NONLINEAR CONTROL
BASED ON T-S FUZZY MODEL
Consider T-S fuzzy systems
If
z1 (t )
is Ai1 and
z2 (t )
is Ai 2 and … and
zq (t )
is Aiq , then
 x (t )  Ai x (t )  Bi u (t )

 y (t )  Ci x (t )
For i  1, 2, , N , where
Ai  R nn , B  R nm
and
Ci  R pn
(1)
. N is the number of the
subsystems and we call system (1) the ith subsystem.
z (t )=  z1 (t )
z2 (t )  zq (t ) 
is a vector of premise variables. Aij (i  1, N ; j  1, , q ) are fuzzy sets. Denote
 A () the membership function determining the membership of z j (t ) in the
ij
fuzzy set Aij .
T
A T-S fuzzy system actually represents a nonlinear system. Fuzzy system (1) can
be represented by equivalently
N

 x (t )   hi ( z (t ))[ Ai x (t )  Bi u (t )]

i 1

N
 y (t )  h ( z (t ))C x(t )

i
i

i 1
where
(2)
q
hi ( z (t )) 

j 1
q
N
Aij
 
i 1 j 1
( z j (t ))
(3)
Aij
( z j (t ))
N
We call hi ( z (t )) the fuzzy weights. Obviously, we have
 h ( z (t ))  1.
i 1
i
Consider affine nonlinear system
x  f ( x )  g ( x)u
(3)
n m
where f ( x )  R n and g ( x )  R are two nonlinear function.
Theorem 1: Suppose that f (x) is continuous and differentiable, then for any  0，
there exists a T-S fuzzy model of system (1) or (2) such that
N
f ( x)  g ( x )u   hi ( Ai x  Bi u )  
i 1
for any x and u in a suitable neighbourhood.
The computation of matrix Ai and Bi
Step 1: Choose the operating points of x0i (i  1, , N ) . Usually, the equilibrium
point x0 : f ( x0 )  0 should be considered.
Step 2: (1) For the equilibrium point x0,
Ai 
where
f ( x )
x
and
x0
 f1 ( x )
 x
1

 f 2 ( x )
f ( x ) 
  x1
x
 

 f n ( x )
 x1
Bi  g ( x0 )
f1 ( x)

x2
f1 ( x ) 
xn 

f 2 ( x )
f 2 ( x ) 

x2
xn 



 

f n ( x )
f n ( x ) 

x2
xn 
(4)
(2) for other operating points:
al ,i  f l ( x ) 
i
0
 a1,Ti 
 
Ai    
 anT,i 
 
where
f l ( x0i )  x0iT f l ( x0i )
i 2
0
x
and
Bi  g ( x0i )
x0i , (l  1, 2, , n)
 f l ( x ) 
 x 
1


 f l ( x ) 
f l ( x )   x2 
  


 f l ( x ) 
 xn 
(5)
Example 1: Consider free-body of a cart with an inverted pendulum system
Fig. 1: Free-bosy of a cart with an inverted pendulum system
The dynamic system of the inverted pendulum system can be described in
form
x  f ( x )  g ( x )(u   ( x ))
(6)
with
0
x2









a
cos
x
g
sin
x
1
1




2
2
 4l / 3  mla cos x1 
 4l / 3  mla cos x1 
f ( x)  

 , g ( x)  
0
x
4






 (1 / 2) mag sin(2 x1 ) 
4a / 3




2
2
4
/
3

ma
cos
x
4
/
3

ma
cos
x

1 

1 
 ( x )  f c  mlx22 sin x1
and
For the nonlinear system (6), we propose the following two rules to construct
a T-S fuzzy system:
Rule1: If x1(t) is about 0, then
x  A1 x  B1u
Rule 2: If x1(t) is about  / 4 , then
x  A2 x  B2u
That is we choose two operating points:
x   0 0 0 0
1
0
T
and
x    / 4 0 0 0 
2
0
T
Next, we compute the matrices of A1, A2, B1 and B2 based on (4) and (5)
2
For the operating point x0    / 4 0 0 0 , based on (5), we have
T
 f 2 
 f 2 
0 
0 
 x 
 x 
1 
0
 1
 1
f1 ( x )    , f 2 ( x )   0  , f 3 ( x )    , f 4 ( x )   0 
0 
0 
 
 
0
0 
 
 



1
0 

 0 
 0 
Therefore,
a1,2  f1 ( x ) 
2
0
Since,
f1 ( x02 )  x02T f1 ( x02 )
2 2
0
x
x  0 1 0 0
2
0
T
a2,2  f 2 ( x ) 
2
0
f 2 ( x02 )  x02T f 2 ( x02 )
2 2
0
x
x02
g sin x1
f 2 ( x) 
4l / 3  mla cos 2 x1
The computation results are:
 0
 17.2941
A1  
 0

 1.7249
1 0 0
 0
14.3077
0 0 0 
, A2  
 0
0 0 1


0 0 0
 0.9723
1 0 0
 0 
 0

 0.1765 
 0.1147 
0 0 0 
 , B2  

, B1  
 0 
 0

0 0 1





0 0 0
0.1176
0.1081




We use the membership functions of the form
1  1 / (1  e 14( x1  /8) )
h1 ( x1 (t )) 
1  e 14( x1  /8)
and
h2 ( x1 (t ))  1  h1 ( x1 (t ))
Then the T-S fuzzy model consisting two subsystems for nonlinear system (6)
can be represented as
x  h1 ( x1 )( A1 x  B1 (u   ))  h2 ( x1 )( A2 x  B2 (u   ))
2
  hi ( x1 )( Ai x  Bi (u   ))
i 1
(A1)
Nonlinear Control Scheme Based on T-S Fuzzy Model
Consider a homogenous linear system
x (t )  Ax (t )
(7)
where x  R n is sate vector and A  R nn is a square constant matrix.
Definition 1: If all the eigenvalues of matrix A have negative real part,
then we call matrix A is a (asymptotical) stable matrix, and the system (7)
is asymptotically stable at equilibrium state x = 0.
Theorem 1: Square matrix A is stable if and only if for any symmetrical
positive definite matrix Q, the following Lyapunov (matrix) equation
AT P  PA  Q
(8)
has a symmetrical positive definite matrix solution of P.
An equivalent statement of Theorem 1 is: Square matrix A is stable if and
only if Lyapunov inequality AT P  PA  0 has a symmetrical positive definite
matrix solution of P.
Now consider a T-S fuzzy homogenous model described by
N
x (t )   hi ( z (t )) Ai x (t )
i 1
(9)
Theorem 2: A sufficient condition for the T-S fuzzy system (9) being
asymptotically stable at 0 is that there exists a common symmetric positive
definite matrix P such that
AiT P  PAi  0
(10)
holds for i  1, 2, , N .
Remark: A necessary condition for the existence of a common symmetrical
positive definite P satisfying (10) is that each Ai is asymptotically stable.
Theorem 3: There exists a common symmetric positive definite matrix P such
that (10) holds, then the matrices
s
A
k 1
ik
are asymptotically stable, where ik  {1, 2, , N } and s  1, 2, , N .
Example: Given a homogenous T-S fuzzy system (9) with two subsystems:
 1 4 
 1 0 
A1  
, A2  


0

2
4

2




Obviously, both A1 and A2 are asymptotically stable, but there is no common
positive definite matrix P such that (10) holds because
 1 4   1 0   2 4 
A1  A2  





0

2
4

2
4

4

 
 

is unstable since its eigenvalues are {1.1231, -7.1231}.
Now consider the T-S fuzzy system (2), we plan to design a state feedback
control scheme such that the close-loop system is asymptotically stable.
Assumption 1: The pair { Ai , Bi }(i  1, 2, , N ) is controllable. (i.e. every
subsystem of the T-S fuzzy system is controllable.)
Under Assumption1, the state feedback controller is designed as
N
u   hi ( z (t )) K i x
i 1
(11)
where the control gains Ki are chosen such that Ai  Bi K i (i  1, , N ) is stable.
Substituting (11) into the state equation in (2) leads to the closed-loop system:
N
N
N
N
N
i 1
j 1
i 1
j 1
j 1
x   hi ( z (t ))[ Ai x  Bi  h j ( z (t )) K j x ]   hi ( z (t ))[ h j ( z (t )) Ai x   h j ( z (t )) Bi K j x]
N
N
N
N
i 1
j 1
i 1 j 1
  hi ( z (t )) h j ( z (t ))( Ai  Bi K j )x   hi ( z (t ))h j ( z (t ))( Ai  Bi K j ) x
That is, the closed-loop system is
N
N
x   hi h j ( Ai  Bi K j ) x
i 1 j 1
(12)
Theorem 4: A sufficient condition for the closed-loop system (12) to be
asymptotically stable at x = 0 is that there exists a symmetric positive definite
matrix P such that the following conditions are satisfied:
( Ai  Bi K i )T P  P ( Ai  Bi K i )  0, (i  1, 2, , N )
and
GijT P  PGij  0, (i  j  N )
(12)
(13)
where Gij  ( Ai  Bi K j )  ( Aj  B j Ki )
Example 2: Consider the inverted pendulum nonlinear system (6) and the
corresponding T-S fuzzy model (A1), and based on the T-S fuzzy model (A1),
we design the state feedback as
u  h1 K1  h2 K 2
(14)
The two gains K1 and K2 are chosen such that the eigenvalues of A1  B1 K1 and
A2  B2 K 2 are placed to
a1   1.0970 2.1263 2.5553 3.9090 
T
and
a2   1.5794 1.6857 2.7908 3.6895
T
respectively. Now by MATLAB code:
a1=[-1.0970 -2.1263 -2.5553 -3.9090]’; a2=[-1.5794 -1.6857 -2.7908 -3.6895]’;
K1=place(A1, B1, a1); K2=place(A2, B2, a2);
We obtain
K1   294.8755 73.1208 13.4726 27.3362
T
and
K 2   440.3915 118.5144 19.1611 35.5575
T
Let G12  A1  B1 K 2  A2  B2 K1 and using LMI toolbox in MABLAB to solve the
following LMIs
( A1  B1 K1 )T P  P ( A1  B1 K1 )  0
( A2  B2 K 2 )T P  P ( A2  B2 K 2 )  0
G12T P  PG12  0
provides a common solution of P:
54.9580 15.6219 6.9389 12.1165
15.6219 4.5429 2.1011 3.5488 

P
 6.9389 2.1011 1.3972 1.7978 


12.1165
3.5488
1.7978
2.9375


So, the controller (14) can be applied to both the fuzzy model and the
nonlinear system.
Response of state x1
Response of state x2
Response of state x3
Fig. 2: The controller applied to the T-S fuzzy model
Response of state x4
Response of state x1
Response of state x2
Response of state x3
Fig. 3: The controller applied to the nonlinear system
Response of state x4