Stable indirect fuzzy adaptive control

Abstract

This paper investigates new fuzzy model-based observer adaptive control for multi-input multi-output continuous-time nonlinear systems. The proposed adaptive scheme uses Takagi–Seguno (TS) fuzzy models to estimate the plant states and dynamics. Using stability arguments, it is shown that the proposed scheme is globally asymptotically stable. The observation and tracking errors are shown to converge asymptotically to zero, despite the presence of external disturbances and approximation errors. The performance of the developed approach is illustrated, by simulation, on two-link robot model.

Full text

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ARTICLE IN PRESS
Fuzzy Sets and Systems()–
www.elsevier.com/locate/fss
Stable indirect fuzzy adaptive control
N. Goleaa, A. Goleaa;b; ∗, K. Benmahammedc
aElectrical Engineering Institute, Oum-El-Bouaghi University, Oum-El-Bouaghi, 04000 Algeria
bElectrical Engineering Institute, Biskra University, 07000 Biskra, Algeria
cElectronics Institute, S? etif University, 19000 Algeria
Received 15 September 1999; received in revised form 8 October 2001; accepted 28 May 2002
Abstract
This paper investigates new fuzzy model-based observer adaptive control for multi-input multi-output
continuous-time nonlinear systems. The proposed adaptive scheme uses Takagi–Seguno (TS) fuzzy mod-
els to estimate the plant states and dynamics. Using stability arguments, it is shown that the proposed scheme
is globally asymptotically stable. The observation and tracking errors are shown to converge asymptotically
to zero, despite the presence of external disturbances and approximation errors. The performance of the de-
veloped approach is illustrated, by simulation, on two-link robot model. c ? 2002 Elsevier Science B.V. All
rights reserved.
Keywords: Fuzzy control; Fuzzy systems model; Adaptive control; Observer; Nonlinear systems
1. Introduction
Fuzzy adaptive control is generally applicable to plants that are mathematically poorly modeled
and where heuristics do not provide enough information to specify all the parameters of the con-
trol problem. Conceptually, adaptive fuzzy systems combine linguistic information from experts with
numerical information from sensors [9,17]. Despite their learning capabilities and practical implemen-
tations, the earlier fuzzy adaptive systems su?er from the lack of stability analysis, i.e. the stability of
closed-loop system is not guaranteed and the learning process do not lead to a well-de?ned dynamic
[9]. Recently, an important class of a fuzzy adaptive systems have been developed, and their stability
is guaranteed using the Lyapunov theory (see [3,4,12,7,14,15,8,17] and references therein). Early
works were concerned with the single-input single-output (SISO) nonlinear systems case with full
information on the state Variables. Lastly, the multi-input multi-output (MIMO) nonlinear systems
problem was also investigated, and some schemes were proposed [7,14,8], and an observer-based
∗Corresponding author. Electrical Engineering Institute, Oum-El-Bouaghi University, Oum-El-Bouaghi, 04000 Algeria.
Tel.: +213-4-745-036; fax: +231-4-861-331.
0165-0114/02/$-see front matter c ? 2002 Elsevier Science B.V. All rights reserved.
PII: S0165-0114(02)00279-8

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2N. Golea et al./Fuzzy Sets and Systems()–
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fuzzy adaptive control scheme was proposed in [15]. Apart from few works [9,8], most of the them
used the Mamdani model to approximate the plant or the controller dynamics. Both of Mamdani
and TS fuzzy models [13] are proven to be universal approximators [5,9,10,17,18], however, the
major advantage of TS fuzzy model is its representation power, i.e. it is capable of describing a
highly nonlinear plants using few rules. Moreover, since the output of the model has an explicit
functional form, analytical knowledge about the plant dynamic can be incorporated in the rules, and
it is possible to analyze TS fuzzy system behavior using conventional tools.
This investigation proposes a stable indirect fuzzy adaptive control for MIMO continuous-time
nonlinear systems. A TS fuzzy model-based observer is developed to approximate the nonlinear plant
dynamic and estimate its state variables. This adaptive scheme presents the advantages that qualitative
and analytic information about the plant operating can be used to design the fuzzy model rules, and
few rules (i.e., parameters) are to be tuned which allows fast control update, which is a limitings
factor for some applications. The proposed adaptive scheme achieves asymptotic tracking of a stable
reference model, and the tracking and observation errors are shown to converge asymptotically to
zero. The performance of this approach is evaluated on a two-link robot model.
2. Problem statement
We consider the MIMO nonlinear systems class given by
˙ xi= Aixi+ bi[fi(x;u) + di];
yi= cT
i = 1:::p;
(1)
ixi;(2)
where xi∈Rniis the ith subsystem state vector, xT=[xT
n2+ ··· + np, uT=[u1 ::: up]∈Rpis the input vector, and yi is the ith measured output, fi is a
smooth unknown nonlinear function, di is bounded external disturbance, and Ai, bi; ciare de?ned
as
?0
For system (1) is it assumed that @fi=@ui¿0 in the relevant control region, this is equivalent to the
assumption of known input gain sign. The case of @fi=@ui¡0 can also be handled. The nonlinear
a?ne systems can be viewed as special case of the more general class (1).
The stable, LTI and controllable reference models are de?ned by the following state equations:
1::: xT
p]∈Rnis the state vector, with n=n1+
Ai=
Ini−1
00
?
ni×ni
;bT
i= [0:::01]1×ni;cT
i= [10 :::0]1×ni:
˙ xmi= Amixmi+ bmiri;
where xmi∈Rniis the state vector, ri is a bounded reference input, and Ami, bmiare given by
?0
with ami∈Rni.
The control problem can be stated as that of designing the control inputs u such that the states of
the plant (1) follow those of the reference model (3), under the condition that all involved signals
i = 1:::p;(3)
Ami=
Ini−1
−ami
?
ni×ni
;bT
mi= [0 :::0bnim]1×ni

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N. Golea et al./Fuzzy Sets and Systems()–3
in the closed-loop remain bounded. Since the nonlinear function is not known, the system states are
not available, and the input do not appear explicitly in (1), a TS fuzzy model-based observer will
be used to realize the control objective.
3. Fuzzy adaptive approach
3.1. Fuzzy modelling
The fuzzy models to be considered here, are multi-input single-output TS fuzzy system character-
ized by a set of If–Then fuzzy rules of the form
Rk
i: If ziis Zi
kThenˆfi= ai
kx + bi
ku;k = 1:::mi;(4)
where mi is the ith fuzzy model rules number, ai
parameter vectors, and zi∈Rqiis the fuzzy model input vector, assumed here to be composed of
measured signals only. The fuzzy sets Zi
The ?nal output of the fuzzy model (4) is inferred as follows:
k∈Rnand bi
k∈Rpare the kth rule consequence
koperate a fuzzy partition of the fuzzy model input space.
ˆfi=
?mi
k=1?i
k(zi)(ai
?mi
kx + bi
k(zi)
ku)
k=1?i
;(5)
where ?i
at least one active rule, i.e.?mi
ˆfi= ?i(?ix + ?iu);
where
k(zi) is the grade of membership of ziin Zi
k. In this paper, it assumed that there exist always
k=1?i
k(zi)¿0.
The fuzzy model output (5) can also be written in following matrix form:
(6)
?i=



ai
ai
...
ai
1
2
mi



mi×n
;?i=



bi
bi
...
bi
1
2
mi



mi×p
and ?iis the normalized ?ring strengths vector given by
?i=
1
?mi
k=1?i
k
[?i
1
?i
2
:::?i
mi]:
Following the universal approximation results [2,5,10], the fuzzy model (6) is rich and able to
approximate the nonlinear function fi(:) on a compact operating space to any degree of accuracy.
Next, we de?ne the optimal fuzzy model parameters ?∗
iand ?∗
i be such that
sup
x;u∈Xc
|fi(x;u) −ˆfi(x;u;?∗
i;?∗
i)| ¡ ?i;(7)

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where ?iis an arbitrary positive constant, and Xcis a compact region of the system space. It follows
that, the nonlinear system (1) can be represented, using the optimal fuzzy model as
˙ xi= Aixi+ bi[?i(?∗
where !i is the minimum approximation error achieved by the optimal fuzzy model.
Using the actual parameters estimate, (8) can be rewritten as
˙ xi= Aixi+ bi[?i(?ix + ?iu) + ?i(˜?ix +˜?iu) + ?i];
where ?i=di+ !i, and ˜?i=?∗
ix + ?∗
iu) + di+ !i]; (8)
(9)
i− ?i, ˜?i=?∗
i− ?i are the parameters estimation errors.
3.2. Observer-based fuzzy control
To estimate the state vector of the nonlinear system (1), we de?ne, based on the actual estimated
fuzzy model, the following state observers:
˙ˆ xi= Aiˆ xi+ bi[?i(?iˆ x + ?iu) + usi] + li˜ yi;
ˆ yi= cT
i = 1:::p;
iˆ xi; (10)
where ˆ xiis the estimated state vector and ˜ yi=(yi− ˆ yi) is the ith output estimation error. The vector
li∈Rnispeci?es the ith observer dynamic, it is chosen such that [Ai− licT
[1,6]. The additional term usiis introduced to re?ect the uncertainty term in (9).
Subtracting (10) from (9) yields the following observation error dynamic:
˙˜ xi= [Ai− licT
where ˜ xi=xi− ˆ xiis the state estimation error.
To eliminate the unavailable state vector x from (11), we use the fact that
˜?ix =˜?iˆ x + ?∗
i] is a Hurwitz matrix
i]˜ xi+ bi[?i?i˜ x + ?i(˜?ix +˜?iu) − usi+ ?i]; (11)
i˜ x − ?i˜ x:(12)
Then, introducing (12) in (11) yields
˙˜ xi= [Ai− licT
The term in (13) involving the optimal parameters re?ects the uncertainty resulting from the com-
bination of the errors on the states and the parameters.
Using the fact that ˜ yi=cT
˜ yi= cT
i]˜ xi+ bi[?i?∗
i˜ x + ?i(˜?iˆ x +˜?iu) − usi+ ?i]: (13)
i˜ xiand (13), the transfer function is given by
i[sIni− Aoi]−1bi[?i?∗
i˜ x + ?i(˜?iˆ x +˜?iu) − usi+ ?i](14)
where s is the Laplace operator, and
Aoi= Ai− licT
Let’s de?ne the polynomial Hi(s)=sni−1+ ?2sni−2+ ··· + ?ni−1s + ?ni, such that H−1
stable transfer function. Then, devising and multiplying (14) by Hi(s) yields
˜ yi= Gi(s)[?f
i:(15)
i
(s) is proper
i?∗
i˜ x + ?f
i(˜?iˆ x +˜?iu) − uf
si+ ?f
i];(16)

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where
Gi(s) = Hi(s)cT
i[sIni− Aoi]−1bi
(17)
and
?f
uf
?f
i= H−1
si= H−1
i= H−1
i
(s)?i;
(s)usi;
(s)?i:
i
i
Hence, (16) can be rewritten in the following state-space form:
˙˜ xi= Aoi˜ xi+ boi[?f
i?∗
i˜ x + ?f
i(˜?iˆ x +˜?iu) − uf
si+ ?f
i];
˜ yi= cT
i˜ xi;(18)
where bT
The estimated tracking error ˆ ei=xmi− ˆ xi, using (3) and (10), is given by
˙ˆ ei= Amiˆ ei− bi[?i(?iˆ x + ?iu) + amiˆ xi− bnimri+ usi] − licT
Using the estimated parameters and states, we de?ne the following certainty equivalent control inputs:
oi=[0 ::: 0 1 ?2 ::: ?ni]∈Rni.
i˜ xi: (19)
ui=
p
?
j=1
?ij[bnjmrj− ?j?jˆ x − amjˆ xj− usj];i = 1:::p; (20)
where [?ij]:=B−1and
B =



b11 ··· b1p
...
bp1··· bpp
...
...



(21)
with
bij= ?i?i
j; (22)
where ?i
tracking error dynamics:
˙ˆ ei= Amiˆ ei− licT
Eq. (23) indicates that, under the control inputs (20), the estimated tracking errors are governed by
stable state-space equations whose inputs are the observation errors.
jis the jth column of ?i. The substitution of (20) in (19) yields the following estimated
i˜ xi;i = 1:::p:(23)
4. Stability analysis
To establish the stability of the proposed adaptive approach, the following lemma and assumptions
are used.

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Assumption 1. The parameters ?∗
bounds.
i
are upper bounded by tr[?∗
i?∗T
i ]6M?i, where M?iare known
Assumption 2. The external disturbances and the approximation errors are upper bounded by |di|¡d0i
and |!i|¡!0i, respectively, where d0i and !0i are known constants.
Lemma 1 (Vidyasagar [16]). If the transfer function Gi(s)=cT
(SPR), then there exists two symmetric positive de?nite matrices Pi and Qi such that the following
equations are veri?ed:
i(sIni−Aoi)−1boiis strictly positive real
PiAoi+ AT
oiPi= −Qi;
i:
(24)
bT
oiPi= cT
(25)
In our case, since Aoi is Hurwitz, and the pairs (Aoi;boi); (Aoi;cT
respectively, Hi(s) can be always chosen such that the SPR condition is veri?ed. In what follows,
we consider that this condition is ful?lled.
Consider the following Lyapunov function:
i) are controllable and observable,
V =
p
?
i=1
Vi
(26)
with
Vi=1
2˜ xT
iPi˜ xi+
1
2?i1tr[˜?i˜?
T
i] +
1
2?i2tr[˜?i˜?
T
i];(27)
where Pi are symmetric positive de?nite matrices solution of (24) and (25) for given symmetric
de?nite positive matrices Qi.
The di?erentiation of (27) along (18) gives
˙ Vi=−1
2˜ xT
iQi˜ xi+
1
?i1tr[˜?i˙˜?
T
i] +
1
?i2tr[˜?i˙˜?
T
i]
+ ˜ xT
iPiboi[?f
i?∗
i˜ x + ?f
i(˜?iˆ x +˜?iu) − uf
si+ ?f
i]:(28)
Exploiting (2) and (25), (28) can be arranged as
˙ Vi=−1
2˜ xT
iQi˜ xi+ ˜ xT
ici?f
i?i˜ x − ˜ yi(uf
si− ?f
1
?i2tr[˜?i(˙˜?
i)
+1
?i1tr[˜?i(˙˜?
T
i+ ?i1ˆ x?f
i˜ yi)] +
T
i+ ?i2’f
iku ˜ yi)]:(29)
New, let us de?ne the following parameters update laws:
˙?i= ?i1(?f
i)Tˆ xT˜ yi; (30)
˙?i= ?i2(?f
i)TuT˜ yi:(31)

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Then, introducing (30) and (31) in (29), and using the fact that˙˜?i= −˙?i and˙˜?i= −˙?i yields
˙ Vi= −1
The switching control terms are chosen as
2˜ xT
iQi˜ xi+ ˜ xT
ici?f
i?∗
i˜ x − ˜ yi(uf
si− ?f
i): (32)
usi= ?0isgn( ˜ yi);(33)
where ?0i=!0i+d0iis the upper bound on the ith uncertainty term. Then, substituting (33) in (32)
yields
˙ Vi6 −1
2˜ xT
iQi˜ xi+ ˜ xT
ici?f
i?∗
i˜ x:(34)
Summing (34) over i=1:::p, we get
˙ V 6 −1
2
p
?
i=1
˜ xT
iQi˜ xi+
p
?
i=1
˜ xT
ici?f
i?∗
i˜ x;(35)
which can be arranged as
˙ V 6 −1
2˜ xTQ˜ x + ˜ xTC??∗˜ x; (36)
where Q=block-diag[Q1;:::;Qp], ?=block-diag[?f
1;:::;?f
p] and
?∗=



?∗
...
?∗
1
p


:
Applying norms to (36) yields
˙ V 6 −1
2?min(Q)|˜ x|2+ M?|˜ x|2; (37)
where we have used the fact that: |C|=1 and |?|61, ?min(Q)= mini{?min(Qi)}, and M?=
?p
?min(Q) − 2M?¿ ? ¿ 0
for some positive constant ?, we get
i=1M?i.
If the matrices Qi are chosen such that
(38)
˙ V 6 −1
The following theorem resumes the stability results for this approach.
2?|˜ x|2: (39)
Theorem 1. The feedback system composed of the nonlinear system (1) and (2), the reference
models (3), the fuzzy model-based observers (10), the control inputs (20) and the update laws

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(30) and (31), is globally asymptotically stable, and the tracking and observation errors converge
to zero.
Proof. From (39) we have that ˙ V is always negative in the ˜ x space if ˜ x?=0, which implies that
˙ Vi; ˜ xi;˜?i and ˜?i∈L∞ for i=1:::p. From the boundedness of ˜ xi, and since (23) is stable linear
system it follows that ˆ ei∈L∞, which implies that ˆ xi∈L∞ for i=1:::p (since xmiis bounded).
Thus, all variables in the right-hand side of (18) are bounded, then˙˜ xi∈L∞, therefore ˜ xiis uniformly
continuous. The integral of (39) yields
V(∞) − V(0) 6 −?
2
?∞
0
|˜ x|2dt(40)
since V(∞)¿0, the inequality (40) implies that
?∞
hence, ˜ xi∈L2, which with the Barbalat’s lemma [11] yields that limt→∞˙ V =0 and limt→∞ ˜ xi=0,
i.e. limt→∞ˆ xi=xifor i=1:::p.
From (23), the convergence of ˜ xiimplies the convergence of ˆ ei, i.e. limt→∞ ˆ ei=0. Hence, since
ei= ˆ ei− ˜ xiit follows that the tracking errors converge to zero, limt→∞ ˆ ei=0 for i=1:::p.
0
|˜ x|2dt 62
?V(0);(41)
Remark 1. Since the additional switching terms usiare discontinuous, control chattering may occur,
which is undesirable in practice because it involves high control activity and may excite high-
frequency plant dynamics. To overcome this problem, the switching control terms can be smoothed
as
usi= ?0i
?
1 +?i
?i
?
˜ yi
| ˜ yi| + ?i;(42)
where ?i;?i¿0 are design parameters. The constant ?iis selected based on engineering consideration
to achieve the admissible tracking error amplitude. The ratio ?i=?i determines the gain amplitude.
Replacing (42) in (32) gives
˙ Vi= −1
2˜ xT
iQi˜ xi+ ˜ xT
ici?f
i?∗
i˜ x − | ˜ yi|
?
?0i
?
1 +?i
?i
?
| ˜ yi|
| ˜ yi| + ?i
− ?f
isgn( ˜ yi)
?
: (43)
If | ˜ yi|¿?ithen the third term in (43) is ¿0, which yields the same result as in (34). On the other
hand, if | ˜ yi|¡?i (43) becomes
˙ Vi6 −1
Hence summing (44) over i=1:::p, and using the same notation as in (36) yields
2˜ xT
iQi˜ xi+ ˜ xT
ici?f
i?∗
i˜ x + ?i|?f
i|:(44)
˙ V 6 −1
2˜ xTQ˜ x + ˜ xTC??∗˜ x +
p
?
i=1
?i|?f
i|: (45)

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If the condition (38) holds, we get
˙ V 6 −?
2|˜ x|2+
p
?
i=1
?i|?f
i|: (46)
Then ˙ V is negative outside the region de?ned by

?(˜ x) =

˜ x?|˜ x| 6
?
2?p
i=1?i?0i
?


:(47)
This result indicates that the observation errors converge to the bounded region de?ned by (47).
Conducting a similar analysis as above, we can show that ˆ e and e are also bounded.
Remark 2. The above stability result is achieved under the assumption that the fuzzy models are
well designed such that B is always invertible. If the above condition is not ful?lled, the control
inputs (20) may be large at certain time, and the internal dynamic of (19) will be instable. More,
if the switching control terms (42) is used the parameters boundedness is no longer ensured. To
prevent this situation, the update laws (30) and (31) should be modi?ed.
To assure the boundedness of the parameters ?iand ?iand the invertibility of the matrix B, the
following additional assumption is required.
Assumption 3. The matrix B elements are bounded by |bij|¡bH
constants.
Using the projection algorithm [17], the update laws (30) and (31) are modi?ed as
ijfor i;j=1:::p, where bH
ijare known
˙?i=











?i1(?f
i)Tˆ xT˜ yi
if
(tr[?i?T
or (tr[?i?T
i] ¡ M?i)
i] = M?iand ?f
i?iˆ x ˜ yi6 0);
?i1(?f
i)Tˆ xT˜ yi− ?i1
?f
tr[?i?T
i?iˆ x ˜ yi
i]?i
if tr[?i?T
i] = M?iand ?f
i?iˆ x ˜ yi¿ 0:
(48)
˙?
i
j=



?i2(?f
i)Tuj˜ yi
if (|?i
if |?i
j| ¡ bH
ij) or (|?i
j| = bH
ijand ?f
i?i
juj˜ yi6 0);
?i2(?f
i)Tuj˜ yi− ?i2
?f
i?i
|?i
juj˜ yi
j|2
?i
jj| = bH
ijand ?f
i?i
juj˜ yi¿ 0:
(49)
The stability of fuzzy adaptive system, with the modi?ed update law, can be veri?ed using the same
analysis. It can be shown that, tr[?i?T
?i(0) and ?i
i]6M?iand |?i
j| 6 bH
ijfor i;j=1:::p, if the initial values of
j(0) are chosen properly.
5. Design example
The proposed fuzzy adaptive system is applied to the two-link robot model described by
M(x3)
?˙ x2
˙ x4
?
+ C(x)
?x2
x4
?
+ G(x1;x3) =
?u1
u2
?
;(50)

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10N. Golea et al./Fuzzy Sets and Systems()–
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0
1
-3
0
3
xj (rad)
?zk (xj)
Fig. 1. Fuzzi?cation for the fuzzy models inputs x1;x3.
where
M =
?1
3m1l2+4
1
3m2l2+1
3m2l2+ m2l2cos(x3)
2m2l2cos(x3)
1
3m2l2+1
2m2l2cos(x3)
1
3m2l2
?
;
C =
?−m2l2sin(x3)x4−1
2m2l2sin(x3)x4
0
1
2m2l2sin(x3)x2
?
G =
?1
2m1glcos(x1) +1
2m2glcos(x1+ x3) + m2glcos(x1)
1
2m2glcos(x1+ x3)
?
:
xT=[x1 x2 x3 x4] is the vector of the angular positions and velocities and u is the vector of control
torques. The physical parameters values are l=1 m, m1=m2=1 kg and g=9:81 m=s2.
The dynamic of (50) can be reformulated as in (1), with
?f1(x;u)
f2(x;u)
?
= M−1
?
−C
?x2
x4
?
− G + u
?
(51)
Taking the angular positions x1 and x3 as the measured outputs, we construct the following fuzzy
approximations of (51):
If z1is Z1
If z2is Z2
kthenˆf1= a1
kthenˆf2= a2
kˆ x + b1
kˆ x + b2
ku
ku
;k = 1:::9;(52)
where Z1
in Fig. 1, with the input vectors z1=z2=[x1 x3]. The outputs of (52) are given by
ˆf1= ?1(?1ˆ x + ?1u);
ˆf2= ?2(?2ˆ x + ?2u);
where ?1(?2) is the 1×9 vector of the normalized strengths, ?1(?2) and ?1(?2) are, respectively,
9×4 and 9×2 consequences parameter matrices. Based on the fuzzy approximations (53) and (54),
we construct the state observers as in (10) with lT
we take H1(s)=H2(s)=(s + 4). The reference models are chosen as am1;2=[1 2], bm1;2=1.
The parameters of both the fuzzy models are updated using the update laws (48) and (49), with the
update gains ?11=?22=50 and ?12=?21=5. Based on physical considerations on the robot model,
kand Z2
kare fuzzy sets constructed by combining the fuzzy sets of the fuzzi?cation depicted
(53)
(54)
1=lT
1=[20 100]. To ensure the SPR condition

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N. Golea et al./Fuzzy Sets and Systems()–11
0102030405060 708090100
−4
−2
0
2
4
Link 1
position (rad)
0 1020 30405060 7080 90 100
−4
−2
0
2
4
velocity (rad/sec)
010 2030 4050 6070 8090 100
−40
−20
0
20
torque (Nm)
Fig. 2. Link 1 tracking performance: (—) robot; (:::) reference.
the upper bounds on the fuzzy models parameters are ?xed to M?1=M?2=1, bH
bH
=0:8 and ?1
!01=!02=1 and the switching terms are selected as in (42) with ?1=?1=0:003 and ?2=?2
=0:006.
In this simulation, the initial states values are taken as x(0)=[1 2 0:5 −1]; xm(0)=0 and
ˆ x(0)=0 for robot model, the fuzzy model-observers and the reference models. Figs. 2 and 3 show
the tracking performance for link 1 and link 2, respectively. It can be seen that the tracking error is
reduced rapidly without large initial picks. After the initial pick to compensate for the initial errors,
the developed torques are seen be smooth. The observation dynamics for the links 1 and 2, are de-
picted in Figs. 4 and 5, respectively. It is clear that the observed states converge rapidly to the real
ones, with small oscillations at the beginning due to the large initial errors and the switching terms.
11=1:73, bH
1(0)=0:2, ?2
22=13:93,
12=bH
21=4:38. The fuzzy model parameters are initialized as ?1(0)=?2(0)=0, ?1
2(0)=?2
2(0)
1(0)=0. The upper bounds on the approximation errors are estimated to be
6. Conclusion
A new fuzzy indirect adaptive control for MIMO nonlinear continuous-time systems is developed.
The proposed TS fuzzy model-based observer was shown to be e?cient in estimating the nonlinear

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12N. Golea et al./Fuzzy Sets and Systems()–
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01020 30405060708090100
−4
−2
0
2
4
Link 2
position (rad)
0 10203040 506070 8090 100
−4
−2
0
2
4
velocity (rad/sec)
0 10 2030 4050607080 90100
−15
−10
−5
0
5
10
t (sec)
torque (Nm)
Fig. 3. Link 2 tracking performance: (—) robot; (:::) reference.
0102030 405060708090100
−4
−2
0
2
4
Link 1
position (rad)
0 10203040506070 80 90100
−4
−2
0
2
4
velocity (rad/sec)
Fig. 4. Link 1 observer performance: (—) robot; (:::) observer.

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0 10 20 304050 60708090 100
−4
−2
0
2
4
Link 2
position (rad)
010203040 5060708090 100
−4
−2
0
2
4
velocity (rad/sec)
Fig. 5. Link 2 observer performance: (—) robot; (:::) observer.
system dynamics and states. The advantage of using TS fuzzy model is that few rules are to be
learned and analytic knowledge, if it exists, can be incorporated in the rules consequences. The
stability analysis has shown that this adaptive scheme achieves asymptotic tracking of stable reference
model and asymptotic state estimation. Furthermore, this algorithm is proved to be robust against
external disturbances and approximation errors.
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