Let us consider an example of dc motor controlling an inverted pendulum via
w x w x
a gear train 22 . Fuzzy modeling for the nonlinear system was done in 3 ,
w23 and 24 . The fuzzy model is as follows:x w x Plant Rule 1
IFx t1Ž .is M1,
xŽ .t sA xŽ .t qB u tŽ .,
˙ 1 1
THEN ẵy tŽ .sC x1 Ž .t . Ž3.78. Plant Rule 2
IFx t1Ž .is M2,
xŽ .t sA xŽ .t qB u tŽ .,
˙ 2 2
THEN ẵy tŽ .sC x2 Ž .t . Ž3.79. Here,
xŽ .t s x t1Ž . x2Ž .t x3Ž .t T,
0 1 0 0
A1s 9.8 0 1 , B1s 0 ,
0 y10 y10 10
w x
C1s 1 0 0
0 1 0 0
A2s 0 0 1 , B2s 0 ,
0 y10 y10 10
w x
C2s 1 0 0 .
Ž . Ž . Ž . Ž .
The angle of the pendulum is x t1 , x t2 s˙x t1 , and x t3 is current of the motor. The M1 and M2 are fuzzy sets defined as
°sin x t1Ž .
, x t1Ž ./0,
~ x tŽ .
M x t1Ž 1Ž ..s 1
¢ 1, x tŽ .s0,
1
M2Žx t1Ž ..s1yM x t1Ž 1Ž ...
This fuzzy model exactly represents the dynamics of the nonlinear me- chanical system under yFx t1Ž .F. Note that the fuzzy model has a
common B matrix, that is, B1sB2. The fuzzy controller design of the common B matrix cases is simple in general. To show the effect of the LMI-based designs, we consider a more difficult case, that is, we change B2 as follows:
0 B2s 0 .
20
3.8.1 Design Case 1: Decay Rate
We first design a stable fuzzy controller by considering the decay rate. The design problem of the CFS is defined as follows:
maximize ␣
X,Y,M1, . . . ,Mr
Ž . Ž .
subject toX)0,YG0, 3.39 and 3.40 .
Fig. 3.3 Design examples 1 and 2.
We obtain
␣ s5.0,
w x
F1s 282.3129 62.4176 3.2238 ,
w x
F2s 110.4644 24.9381 1.2716 , 105.108 20.4393 1.05294
y1
P sX s 20.4393 4.29985 0.23680 )0, 1.05294 0.23680 0.01567
1432.034 299.8039 16.26773
y1 y1
Q sX YX s 299.8039 63.19188 3.449801 G0.
16.26773 3.449801 0.190786
Ž . w Ž .x The dotted line in Figure 3.3 shows the responses of y t sx t1 and u tŽ ..
3.8.2 Design Case 2: Decay RateHConstraint on the Control Input 5 Ž .5
It can be seen in the design example 1 that maxt u t 2s624. In practical design, there is a limitation of control input. It is important to consider not only the decay rate but also the constraint on the control input. The design problem that considers the decay rate and the constraint on the control input
Ž . w xT
is defined as follows, where s100 and x0 s 0 10 0 : maximize ␣
X,Y,M1, . . . ,Mr
Ž . Ž . Ž . Ž .
subject to X)0, YG0 3.39 , 3.40 , 3.46 , and 3.47 . The solution is obtained as
␣ s4.23,
w x
F1s 38.3637 9.9338 0.7203 ,
w x
F2s 18.2429 6.4771 0.5118 ,
0.1578 0.03847 0.002738 0.03847 0.009995 0.000742
P s )0,
y5
0.002738 0.000742 5.831=10
y5
0.001250 0.000281 4.275=10
y6
Q s 0.000281 0.0001215 6.332=10 G0.
y5 y6 y6
4.275=10 6.332=10 1.976=10
Ž .Ž Ž .. Ž . The real line in Figure 3.3 shows the responses of y t sx t1 and u t . The
5 Ž .5
designed controller realizes the input constraint maxt u t 2s99.3-. 3.8.3 Design Case 3: StabilityHConstraint on the Control Input It is also possible to design a stable fuzzy controller satisfying the constraint on the control input, where s100.
Ž . Ž . Ž . Ž .
Find X)0, YG0, and Mi is1, . . . ,r satisfying 3.23 , 3.24 , 3.53 ,
Ž .
and 3.54 .
The solution is obtained as
w x
F1s 13.0065 3.6948 0.1786 ,
w x
F2s 7.7309 2.7900 0.1163 , 0.0335 0.0106 0.0015 Ps 0.0106 0.0036 0.0005 )0,
0.0015 0.0005 0.0001 0.0522 0.0203 0.0040 Qs 0.0203 0.0082 0.0016 G0.
0.0040 0.0016 0.0003
Ž . w Ž .x The dotted line in Figure 3.4 shows the responses of y t sx t1 and
Ž . 5 Ž .5
u t. It can be found that maxt u t 2s38.1-.
Fig. 3.4 Design examples 3 and 4.
3.8.4 Design Case 4: StabilityHConstraint on the Control InputH Constraint on the Output
The response of the control system in the design example 3 has a large
Ž 5 Ž .5 .
output error maxt y t 2s2.16 since the constraint on the output is not considered in the fuzzy controller design. To improve the response, we can design a fuzzy controller by adding the constraint on the output.
Ž . Ž . Ž . Ž .
Find X)0, YG0, and Mi is1, . . . ,r satisfying 3.23 , 3.24 , 3.46 , Ž3.47 , and 3.54 where. Ž . s100 and s2.
The solution is obtained as
w x
F1s 59.2819 9.3038 0.5580 ,
w x
F2s 33.7254 7.4115 0.4122 , 0.5478 0.0519 0.0034 P s 0.0519 0.0098 0.0006 )0,
0.0034 0.0006 0.0001 0.9936 0.0334 0.0075 Q s 0.0334 0.0118 0.0008 G0.
0.0075 0.0008 0.0001
Ž . w Ž .x Ž .
The real line in Figure 3.4 shows the responses of y t sx t1 and u t . 5 Ž .5 The response of the control system satisfies the constraints maxt u t 2s93
5 Ž .5
- and maxt y t 2s1.25-.
REFERENCES
1. K. Tanaka, T. Taniguchi, and H. O. Wang, ‘‘Model-Based Fuzzy Control of TORA System: Fuzzy Regulator and Fuzzy Observer Design via LMIs that Represent Decay Rate, Disturbance Rejection, Robustness, Optimality,’’ Seventh IEEE International Conference on Fuzzy Systems, Alaska, 1998, pp. 313᎐318.
2. K. Tanaka, T. Ikeda, and H. O. Wang, ‘‘Design of Fuzzy Control Systems Based on Relaxed LMI Stability Conditions,’’ 35th IEEE Conference on Decision and Control, Kobe, Vol. 1, 1996, pp. 598᎐603.
3. K. Tanaka, T. Ikeda, and H. O. Wang, ‘‘Fuzzy Regulators and Fuzzy Observers,’’
Ž .
IEEE Trans. Fuzzy Syst., Vol. 6, No. 2, pp. 250᎐265 1998 .
4. K. Tanaka and M. Sugeno, ‘‘Stability Analysis of Fuzzy Systems Using Lyapunov’s Direct Method,’’Proc. of NAFIPS’90, pp. 133᎐136, 1990.
5. R. Langari and M. Tomizuka, ‘‘Analysis and Synthesis of Fuzzy Linguistic Control Systems,’’ 1990 ASME Winter Annual Meeting, 1990, pp. 35᎐42.
6. S. Kitamura and T. Kurozumi, ‘‘Extended Circle Criterion and Stability Analysis of Fuzzy Control Systems,’’ in Proc. of the International Fuzzy Eng. Symp.’91, Vol. 2, 1991, pp. 634᎐643.
7. K. Tanaka and M. Sugeno, ‘‘Stability Analysis and Design of Fuzzy Control
Ž .
Systems,’’Fuzzy Sets Systs. Vol. 45, No. 2, pp. 135᎐156 1992 .
8. S. S. Farinwata et al., ‘‘Stability Analysis of The Fuzzy Logic Controller Designed by The Phase Portrait Assignment Algorithm,’’ Proc. of 2nd IEEE International Conference on Fuzzy Systems, 1993, pp. 1377᎐1382.
9. K. Tanaka and M. Sano, ‘‘Fuzzy Stability Criterion of a Class of Nonlinear
Ž .
Systems,’’Inform. Sci., Vol. 71, Nos. 1 & 2, pp. 3᎐26 1993 .
10. K. Tanaka and M. Sugeno, ‘‘Concept of Stability Margin or Fuzzy Systems and Design of Robust Fuzzy Controllers,’’ in Proceedings of 2nd IEEE International Conference on Fuzzy Systems, Vol. 1, 1993, pp. 29᎐34.
11. H. O. Wang, K. Tanaka, and M. Griffin, ‘‘Parallel Distributed Compensation of Nonlinear Systems by Takagi and Sugeno’s Fuzzy Model.,’’Proceedings of FUZZ- IEEE’95, 1995, pp. 531᎐538.
12. H. O. Wang, K. Tanaka, and M. Griffin, ‘‘An Analytical Framework of Fuzzy Modeling and Control of Nonlinear Systems,’’ 1995 American Control Confer- ence, Vol 3, Seattle, 1995, pp. 2272᎐2276.
13. S. Singh, ‘‘Stability Analysis of Discrete Fuzzy Control Systems,’’ Proceedings of First IEEE International Conference on Fuzzy Systems, 1992, pp. 527᎐534.
14. R. Katoh et al., ‘‘Graphical Stability Analysis of a Fuzzy Control System,’’
Proceedings of IEEE International Conference on IECON’93, Vol. 1, 1993, pp. 248᎐253.
15. C.-L. Chen et al., ‘‘Analysis and Design of Fuzzy Control Systems,’’Fuzzy Sets and
Ž .
Syst., Vol. 57, pp. 125᎐140 1993 .
16. F. Hara and M. Ishibe, ‘‘Simulation Study on the Existence of Limit Cycle Oscillation in a Fuzzy Control System,’’ Proceedings of the Korea-Japan Joint Conference on Fuzzy Systems and Engineering, 1992, pp. 25᎐28.
17. H. O. Wang, K. Tanaka, and M. Griffin, ‘‘An Approach to Fuzzy Control of Nonlinear Systems: Stability and Design Issues,’’IEEE Trans. Fuzzy Syst., Vol. 4,
Ž .
No. 1, pp. 14᎐23 1996 .
18. K. Tanaka and M. Sano, ‘‘A Robust Stabilization Problem of Fuzzy Controller Systems and Its Applications to Backing up Control of a Truck-Trailer,’’ IEEE
Ž .
Trans. Fuzzy Syst., Vol. 2, No. 2, pp. 119᎐134 1994 .
19. S. Kawamoto et al. ‘‘An Approach to Stability Analysis of Second Order Fuzzy Systems,’’ Proceedings of First IEEE International Conference on Fuzzy Systems, Vol. 1, 1992, pp. 1427᎐1434.
20. A. Ichikawa et al.,Control Hand Book, Ohmu Publisher, 1993, Tokyo in Japanese.
21. K. Tanaka , T. Taniguchi, and H. O. Wang, ‘‘Trajectory Control of an Articulated Vehicle with Triple Trailers,’’ 1999 IEEE International Conference on Control Applications, Vol. 2, Hawaii, August 1999.
22. J. G. Kushewski et. al., ‘‘Application of Feedforward Neural Networks to Dynam- ical System Identification and Control,’’IEEE Trans.Control Sys.Technol., Vol. 1,
Ž .
No. 1, pp. 37᎐49 1993 .
23. K. Tanaka and M. Sano, ‘‘On Design of Fuzzy Regulators and Fuzzy Observers,’’
Proc. 10th Fuzzy System Symposium, 1994, pp. 411᎐414 in Japanese.
24. S. Kawamoto, et. al., ‘‘Nonlinear Control and Rigorous Stability Analysis Based on Fuzzy System for Inverted Pendulum,’’Proc. of FUZZ-IEEE’96, Vol. 2, 1996, pp. 1427᎐1432.
Copyright䊚2001 John Wiley & Sons, Inc.
Ž . Ž .
ISBNs: 0-471-32324-1 Hardback ; 0-471-22459-6 Electronic
CHAPTER 4