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Published in IET Control Theory and Applications Received on 10th February 2009
doi: 10.1049/iet-cta.2012.0410
ISSN 1751-8644
Comment on ‘evolutionary design of static output
feedback controller for Takagi–Sugeno fuzzy systems’
W.-H. Ho
1J.-H. Chou
21Department of Medical Information Management, Kaohsiung Medical University, 100 Shi-Chuan 1st Road,
Kaohsiung 807, Taiwan
2Institute of System Information and Control, National Kaohsiung First University of Science and Technology,
1 University Road, Yenchao, Kaohsiung 824, Taiwan E-mail: [email protected]
Abstract: In this note, a problem that must be paid attention to is highlighted here on using the sector non-linearity in the
fuzzy model construction. The sector non-linearity approach can guarantee an exact fuzzy model construction. However, it needs carefully checking to prevent that the physical constraints are violated when the sector non-linearity approach is used. Based on the exact fuzzy model obtained from the sector non-linearity approach, an approach of designing static output feedback controller is presented in a recent study. However, it can be found that the designed control result is erroneous in that published study, owing to that a physical constraint is violated.
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Introduction
Based on the approach of using the sector non-linearity in the fuzzy model construction, both the fuzzy sets of premise part and the linear dynamic model of consequent part in the Takagi–Sugeno (TS) fuzzy control model can be easily derived from the physical model of the given non-linear control system [1]. This approach guarantees an exact fuzzy model construction for the given non-linear control model. That is, the advantage of using the approach of sector non-linearity is that there is no approximation error between the original non-linear control system and its TS-fuzzy-model-based control system [1]. Each fuzzy rule for the exact TS fuzzy control system has a linear dynamic model as the consequent part which expresses the local dynamics of each fuzzy rule. Then, the overall fuzzy model is achieved by blending these rules. The advantage of controller synthesis for such a fuzzy model is that the linear control methods can be used. However, it needs carefully checking to prevent that the physical constraints are violated when the sector non-linearity approach is used. Recently, based on the exact TS fuzzy model from the sector non-linearity approach, Chung et al. [2] proposed an integration method, which combines the genetic algorithm (GA) [3, 4] with the linear-matrix-inequality (LMI) approach [5], to design the static output feedback parallel-distributed-compensation controller. However, in the following section, it can be seen that the designed control result is erroneous because of that a physical constraint is violated.
2
Numerical example
Consider a non-linear system given by Chung et al. [2] as ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ˙x1(t)= x2(t)+ sin(x3(t))+ (x21(t)+ 1)u(t) ˙x2(t)= x1(t)+ 2x2(t) ˙x3(t)= x21(t)x2(t)+ x1(t) ˙x4(t)= sin(x3(t)) y1(t)= −3x2(t)+ sin(x3(t)) y2(t)= −1.5x1(t)+ x4(t) (1)
where x1(t)∈ [−a, a] = [−0.8, 0.8] and x3(t)∈ [−b, b] =
[−0.6, 0.6] assumed by Chung et al. [2].
Based on the approach of using the sector non-linearity in the fuzzy model construction, both the fuzzy set of premise part and the linear dynamic model of consequent part in the exact TS fuzzy control model can be derived from the given non-linear control model [1]. The model-based control system and the resulting TS-fuzzy-model-based dynamic system for the non-linear control system (1) can be obtained as that given in the work of Chung et al. [2]. Then, the closed-loop TS-fuzzy-model-based dynamic system with the static output feedback PDC controller is ˙x(t) = 4 i=1 4 j=1 4 k=1 h1(z(t))hj(z(t))hk(z(t))(Ai− BiGjCk)x(t) (2a)
IET Control Theory Appl., 2012, Vol. 6, Iss. 9, pp. 1325–1327 1325
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References
1 Tanaka, K., Wang, H.O.: ‘Fuzzy control systems design and analysis: a linear matrix inequality approach’ (John Wiley and Sons, New York, 2001)
2 Chung, H.Y., Wu, S.M., Yu, F.M., Chang, W.J.: ‘Evolutionary design of static output feedback controller for Takagi–Sugeno fuzzy systems’, IET Control Theory Appl., 2007, 1, pp. 1096–1103
3 Goldberg, D.E.: ‘Genetic algorithms in search, optimization and machine learning’ (Addison-Wesley, Massachusetts, 1989)
4 Tsai, J.T., Chou, J.H., Liu, T.K.: ‘Tuning the structure and parameters of a neural network by using hybrid Taguchi-Genetic algorithm’, IEEE Trans. Neural Netw., 2006, 17, pp. 69–80
5 Gahinet, P., Nemirovski, A., Laub, A.J., Chilali, M.: ‘LMI control toolbox’ (The Math Works Inc., Massachusetts, 1995)
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