 Open Access
 Total Downloads : 167
 Authors : Youssef Baba, Mostafa Bouzi, Ismail Lagrat, Mounir Derri
 Paper ID : IJERTV4IS090481
 Volume & Issue : Volume 04, Issue 09 (September 2015)
 Published (First Online): 25092015
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Design of Adaptive Fuzzy Sliding Mode Control for a Travelingwave Ultrasonic Motor
Youssef Baba, Mostafa Bouzi, *Ismail Lagrat, Mounir Derri
Univ Hassan 1 ,Laboratory of Mechanical Engineering, Industrial Management and Innovation Faculty of Science and Technology, Settat, Morocco
*National School of Applied Sciences, Khouribga, Morocco
Abstract In this paper, we propose an adaptive fuzzy sliding mode control scheme for a travelingwave ultrasonic motor. Because external disturbances and parameter variations, it is difficult to design a conformable modelbased control scheme. In order to deal with this problem, the proposed control law is combined a fuzzy logic and the sliding mode control (SMC). Firstly, the equivalent controller is obtained by a fuzzy strategy. Secondly, we use a boundary layer approach to avoid chattering problem and satisfactory trajectory tracking. An on line adaptive tuning algorithm for the consequent parameters in the fuzzy rules is designed. Simulation studies have shown that the presented adaptive design of fuzzy sliding mode controller performs very well in the presence of unknown disturbances.
KeywordsSliding Mode Control; Fuzzy Control; Traveling
introduces the proposed adaptive fuzzy sliding mode controller. Simulation results are presented in Section 4. Section 5 offers our concluding remarks.

USR60 MODEL
Travelingwave ultrasonic motors are complex electromechanical devices in which a mechanical resonant vibration is excited in the stator through proper forcing piezoelectric ceramics. This stator vibration is transformed into a rotation through friction contact between the stator and rotor.
The model of piezoelectric and stator can be described by the following equation
Wave Ultrasonic Motor.
M D C v Fd
(1)
I. INTRODUCTION
The Piezoelectric travelingwave ultrasonic motor (TUSM) has excellent performance and many useful features such as high holding torque, high torque at low speed, quiet operation, simple structure, compact size, and no electromagnetic interferences [1].
Nevertheless, the control characteristics of TUSM are complex and nolinear. The motor parameters are time varying due to increase in temperature and change of motor drive operating conditions [26]. In order to overcome these problems, several control systems have been proposed [68]. But the complexity of these algorithms is far beyond the fixed parameters PID control requiring higher online calculation ability and the resulting increase of the cost of hardware as well as software of the system. In [911] authors tray to use the nonlinear characteristic of fuzzy and neural control to deal with the nonlinear problems of TUSM control, except that we must find a compromise between the complexity of the algorithm proposed and its real time implementation.
In this work, an adaptive fuzzy sliding mode control (AFSMC) scheme is proposed for a travelingwave ultrasonic motor type USR60. The proposed control law is based to combine a fuzzy logic and the SMC. Firstly, in order to realize the control law without the model of system, a fuzzy logic
with represents the modal amplitude of the vibrating system (ceramics and stator), M is the total mass matrix of system(ceramics and stator), D is the structural damping
matrix assumed to be diagonal, and C is the total stiffness
matrix. H is the electromechanical coupling matrix and v is
the voltage excitation vector. The term Fd is a nonlinear modal force vector to consider the interaction between the stator/rotorcontact.
TABLE I. SPECIFICATIONS OF USR60
Drive frequency [kHz] 40.0
Drive voltage [Vrms] 100
Rated torque [Nm] 0,32
Rated speed [rpm] 130
Rotor inertia [Nms2] 7,2.106
Rotor damping in spinning direction 0.05
In dealing with the dynamics of the rotor, two degrees of freedom must be taken into account: first the rotation of the rotor and second the motion in zdirection. The motion in z direction is represented by the quantity w . The dynamics of the vertical rotor motion is obtained by the following equation
controller is designed to estimate the equivalent controller.
mr w dz w Fz – Fn
(2)
The fuzzy parameters are estimated online by the adaptive laws. Secondly, a sliding mode control is used, but this control
with mr is the mass of the rotor, d z is the damping of the
scheme suffers from chattering problems. In order to
vertical motion, and Fn is the applied axial force. The
guarantee the stability of the sliding mode system, the boundary of the uncertainties has to be estimated.
The paper is organized as follows. In section 2, a mathematical model of USR60 is presented and a reference model is proposed in order to control the motor. Section 3
equation of rotational motion is calculated by
Jr dr Tr TL
(3)
where Jr is the rotor inertia, dr denotes the damping in spinning direction, and TL is the applied torque.
In Fig.1 the Matlabsimulink model of USR60 is described. Fig.2 shows the relation between speed and torque measurements when the driving frequency is 40 kHz .The speed versus drive frequency for different applied load torques is represented in Fig.3. The speed of the TUSM has its maximum at the mechanical resonant frequency of the motor. So, any deviation from this frequency degrades the motor performance. However, this effect seems more serious for frequency decrements. The characteristic curves in Fig.2 and Fig.3 are derived from calculations using the Matlabsimulink model represented in Fig.1.

DESIGN OF ADAPTIVE FUZZY SLIDING MODE CONTROL

Design of Sliding Mode controller
In order to apply sliding mode control, let us write (3) as
– a bu h(t)
Fig. 2. Speedtorque characteristics
where
a, b 0
h(t) T J 1 J 1J a bu
(4)
is the equivalent
L r r
disturbance torque,
J ,
a , and b
are the parameter
variations from normal value. Fig.4 shows the block diagram
of motor speed control using AFSMC, where c denotes the
given speed, denotes the actual speed. Considering e c as speed tracking error, the timevarying surface of the sliding mode control is introduced as [1213]
s e e
(5)
with is a strictly positive constant, whose choice we shall interpret later. If sliding mode exists, the following condition satisfies
s e e 0
Therefore, the control law is obtained as [13]
u ue us
(6)
(7)
Fig. 3. Speedfrequency characteristics
The term us (t) is called the hitting controller and obtained as
where
ue (t) is a solution of (6) and called the equivalent
us – K0sgn(s)
(8)
controller.
where sgn is the sign function
sgn(s) 1 if s 0
sgn(s) 1 if s 0
In order to ensure the existence condition of sliding mode using (8), the condition ss 0 must be satisfied [1213].
ss s (e e)
s(a( – a) ( – a) d (t) – b( – a)K0 sgn(s)) where ( a)d (t) h(t) ( a)h(t) c c assumed to be bounded.
So with
(9)
and
bK0  a d (t) 
(10)
the value of s and s have opposite signs and the state reaches
the sliding line s 0
after a finite time interval. Inequality
Fig. 1. Matlabsimulink model of USR60
(10) determines the frequency needed for enforcing the sliding mode; as a result, the control error is steered to zero.
While control law (7) achieves the target dynamics (5) exactly, the presence of the switching term K0 sgn(s) impliess
s
that in practice undesirable control chattering will occur. To suppress the chattering and obtain a bandwidthlimited
where is the membership functions width.
The local control of fuzzy system is defined as
e
1 2
ui i s i s
(13)
controller that best approximates the exact behavior described
The local control actions i
and i
which are contained in
above, the switched action K0 sgn(s) is replaced by a smooth
the parameters vector , T are calculated online by
interpolation in a boundary layer neighboring the sliding surface as [13]
1 2
the following least squares algorithm
us K0sat(s)
The sat function is defined as
sat(s) 1s if s
(11)
(i) (i 1) P(i1)* (i1)*((i) (i1)T *(i1))
(i) (i1)T *P(i1)* (i1)
P(i1)* (i1)* (i1)T *P(i1)
sat(s) sgn(s)
if otherwise
P(i) P(i 1)
(i) (i1)T *P(i1)* (i1)
where is the boundary layer thickness.
with P is a covariance matrix,
is an estimation of the

Adaptive Fuzzy Sliding Mode control
According to the universal approximation of TSfuzzy
9
systems [14], an optimal fuzzy controller u exists such that
vector parameters , and denotes the forgetting factor. By fuzzification, the fuzzy inputs s and s are obtained. The fuzzy equivalent controller is obtained by defuzzification
e
i
the approximation error of fuzzy controller can be defined as
Ai Bi ue T
ue ue
(12)
u
i1
(14)
e 9 i i
where is the approximation error and assumed to be bounded. In this work, there are nine rules in a fuzzy base and they have the following form
where [s s]T .
A B i1
1 2
Rule i : if s is Ai and s is Bi then ue is i s
Finally the control law in (7) becomes
T
1 i 9
u K0sat(s)


SIMULATION RESULTS
(15)
In order to evaluate the performance of our control scheme, the simulation results of the proposed controller are achieved. The simulation study of the system was implemented using Matlab. The specification of USR60 is shown in Table I. Choose the sliding surface as s e5e . The
control parameters: K0 0.02 , and 0.01 . The initial
Fig. 4. Speed control block diagram
e
where s and s are the inputs variables of the fuzzy system and u is its output variable. The linguistic terms Ai and Bi are defined as
Ai , Bi{N (negative), Z (zero), P(positive)}
and they are characterized by their corresponding membership functions
values: (1) 0 , and P(1) 1000I . The unknown disturbances are modeled by the randn function from Matlab library. Fig.6 shows the speed tracking response by
applying the control law (15) represented in Fig.5. The
dynamic response is good. The speed characteristics of motor changed when the load torque is applied or small parameter variations occur, but the control gains of the fuzzy sliding mode controller also changed to compensate the parameter
variations. Adaptive parameters 1 and 2 are represented in Fig.7 and Fig.8 respectively.
N (s) exp(0.5 2 (s 1)2 ); N (s) exp(0.5 2 (s 1)2 )
P (s) exp(0.5 2 (s 1)2 ); P (s) exp(0.5 2 (s 1)2 )
Z (s) exp(0.5 2s2 ); Z (s) exp(0.5 2s2 )
Fig. 5. Total control law u
Fig. 8. Evolution of 2
It is clear that the proposed AFSMC control scheme introduces excellent performance where the controller variables track their reference values exactly in a very short time.

CONCLUSIONS
In this paper, an adaptive fuzzy sliding mode structure has been proposed for speed control of a travelingwave ultrasonic motor USR60. The strategy of control is based to combine the fuzzy logic and the sliding mode control to guarantee the stability and the tracking performance. The main advantages of the proposed speed controller are robustness to parameter variations and external load disturbances. Simulation results confirm the abovementioned claims for the control scheme in TUSM control drive.
Fig. 6. Speed tracking response
Fig. 7. Evolution of 1
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