 Open Access
 Total Downloads : 236
 Authors : Pham Van Thiem, Lai Khac Lai, Nguyen Thi Thanh Quynh
 Paper ID : IJERTV5IS040050
 Volume & Issue : Volume 05, Issue 04 (April 2016)
 Published (First Online): 04042016
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Tracking Control of Dynamic Nonlinear Systems via Improved Adaptive Fuzzy Control
Pham Van Thiem, Lai Khac Lai
Electronics Faculty ,
Thai Nguyen University of Technology, Thai Nguyen City,
Viet Nam
Nguyen Thi Thanh Quynh
Centre de Recherche en Sciences et Technologies de IInformation et de la Communication (CRESTIC), UniversitÃ© de Reims Champagne Ardenne,
France
AbstractIn this paper, an Adaptive Fuzzy combined with Model Adaptive Reference System (MARS) is proposed to a single inputsingle output nonlinear system. The main goals of this paper are estimation and control of second order nonlinear system. Firstly, adaptive fuzzy system is applied to design the estimator and the controller. Next, a performance of system is improved by adjusting parameters of the controller via MARS. Finally, two examples are presented to illustrate of the proposed methods. By applying Lyapunov stability theory the adaptive law that is derived in this study is robust and convergent quickly. The simulation results and analysis show that the proposed method have better than Adaptive Fuzzy in the sense of robustness against disturbance.
KeywordsAdaptive Fuzzy System; Model Reference Adaptive Systems (MARS); Coupled Tank Liquid Level; Robot Arm.

INTRODUCTION
The model of object is usually obtained by identification methods, then we use this model to design controller. If the characteristic of an object is not changing in all working process, the closed loop system with this controller is good for demanded designing. However, the parameters of object are unpredictably time variant in practice. Consequently, we need to have an online identification method based on adaptive algorithm which can be adapted by the changing parameters. To do this, the neural network is used [1, 2] to identify the dynamic systems with the back propagation algorithm and found on adaptive fuzzy control which is proposed by LiXin
Wang [3, 4].
in order to transform a given nonlinear system with outputs into a controllable and observable linear one.
With the nonlinear system [5, 6, and 9], K. Khalil use the exact feedback linearization approach to define a control law with assumption that the modeled object is known clearly and states are measured. If we dont know exactly parameters of object, two trends will appear. The first trend consists in introducing a direct adaptive fuzzy control [3, 4] based on Lyapunov theory. The second trend, with which we are concerned, is carried out in controller design based on universal approximation [3, 10], the indirect adaptive fuzzy control. On the other hand, the controller in [3, 4, and 5] guarantees still the stable closed loop system, but it does not concentrate to performance of closed loop system, for example: the settling time, overshoot and error between the output signal and referenceTherefore, to improve performance of system, concretely, the redue error between reference trajectory and output trajectory, and rise the settling time of transient response.. Parameters of controller will be adjusted by adaptive law found on model adaptive reference system (MARS).
The remainder of this paper is organized as follows. In Section II, we provide the exact linearization of nonlinear. Section III estimation and control based on adaptive fuzzy mode control. Section IV then consider the improved adaptive fuzzy mode control. Simulation results are given in Section V and Section VI concludes this work.

EXACT LINEARIZATION OF NONLINEAR WITH OUTPUT
Consider the SISO nonlinear system given as:
1 2 n
1 2 n
dx T n
The nonlinear systems has attracted widespread attention in
f (x) h(x).u
x (x x … x ) R
(1)
the recent decades [5, 6]. There are many controlled methods derived from linear controller as gain scheduling, Jacobi matrix [7]. However, a question is that of when there exists a global change of coordinates, a diffeomorphism, in the state
space that carries the nonlinear system into linear system.
dt u,y, g(x) R; f (x),h(x ) Rn
y g(x)
We always find the coordinates transformation in order to transform the nonlinear system into linear system.
Theorem 1: Consider the SISO nonlinear system (1) with
Krener [8] showed the importance of the Lie algebra of a vector fields associated with the system in studying such a question and gave an answer to this problem. D. Cheng [9] illustrated that we always find the coordinates transformation
r is the characteristic number at x . We assumern thespace, under the diffeomorphism:
f f
f f
z m(x) (g(x), L g(x),…, Ln1g(x))T
in (2)
the nonlinear system will be transformed into linear system:
z Az bw
(3)
where bl is the point at which output membership function
A
A
achieves its maximum value and are input membership
where
Bl
functions. If we fix the
l j
(x ) and view thebl as adjustable
0 1 … 0
0
Al j
j
j
A = ; b ; w F(x) G(x).u
parameters, then (5) can be written as:
0
0
1
1
0
0
0
0
b(x) T x
(10)
T 1 M
0 0 0 0
1
where
(b
,…,b
) is a parameter vector of the output
Proof (see Appendix)
membership functions, and
x (1(x ),…, M (x ))T is a

ESTIMATION AND CONTROL VIA ADAPTIVE FUZZY MODE regressive vector with the regression l (x ) defined as:
CONTROL (AFC)
Without loss of generality, we consider the dynamic
(x)
2
j 1
j
Aij
(x j )
(11)
nonlinear has two state variables linearized by Theorem 1 and
M 2
j (x )
given a structure as:
l 1
j 1
Aij j
x x
Assume that the functions F(x) andG(x) describing the
1 2
2
2
x
F(x) G(x)u
(4)
system dynamics are unknown, we can be estimated the function F(x) andG(x) by fuzzy logic system with the
y x1
adjusted parameters of output membership functions based on
1 2
1 2
G
G
where x (x x )T R2; u,y R; F(x) R2;G(x) R2
and F(x);G(x) C are smooth functions. The control target
adaptive law.
F(x) T
x; (x) T
x
(12)
of system (4) is stable tracking a prior trajectory y
(t) , called
F F G G
where the parameter vector T ;T are updated online so that
m F G
reference trajectory, as means:
the approximate error between F(x);G(x) and F(x);G(x) is
lime(t) 0 ,
t
e(t)
is finite (5)
minimal. Define the optimal parameter vector as:
where e(t) y
(t) y(t) is tracking error.
* arg min(sup  T (x) F(x))
F F F
F F F
m F x
(13)
From (4), we express: * arg min(sup  T (x) G(x))
x
x
y x1 2
If
G
F
* ;
x
* then
G G
F(x) F(x), G(x) G(x) that
y x2 F(x) G(x)u
y x
(6)
F F G G
means the fuzzy system (9) can approximate smooth nonlinar
1
We assume that the parameters of system (6) are known exactly. Then, the state feedback controller shown as:
functions with the arbitrary small error if the number of fuzzy rules is large enough [10]. The state feedback controller (7)
can be expressed:
u 1 (y F(x) k e k e)
(7) 1
G(x) m p d
u* F(x) y
k e k e
(14)
where kp and kd are coefficients of Hurwitz polynomial:
d p
d p
h(s) s2 k s k (8)
This controller will be made stable closed loop system as means (4).
In formula (7), the controller depends on functions F(x),G(x) . If they are unknown, the controller cannot be implemented. Consequently, we need to estimate an online
G(x) m p d
From (10), we see that if the parameters of systems have change by the time, the functions F(x),G(x) will change and
the control signal u will be suitable adjusted for alterative parameters of object.
Theorem 2: Consider the nonlinear system (4), the state
feedback controller (10) and F(x),G(x) in (12) are applied
with the T and T updated by adaptive law:
function F(x),G(x) based on adaptive fuzzy mode algorithm. F G
Q1 (x)s (e); Q1 (x)s (e)u
1 2
1 2
In the [3], consider the MISO fuzzy logic control system has two inputs x (x x )T R2 and an outputb . This
j
j
fuzzy logic system with the centeraverage defuzzifier, product inference, and singleton fuzzifier is of the following form:
F F F f G G G f
The closed loop system will be asymptotic stable.
Proof:
The second derivative of the output error is expressed:
y F(x) G(x)u
M bl
l 1
2
j 1 Al
(x )
F(x) F(x)G(x) G(x)u F(x) G(x)u
m p d
e k e k e T (x) *T (x) T (x) *T (x)u
m p d
e k e k e T (x) *T (x) T (x) *T (x)u
M 2
M 2
l
(x )
l
(x )
b(x) j
(9)
y y
k e k e F(x) F(x)G(x) G(x)u
l 1
j 1 Aj j
p d F F F F G G G G
s T *T (x) T *T (x)u
f F F F G G G
F F G G
F F G G
T (x) T (x)u
T T *T , T T *T are the parameter error * 1

F F

G G
u F(x) y
k e k e
Consider the Lyapunov candidate function:
G(x) m p d
V 1 s 2 1 TQ
T 1 TQ
T
; Q 0; Q 0


IMPROVED ADAPTIVE FUZZY MODE CONTROL(IAFC)
2 f 2
F F F
2 G G G F G
i i
i i
whereQ Rdd (d dim ) is a positive definite matrix. Take
i i
i i
the derivative ofV with respect to time and notice that , we have:
In formula (14), two parameters kp ;kd of the state feedback controller are chosen in (8) that they still guarantee the stable closed loop system. However, it does not concentrate to
performance of closed loop system, for example: the settling
V s s TQ TQ
time, overshoot and error between the output signal and
f f F F F G G G
V s
T (x ) T (x )u TQ TQ
referenceTherefore, to improve performance of system,
f F F G G F F F G G G
concretely, the decrease error between reference trajectory and
V T s (x ) Q T s (x )u Q 0
F f F F F G f G G G
Choose the parameters update law as follow:
output trajectory, and rise the settling time, two parameters
kp ;kd will be adjusted by adaptive law based on model
Q1s (x); Q1s (x)u
F F f F G G f G
with the error surface is defined:
adaptive reference system
The controller (14) can be rewritten as:
s (e) k e k de
(15)
u*
1 (x) y
T
(18)
f 1 2 dt
F
G(x)
m k e
where T (k k ); (e e)T . The T will be updated by
k p d e k
the adaptive law based on reference system in Theorem 3.
Theorem 3: Consider the nonlinear system (1), the state
feedback controller (18) and F(x),G(x) in (12) are applied
with the T ; T and T updated by adaptive law:
a b k
Q1 (x); Q1 (x)u; Q1
F F F G G G k k e
The closed loop system will be asymptotic stable.
Proof:
The second derivative of the output error between output and reference is expressed:
y F(x) G(x)u
F(x) F(x)G(x) G(x)u F(x) G(x)u
Fig. 1. The overall scheme of identification and control via adaptive
y y
T
x) F(x)
x) G(x) u
fuzzy system (AFC)
m k e
F(
G(
2m
2m
2
2
F F F F G G G G
F F F F G G G G
k e
k e
x x T (x) *T (x) T (x) *T (x)u T
where SVF is state variable filter. The derivative of the error can be created by using SVF. The parameters of this state variable filter are chosen in such as the parameters of the reference model.
T *T (x) T *T (x)u T
F F G G
F F G G
k e
k e
F F F G G G k e
F F F G G G k e
T (x) T (x)u T
T T *T , T T *T are the parameter error
The following steps to design the estimation and control via

F F

G G
adaptive fuzzy system:
Step1: The regressive vector l (x ) as:
Consider the Lyapunov candidate function:
V 1 2 1 TQ T 1 TQ T 1 TQ
2 2 F F F 2 G G G 2 k k k
2
2
j (x )
Q 0; Q 0; Q 0
M
M
2
2
(x )
(x )
whereQ
whereQ
Rdd (d dim ) is a positive definite matrix. Take
Rdd (d dim ) is a positive definite matrix. Take
(x)
j 1 Aij j
(16)
F G k
l 1 j 1 Aj j i i
i i
i i
ij
Step 2: The adaptive law:
the derivative ofV with respect to time and notice that ,
Q1s (x); Q1s
(x)u
(17)
we have:
F F f F G G f G
V TQ TQ
TQ
Step 3: Estimation the function:
F F F G G G
k k k
V T (x) T (x)u T TQ
TQ
F(x) T x ; G(x) T
x
F F G G
k e F F F G G G
F F G G
TQ
Step 4: Define the error surface:
k k k
T
T
V F F (x) QF F G G (x )u QG G
s (e) k e k de
T (Q ) 0
f 1 2 dt
k k k e
Step 5: The state feedback controller:
Choose the parameters update law as follow
(x) Q 0 Q1
(x)


ILLUSTRATIVE EXAMPLE

F F F F F
(x)u Q 0 Q1
(x)u
In this section, we present two examples of tracking

G G
G G G
reference set point with a 1 DOF robot arm and a coupled tank
Q 0
Q1
k k e
k k e
liquid level system. The simulation illustrates the convergence
where a weighting matrixQk given as:
of error under our proposed AFC and IAFC approach.
q
0
k
1 q
0 e

1 DOF Robot Arm
Q k 11
;
p> Q1 p
k 11
Consider the 1 DOF robot arm [1] has the dynamic
k 0
qk 22
1
k k e
k
0 q
k 22 e
equation shown as:
x1 x2
d
d
kd (t) kd (0)
(ym y)dt
g d 1
qk 22
1
x
2
2
l
sin(x1 )
ml2
x
2
u
ml 2
(22)
k (t) k (0)
(y y)dt;
y
y (19)
' x
p p q
m m
k 11
y 1
and kp (0) ; kd (0) are chosen in(8) to guarantee the stability of the closed loop system.
where x (x x )T ( )T ; is the angle of robot arm and
1 2
1 2
is the velocity of robot arm,l 0.85m is length of arm.
d 0.85kg / m2 is the coefficient of friction, m 0.9kg is
weight of robot arm, and g 9.8m / s2 is the gravity acceleration .
x
x
g d 1
we have
F (x )
sin(x )
2
1
2
1
2
and G(x) .
2
l ml ml
We will identify and design the controller following steps as:
Choose the inputs membership function of F(x); G(x) have
shaped in Gaussian over the interval [1 1]:
x exp x
A
A
cl 2 / 2 l 2
(23)
j
j
l j
i j j
and F(x); G(x) are identified in Theorm 2 for AFC and
Theorm 3 for IAFC:
Fig. 2. The overall scheme of identification and control via improved adaptive fuzzy system (IAFC)
Reference Model: Explicit output reference, derivative output, and acceleration profile set point signals are created using the reference model, which is described by the transfer function:
H
H
w 2
Fig. 3. Identification nonlinear system via adaptive fuzzy system
ref
n
n n
n n
s 2 2zw s w 2
(20)
We choose the fix coefficients kp 1; kd 5 of AFC and
The parameters of the reference model are chosen such as the higher order dynamics of the system will not be excited
initial parameters kp (0) 1; kd (0) 5 of IAFC. The results of simulation with the two algorithms AFC and IAFC contain
[11].The following steps to design the identification andcontrol based on improved adaptive fuzzy system:
f
f
From Step1 to Step 3 is same the following steps in Section 3 noticed that we replaces with .
Step 4: The parameters of controller are updated by adaptive law as:
two parts: the estimation the smooth function F(x); G(x) and error between output and reference trajectory:
k k e m
k k e m
Q1 ; y
y
(21)
Step 5: The state feedback controller:
u *
1 (x ) y
T
F
G (x )
m k e
Fig. 4. Estimation two nonlinear functions of systems
b2 the cross sectional area of outlet pipe in tank 2
0.5cm2
a12 the crosssectional area interaction pipe between tank 1and tank 2
0.5cm2
b12 the value ratio of interaction pipe between tank1 and tank 2
1.5
g acceleration of gravity
981cm2/sec
a2 the value ratio of outlet pipe of tank 2
1.5
We use the Theorem 1 in order to convert (24) into (4) where:
Fig. 5. Tracking error with IAFC (1st line) and with AFC (2nd line)
F(x) L2 g(x);G(x) L L g(x)
f h f
f h f
Assuming that the function f (x) and
(25)
h(x) describing the
Comparison of angle control simulation between AFC and
system dynamics are unknow, so that we employ adapative fuzzy system to online identify F(x) andG(x) . The control
IAFC is present. With F(x); G(x)both AFC and IAFC are able
to identify (see Fig.4). Response of both AFC and IAFC are almost same. However, it is clearly that the tracking errors for the AFC due to reference trajectory are larger than those by IAFC (see Fig.5), concretely, the tracking errors for IAFC is
about 100 times as small as those for AFC. The adaptive
target guarantee for liquid level of Tank 2 at the setpoint. Opened valve ratio of pump 1 is adjusted by control law, and
pump 2 open arbitrary.
Fistly, we will estiamte the F(x); G(x) by the adaptive
fuzzy system. The inputs membership function are shaped in
k p d
k p d
gains T (k k ) of the I
AFC
automatically reach stationary
Gaussian over the interval [0 100] as (19) and T is parameter vector of output membership function which is updated by
values (see Fig.6).
Fig. 6. Adaptive gains of the IAFC

Coupled Tank Liquid Level System
In [12], the modeled equation of coupled tank liquid level system and parameters given as:
1 2 1 2
1 2 1 2
x (x x )T (h h )T
adaptive law in Thoerm 2 for AFC and Theorm 3 for IAFC as Fig. 3. We choose the fix coefficients kp 1; kd 5 of AFC
and initial parameters kp (0) 1; kd (0) 5 of IAFC given as:
Fig. 7. Simulation control structure with MATLAB
b a 2g x x
12 12 1 2
k
d x1 A
1 A u(t)
dt x 1
1
2 b a
2g x x
b a
2gx
0
(24)
2
2
y x
dx
A2
12 12 1 2 2 2 2
dt
f (x) h(x).u
Fig. 8. Estimation two nonlinear functions of systems
y g(x)
Parameter
Value
A1, A2 the crosssectional area of tank 1 and tank 2
100cm2
Qin the flow rate of liquid into tank 1
u(cm3/sec)
Qout the flow rate of liquid into tank 2
0.8cm3/sec
p, p the height of liquid in tank1 and tank 2
cm
Parameter
Value
A1, A2 the crosssectional area of tank 1 and tank 2
100cm2
Qin the flow rate of liquid into tank 1
u(cm3/sec)
Qout the flow rate of liquid into tank 2
0.8cm3/sec
p, p the height of liquid in tank1 and tank 2
cm
TABLE I. THE PARAMETERS OF COUPLED LIQUID LEVEL SYSTEM
Fig. 9. Tracking error with AFC ( – ) and with IAFC ( )
the influence of load disturbances and measurement noise. Strong properties achieved via the proposed method confirm that improved adaptive fuzzy system is an attractive approach for controlling single inputsingle output nonlinear systems.
VII. ACKNOWLEDGMENTS
This research was supported by the MOET (The Ministry of Education and Training), Viet Nam, under grant no. B2016 TN0101.
APPENDIX
The relative order of single inputsingle output (SISO) system is defined:
k
k
0, 0 k r 2
L L g(x)
Fig. 10. Tracking output with both of AFC and IAFC
h f
0,k r 2
In this coupled tank liquid level system, we consider that
dy g(x) x g(x) f (x) h(x).u L g(x) L L0 g(x)u
this system is affected by disturbaced(t) which has changing dt x x
f h f
2
L g(x)
0
L g(x)
frequency by the time as Fig.10. With F(x); G(x)both AFC and
dy
dt
f x
x
f f (x) h(x).u
x
IAFC coul be evaluated when there is disturbance effect on
L2 g(x) L L1 g(x)u
the system (see Fig.8). Comparison of liquid level control simulation between AFC and IAFC is present. Response of both
AFC and IAFC are almost same (see Fig.10). However, it is obviously that the tracking errors for the IAFC due to reference
dyr 1
Lr 2g(x)
f h f
0
Lr2g(x)
liquid level are less than those by AFC (see Fig.9). The
f
dt x
x
f f (x) h(x).u
x
adaptive T (k k ) of the IAFC automatically alter when
Lr1g(x) L Lr 2g(x)u
k p d
f h f
disturbance inluence on system (see Fig.11).
r r 1
L
L
0
r 1
L
L
dy
g(x)
f x
f g(x) f (x) h(x).u
dt x
x
Lr g(x) L Lr 1g(x)u
f h f
0
with n r , we chose the coordinate transformation as:
1 2
1 2
n k
n k
f
f
z m(x) m (x), m (x),…,m (x)T ; m (x) Lk 1g(x)
dz1 m1 x g(x) dx L g(x) z
dt x x dt f 2
dz2 m2 x Lf g(x) dx L2 g(x) z
dt x
x dt f 3
dz m
Lr 2g(x) dx
Fig. 11. Adaptive gians of the IAFC
n1 n1 x f
Lr 1g(x) z
dt x
dz m
x dt f n
Lr 1g(x) dx


CONCLUSION
n n x f
dt x x
Lr g(x) L Lr 1g(x)u
f h f
f h f
dt
Lr g(m1(z)) L Lr1g(m1(z))u
In the paper we have presented adaptive fuzzy system combined with MARS, improved adaptive fuzzy system,
f
F (x )
hf
G (x )
offers a potential on deliver more accurate and high overall performance in the presence of all the preceding issues. We investigate the effect of the identifier and controller from the simulation results. Compare to the case with AFC and IAFC in two illustrated examples (1DOF robot arm and coupled tank liquid level), for instance, can do the following (see Fig.4, Fig.5, Fig.8, and Fig.9): (a) Improve the transient behavior of the system; (b) Decrease the sensitivity to plant parameter changes; (c) Eliminate steadystate errors; and (d) Decrease
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