A fuzzy adaptive sliding mode controller for uncertain nonlinear multi ...

MATEC Web of Conferences 161, 02013 (2018) https://doi.org/10.1051/matecconf/201816102013 13th International Scientific-Technical Conference on Electromechanics and Robotics “Zavalishin’s Readings” - 2018

A fuzzy adaptive sliding mode controller for uncertain nonlinear multi motor systems Tran Xuan Tinh1, Pham Tuan Thanh1, Tran Van Tuyen1, Nguyen Van Tien1 and Dao Phuong Nam2,* 1Le

Quy Don University (Military Technical Academy), Vietnam School of Electrical Engineering, Hanoi University of Science and Technology, Vietnam

2

Abstract. Multi-motor drive systems are nonlinear, multi-input multi-output (MIMO) and strong-coupling complicated system, including the effect of friction and elastic, backlash. They have been widely used in many modern industries. The control law for this dive system much depend on the determining of the tension being hard to obtain this tension in practice based on a load cell or a pressure meter due to the accuracy of sensors or external disturbance. An emerging proposed technique in the control law is the use of adaptive sliding mode control scheme to stabilize closed system. However, the control system would be affected by chattering phenomenon. In order to eliminate this term, fuzzy technique is proposed by adjusting equivalent coefficients. The theory analysis and simulation results point out the good performance of the proposed fuzzy adaptive sliding mode control for the drive system.

1 Introduction Multi-motor drive systems have been investigated by many researchers in the recent times. The neural network technique based control law have been proposed by Yaoji Me et al. (2013) (see [1-9]). However, it is hard to find the equivalent networks as well as corresponding learning rules. Besides, the model of this system is approximately described as a linear system to use the transfer function to design the control law. Furthermore, the tracking ability or the stabilization of the whole system are not still solved under the effects of neural network based observer. In the multi-motor drive control systems, it is necessary to obtain the belt tension to design the suitable state feedback control law. However, it is hard to measure this belt tension based on sensors, … and the high gain technique based observer is proposed in our work. Besides, the state feedback control design based on sliding mode control technique ensure that it is easy to remove efficient of disturbance and uncertainties. Therefore, an adaptive sliding mode controller is proposed to obtain tracking effectiveness. Moreover, fuzzy technique is considered to eliminate the chattering phenomenon disappearing by sliding mode control. The stability of closed system is obtained and verified by theory analysis, simulations.

2 Problem statements Due to the effects by backlash and elastic (Fig. 1) and parameters (Table 1), we extend the model in [1] to obtain the following dynamic equation (2, 3) and the corresponding transfer function diagram (Fig. 2):

1  (1) ωr1 = J [ c1 . f11 (Δϕ1 ) + b1Δω1 f12 (Δϕ1 ) − (TL1 + r1F ) ] L1   1 ω r 2 = J [ c2 . f 21 (Δϕ2 ) + b2 .Δω2 f 22 (Δϕ 2 ) − (TL 2 + r2 F ) ] . L2     1 F)   F = C12  r1ωr1 − r2ωr 2 (1 + C12 .l    Table 1. Parameters of a Multi-Motor System.

K=E

V

Transfer function

E

Young’s Modulus of belt

V

Expected line velocity

T=

L0 AV

Time constant of tension variation

L0 , A

Distance between racks, Section area (m2)

n pi

Number of pole-pairs in the ith Motor

J1, J2, JL1, JL2 T , TL ,

Inertia moment of Motors and Loads (kgm2)

ϕr

Motor, Load torque (Nm), Flux of rotor (Wb)

Lr

Self-induction of rotor (H)

c1, c2, b1, b2

Stiffness and friction coefficient

Δω1 , Δω2 The errors of angle speed

in presence of backlash, elastic

We denote: *

Corresponding author: [email protected]

© The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/).

MATEC Web of Conferences 161, 02013 (2018) https://doi.org/10.1051/matecconf/201816102013 13th International Scientific-Technical Conference on Electromechanics and Robotics “Zavalishin’s Readings” - 2018

ud 1 = ud 2

1 [c1 f11 (Δϕ1 ) + b1Δω1 f12 (Δϕ1 )] ; J L1

1 = [c2 f 21 (Δϕ2 ) + b2 Δω2 f 22 (Δϕ2 )] J L2

−ϕ0 ϕ0

(2) T1 −

1 ω1 J1 s

−

Δω1

−ϕ0 1 ϕ0 Ts Δϕ 1

K f1 +

K e1

Ts1

T21

0

0 k6 0

v1

(4)

ϕ0

Ke2

−TL 2

1

r2

ωT 2

1 s

J L2s

ϕr 2

F ] in presence of friction and

Assumption 2: There are real positive numbers that

ωr1max , ωr 2 max , TL1max , TL 2 max , ud 1max , ud 2 max such ωr1 , ωr 2 , TL1 , TL 2 , ud 1 , ud 2 are bounded by these values;

In this section, the main work is to find a state feedback control law based on the adaptive sliding mode control technique for the class of multi motor systems. The proposed control law based on the following theorems as follows: Theorem 1: The adjusting mechanism (5):

v2

ωr 2

ωr1 Ts1

Ts 2

θat = ηaαψ a (x)s,  t θb = ηbαψ b (x)su eq ,

c2 , b2

α

(5)

with s = α e + ξ ,

α ω2

(

)

ueq (t) = Bˆ −1 (x) − Aˆ (x) + x d − β sgn(s) =

T2 , J 2

−1

ueq = Bˆ T (x, θ bt ) ε 0 I m + Bˆ (x, θ bt ) Bˆ T (x, θ bt )  ; (6)

Đ2

( − Aˆ (x,θ ) + x t a

AC

Ts 2

+

+

elastic. The following assumptions must be satisfied in order to design the control law: Assumption 1: xT = [ωr1 ωr 2 F ] is measurable;

TL 2 , J L 2

Đ1

T12

−ϕ0

− v2

1 l.C12

3 Fuzzy adaptive sliding mode control design

TL1 , J L1

ω1

−

Δϕ 2

tracked by xT = [ωr1 ωr 2

 ;  k8 x1 + k9 x2 

F

T1 , J1

ω2

1 Ts

+ v1 −

The control objective is to find the control input vector u T = [ TL1 TL 2 C12 ] to obtain that the desired value are

0 0

r2

c1 , b1

Δω2

C12 s

Fig. 2. The corresponding transfer-function diagram of the TwoMotor Drive System.

 ud 1  D = ud 2   0  r1

1 J2s

ϕ r1

Kf2

where k2   k3 k5  ; B =  0 k7 x2   0

−

1 s

−ϕ0 ϕ0

The model is described by eq. (3) belongs to the class of nonlinear systems as follows:

0 k4

F12

r2 T2

ω r1

r1

F12

1 1    ωr1 = −c1ωr1 − J TL1 − J r1F+ ud 1 L1 L1   1 1 TL 2 − r2 F+ ud 2 . (3) ω r 2 = −c2ωr 2 − J L2 J L2   1  F = C12 r1ωr1 − C12 r2ωr 2 − ωr 2 F l 

 k1 A =  0  0

1 J L1 s

−

r1

are components including backlash and elastic to obtain the following equations:

 x (t) = A(x, t) x + B (x, t)u + D (x, t) ,  y = x

−TL1

AC

d

− β sgn(s)

)

pa

A ( x ) ≈ Aˆ ( x, θ at ) = θ aψ a (x) =  θ aiψ ai (x);

Fig. 1. The Two-Motor Drive System.

i =1

Remark 1: The dynamic equations (1,2, 3) and Figures 1, 2 are described by the effect of friction, backlash, elastic and pointed out the nonlinear property of multi-motor systems.

pb

ˆ ( x, θ t ) = θ ψ (x) =  θ ψ (x). B( x) ≈ B b b b bi bi

(7)

i =1

ensure some results as follows: 1. All signals of closed loop will be bounded and θ a ,θb converge to

2

MATEC Web of Conferences 161, 02013 (2018) https://doi.org/10.1051/matecconf/201816102013 13th International Scientific-Technical Conference on Electromechanics and Robotics “Zavalishin’s Readings” - 2018

{ = arg min {sup B ( x, t ) − Aˆ ( x, θ ) }

θ a* = arg min sup A ( x, t ) + D ( x, t ) − Aˆ ( x, θ ai ) θb*

θ ai

x∈Ω

θbi

x∈Ω

}

(

) (

)

 ε a ( x, t ) −ψ aθa + ε b ( x, t ) −ψ bθb ueq ( t )  + V = sα   − B ( x, t ) uc ( t ) − β sgn(s)   

ai

+θa ηaαψ a (x) s + θb ηbαψ b (x) su eq

as t → ∞ ˆ ( x, θ t ) ; B* ( x,θ * ) = Bˆ ( x,θ t ) then 2. If A* ( x,θ a* ) = A a b b

(

)

V = sα ε a ( x, t ) + ε b ( x, t ) ueq ( t ) − B ( x, t ) uc ( t ) − β sgn(s) =

the

errors converge to zero in finite time; 3. If A* ( x,θ a* ) ≠ Aˆ ( x,θ at ) ;B* ( x,θb* ) ≠ Bˆ ( x,θbt ) then closed loop

= sα ε a ( x, t ) + ε b ( x, t ) ueq ( t ) − ε a − ε b ueq − β sgn(s) ≤

system converges to the neighborhood of sliding surface in finite time; Proof: The Lyapunov candidate function is selected as follows:

≤ −α s β sgn(s)

V=

1 2 1 1 2 1 1 2 s + θa + θb 2 2 ηa 2 ηb

(

∀x ∈ Ω . Theorem 2: u ( t ) = ueq ( t ) − uc ( t ) ; t a

(9)

c

)

( ) ( ) ( ( x,θ ) − Bˆ ( x,θ ) = B ( x, t ) − Bˆ ( x,θ ) b

)

) ( )) ( ) ( )) u

B* ( x,θb* ) − Bˆ ( x, θb ) = ψ b (θb* − θb ) = −ψ bθb

(

) (

with

−1

b

eq

a

) (

)

Therefore, s = α e + ξ → 0 in finite time Remark 1. it is necessary to ensure that the time of convergence of sliding surface is finite. The fact is described based on the following example: We consider the system as follows:

b

 ε ( x, t ) + A* x,θ * − Aˆ x,θ a a  a  s = α  + ε b ( x, t ) + B* x,θb* − Bˆ x,θb   − B ( x, t ) uc ( t ) − β sgn(s)  On the other side, A* ( x,θ a* ) − Aˆ ( x, θ a ) = ψ a (θ a* − θ a ) = −ψ aθa

−1

≤ − s αβ sgn(s)

Therefore, we obtain:

(

input

≤ − sT αβ sgn(s)

ε a ( x, t ) + A* x,θ a* − Aˆ x, θ a = A ( x, t ) + D ( t ) − Aˆ x, θ a

(

control

d

(

However, we have the relation:

(

obtain

s = α e + ξ ensure the nonlinear system (6) stability in finite time. Proof: We obtain the result:  ε a ( x, t ) −ψ aθa + ε b ( x, t ) −ψ bθb ueq ( t )   sT s = sT α   − B ( x, t ) uc ( t ) − β sgn(s)   ε a ( x, t ) + ε b ( x, t ) ueq ( t )   ≈ sT α   −ε a − ε b ueq − β sgn(s)   

 A ( x, t ) + B ( x, t ) u ( t ) + D ( x, t )   =α  − Aˆ ( x, t ) − Bˆ ( x, t ) ueq ( t ) − β sgn(s)     A ( x, t ) + D ( x, t ) − Aˆ ( x, t )   =α   ˆ + B ( x, t ) − B ( x, t ) ueq ( t ) − B ( x, t ) uc ( t ) − β sgn(s)   

* b

The

( − Aˆ (x,θ ) + x − β sgn(s) ) ; u ( t ) = B s (ε u + ε )

s = α e = α ( x − xd )

ε b ( x, t ) + B*

we

ueq = Bˆ T (x, θbt ) ε 0 I m + Bˆ (x, θbt ) Bˆ T (x, θ bt ) 

and:

)

and

V ≤ 0

(8)

  V = ss + θaθa + θbθb

(

α > 0;s.sgn(s) ≥ 0

where

where θa = θ a − θ *a ;θb = θb − θ *b , we have:

(

)

   eq ( t )  (10)   

ìï dx ïï = Ax + Bs ïïí dt ïï ds ïï = Cx + Ds ïî dt

where: A Î n´n , B Î n´r , C Î r´n , D Î r´r , s is the sliding é A Bù ú is not surface. Selecting A is Hurwitz matrix and ê êëC Dúû Hurwitz matrix. We obtain that although s converges to 0 in infinite time, x does not converge to 0. Remark 2. The proposed control law ensure that the system trajectory converge to sliding surface in finite time. Remark 3. We almost utilized theorem 1 to design adaptive sliding mode control technique for multi motor systems.

and we obtain:

)

 ε a ( x, t ) −ψ aθa + ε b ( x, t ) −ψ bθb ueq ( t )   s = α   − B ( x, t ) uc ( t ) − β sgn(s)    and:

3

MATEC Web of Conferences 161, 02013 (2018) https://doi.org/10.1051/matecconf/201816102013 13th International Scientific-Technical Conference on Electromechanics and Robotics “Zavalishin’s Readings” - 2018

In order to eliminate the effect of chattering phenomenon, fuzzy technique (by Tagaki – Sugeno – Kang) would be proposed to adjust the coefficient α depending on the sliding surfaces s and s , table 2, 3 and figure 3: Table 2. Rule Matrix of control.

Table 3. Properties of controller.

AND method

MIN

OR method

MAX

Implication

MIN

Aggregation

MAX

Defuzification

Weighted average

Fig. 4. The behaviour of the first motor’s speed in presence of disturbance.

Fig. 5. The behaviour of the second motor’s speed in presence of disturbance.

Fig. 3. Fuzzification.

4 Simulation results In this section, we consider several simulation results to demonstrate the effectiveness of the proposed sliding mode control law based on the two-motor system having parameters as follows: n p1 = n p 2 = 4, J1 = J 2 = 500 Kgm 2 , Lr1 = 0.2 H , Lr 2 = 0.3H , ωr1d = ωr 2 d = 700v / p, Fd = 250 N . Fig. 6. The behaviour of the first motor’s speed without disturbance.

Figures 3, 4 show the tracking performance behaviour of velocity based on fuzzy adaptive sliding mode control law in presence of disturbance (figures 4, 5, 8). Figures 6, 7 show the high tracking performance behaviour of velocity based on adaptive sliding mode control law without disturbance.

4

MATEC Web of Conferences 161, 02013 (2018) https://doi.org/10.1051/matecconf/201816102013 13th International Scientific-Technical Conference on Electromechanics and Robotics “Zavalishin’s Readings” - 2018

6. 7. 8. 9.

Fig. 7. The behaviour of the second motor’s speed without disturbance.

Fig. 8. The speed error between 2 motors.

5 Conclusion This paper described a fuzzy adaptive sliding mode control law the two-motor system in presence of elastic and backlash, friction. The effectiveness of the proposed control scheme was pointed out by theoretical analysis and simulation results.

References 1. 2. 3. 4. 5.

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