Free Chattering Fuzzy Sliding Mode Controllers
to Robotic Tracking Problem
R. AZARAFZA *1, SAEED M. HOSEINI2, MOHAMMAD
FARROKHI3
*1
Islamic Azad University, Sanandaj Branch,Department of Mechanical
Engineering, Sanandaj, Iran.([email protected])
2
Islamic Azad University, Science & Research Branch, Borujerd , Iran.
([email protected])
3
Department of Electrical Engineering Iran University of Science and Technology,
Tehran ,Iran. ([email protected])
Abstract: Sliding mode control (SMC) is a powerful approach to solve the
tracking problem for
dynamical systems with uncertainties. However, the
traditional SMCs introduce actuator chattering phenomenon which performs a
desirable behavior in many physical systems such as servo control and robotic
systems, particularly, when the zero steady state error is required. Many methods
have been proposed to eliminate the chattering from SMCs which use a finite DC
gain controller. Although these methods provide a free chattering control but they
deals with only the steady state error and are not able to reject input disturbances.
This paper presents a fuzzy combined control (FCC) using appropriate PID and
SMCs which presents infinite DC gain. The proposed FCC is a free chattering
control which guarantees a zero steady state error and rejects the disturbances. The
stability of the closed loop system with the proposed FCC is also proved using
Lyapunov stability theorem. The proposed FCC is applied to a two degree of
freedom robot manipulator to illustrate effectiveness of the proposed scheme.
Keywords: Chattering phenomenon, Fuzzy controller, Lyapunov stability, PID
controllers, Sliding mode controllers.
*
Corresponding author


1.
Introduction
Sliding mode control (SMC) approaches have been recognised as powerful method
for designing robust controllers for complex high-order nonlinear dynamic plants
operating under various uncertainty conditions. The major advantage of SMC is the
low sensitivity to plant parameter variations and disturbances which relaxes the
necessity of exact modelling [1]. In the traditional SMC theory, the control
switches between two different sub-controllers repeatedly, leading to an
undesirable phenomenon, the so-called chattering. However, in practice, this type
of switching is impossible due to finite time delays and physical limitations.
Many methods have been proposed for eliminating or reducing the chattering
including the boundary layer, continuous approximations and higher order SMC
approaches. Using the boundary layer approach a continuous control associated
with the original SMC is introduced to enforce the trajectories to remain nearby the
sliding surface within a specified region, well-known as a boundary layer. If
systems uncertainties are large, an SMC designed based on the boundary layer
techniques, requires a high switching gain with a thicker boundary layer to
eliminate the influence of the chattering. However, using a boundary layer with
large thickness, a sliding motion may not occur as the system trajectories may not
be sufficiently close to the sliding surface. Therefore, the boundary layer SMC with
large thickness may not guarantee the insensitivity with respect to the matched
uncertainties. However, this method yields a finite steady state error due to finite
non-switching gain of the controller in the steady state [2].
Another approach which prevents the occurrence of chattering is based on the
generation of an ideal sliding mode in the auxiliary loop including using an
observer. The observer-based chattering suppression obviously requires additional
effort in control design; the plant parameters must be known to obtain a proper
observer. However, including an observer in the control system may bring extra
benefits such as identification of uncertainties and disturbances, in addition to its
value in estimating unavailable states [1, 3]. Since the magnitude of chattering is
proportional to the switching gain, some chattering reduction approaches are based
on reducing the values of the switching gain to decrease the amplitude of chattering
preserving the existence of sliding mode. Some of these approaches may benefit
from the idea of proposing a state-dependent gain method [4, 5, 6]. In this method
the amplitude of the discontinuous control input is significantly reduced as the
states stabilise, however chattering arises in the presence of unmodelled dynamics.
The switching gain also can be adjusted in other manners; e.g. it may be a function
of the non-switching part of SMC (continuous part, say the equivalent control).
This methodology also looks promising since the equivalent control decreases as
the sliding mode occurs along the discontinuity surface. In this method, the input
signal contains chattering although its amplitude decreases as the system trajectory
is sufficiently nearby the sliding surface [1].
Some methods embed an SMC in a fuzzy logic controller (FLC). In these methods,
a set of the fuzzy linguistic rules based on expert knowledge are used to design the
switching control law using either the output error or the change of output error as
input of fuzzy controller [7]. Many other methods, apply the distance between the
states and the sliding surface as the input of FLC. In these methods, since the
switching performance do not present, the system gain in the steady state is finite
and the steady state error may still exist [8].
To remove this obstacle, an FLC is proposed in [9] that combines an SMC with a
feedback linearisation with integral action. This method assures the infinite gain of
controller in the neighbourhood of the sliding surface so that the zero steady state
error is achieved. This method deals only with the stability problem of SISO
systems. Moreover, the method of stability proof is restricted to achieving the same
phrase for the time derivative of the Lyapunov function when each of control law is
applied to the system, this prevents to consider the uncertainties in the theoretical
results.
In this paper, a combination of SMC and proportional plus integrator with
derivation (PID) controller is proposed to solve the tracking problem of a class of
nonlinear MIMO system. First, an SMC with integral action is studied in the
presence of structured uncertainties of system. Then it is shown that in the
neighbourhood of the sliding surfaces, the system can be considered as a linear
time variant system. A suitable PID control is proposed to guarantee the asymptotic
tracking and input disturbance rejection of the closed-loop system in this region.
These control laws are combined using a suitable fuzzy rules and input variable to
introduce the fuzzy combined controller (FCC). The system stability with FCC is
shown using the Lyapunov stability theorem.
This paper is organised as follows: Section 2 describes the class of nonlinear
systems which is considered in this paper. The procedures of the SMC and PID
control design are addressed in Section 3. Also this section introduces the fuzzy
control system with FCC and provides analytical results on the stability of the
closed-loop system. In Section 4, the proposed FCC is applied to a two degree of
freedom robot manipulator (2DOFRM) to support the theoretical results
effectiveness of the proposed scheme. Conclusions are given in Section 5.
2.
Problem Statement
Consider the nonlinear MIMO system in the following form:
x  x
2i
 2i 1
n

 x2i  f 2i  x    g 2i ,i  x  ui
i 1

T
y  x , , x
2 n 1 
  1
where x   x1 ,
i  1, 2,
,n
(1)
, x2n  x  R2n , is the state vector and the operating region  x
T
is a compact set. Also, u  Rn and y  R n are the control input and the system
output, respectively. The mappings f2i : R2n  R and g2i ,i : R2n  R are partially
known
Lipschitz
continuous
functions
of
their
arguments
and
g 2i ,i  x   0, x  x , which means g 2i ,i  x  is either positive or negative on the
compact set  x . In fact, the system (1) presents the general form of n degree of
freedom dynamics of robot manipulators with rigid links [10]. For the sake of
complexity reduction and without loss of generality, here the proposed control
method is derived for n=2, although the results can be straightforwardly extended
to any system in the form of (1) with a higher order. For n=2, the system dynamics
is
x  x
2
 1
 x2  f 2  x   g 21  x  u1  g 22  x  u2

 x3  x4
x  f x  g x u  g x u
4 
41   1
42   2
 4
T
y  x , x
 1 3

(2)
The goal is to design a free chattering combined control such that the outputs track
the desired reference signals y d  [ y1d , y3d ] with zero tracking error. Moreover, the
closed-loop system eliminates the effect of the disturbance on the outputs. Next
section presents various features of the proposed control method.
3
Controller Design Method
In this section, first an SMC with integral action is studied in the presence of
modelling errors. Then a suitable PID controller is proposed to stabilise the error
dynamics when the system trajectories are near the sliding surface. Finally, to
remove the chattering phenomenon without loosing the advantages of SMC, a
fuzzy control is designed using the PID and SMCs.
3.1 SLIDING MODE CONTROLLERS
Let the uncertainties on f 2 and f 4 be in additive form as follow
fˆ2i  f 2i  F2i
i  1,2
(3)
where fˆ2  x  and fˆ4  x  are estimates of f 2  x  and f 4  x  respectively. Also,
consider the uncertainties on the input matrix in the multiplicative form as
ˆ ( x)  Δ  G
ˆ (x);    ; i  1,2, j  1,2
G  x  G
ij
ij
g
where G   21
 g 41
(4)
g 22 
and Ĝ is an estimate of G .
g 42 
 12 

ASSUMPTION 1: The maximum eigenvalue of the matrix Γ   11
 is less
 21  22 
than 1, that is max  Γ   1 . Note that since Γ is a matrix with nonnegative real
entries, then it has a nonnegative real eigenvalue which, dominates the absolute
value of other eigenvalue of Γ .
Consider the system (2), and define the error state vector as
e  e1 , e2 , e3 , e4  :  x1  y1d , x2  y1d , x3  y3d , x4  y3d 
T
T
(5)
Then the error dynamics can be written in the following form
e1  e2

e2  f 2  e, y d   g 21  e, y d  u1  g 22  e, y d  u2  y1d

e3  e4
e  f  e, y   g  e, y  u  g  e, y  u  y
4
d
41
d
1
42
d
2
3d
 4
(6)
where y d :  y1d , y1d , y3d , y3d  , in which y1d and y3d the desired trajectories x1
T
and x3 , respectively.
Now let define the sliding surfaces  1 and  3 as follow
 1  e1   e1   2  e1   d
t
0
 3  e3   e3  
2

t
0
e3   d
(7)
Where  is a positive constant. Sufficient conditions for enforcing the error
trajectories reach on the sliding surfaces in finite times and remain on it afterward,
are that the SMC, u  u1 , u2  is designed such that the following condition is
T
fulfilled :
1d 2
1   32   1 1  3  3

2 dt
(8)
where 1 and  3 are small positive numbers.
Proposition 1. Consider the SMC
  fˆ  E1  k s1 sgn  1  
ˆ 1  2
u SM  G

  fˆ4  E3  ks 3 sgn  3  
(9)
with E1 : e1   2e1  y1d , E3 : e3   2e3  y3d , then there exists the vector gain
k s  [ks1 , ks3 ]T , which satisfies the reaching conditions (8).
Proof. Consider the following Lyapunov function
1
1
V  12   32 ,
2
2
(10)
Using (2) and (7), the time-derivative of (10) becomes
 
 x  E1 
V   1  3   1    1  3   1

 3 
 x3  E3 
  1
 f 
E 
 3    2   Gu   1  
 E3  
  f4 
(11)
Appling the control law (9) yields
 f 
  fˆ  E1  ks1 sgn  1    E1  
ˆ 1  2

V   1  3    2   GG

ˆ  E  k sgn     E3  
  f4 

f

4
3
s
3
3




(12)
ˆ 1  I  Δ into (12) and using (4), the time-derivative of V is
Substituting GG
  f  1  
12    fˆ2  E1  ks1 sgn  1    E1  
11

V   1  3    2   


  f 4    21 1   22    fˆ4  E3  k s 3 sgn  3    E3  




 f 2  fˆ2  11 fˆ2  E1  12 fˆ4  E3  k s1 1  11  sgn( 1 )  k s 3  21 sgn( 3 ) 

  1  3  


ˆ
ˆ
ˆ
 f 4  f 4   21 f 2  E1   22 f 4  E3  k s1 21 sgn( 1 )  ks 3 1   22  sgn( 3 ) 














Then the bound defined in (3) and (4) gives

 F  

k  
V  F2   11 fˆ2  E1   12 fˆ4  E3  1   11  ks1  ks3 12 1
4
21
fˆ2  E1   22 fˆ4  E3   11ks1  1   22
s3
(13)
2
If the following conditions are satisfied
1   11  ks1  F2   11
fˆ2  E1   12 fˆ4  E3   12 ks3  1
1   22  ks3  F4   21 fˆ2  E1   22 fˆ4  E3   11ks1  3
(14)
then the sliding mode reaching conditions (6) are verified
V  1 1 3  3
(15)
□
Note that in general, if u   u SM with 0    1 is applied to plant the above
conditions are replaced with

1  11  ks1  2   11  1   fˆ2  E1  12 fˆ4  E3  12ks3  1
F
1   22  ks3
F
 4 


1  ˆ

fˆ  E    
f  E  k  3

21 2
1  22
3
11 s1 
  4
(16)
Using the Frobenius-Perron Theorem, it can be shown that there exists the positive
vector  k s 1 k s 3  such that inequalities (16) hold. Consider the above inequalities
as follow


  11  12    ks1   1 
I  
    
  21  22    ks 3   3 


Γ


where 1  0 and 3  0 are chosen such that
(17)
1
1  ˆ

ˆ
   11 
 f 2  E1   12 f 4  E3 
 
 


F
1  ˆ

3  4   21 fˆ2  E1    22 
f 4  E3  3

 


1 
F2
(18)
Since Γ is a positive definite matrix, according to the Frobenius-Perron Theorem,
if max  Γ   1 (see. Assumption 1) then there exist unique positive k s1 and k s 3
which satisfy the equation (17). Hence from (17) and (18) it can be concluded that
there exist the gains  ks1
ks 3  which satisfy (16).
When the system trajectory is nearby the sliding surface, the SMC normally deals
with the chattering phenomenon. So a control should be designed to eliminate the
chattering. Based on this fact, a PID controller is proposed to control the system in
the neighbourhood of sliding surface.
3.2. PID Control Design
As stated before, the chattering phenomenon occurs when the system trajectory is
in the neighbourhood of the sliding surfaces, that is σ  1 where  1 , is a small
positive value. To eliminate the chattering, the SMC should be designed such that
switching control performance is avoided. Therefore, a suitable set of fuzzy rules is
applied to activate an appropriate PID controller when the system trajectory is in
the vicinity of the sliding surfaces. This importance is addressed in the next
section. Using the following lemma it can be shown that under condition of
σ  1 where σ : 1  3 
T
, the nonlinear error dynamic (6) can be
approximated as a linear time variant system.
1

Lemma 1: let σ  1 then e  4  2   1 .


Proof: Appling the Laplace transform to (7) gives
1 (s)  sE1 (s)  2 E1 (s)   2
E1 (s)
s
(19)
where E1 ( s)  L e1 (t ) and 1 ( s)  L  1 (t ) . Equation (19) yields
E1 ( s ) 
s
s  
2
1 ( s )
(20)
From σ  1 it can be concluded that  1  1 and  3  1 . Also let define the
 1

auxiliary variable z1 (t ) : L1 
1 ( s)  , which is bounded as
  s   

: Z1 ( s )
t
t
0
0
z1 (t )   e  (t  ) 1 ( ) d  1  e  (t  ) d 
1

1  e  (t  )   1



(21)
Now the bounds on e1 and e1 can be derived using (20) and (21) as follows:


s
e1 (t )  L1  E1 ( s)   L1 
 ( s) 
2 1
s   



 
 L1 
  s   

t

0
Z1 ( s)  Z1 ( s)     e   (t  ) z1 ( )d  z1 (t )
Hence,
e1 (t ) 
z

1  e   ( t  )   z1  2 1



(22)
Also the bound on e1 can be calculated as


 2

s2
 ( s)   L1 
Z ( s)  2 Z ( s)  1 ( s) 
2 1
s   

  s   



e1 (t )  L1  sE1 ( s)   L1 
t
   2e (t  ) z1 ( )d  2 z1 (t )   1 (t )
0
Thus, using  1  1 and (21), it gives
e1 (t ) 
2 z
1  e (t  )   2 z  1  41

(23)
Using similar procedures, the bounds on e2 and e2 can be obtained. Now (5),
(22) and (23) yields
1

e  e1  e1  e2  e2  4  2   1


(24)
1

The trajectories eventually enters into a small ball e  4  2   1 , and if  1 is


selected sufficiently small, then e-trajectories moves within this ball closely ti the
origin. To stabilise the error system when the trajectories are inside this ball, a PID
sub-controller is designed. Since the behaviour of the nonlinear system (6) and its
linearised counterpart are the same, to prove the system stability the following
associate linear system is considered.
e1  e2

T
T
e2  a 2 e  b 2 u

e3  e4
 e  aT e  b T u
4
4
 4
where
aT2  y d  
f 2
e
,
e0
(25)
bT2   g21  0, y d  , g22  0, y d 
aT4  y d  
f 4
e
and
e0
bT4   g41  0, y d  , g42  0, y d  .
To stabilise the above controllable system the following PD controller may be used
k
k
u PD  K e (t )e   11 12
0 0
0
k23
0
e
k24 
In order to guarantee the asymptotic tracking and input disturbance rejection of the
closed-loop system, an integral action is added to the controller. Let define the new
t
t
0
0
state variables as p1 :  e1 ( )d and p3 :  e3 ( )d . Now the augmented openloop plant can be presented as
 p1  0 1
 e  0 0
 1 
d  e2  0 a21
 
dt  p3  0 0
 e3  0 0
  
 e4  0 a41
0
0
0
1
0
0
a22
0 a23
0
0
1
0
0
0
a42
0 a43
0   p1   0
0




0   e1   0
0 
a24   e2  b21 b22 
   
u
0   p3   0
0
1   e3   0
0
  

a44   e4  b41 b42 
: e p
: A p
(26)
: b p
New a PID controller is proposed to stabilise the closed- loop error dynamics:
k
k
u PID  K p (t )e p   11 12
k21 k22
k13
k23
k14
k24
k15
k25
k16 
e
k26  p
(27)
Since  A p , b p  is controllable, it is always possible to place the eigenvalues of
A p  b p K p in the desired locations. Here the control gain K p (t ) is obtained such
that all eigenvalues of the following matrix are negative
A p  b pK p

066



ˆ ( y )K   0
Λ G
d
p

: QG
066
T
(28)
where
 2 2 1

013
Λ
 . Note that there are sufficient free
2
 2 1
 013
parameters kij
i  1,2
j  1,
,6 to hold the condition (28). Hence the stable
closed loop dynamics can be presented as
e p   A p  b pK p e p
(29)
A cl
Figure 1 shows the closed-loop system with the added integral control.
Figure 1. Closed-loop system with an integral action
Now it is shown that the system trajectories (29), asymptotically tend to the origin
along the sliding surface  1   3 .
Proposition 2: The time derivative of the Lyapunov function (10) is negative on
closed loop dynamic (29).
Proof: The Lyapunov function (8) can be rewritten as
1 1 0 
V  σT 
σ
2 0 1 
(30)
where σ  1  3  , in which  1 and  3 are defined in (7) and can be
T
represented as
 1  
2
 3   2
 p1 
2 1  e1    2 2 1 013  e p ,
 e2 
 p3 
2 1  e3   013  2 2 1 e p
 e4 
(31)
Substituting (31) into (30), yields
1
V  eTp ΛT Λ e p
2
P
(32)
where P is a symmetric positive definite matrix. Using (29), the time-derivative of
(32) becomes
1
1
1
1
1
V  eTp Pe p  eTp Pe p  eTp ATcl Pe p  eTp PAcl e p  eTp  ATcl P  PAcl  e p
2
2
2
2
2
The stability of A cl ensures existence of the symmetric positive-definite solution
P of the following algebraic Riccati equation
PAcl  ATcl P  Q
(33)
where Q is an arbitrary symmetric positive-definite matrix. Hence,
2
1
1
V   eTp Qe p   min (Q) e p  0
2
2
(34)
3.3.Fuzzy Combination of PID and SMC
To remove the chattering without loosing the advantages of SMC such as
asymptotic tracking and robustness, a fuzzy combination of PID and SMC is used.
In this approach, based on the rules, a fuzzy system decides about the activation of
the PID and SMCs.
In this method, a continuous fuzzy switch makes smooth
changes between these two controllers based on the following fuzzy IF-THEN
rules:
Rule 1: IF σ is S, THEN u  u PID
Rule 2: IF σ is L, THEN u  u SM
(35)
where S and L are fuzzy sets defined on input fuzzy variable σ , witch is applied
to fuzzy controller. Also u SM and u PID are the outputs of the fuzzy inference engine
for the above fuzzy rules. In the above fuzzy rules, there exists at least one nonzero
degree of membership S ( ), L ( )  0,1 corresponding to each rule as depicted
in Figure 2 in which  2 is a positive small number. Appling the weighted sum
defuzzification method, the overall output of the fuzzy controller can be written as
u
S ( )u PID  L ( )u SM
S ( )  L ( )
(36)
Figure.2. Membership function of fuzzy variable σ
Using the following property of input matrix of robot manipulator system; G, the
stability of the closed-loop system over the entire operation range of the fuzzy
logic control system is studied.
Lemma 2: Consider the estimated input matrix of a 2DOFRM system as
ˆ (x)  H
ˆ 1 where
G
ˆ (x)   c11  d11 cos x3 c12  d12 cos x3 
H
c22
c12  d12 cos x3

and x3 is the angle of second joint of manipulator as depicted in Figure 3. If
σ   2 and
2

1 , then the input matrix can be considered as
ˆ (x)  G
ˆ (y )  Δ (y , e)
G
d
G
d
ΔG  2w
2

where w is a positive constant.
Proof: Using (5) and expanding Ĥ , gives
 c11  d11 cos  e3  y3d  c12  d12 cos  e3  y3d  

c22
c12  d12 cos  e3  y3d 

ˆ ( x)  
H
 c11  d11 cos e3 cos y3d

c12  d12 cos e3 cos y3d
 d11 sin e3 sin y3d

 d12 sin e3 sin y3d
Since
2

c12  d12 cos e3 cos y3d 

c22

d12 sin e3 sin y3d 

0

1 , cose3  1 and sin e3  e3 , therefore,
ˆ ( x)  H
ˆ (y d )   d11 sin y3d
H
 d12 sin y3d
d12 sin y3d 
 e3
0

T
ˆ (x)  H
ˆ 1 , yields
Applying the inverse of sum of matrix formula, and G
(37)
ˆ (x)  G
ˆ (y )  Δ (y , e )
G
d
G
d
3


ˆ (y ) I  TG
ˆ ( y )e
where ΔG (y d , e3 )  G
d
d
3

1

ˆ (y ) e .
TG
d
3
Δ1
As it was shown in Lemma 1, σ   2 ensures that e3 
ΔG  Δ1 e3  2w
2 2

. So
2
where w is defined as the upper bound of Δ1 .

□
THEOREM 1: Consider the fuzzy control (36), with the membership function as
depicted in Figure 2, and let max    be sufficiently small then the closed-loop
system is asymptotically stable if
2
is selected such that


ˆ (y )  2 Δ 1        2  Λ K
min  QG    max    G
d
1
max
p
 

Proof: It can be seen from Figure 2 that, for any value of σ , only one of the
following two cases will occur:
1) Either Rule 1 or Rule 2 is active. In this case σ  1 or σ   2 .
If σ  1 then u  u PID . According to Proposition 1,
1
V   min (Q) e p
2
2
(38)
Also from (30) and (31) , σ  eTp Pe p , so
2
min  P  e p
2
 σ  max  P  e p
2
2
(39)
Then (37) and (38) yields
V 
min (Q)
σ
2max  P 
2
(40)
σ  1 guarantees that V  0 except when σ  0 which implies V  0 . In this
case from (7), it can be concluded that e p  0 . On the other hand, when
σ   2 , u  u SM . Therefore from (15)
V  1 1 3  3  min σ
where min  min 1 ,3  . Since σ  0 , (40) shows V  0
(41)
2) Either Rules 1 and 2 are active simultaneously, i.e. 1  σ   2 .
In this case, the overall control is the convex combination of the PID and SMCs
which is applied to the system
u   uSM  1    uPID where 0   
L
S  L
1.
Consider the Lyaponuv function (10). Using (2) and (7), the time-derivative of V
becomes
 
 x  E1    f 2 
E 
V   1  3   1    1  3   1
      Gu SM   1  

 3 
 x3  E3    f 4 
 E3  
 1     1  3  Gu PID
Appling the SMC (9) with the conditions (16), yields
V  1  1  3  3  1    eTp ΛT Gu PID .
(42)
Substituting from (4), (37) and (27)


ˆ (y )      G
ˆ (y )     K e
 1    e Λ  G
ˆ ( x)    G
ˆ K e
V  1  1  3  3  1    eTp ΛT G
p p
 1  1  3  3
T
p
T
d
G
d
G
p p
ˆ (y )K , where Q is a positive matrix and the bounds
Using (28), QG  ΛT G
G
d
p
defined in (4) and (7), gives
V  1 1  3  3  1   e e p
2
(43)

ˆ (y )  2 Δ 1        2  Λ K .
where e  min  QG    max    G
d
1
max
p
 

Assume that the uncertainty Δ , defined in (4), and respectively max    is
sufficient small. Moreover let select  2
 enough small such that the following
condition holds

ˆ (y )  2 Δ 1        2  Λ K
min  QG    max    G
d
1
max
p
 

(44)
Then, (43) guarantees the stability of the closed-loop system over the entire of the
operation region of the fuzzy logic control system.
4.
Example
The performance of the proposed controller is shown through simulations using the
two degree of freedom robot manipulator (2DOFRM). See Figure 3. The dynamics
of this system is described by the following differential equations [2]:
H(q)q  M(q, q)q  N(q)  τ
with q  [q1 , q2 ]T being the two joint angle and τ  [1 , 2 ]T being the joint inputs .
H is the mass matrix, and M is the vector associated with the Coriolis and
centrifugal forces. The elements of H and M are as follows:
H11  I1  I 2  m1lc21  m2l12  m2lc22  2m2l1lc 2 cos q2
H 22  m2lc22  I 2 ,
H12  H 21  m2lc22  I 2  m2l1lc 2 cos q2
M 11  m2l1lc 2 q2 sin q2 ,
M 12  m2l1lc 2  q1  q2  sin q2
M 21  m2l1lc 2 q1 sin q2 ,
M 22  0
N1  m1lc1 g cos q1  m2 g  lc 2 cos  q1  q2   l1 cos q1 
N 2  m2lc 2 g cos  q1  q2 
where m1 , m2 , l1 , l2 , I1 , I 2 are masses, lengths and inertia moments of the arms,
respectively, and lc1  0.5l1 , lc 2  0.5l2 .
Defining the state vector as x :  q1 , q1 , q2 , q2  and the input vector as u : 1 , 2 
T
the above system can be rewritten as
x  f (x)  g1 (x)u1  g 2 (x)u2
where
x2





h
M
x

N

h
M
x

N

h
M
x
11 
11 2
1
12 
21 2
2
11 12 4 

f ( x) 


x4


 h12  M11 x2  N1   h22  M 21 x2  N 2   h12 M 12 x4 
0
0 
h 
h 
h 
h
11 

g1 (x) 
, g 2 (x)   12  with H 1   11 12 
0
0 
 h21 h22 
 
 
 h12 
 h22 
T
Figure 3. A two degree of freedom robot manipulator model
The simulations have been carried out using the following parameters and initial
conditions
m1  1.2kg , m2  0.82kg , l1  0.73m, l2  0.65m, I1  0.0833kgm2 ,
I1  0.0652kgm2 x1 (0)  0.1rad , x2 (0)  0 rad / s, x3 (0)  0.8 rad ,
x4 (0)  0.1rad / s . Also the parameters uncertainties are considered as
mˆ 1  1.05m1 , mˆ 2  1.05m2 , lˆ1  1.04l1 , lˆ2  0.97l2 ,
Iˆ1  1.05I1 , Iˆ2  0.95I 2 . Select
  10 and the fuzzy controller parameters are chosen as 1  0.1,  2  1 .
First the SMC is applied to the system with k s1  k s 2  1 as the Figures 4 and 5
show the tracking error exists due to the plant uncertainties, in spite of the control
signal (see. Figure 6) is chattering free. To compensate the tracking error, the
sliding gains are increased to ks1  ks 2  5 . In this case, the asymptotic tracking is
achieved, but the control signals contain chattering. To remove the chattering the
fuzzy combined controller is applied. The results show both the zero tracking error
is obtained, and the control signal is chattering free. See Figure 6.
The validity of the necessary condition stated in Assumption 1, is verified in Figure
7.
The disturbance rejection ability of the proposed combined controller is examined
by applying an step disturbance with amplitude of 10 N/m at t  13sec. As Figure
8 shows, the SMC with the gain ks1  ks 2  5 , cannot reject the disturbance and so
the tracking error is increased, this cause the system trajectories run away from the
sliding surfaces. To achieve the disturbance rejection property under the SMC, it is
required to increase the control gains to ks1  ks 2  35 . These high sliding mode
gains guarantees the perfect tracking, but it the chattering of control signal is very
large and cannot be neglected (see. Figure 9), while, using the propose FCC with
ks1  ks 2  5 , as the sliding mode gains for the SMC part rejected the disturbance
with free chattering control signal.
5.
Conclusions
A fuzzy combined control for a class of nonlinear MIMO system was proposed in
this paper. The proposed method relays on the combination of conventional PID
and SMCs. The proposed controller has the advantages of both controllers
including the robustness of SMC and the smooth control signal, zero steady state
error and the disturbance rejection property of PID control. The overall stability of
FCC has been shown using the Lyapunov direct method. Simulation results, have
been carried out for the 2DOFRM in the presence of parameters uncertainty,
illustrate the good performance of the proposed method.
References
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London: Taylor and Francis, (1999).
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Mir Publishers. (1978).
[5] Parra-Vega V, Liu YH, Arimoto S. Variable structure robot control undergoing
chattering attenuation: adaptive and non-adaptive cases. In: Proceeding of IEEE
international conference robotics and automation; pp. 1824–29, (1994).
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of Nonlinear Non-Minimum Phase Systems, J Intelligent Robotic Systems 56:487–
511, (2009).
[7] Yau, H. and Chen, C. L., Chattering-free fuzzy sliding-mode control strategy for
uncertain chaotic systems Chaos, Solitons and Fractals 30, pp. 709–718, (2006)
[8] Glower, J.S. and Munighan, J., Designing fuzzy controllers from variable structures
standpoint , IEEE Trans. Fuzzy Syst., 5, pp 138-144, (1997).
[9] Wong, L. K., Leung, F., and Tam, K. S., A fuzzy sliding controller for nonlinear
systems, IEEE Trans. On Industrial electronic, 48, no. 1, (2001).
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Manipulators and Nonlinear Systems, Taylor & Francis, 1999.
0.15
SMC with k s1=k s3=1
Reference
SMC with k s1=k s3=5
0.1
0.05
FCC with k s1=k s3=5 in SMC part
q1 (rad)
0
-0.05
-0.1
-0.15
-0.2
-0.25
-0.3
0
5
10
15
20
25
30
time, s
Figure 4.The first joint tracking performance using the SMC and the proposed FCC
1
SMC w ith ks1=ks3=1
0.8
Reference
SMC w ith ks1=ks3=5
0.6
FCC w ith ks1=ks3=5 in SMC part
0.4
q2 (rad)
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0
5
10
15
20
25
time, s
Figure 5.The second joint tracking performance using the SMC and the proposed FCC
30
1(N.m)
50
0
SMC w ith ks1=ks3=5
FCC w ith ks1=ks3=5 in SMC part
SMC w ith ks1=ks3=1
-50
0
5
10
15
20
25
30
25
30
time, s
 2(N.m)
20
10
0
SMC w ith ks1=ks3=5
-10
FCC w ith ks1=ks3=5 in SMC part
SMC w ith ks1=ks3=1
-20
0
5
10
15
20
time, s
Figure 6. SMC signals with various gains and the proposed FCC action.
 max ( (x) )
0.24
0.22
0.2
0.18
0.16
0
2
4
6
8
10
12
14
16
18
20
time, s
Figure 7. Validation of condition max  Γ   1
0.05
eq1 (rad)
0
-0.05
-0.1
FCC w ith ks1=ks3=5 in SMC part
SMC w ith ks1=ks3=35
-0.15
SMC w ith ks1=ks3=5
-0.2
0
5
10
15
20
25
30
0.4
0
-0.2
FCC w ith ks1=ks3=5 in SMC part
-0.4
SMC w ith ks1=ks3=35
-0.6
SMC w ith ks1=ks3=5
e
q2
(rad)
0.2
0
5
10
15
20
25
time, s
Figure 8. Comparison of tracking errors using SMC and FCC in the presence of
disturbance
30
100
 1 (N.m)
50
0
SMC w ith ks1=ks3=35
-50
-100
SMC w ith ks1=ks3=5
FCC w ith ks1=ks3=5 in SMC part
0
2
4
6
8
10
12
14
16
18
time, s
2 (N.m)
20
10
0
SMC w ith ks1=ks3=35
-10
SMC w ith ks1=ks3=5
-20
FCC w ith ks1=ks3=5 in SMC part
0
2
4
6
8
10
12
14
16
18
time, s
Figure 9. SMC signals with various gains and the proposed FCC action in the presence of
disturbance
List of figures caption:
Figure. 1. Closed loop system with integral action
Figure.2. Membership function of fuzzy variable σ
Figure 3. Two degree of freedom robot manipulator
Figure 4. Comparison of first joint tracking performance of Sliding-Mode control with
Fuzzy combined control
Figure 5. Comparison of second joint tracking performance of Sliding-Mode control with
Fuzzy combined control
Figure 6. Control signals of sliding-mode control with different gains and fuzzy
combined controller
Figure 7. validation of condition max  Γ   1
Figure 8. Comparison of tracking errors of SMC and FCC in the presence of input
disturbance
Figure 9. Control signals of SMC different gains and fuzzy combined controller in the
presence of input disturbance
R. Azarafza received his PhD in mechanical engineering from university
of K.N. Toosi University of Technology at 2005. He is assistant
professor at the department of mechanical engineering, Islamic Azad
University, Sanandaj Branch, Sananadaj, Iran. His current research
interest includes Vibration of Shell and Composite.
Saeed Mohammad Hoseini received his BS in Electronic Engineering from
Ferdowsi University of Mashhad and his MSc and PhD in Control
Engineering from Iran University of Science and Technology in 2001 and
2009, respectively. He is currently an Assistant Professor at Malik Ashtar
University of Technology, Iran.. He is the author of several peer-reviewed
journal and conference papers. His research interests include adaptive
control, intelligent control of non-linear system, and flight control.
Mohammad Farrokhi has received his BS from K.N. Toosi University,
Tehran, Iran in 1985, and his MS and PhD from Syracuse University,
Syracuse, New York in 1989 and 1996 respectively, both in Electrical
Engineering. He joined Iran University of Science and Technology in 1996,
where he is currently an Associate Professor of Electrical Engineering. He
has published more than 120 refereed research papers. His research interests
include automatic control, fuzzy systems, and neural networks.