˙K2= −γ2bm? y
(50)
?
φT
2f(? x)r − IK2
K2KT
2φT
|K2|2
2f(? x)r
?
where

Page 7

N. Gol´ ea et al. / Nonlinear model reference adaptive control using Takagi-Sugeno fuzzy systems 53
IK1=














0, if
??K1? < κ1or ?K1? = κ1
??K1? = κ1and
? ytr
and ? yφ2f(? x)K2r ? 0
and ? ytr
?
KT
1φT
1f(? x) ? xT?
? 0
1, if
?
KT
1φT
1f(? x) ? xT?
< 0
IK2=
0, if
?|K2| < κ2or |K2| = κ2
1, if |K2| = κ2and ? yφ2f(? x)K2r < 0
controller, the following lemma is required.
To establish the stability of the observer-basedfuzzy
Lemma [11]: If the transfer function Go(s) is strictly
positive real (SPR), then there exists two symmetric
positive definite matrices P and Q such that the fol-
lowing equations are verified
ATP + PA = −Q
(51)
BP = C
SinceG(s)isHurwitz,andthepairs{Am− LC,B}
and {Am− LC,C} are controllable and observable
respectively, H(s) can be always chosen such that the
Go(s) SPR condition is verified. On the other hand,
we have CT[sI2n− A]B = Go(s) which means that
Eqs (51)–(52) is verified by the way.
(52)
It can be verified, using the same approach of the
precedent section, that the update laws Eqs (49)–(50)
guarantee that ?K1? ? κ1and |K2| ? κ2∀t ? 0.
Consider the Lyapunov function
V =1
2φ2(x)K∗
1
2γ2
2zTPz +
1
2γ1tr
?? KT
1? K1
?
+
? KT
2? K2
(53)
whereP is solutionofEqs (51)–(52),forgivendefinite
positive matrix Q.
The differentiation of Eq. (53) along Eq. (47) yields
˙V = −1
2φ2(x)K∗
2zTQz +1
?
2
˙φ2(x)K∗
2zTPz
−bmzTPB
−zTPB[α0z + ζf] +1
+1
γ2
φ1f(? x)? K1? x + φ2f(? x)? K2r
γ1tr
?
?? KT
1
˙? K1
?
? KT
2
˙? K2
(54)
Then, using Eq. (52), Eq. (54) can be arranged as
˙V = −1
2φ2(x)K∗
?
−zTPB[α0z + ζf]
+1
γ1tr
Further, using trace properties Eq. (55) becomes
˙V = −1
−zTPB[α0z + ζf]
−1
−1
γ2
Further, using the update laws Eqs (49)-(50) and the
same analysis performedin the previous section, it can
be shown that the third and fourth terms in Eq. (56) are
? 0. Hence, Eq. (56) becomes
˙V ? −1
−zTPBα0z − zTPBζf
which is upper bounded by
˙V2? −1
+bzTPz +1
??Bα0
Further, Eq. (58) can be arranged as
˙V ? −1
?
+1
2ζ2
Hence, if the matrices Q and P are chosen such that
Q − (ϕ0+ 2b
for some positive definite matrix Q2, then
˙V ? −1
which is upper bonded by
2zTQz +1
2
˙φ2(x)K∗
2zTPz
?
−bm? y
φ1f(? x)? K1? x + φ2f(? x)? K2r
?? KT
2zTQz +1
(55)
1
˙? K1
?
+1
γ2
? KT
2
˙? K2
2φ2(x)K∗
2
˙φ2(x)K∗
2zTPz
γ1tr
?? KT
2
1
?
˙K1+ γ1bm? yφ1f(? x)T? xT??
˙K2+ γ2bm? yφT
? KT
?
2f(? x)r
?
(56)
2φ2(x)K∗
2zTQz +1
2
˙φ2(x)K∗
2zTPz
(57)
2κ2zTQz +1
2ϕ0κ2zTPz
2zTPBBTPz +1
??and
(58)
2ζ2
???,
f
where the facts that b ?
are used.
????φ1f
????φ2f
??? ? 1
2κ2zT
Q −
?
ϕ0+ 2b
κ2
?
P −
1
κ2PBBTP
?
z
f
(59)
κ2)P −
1
κ2PBBTP ? Q2
(60)
2κ2zTQ2z +1
2ζ2
f
(61)

Page 8

54
N. Gol´ ea et al. / Nonlinear model reference adaptive control using Takagi-Sugeno fuzzy systems
0
2
4
6
8
10
12
14
16
18
20
-1
-0.5
0
0.5
1
0
2
4
6
8
10
12
14
16
18
20
-1
-0.5
0
0.5
1
0
2
4
6
8
10
12
14
16
18
20
-30
-20
-10
0
10
20
Position (rd)
Velocity (rd/s)
Torque (Nm)
t (sec)
Fig. 1. State feedback (– plant, ... reference).
˙V ? −1
2κ2λQ2|z|2+1
2ζ2
f
(62)
where λQ2is the smallest eigenvalue of Q2.
Then,˙V ? 0 outside the region bounded by
1
?κ2λQ2
The properties of the observer-based fuzzy scheme
is summarized by the following theorem.
|z| ?
|ζf|
(63)
Theorem 2. The controlsystem composedby the non-
linear system Eqs (1)–(2) satisfying P1–P2, the refer-
ence model Eq. (3), the fuzzy controller Eq. (40) satis-
fying C1–C3, the state observer (37)–(38), The matri-
ces P and Q satisfying Eqs (51)–(52)and (60),and the
update laws Eqs (49)–(50), guarantees the following:
|? x| ? |xm| + c3|ζ|
|u| ? κ1|xm| + κ2|r| + c5|ζ|
Proof: Let’s define the following Lyapunov function
V1=1
2? eTP? e
(64)
|x| ? |xm| + c4|ζ|
(65)
(66)
(67)
where P is solution of Eq. (17).
The differentiation of Eq. (67) along Eq. (39) yields
˙V1= −1
which is upper bounded by
˙V1? −1
where λQ, λP are the smallest and the largest eigen-
value of Q and P, respectively.
Then,˙V1 ? 0 whenever |? e| is outside the region
|? e| ? 2λP
On the other hand, we have |z| =
which combined with Eq. (70) yields
?
Further,usingEqs (63)and(71)yields the following
upper bound
1
?
2? eTQ? e + ? eTPLC? e
2λQ|? e|2+ λP|L||? e||? e|
(68)
(69)
defined by
λQ|L||? e|
(70)
?
|? e|2+ |? e|2,
|? e|
1 + 4
?λP
λQ|L|
?2
? |z|
(71)
|? e| ?
κ2λQ2
?
1 + 4
?
λP
λQ|L|
?2?|ζf|
(72)

Page 9

N. Gol´ ea et al. / Nonlinear model reference adaptive control using Takagi-Sugeno fuzzy systems 55
0
2
4
6
8
10
12
14
16
18
20
-1
-0.5
0
0.5
1
0
2
4
6
8
10
12
14
16
18
20
-1
-0.5
0
0.5
1
0
2
4
6
8
10
12
14
16
18
20
-30
-20
-10
0
10
20
Position (rd)
velocity (rd/s)
Torque (Nm)
t (sec)
Fig. 2. Observer-based control (– plant, ... reference).
Then, introducing Eq. (72) in Eq. (70) becomes
2λP|L|/λQ
?
From Eq. (73) and the fact that |? x| ? |xm| + |? e| we
|? x| ? |xm|
?
which yields Eq. (64) with
2λP|L|/λQ
?
Now, usingthe fact that |x| ? |? e|+|? x| with Eqs (72)
|x| ? |xm|
+
?
|? e| ?
κ2λQ2
?
1 + 4
?
λP
λQ|L|
?2?|ζf|
(73)
get
(74)
+
2λP|L|/λQ
?
κ2λQ2
1 + 4
?
λP
λQ|L|
?2?|ζf|
c3=
κ2λQ2
?
1 + 4
?
λP
λQ|L|
?2?.
and (74) yields
(75)
2λP|L|/λQ+ 1
?
κ2λQ2
1 + 4
?
λP
λQ|L|
?2?|ζf|
Then Eq. (65) is verified with
c4=
2λP|L|/λQ+ 1
?
?
κ2λQ2
1 + 4
?
λP
λQ|L|
?2?.
UsingEq.(74)withthefactthat|u| ? κ1|? x|+κ2|r|
|u| ? κ1|xm| + κ2|r| + κ1c4|ζf|
Hence Eq. (66) is proved with c5= κ1c4.
Remarks:
gives
(76)
1. Other observers such as the high gain observer
may be used, but in this case the control input
should be saturated to avoid the effect of initial
large values of the estimated states.
2. The satisfaction of Eqs (51)–(52) and (60), can
be expressed as an LMIs problem of the form
P > 0
?
where Ξ = ATP − PA − (ϕ0+ 2b
1
κ2PBBTP. ThefeasibilityofEq.(77)depends
onthe choiceofthe referencemodelEq. (3)and
the observer Eqs (37)–(38) dynamics.
Ξ
CT− PB
0
C − BTP
?
> 0
(77)
κ2)P −

Page 10

56
N. Gol´ ea et al. / Nonlinear model reference adaptive control using Takagi-Sugeno fuzzy systems
0
2
4
6
8
10
12
14
16
18
20
-1
-0.5
0
0.5
1
0
2
4
6
8
10
t (sec)
12
14
16
18
20
-1
0
1
2
Position (rd)
Velocity (rd/s)
Fig. 3. Observer performance (– plant, ... observer).
4. Simulation
The validity of proposedfuzzy adaptivecontroller is
verified on the inverted pendulum, described by
˙ x2= f(x1,x2) + g(x1)u
(78)
with f(x1,x2)
g(x1) =
the angular position, velocity and input force, respec-
tively. Remark that f(x) = a(x)x, with xT= [x1
x2] and a1(x) =
l(4
−
mc=1 Kg, m = 0.1 Kg, l = 0.5 m and g = 9.8 m/s2.
The reference model is given by am= [100 20] and
bm= 100.
The fuzzy adaptive control design steps are:
=
(mc+m)g sinx1−mlx2
l(4
cos x1
3(mc+m)−mcos2x1)and x1, x2and u are
2cos x1sinx1
3(mc+m)−mcos2x1)
,
l(4
(mc+m)g
3(mc+m)−mcos2x1)
sinx1
x1, a2(x) =
mlx2cosx1sinx1
3(mc+m)−mcos2x1). The system parameters are:
l(4
1. Define for x1, x2three fuzzy sets characterized
by: µj1(xj) = 1/(1 + exp(10(xj+ 0.5))),
µj2(xj) = exp(−x2
exp(−10(xj− 0.5))) j = 1,2. Note that the
pendulum velocity x2is replaced by ? x2.
R1
Then uf1= K1j? x,j = 1...9
j = 1...3
j/0.2), µj3(xj) = 1/(1 +
2. Define two fuzzy control if-then rules as
j: If x1is X1
jand ? x2is X2
jThen uf2= kj2r,
j
R2
j: If x1is X1
Then, the control input is given by

6
0
5
10
t(sec)
15
20
0
2
4
0
5
10
t(sec)
15
20
0
2
4
IIK1II
IK2I
Fig. 4. Fuzzy controller parameters evolution.
u = φ1(x1, ? x2)K1? x + φ2(x1)K2r
arethenormalizedmembershipfunctions;K1∈
?9×2and K2∈ ?3×1are the fuzzy controller
adjustable parameters. The Update laws step
sizes are γ1= 500 and γ2= 100.
3. The required bounded are g = 1.12, g = 1.47,
κ1 = 6, κ2 = 3.5, ϕ0 = 2.81, ε1 = 0.1 and
ε2= 0.05.
4. For the state feedback control, using the above
reference model and upper bounds and solving
the LMIs (35) yields P and Q.
5. For the observerdesign, select LT=?400 40?.
4). Solving LMIs Eq. (77), for the selected
reference model and observer together with the
fixed upper bounds, yields P and Q.
where φ1(x1, ? x2) ∈ ?1×9and φ2(x1) ∈ ?1×3
ToensuretheSPRconditionselectH(s) = (s+

Page 11

N. Gol´ ea et al. / Nonlinear model reference adaptive control using Takagi-Sugeno fuzzy systems 57
Thetrackingperformanceforthestatefeedbackcase
is shownin Fig. 1. It is clearthat thetrackingerrorsare
rapidly reduced, and the control input is moderate and
smooth. Figure 2 shows the observer based tracking
performance, where after a short transient period the
tracking errors convergeto bounded small values. The
estimated states are depicted in Fig. 3, from which it
is clear that the estimations converge fast to the real
states. Figure4depictstheevolutionofthenormsofthe
fuzzycontrollerparameters,intheobserverbasedcase.
As predicted by the theory the controller parameters
converge to bounded values.
5. Conclusion
This paper has presented an extension of the model
reference control approach to nonlinear plants using
TS fuzzy systems. Both full state and observer-based
approachwereanalyzedandtheirstabilityproved. The
advantages of this approach over the previously de-
signed are that no supervisory control is needed to en-
sure the closed loop stability, and the knowledgeof the
high-frequency gain derivative bound is not required.
Future work is directed to extend this control design to
the multivariable nonlinear systems.
References
[1]J.J. Buckly, Sugeno type controllers are universal controllers,
Fuzzy Sets and Syst 53 (1995), 299–303.
Y.C. Chen and C.C.Teng, A model reference control structure
using a fuzzy neural networks, Fuzzy Sets and Syst 73 (1995),
291–312.
[2]
[3]C.S. Chen and W.L. Chen, Robust model reference adaptive
control of nonlinear systems using fuzzy systems, Int. J. Syst.
Sci 27 (1996), 1435–1442.
C. Fantuzzi and R. Rovatti, On the approximation capabilities
of the homogeneous Takagi-Sugeno model, in Proc. 5th IEEE
Int. Conf. Fuzzy Syst., New Orleans, LA, 1996, 1067–1072.
N. Gol´ ea and A. Gol´ ea, Fuzzy adaptive control of multivari-
able nonlinear systems, IEEE Conference on Fuzzy Systems
IEEE-FUZZ’02, 2002, 209–213.
N. Gol´ ea, A. Gol´ ea and K. Benmahammed, Fuzzy model ref-
erence adaptive control, IEEETrans.FuzzySyst.10(4) (2002),
436–444.
N. Gol´ ea, A. Gol´ ea and K. Benmahammed, Stable indirect
fuzzy adaptive control, Fuzzy Sets & Systems 137 (2003),
353–366.
S. Kawamoto, K. Tada, N. Onoe, A. Ishigame and T.
Taniguchi, Construction of exact fuzzy system for nonlinear
systems and its stability analysis, 8th Fuzzy systems symp.,
Hirochima, Japan, 1992, 517–520.
S.H.Kawaji, Model following servo control for fuzzy linguis-
tical systems, Proc. IEEE Conf. Fuzzy Syst. 2 (1996), 1420–
1426.
W.-S. Lin and C.-H. Tsai, Neurofuzzy model-following con-
trol of MIMO nonlinear systems, IEE Proc.-Control Theory
Appl. 146 (1999), 157–164.
Y.D. Landau, Adaptive control: The model reference ap-
proach, New York: Marcel Dekker, 1979.
K.M. Passino and S. Yurkovich, Fuzzy Control, Addison-
Wesley, 1998.
J.F. Spooner and K.M. Passino, Stable adaptive control using
fuzzy systems and neural networks, IEEE Trans. Fuzzy Syst.
4 (1996), 339–359.
T. Takagi and M. Sugeno, Fuzzy identification of systems and
its application to modeling and control, IEEE Trans. Syst.,
Man, Cybern. 15 (1985), 116–132.
K. Tanaka and M. Sugeno, Stability analysis and design of
fuzzy control systems, Fuzzy Sets Syst 45 (1992), 136–156.
T.K. Yin and C.S. Goerge Lee, Fuzzy model-reference adap-
tive control, IEEETrans.Syst.Man, Cybern. 25(1995), 1606–
1615.
L.X.Wang, A course in fuzzy systems and control, Englewood
Cliffs, NJ: Prentice-Hall, 1997.
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]