Robust Local Regularity and Controllability of Uncertain TS Fuzzy Descriptor Systems

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Journal of Applied Mathematics Volume 2012, Article ID 825416,14pages doi:10.1155/2012/825416

Research Article

Robust Local Regularity and Controllability of

Uncertain TS Fuzzy Descriptor Systems

Shinn-Horng Chen,

1

_{Wen-Hsien Ho,}

2

_{and Jyh-Horng Chou}

1, 3

1_{Department of Mechanical/Electrical Engineering, National Kaohsiung University of Applied Sciences,} 415 Chien-Kung Road, Kaohsiung 807, Taiwan

2_{Department of Healthcare Administration and Medical Informatics, Kaohsiung Medical University,} 100 Shi-Chuan 1st Road, Kaohsiung 807, Taiwan

3_{Institute of System Information and Control, National Kaohsiung First University of Science and} Technology, 1 University Road, Yenchao, Kaohsiung 824, Taiwan

Correspondence should be addressed to Jyh-Horng Chou,choujh@nkfust.edu.tw

Received 2 October 2012; Accepted 28 October 2012 Academic Editor: Jen Chih Yao

Copyrightq 2012 Shinn-Horng Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The robust local regularity and controllability problem for the Takagi-SugenoTS fuzzy descriptor systems is studied in this paper. Under the assumptions that the nominal TS fuzzy descriptor systems are locally regular and controllable, a sufficient criterion is proposed to preserve the assumed properties when the structured parameter uncertainties are added into the nominal TS fuzzy descriptor systems. The proposed sufficient criterion can provide the explicit relationship of the bounds on parameter uncertainties for preserving the assumed properties. An example is given to illustrate the application of the proposed sufficient condition.

1. Introduction

Recently, it has been shown that the fuzzy-model-based representation proposed by Takagi and Sugeno 1, known as the TS fuzzy model, is a successful approach for dealing with the nonlinear control systems, and there are many successful applications of the TS-fuzzy-model-based approach to the nonlinear control systemse.g., 2–19 and references therein. Descriptor systems represent a much wider class of systems than the standard systems20. In recent years, some researcherse.g., 4–6,8,21–28 and references therein have studied the design issue of the fuzzy parallel-distributed-compensation PDC controllers for each fuzzy rule of the TS fuzzy descriptor systems. Both regularity and controllability are actually two very important properties of descriptor systems with control inputs29. So, before the design of the fuzzy PDC controllers in the corresponding rule of the TS fuzzy descriptor

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systems, it is necessary to consider both properties of local regularity and controllability for each fuzzy rule23. However, both regularity and controllability of the TS fuzzy systems are not considered by those mentioned-above researchers before the fuzzy PDC controllers are designed. Therefore, it is meaningful to further study the criterion that the local regularity and controllability for each fuzzy rule of the TS fuzzy descriptor systems hold30.

On the other hand, in fact, in many cases it is very diﬃcult, if not impossible, to obtain the accurate values of some system parameters. This is due to the inaccurate measurement, inaccessibility to the system parameters, or variation of the parameters. These parametric uncertainties may destroy the local regularity and controllability properties of the TS fuzzy descriptor systems. But, to the authors’ best knowledge, there is no literature to study the issue of robust local regularity and controllability for the uncertain TS fuzzy descriptor systems.

The purpose of this paper is to present an approach for investigating the robust local regularity and controllability problem of the TS fuzzy descriptor systems with structured parameter uncertainties. Under the assumptions that the nominal TS fuzzy descriptor systems are locally regular and controllable, a sufficient criterion is proposed to preserve the assumed properties when the structured parameter uncertainties are added into the nominal TS fuzzy descriptor systems. The proposed sufficient criterion can provide the explicit relationship of the bounds on structured parameter uncertainties for preserving the assumed properties. A numerical example is given in this paper to illustrate the application of the proposed sufficient criterion.

2. Robust Local Regularity and Controllability Analysis

Based on the approach of using the sector nonlinearity in the fuzzy model construction, both the fuzzy set of premise part and the linear dynamic model with parametric uncertainties of consequent part in the exact TS fuzzy control model with parametric uncertainties can be derived from the given nonlinear control model with parametric uncertainties5. The TS continuous-time fuzzy descriptor system with parametric uncertainties for the nonlinear con-trol system with structured parametric uncertainties can be obtained as the following form:

Ri: IF z1is Mi1and . . . and zgis Mig,

then Ei˙xt Ai ΔAixt Bi ΔBiut,

2.1

or the uncertain discrete-time TS fuzzy descriptor system can be described by

Ri: IF z1is Mi1and . . . and zgis Mig,

then Eixk 1 Ai ΔAixk Bi ΔBiuk,

2.2

with the initial state vector x0, where Rii 1, 2, . . . , N denotes the ith implication, N is the

number of fuzzy rules, xt x1t, x2t, . . . , xntT and xk x1k, x2k, . . . , xnkT

denote the n-dimensional state vectors, ut u1t, u2t, . . . , uptT and uk 

u1k, u2k, . . . , upkT denote the p-dimensional input vectors, zii 1, 2, . . . , g are

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n× p consequent constant matrices, ΔAi and ΔBii 1, 2, . . . , N are, respectively, the

parametric uncertain matrices existing in the system matrices Aiand the input matrices Biof

the consequent part of the ith rule due to the inaccurate measurement, inaccessibility to the system parameters, or variation of the parameters, and Miji 1, 2, . . . , N and j 1, 2, . . . , g

are the fuzzy sets. Here the matrices Ei i 1, 2, . . . , N may be singular matrices with

rankEi ≤ n i 1, 2, . . . , N. In many applications, the matrices Ei i 1, 2, . . . , N

are the structure information matrices; rather than parameter matrices, that is, the elements of Eii 1, 2, . . . , N contain only structure information regarding the problem

considered.

In many interesting problemse.g., plant uncertainties, constant output feedback with uncertainty in the gain matrix, we have only a small number of uncertain parameters, but these uncertain parameters may enter into many entries of the system and input matrices 31,32. Therefore, in this paper, we suppose that the parametric uncertain matrices ΔAiand

ΔBitake the forms

ΔAi m k1 εikAik, ΔBi m k1 εikBik, 2.3

where εiki 1, 2, . . . , N and k 1, 2, . . . , m are the elemental parametric uncertainties, and Aik and Biki 1, 2, . . . , N and k 1, 2, . . . , m are, respectively, the given n × n and n × p

constant matrices which are prescribed a priori to denote the linearly dependent information on the elemental parametric uncertainties εik.

In this paper, for the uncertain TS fuzzy descriptor system in2.1 or 2.2, each fuzzy-rule-nominal model Ei˙xt Aixt Biut or Eixk 1 Aixk Biuk, which

is denoted by {Ei, Ai, Bi}, is assumed to be regular and controllable. Due to inevitable

uncertainties, each fuzzy-rule-nominal model {Ei, Ai, Bi} is perturbed into the

fuzzy-rule-uncertain model {Ei, Ai  ΔAi, Bi  ΔBi}. Our problem is to determine the conditions

such that each fuzzy-uncertain model {Ei, Ai  ΔAi, Bi  ΔBi} for the uncertain TS fuzzy

descriptor system 2.1 or 2.2 is robustly locally regular and controllable. Before we investigate the robust properties of regularity and controllability for the uncertain TS fuzzy descriptor system2.1 or 2.2, the following definitions and lemmas need to be introduced first.

Definition 2.1see 33. The measure of a matrix W ∈ Cn×nis defined as

μW≡ lim

θ→ 0

_{I θW}_{− 1}

θ ,

2.4

where · is the induced matrix norm on Cn×n.

Definition 2.2see 34. The system {Ei, Ai, Bi} is called controllable, if for any t1 > 0 or

k1 > 0, x0 ∈ Rn_{, and w}_{∈ R}n_{, there exists a control input ut or uk such that xt}

1 w

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vii 2 k−1 εikφik 0 < 1, for εi1∈ −1 1.1, εi2 ∈ 0 10, 3.4g viii 2 k−1 εikφik≤ 0.1200 < 1, for εi1 ∈ −1 1.1, εi2∈ −1.2 0. 3.4h

From the results in3.3a–3.3h and 3.4a–3.4h, we can conclude that the uncertain TS fuzzy descriptor system3.1a and 3.1b is locally robustly regular and controllable.

4. Conclusions

The robust local regularity and controllability problem for the uncertain TS fuzzy descriptor systems has been investigated. The rank preservation problem for robust local regularity and controllability of the uncertain TS fuzzy descriptor systems is converted to the nonsingularity analysis problem. Under the assumption that each fuzzy rule of the nominal TS fuzzy descriptor system has the full row rank for its related regularity and controllability matrices, a sufficient criterion has been proposed to preserve the assumed properties when the elemental parameter uncertainties are added into the nominal TS fuzzy descriptor systems. The proposed sufficient conditions in 2.9a–2.9c can provide the explicit relationship of the bounds on elemental parameter uncertainties for preserving the assumed properties. One example has been given to illustrate the application of the proposed sufficient conditions. On the other hand, the issue of robust global regularity and controllability with evolutionary computation 36 for the uncertain TS fuzzy descriptor systems will be an interesting and important topic for further research.

Acknowledgment

This work was in part supported by the National Science Council, Taiwan, under Grants nos. NSC 100-2221-E-151-009, NSC 101-2221-E-151-076, and NSC 101-2320-B-037-022.

References

1 T. Takagi and M. Sugeno, “Fuzzy identification of systems and its applications to modeling and control,” IEEE Transactions on Systems, Man and Cybernetics, vol. 15, no. 1, pp. 116–132, 1985.

2 R. Babuska, Fuzzy Modeling for Control, Kluwer, Boston, Mass, USA, 1998.

3 S. S. Farinwata, D. Filev, and R. Langari, Fuzzy Control: Synthesis and Analysis, John Wiley & Sons, Chichester, UK, 2000.

4 T. Taniguchi, K. Tanaka, and H. O. Wang, “Fuzzy descriptor systems and nonlinear model following control,” IEEE Transactions on Fuzzy Systems, vol. 8, no. 4, pp. 442–452, 2000.

5 K. Tanaka and H. O. Wang, Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality

Approach, John Wiley & Sons, New York, NY, USA, 2001.

6 Y. Wang, Q. L. Zhang, and W. Q. Liu, “Stability analysis and design for T-S fuzzy descriptor systems,” in Proceedings of the 40th IEEE Conference on Decision and Control (CDC ’01), pp. 3962–3967, Orlando, Fla, USA, December 2001.

7 S. C. Tong, T. Wang, Y. P. Wang, and J. T. Tang, Design and Stability Analysis of Fuzzy Control Systems, Science Press, Beijing, China, 2004.

8 Y. Wang, Z. Q. Sun, and F. C. Sun, “Robust fuzzy control of a class of nonlinear descriptor systems with time-varying delay,” International Journal of Control, Automation and Systems, vol. 2, no. 1, pp. 76–82, 2004.

(5)

9 T.-S. Lee, Y.-H. Chen, and J. C.-H. Chuang, “Robust control design of fuzzy dynamical systems,”

Applied Mathematics and Computation, vol. 164, no. 2, pp. 555–572, 2005.

10 J. S. Ren and Y. S. Yang, “Robust control of chaotic system based on T-S fuzzy model,” Systems

Engineering and Electronics, vol. 27, no. 3, pp. 474–478, 2005.

11 K. Y. Lian, J. J. Liou, and C. Y. Huang, “LMI-based integral fuzzy control of DC-DC converters,” IEEE

Transactions on Fuzzy Systems, vol. 14, no. 1, pp. 71–80, 2006.

12 Z. Li, Fuzzy Chaotic Systems: Modeling, Control and Applications, Springer, Berlin, Germany, 2006. 13 W. H. Ho and J. H. Chou, “Design of optimal controllers for Takagi-Sugeno fuzzy-model-based

systems,” IEEE Transactions on Systems, Man, and Cybernetics Part A, vol. 37, no. 3, pp. 329–339, 2007. 14 W. H. Ho, J. T. Tsai, and J. H. Chou, “Robust-stable and quadratic-optimal control for

TS-fuzzy-model-based control systems with elemental parametric uncertainties,” IET Control Theory and Applications, vol. 1, no. 3, pp. 731–742, 2007.

15 M. R. Hsu, W. H. Ho, and J. H. Chou, “Stable and quadratic optimal control for TS fuzzy-model-based time-delay control systems,” IEEE Transactions on Systems, Man, and Cybernetics Part A, vol. 38, no. 4, pp. 933–944, 2008.

16 W. H. Ho, J. T. Tsai, and J. H. Chou, “Robust quadratic-optimal control of TS-fuzzy-model-based dynamic systems with both elemental parametric uncertainties and norm-bounded approximation error,” IEEE Transactions on Fuzzy Systems, vol. 17, no. 3, pp. 518–531, 2009.

17 Z. Hidayat, Z. Lendek, R. Babuˇska, and B. De Schutter, “Fuzzy observer for state estimation of the METANET traﬃc model,” in Proceedings of the 13th International IEEE Conference on Intelligent

Transportation Systems (ITSC ’10), pp. 19–24, Funchal, Portugal, September 2010.

18 C. W. Chen, “Stability analysis and robustness design of nonlinear systems: an NN-based approach,”

Applied Soft Computing Journal, vol. 11, no. 2, pp. 2735–2742, 2011.

19 W. H. Ho, “Takagi-Sugeno fuzzy model of nonlinear HIV dynamics: chebyshev-series approach integrated with genetic algorithm,” International Journal of Innovative Computing Information and

Control, vol. 8, pp. 1439–1452, 2012.

20 G. C. Verghese, B. C. L´evy, and T. Kailath, “A generalized state-space for singular systems,” IEEE

Transactions on Automatic Control, vol. 26, no. 4, pp. 811–831, 1981.

21 Y. Wang, Q. L. Zhang, and X. D. Liu, “Robustness design of uncertain discrete-time fuzzy descriptor systems with guaranteed admissibility,” in Proceedings of the American Control Conference, pp. 1699– 1704, Anchorage, Alaska, USA, May 2002.

22 S.-W. Kau, Y.-S. Liu, C.-H. Lee, L. Hong, and C.-H. Fang, “Static output feedback fuzzy controller design for T-S fuzzy descriptor systems,” International Journal of Fuzzy Systems, vol. 6, no. 4, pp. 187– 193, 2004.

23 B. Y. Zhu, Q. L. Zhang, and K. N. Da, “State feedback robust H∞control for T-S fuzzy descriptor

systems,” Journal of Northeastern University. Natural Science, vol. 25, no. 5, pp. 453–456, 2004.

24 X. Shi and Z. W. Gao, “Stability analysis for fuzzy descriptor systems,” Systems Engineering and

Electronics, vol. 27, no. 6, pp. 1087–1089, 2005.

25 B.-Y. Zhu, Q.-L. Zhang, and S.-C. Tong, “Passivity control for uncertain T-S fuzzy descriptor systems,”

Acta Automatica Sinica, vol. 32, no. 5, pp. 674–679, 2006.

26 K. Tanaka, H. Ohtake, and H. O. Wang, “A descriptor system approach to fuzzy control system design via fuzzy Lyapunov functions,” IEEE Transactions on Fuzzy Systems, vol. 15, no. 3, pp. 333–341, 2007. 27 W. H. Tian and H. G. Zhang, “Optimal guaranteed cost control of uncertain fuzzy descriptor systems

with time-varying delay,” Journal of Northeastern University, vol. 29, no. 4, pp. 464–468, 2008.

28 J. R. Wu, “The generalized quadratically stability for a fuzzy descriptor system,” in Proceedings of the

International Conference on Machine Learning and Cybernetics, pp. 855–859, Baoding, China, July 2009.

29 L. Dai, Singular Control Systems, vol. 118, Springer, Berlin, Germany, 1989.

30 Y. H. Yuan and G. M. Zhang, “Research on T-S fuzzy descriptor systems: a review,” Acta Automatica

Sinica, vol. 36, no. 7, pp. 901–911, 2010.

31 J. Chen and Z. Ren, “A comparison of small gain versus Lyapunov type robust stability bounds,”

International Journal of Robust and Nonlinear Control, vol. 11, no. 15, pp. 1407–1414, 2001.

32 C. H. Hsieh and J. H. Chou, “Robust stability analysis of TS-fuzzy-model-based control systems with both elemental parametric uncertainties and norm-bounded approximation error,” JSME International

Journal, Series C, vol. 47, no. 2, pp. 686–693, 2004.

33 C. A. Desoer and M. Vidyasagar, Feedback Systems: Input-Output Properties, Academic Press, New York, NY, USA, 1975.

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34 E. L. Yip and R. F. Sincovec, “Solvability, controllability, and observability of continuous descriptor systems,” IEEE Transactions on Automatic Control, vol. 26, no. 3, pp. 702–707, 1981.

35 D. Cobb, “Controllability, observability, and duality in singular systems,” IEEE Transactions on

Automatic Control, vol. 29, no. 12, pp. 1076–1082, 1984.

36 W.-H. Ho, J.-H. Chou, and C.-Y. Guo, “Parameter identification of chaotic systems using improved diﬀerential evolution algorithm,” Nonlinear Dynamics, vol. 61, no. 1-2, pp. 29–41, 2010.

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Robust Local Regularity and Controllability of Uncertain TS Fuzzy Descriptor Systems

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