Locally Optimal
Takagi-Sugeno Fuzzy Controllers
Amir massoud Farahmand
Mohammad javad Yazdanpanah
[email protected]
[email protected]
Department of Electrical and Computer Engineering
University of Tehran
Tehran, Iran
Fuzzy Control


Successful in many applications
Ease of use


Intuitive and interpretable
Powerful nonlinear controller
Department of Electrical and Computer Engineering
University of Tehran
Takagi-Sugeno Plant Model
R i : IF x1 (t ) is M1i and ... and xn (t ) is M in THEN x (t)  A i x(t )  Biu (t )
r
x (t )   hi  x(t )  Ai x(t )  Biu (t ) 
i 1
,
hi ( x) 
wi ( x)
r
 w ( x)
i 1
r
wk ( x)    M k ( xi (t ))
i 1
i
i
Theorem 1. The continuous uncontrolled T-S fuzzy system is globally
quadratically stable if there exists a common positive definite matrix P such that
AiT P  PAi  0, i  1,..., r
Department of Electrical and Computer Engineering
University of Tehran
Parallel Distributed
Compensation
Control rule i : IF x1 (t ) is M1i and ... and xn (t ) is M in THEN u(t)  -Ki x(t)
r
u   hi ( x) K i x(t )
i 1
r
r


x (t )   hi ( x) h j ( x) Ai  Bi K j x(t )
i 1 j 1
 G  G ji 
  h i ( x)Gii x(t )   hi ( x)h j ( x). ij
x(t )

2
i 1
i 1 j 1


r
r
r
2
T
Stability condition:
Gij  Ai  Bi K j
 Gij  G ji 
 G  G ji 

 P  P ij
  0, i, j  1,..., r
2
2




Department of Electrical and Computer Engineering
University of Tehran
Locally Optimal Design
Linearization
x  f ( x, u )
Ai , Bi
Locally optimal design
x  Ai x  Bi u
K i  R 1 B T P



J x(t ), u (t )    x T Qx  u T Ru .dt
0
Department of Electrical and Computer Engineering
University of Tehran
Experiments: Problem description
Nonlinear Mass-SpringDamper system
Mx(t )  g x(t ), x(t )  f x(t )   x(t )u(t )

 x2
x (t )  
3
2

x

0
.
01
x

0
.
1
x

1
.
4387

0
.
13
x
1
1
2
 2

0

f x, u 
A
 
x ux  0x0   0.01  0.3x12
1 

 1 x
0


u 

0

f  x, u 

B
 
2
u x  x0 1.4387  0.13x2 
x0
Department of Electrical and Computer Engineering
University of Tehran
Experiments: Fuzzy Settings
The dynamics of the plant is
approximated using Gaussian
membership function
 ( x  xc ) 2 

M k ( x(t ))  exp  
2


2



Approximation error
Department of Electrical and Computer Engineering
University of Tehran
Experiments: Stabilization (I)
2
0.5
5
0
4
1
-0.5
0
-1
3
-1
2
U
X2
X1
-1.5
-2
-2
-2.5
1
-3
-3
-3.5
0
-4
-4
-5
-1
-4.5
0
1
2
3
4
T
5
6
7
0
1
2
3
4
5
6
0
1
7
2
3
4
5
T
T
Comparison of T-S controller (bold) and linear controller (dotted) with different initial conditions
Both TS and linear controller are stable in this
case. However, the behavior of fuzzy controller is
smoother and with lower overshoot.
Department of Electrical and Computer Engineering
University of Tehran
6
7
Experiments: Stabilization (II)
10
2
8
0
6
-2
2
1
0
4
-4
U
x2
x1
-1
-2
-3
2
-6
0
-8
-4
-5
-2
0
1
2
3
4
5
6
T
7
-10
0
1
2
3
4
5
6
7
-6
0
1
2
T
The linear controller is not stable in this case,
but the fuzzy controller can handle it easily.
Response of T-S controller to (10 0)'
Department of Electrical and Computer Engineering
University of Tehran
3
4
T
5
6
7
Experiments: Performance
Comparison
Linear
TS
Q=I, R=1
5.80
5.47
Q=10I,R=
1
9.06
8.44
Q=I, R=10
5.58
5.62
,,
Department of Electrical and Computer Engineering
University of Tehran
Experiments: Performance
Comparison
Q=10I,I=1
Q=I,I=10
-4
-4
-3
-3
-3
-2
-2
-2
-1
-1
-1
0
0
1
1
1
2
2
2
3
3
3
x1
x1
Q=I,I=1
-4
0
4
4
-4
-3
-2
-1
0
x2
1
2
3
4
4
-3
-2
-1
0
x2
1
2
3
-4
-3
-2
-1
0
1
Fig. 3. Performance region comparison for different performance indices: (Q=1, R=1), (Q=10, R=1), and (Q=1, R=10), from left to right,
respectively (dark region means linear one has better performance).
Department of Electrical and Computer Engineering
University of Tehran
2
3
4
Conclusions and Suggestions

Conclusions
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Stable Fuzzy Controller
Local Optimality
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How close is it to the global optimal solution?!
Suggestions
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Comparison with other T-S controllers
Modeling error and stability (polytopic systems)
Considering the effect of membership functions
explicitly
Department of Electrical and Computer Engineering
University of Tehran