2015 19th International Conference on System Theory, Control and Computing (ICSTCC), October 14-16, Cheile Gradistei, Romania
Adaptive fuzzy synergetic control of nonlinear systems with unknown backlash-like hysteresis Kamel Guesmi
Aissa Rebai Process Control Laboratory National Polytechnic School of Algiers-Algeria
[email protected]
Ahstract-The hysteresis non-linearity exists in many physical systems and materials, such as electrical and mechanical actu ators, ferromagnetic and ferroelectric materials. However, this property makes the analysis and control of such systems and devices difficult. In order to control this property, a new Adaptive fuzzy synergetic control strategy is proposed in this paper. Indeed, to highlight the problem, the hysteresis behavior of the system is described using a backlash-like model. Then a synergetic control scheme is proposed to deal with the problem of controlling non-linear hysteretic systems. It's formed of a fuzzy system to approximate the unknown system dynamics with an adaptive synergetic controller to achieve the desired performances. The proposed approach is validated through simulation along with its satisfactory performances.
Index Terms-Nonlinear systems, Hysteretic systems, Hystere sis, Synergetic control theory, Adaptive control.
I.
Djamal Gozim
Boualem Hemici
CReSTIC, IUT de Troyes, Science and Technology Faculty Process Control Laboratory 09 rue de Quebec, 10026 National Polytechnic School Ziane Achour University Troyes, France Djelfa, Algeria of Algiers-Algeria
[email protected] [email protected] [email protected]
INTRODUCTION
In last decades, much attention has been focused on the study of hysteresis due to the existence of this property in a wide range of physical systems, devices and materials, which severely limit system performance. It can be found, for exam ple, in ferroelectric and ferromagnetic materials, mechanical systems, electrical actuators and electronic relays circuits [1] [8]. Because of the hysteresis phenomenon complexity, the modeling of this nonlinearity is still a big challenge. Thus, there exist many approaches proposed in the literature to describe the hysteresis nonlinearity such as Preisach model [9]-[11], Duhem model [12], Bouc-Wen model [13]-[16], Prandtl- Ishlinskii model and its generalizations [17]-[19], Jiles-Atherton [20], Duffing-like equation [21], Backlash-like model [22], minimum variance scheme [23], fractional order models [24] and fuzzy models [25]. A survey on these models and others can be found in [26]. Among all of the above mentioned models, the backlash hysteresis model is considered in this paper due to its sim plicity. To simplify, the hysteresis nonlinearity is sometimes omitted in the controller design. However, this ignorance of nonlinear phenomenon can lead to the instability of the system. In this context, a particular attention is given to the question of control of hysteretic systems based on backlash-like hys teresis model. Indeed, many works, in the literature, deal with the control of systems with unknown backlash-like hysteresis
978-1-4799-8481-7/15/$31.00 ©2015 IEEE
[26]-[34]. In [26], an adaptive controller is developed and in [27], the authors proposed an adaptive fuzzy control scheme to deal with the question. In [28], an adaptive dynamic surface control strategy is given and in works such [29]-[31], an adaptive backstepping control strategies are adopted. In another research direction, the question is treated based on intelligent approaches like in [33], where authors proposed an adaptive fuzzy-backstepping controller and in [22], [34], where approaches based on neural-network are developed. However, the main disadvantages of these techniques is the multiple constraints on system uncertain parameters and on the backlash-like term. Furthermore, without a proper initialization of these techniques, they tend to have a weak performance during the learning phase. In addition to above mentioned works, fuzzy logic control is used in various fields due to its high performance [35]-[38]. In this context, fuzzy and synergetic theories are combined in this paper to design a new fuzzy adaptive synergetic controller to mitigate the hysteretic phenomenon without the above re strictions. An adaptive fuzzy system is used to approximate the unknown system dynamics and the system stability is analysed by Lyapunov theory. Also, the proposed control strategy is compared to the adaptive backstepping control method, and the results show that the proposed controller has better efficiency, with an excellent tracking performance. This paper is organized as follows: the control problem and the hysteresis model are stated in Sect. II. The design of synergetic controller is summarized in Sect. III. Fuzzy systems are introduced and adaptive fuzzy synergetic controller is synthesized in Sect. IV. The proposed approach validation and their performances evaluation, through simulation results, are given in Sect. V. II.
PROBLEM FORMULATION
Consider the following second order SISO nonlinear system described by
(1)
213
6 ,---,---�---,----,--� 1.5
4 2
0.5
-0.5
-2 -4
-1
. ..... .. --......... -- ...... ........
......... ... .........
-1.5
_ 6 L-4
�__-L__�____�__-L__�____L-__�
__
-2
-3
-1
2
o ultl
Fig. 1: Hysteretic curve given by (5) and (6) with
2.356 and c =0.453 for u(t) =3.5sin(2.4t). T
4
3 a
5
= B=
2 is the state vector, u E · and where x = [Xl, X2] E y E · are the input and output of the system respectively. f(x) and g(x) > 0 are two unknown non-linear functions. The function w(u) is the system input and the output of the backlash-like hysteresis [31], which is described as follows •
w(u(t)) = cu(t) + d(u(t))
(3)
(4)
with Wo and Uo the initial conditions considered equal to zero in this paper. Solving (3) and (4), w(u(t)) can be expressed as follows when U::::.-. 0,
_c
� B [1 _ ( 2e-au, _e-2au, ) e-au(t) ]
(5)
and when
C
� B [1- (
2e-au, _e-2aU' ) eau(t)
]
(6) where Us is the upper bound of u. An example of hysteretic curve generated by (5) and (6) is given in Fig. l. J.-
From (5) and (6), we have
w(u(t)) = cu+ sgn(u)B;;c e-sgn(u)au(t) ]
[1 - (
2e-au, - e-2au, )
x
B
� c [1- ( 2e-au, _e-2aU, ) e-sgn(,;)au(t) ]
u ::::.-. 0 we have
B- c = -d(u) --++=
uSimilarly, if
a
lim
u---+-=
c-B d(u) = -a
From (7) and (8), we can conclude that
d(u)
0
(11)
where T determines the convergence rate. To design the controller for system (I), we suppose that the following assumptions are satisfied.
Assumption 1.- The term d(u) with different values of a is represented in Fig. 2. It can be seen from Fig. 2 that the term d(u) varies slightly, and almost equal to d(u) =sgn(u)B;;c. Therefore, we can consider that d(u) is constant. i.e.,
d(u) = -sgn(u)d (7)
a
(8)
Obviously, if
u < 0,
w(u(t)) =cu(t)+
Remark
d(u) =sgn(u)
III.
d(u) = [ wo - cuo] e-a(u-uo)sgn(u)
w(u(t)) = cu(t)
with different values of
Then,
(2)
with
[ B - c] e-a�sgn(u)d�
d(u)
lim
where a, candB are positive constants satisfying c > B. u(t) is the system input. Then, the explicit solution of (2) is given by
+e-ausgn(u) J' u Uo
Fig. 2: The term
1,
dw du du =a I I (cu - w) + B dt dt dt
10
Time (s)
(12)
with d > O.
214
Assumption 2: We assume the existence of functions and that verify
g2(X)
0
(31)
1 1 8fT8f. -0.2 .
til
-0.4 . -0.6 -
0 8 '--_-'---'-__---'___-'-__ 4 o 2 6 Time(s)
__----'
----1.
.
8
10
Fig. 5: Output trajectories
proposed approach and those obtained by the backstepping adaptive control. Figure 6 depicts the tracking errors stated for the two controllers. From Fig. 6, it can be seen that the proposed controller presents two times smaller steady-state error (4%), with a dynamic ten times faster (tT 0.8s VS =
tT
=
3.58).
From the discussed example, it can be noticed that the pro posed control scheme can lead to better tracking performance while avoiding the aforementioned problems and constraints. It presents faster convergence performance than the adaptive backstepping control and the smallest steady state error, and also more better robust characteristics for the parameters variation. VI.
CONCLUSION
In this work, a new adaptive fuzzy synergetic controller is developed for a class of nonlinear hysteretic systems. Then, Asymptotic stability is demonstrated through Lyapunov stability method. Finally, The simulation results validated the
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