Locally Optimal Takagi-Sugeno Fuzzy Controllers

Amir massoud Farahmandamir@cs.ualberta.ca

Mohammad javad Yazdanpanahyazdan@ut.ac.ir

Department of Electrical and Computer EngineeringUniversity of Tehran

Tehran, Iran

Department of Electrical and Computer Engineering

University of Tehran

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)()(A(t)x THEN M is )( and ... and M is )( IF :R i

in

i11i tuBtxtxtx in +=&

( )( )∑=

+=r

iiii tuBtxAtxhtx

1

)()()()(&

∑=

= r

ii

ii

xw

xwxh

1

)(

)()(

,

∏=

=r

iiMk txxw k

i1

))(()( μ

Theorem 1. The continuous uncontrolled T-S fuzzy system is globally quadratically stable if there exists a common positive definite matrix P such that

r1,...,i ,0 =<+ iTi PAPA

Department of Electrical and Computer Engineering

University of Tehran

Parallel Distributed Compensation

x(t)-Ku(t) THEN M is )( and ... and M is )( IF :rule Control iin

i11i =txtx n

∑=

−=r

iii txKxhu

1

)()(

r1,...,ji, ,022

=<⎟⎟⎠

⎞⎜⎜⎝

⎛ ++⎟⎟

⎠

⎞⎜⎜⎝

⎛ + jiijT

jiij GGPP

GG

jiiij KBAG −=

Stability condition:

Department of Electrical and Computer Engineering

University of Tehran

Locally Optimal Design

( ) [ ]∫∞

+=0

.)(),( dtRuuQxxtutxJ TT

uBxAx ii +=& PBRK Ti

1−=

),( uxfx =&Linearization

ii BA ,

Locally optimal design

Department of Electrical and Computer Engineering

University of Tehran

Experiments: Problem description

Nonlinear Mass-Spring-Damper system( ) ( ) ( ) )()()()(),()( tutxtxftxtxgtxM &&&& φ=++

( ) ⎟⎟⎠⎞

⎜⎜⎝

⎛

−+−−−=

uxxxx

xtx

22

3112

2

13.04387.11.001.0)(&

( )0

0 13.001.0

10,210 xu

xx xx

uxfA ⎟⎟

⎠

⎞⎜⎜⎝

⎛−−−

=∂

∂=

==

( )

00

2213.04387.1

0,

xxx xu

uxfB ⎟⎟

⎠

⎞⎜⎜⎝

⎛

−=

∂

∂=

=

Department of Electrical and Computer Engineering

University of Tehran

Experiments: Fuzzy Settings

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ −−=

2

2

2

)(exp))((

σμ c

M

xxtxk

The dynamics of the plant is approximated using Gaussian membership function

Approximation error

Department of Electrical and Computer Engineering

University of Tehran

Experiments: Stabilization (I)

0 1 2 3 4 5 6 7-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

T

X2

0 1 2 3 4 5 6 7-1

0

1

2

3

4

5

T

X1

0 1 2 3 4 5 6 7-5

-4

-3

-2

-1

0

1

2

T

U

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

Experiments: Stabilization (II)

0 1 2 3 4 5 6 7-2

0

2

4

6

8

10

T

x1

0 1 2 3 4 5 6 7-10

-8

-6

-4

-2

0

2

T

x2

0 1 2 3 4 5 6 7-6

-5

-4

-3

-2

-1

0

1

2

T

U

Response of T-S controller to (10 0)'

The linear controller is not stable in this case, but the fuzzy controller can handle it easily.

Department of Electrical and Computer Engineering

University of Tehran

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

x2

x1

Q=10I,I=1

-3 -2 -1 0 1 2 3

-4

-3

-2

-1

0

1

2

3

4

x2

x1

Q=I,I=1

-4 -3 -2 -1 0 1 2 3 4

-4

-3

-2

-1

0

1

2

3

4

Q=I,I=10

-4 -3 -2 -1 0 1 2 3 4

-4

-3

-2

-1

0

1

2

3

4

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

Conclusions and Suggestions Conclusions

Stable Fuzzy Controller Local Optimality

How close is it to the global optimal solution?!

Suggestions Comparison with other T-S controllers Modeling error and stability (polytopic systems) Considering the effect of membership functions

explicitly