About this Journal Submit a Manuscript Table of Contents
Mathematical Problems in Engineering
Volume 2012 (2012), Article ID 105074, 12 pages
http://dx.doi.org/10.1155/2012/105074
Research Article

Fuzzy Variable Structure Control for Uncertain Systems with Disturbance

1School of Electrical and Information Engineering, Xihua University, Chengdu 610096, China
2School of Applied Mathematics, University of Electronic Science and Technology of China, Chengdu 610054, China
3School of Electrical and Electronic Engineering, The University of Adelaide, Adelaide SA 5005, Australia
4School of Engineering and Science, Victoria University, Melbourne, VIC 8001, Australia
5Department of Engineering, Faculty of Engineering and Science, University of Agder, 4898 Grimstad, Norway
6School of Automation, Chongqing University, Chongqing 400044, China

Received 2 September 2012; Accepted 26 October 2012

Academic Editor: Mohammed Chadli

Copyright © 2012 Bo Wang 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.

Abstract

This paper focuses on the fuzzy variable structure control for uncertain systems with disturbance. Specifically, the fuzzy control is introduced to estimate the control disturbance, the switching control is included to compensate for the approximation error, and they possess the characteristic of simpleness in design and effectiveness in attenuating the control chattering. Some typical numerical examples are presented to demonstrate the effectiveness and advantage of the fuzzy variable structure controller proposed.

1. Introduction

Since the pioneering works of Utkin in 1977 [1], the variable structure control (VSC) has generated considerable interests in control field. Up to now many researches on VSC have been carried out [216]. Based on VSC theory, [2] developed an adaptive fuzzy control system design method for uncertain Takagi-Sugeno fuzzy models with norm-bounded uncertainties. By using a high-gain observer, [3] presented an output feedback model-reference variable structure controller to achieve global exponential stability with respect to a small residual set without generating peaking in the control signal. In [4], the subordinated reachability of the sliding motion is introduced to realize the control on a class of uncertain stochastic systems with time-varying delay. Via introducing a pseudo-inversion, the authors in [5] discussed the adaptive control for the uncertain discrete time linear systems preceded by hysteresis nonlinearity. In [6], a sufficient condition for existence of reduced-order sliding mode dynamics was derived to realize the sliding mode control of a continuous-time switched stochastic system. For VSC, one of the most intriguing properties is the insensitivity to parameter uncertainties and external disturbance for the switching action between sliding modes, which can lead to the generation of chattering phenomenon and make a difference to system control performance. Therefore how to solve this problem is always a challenging topic for VSC.

Uncertainties and disturbances exist in many kinds of systems; this makes the practical control problem complicated and has received much attention from scholars [1722]. VSC method is one of the effective solutions, and conventionally the switching term is built based on the upper norm bound of control disturbance to satisfy the system control condition. Therefore there exists the difference between real control disturbance and its upper norm bound. The maximum switching amplitude can be double disturbance error upper bound. For ease of use, the upper norm bound sometimes is taken as a constant by experience. This may lead to the serious chattering problem. Widely acknowledged, an effective solution is to build a unit to obtain the estimate of time-varying control disturbance. Up to now, there exist some feasible methods, such as neural networks and genetic algorithm, to tackle the problem. However in real application, those approaches are too complicated and need much more control information. Corresponding control cost problem cannot be ignored.

Recently, fuzzy method gets wide attention in the control field, corresponding research can be seen in [2333] and the references therein. It is also introduced to VSC area for its characteristic of simpleness in design, and effectiveness in attenuating chattering. In this paper, a fuzzy auxiliary controller will be built to approximate the control disturbance based on just one feedback signal and a switching control term will be designed to compensate for the approximation error. Some typical simulation examples will be concerned afterward to illustrate the effectiveness of the controller given.

Notations used in this paper are fairly standard. Let be the -dimensional Euclidean space, represents the set of real matrix, denotes the th derivative of , and the notation means that is real symmetric and positive definite, denotes the operator , and denotes the saturated function.

2. Problem Statement

In this paper, the following high order uncertain single-input single-output (SISO) system with disturbance is considered: where is the system state vector, is the nonlinear function, is the nonlinear uncertainties, is the external disturbance, is the nonzero coefficient of control input, and is the control input.

Define the tracking error where , , and is the desired trajectory with where denotes the th derivative of . Then the error dynamic system can be expressed by where is the control disturbance.

The problem to be addressed in this paper is to design a controller such that the tracking error variable satisfies In this paper, the following lemma is needed

Lemma 2.1 (see [34]). If is a uniformly continuous function for and if exists and is finite, then

3. Design of Fuzzy Variable Structure Controller

In this section, the FVSC method is introduced to realize the control for uncertain system with disturbance. First, the following sliding surface is introduced: where is chosen such that the distribution of the roots of characteristic equation is on the left side of complex plane to make the following system stable: Then, we have Based on Lyaponov method and VSC theory, the following theoretical result can be obtained.

First, a fuzzy auxiliary controller is built to estimate the control disturbance . Corresponding fuzzy rules are given byIF THEN should be increased,IF THEN should be decreased,

where The term under consideration can take a greater value. If it is too big, this may lead to some serious control problem in practice. Therefore in this paper, based on the integral method, the small value is recommended to replace by for their relations as follows: where is the proportionality coefficient.

Let denote the fuzzy input , and denotes the fuzzy output . The fuzzy sets of the input and the output are defined, respectively, as where NB is negative and large, NM is negative and medium, ZO is zero, PM is the positive and medium, and PB is positive and large.

Select the following fuzzy rules:R1: IF    is PB  THEN    is PB,R2: IF    is PM  THEN    is PM,R3: IF    is ZO  THEN    is ZO,R4: IF    is NM  THEN    is NM,R5: IF    is PB  THEN    is NB.

Hence based on the proposed fuzzy auxiliary controller, the following theoretical result can be concluded.

Theorem 3.1. For , system (2.1) can track the desired trajectory (2.3) based on the following fuzzy variable structure controller:

Proof. Choose the Lyapunov functional candidate as The time derivative of along trajectories of error model (2.4) is as Substituting (3.7) into (3.9), we have where . For , we have . Integrating both sides of (3.9) from 0 to leads to Since is positive and is finite, the following inequality can be concluded: Based on Lemma 2.1, we can obtain Hence This means the system control can be achieved based on the fuzzy VSC proposed. The proof of Theorem 3.1 is thus completed.

Remark 3.2. The fuzzy auxiliary controller is constructed based on the feedback signal , the employed fuzzy rule is simple, and essentially used to keep at zero. Hence it can be concluded that . This completes our proof.

We can see that the fuzzy auxiliary controller and the sliding mode controller come together to realize the effective control on system (2.1).

4. Numerical Example

In this section, we will verify the proposed methodology by giving an illustrative example. First consider the following disturbed system where For simulation purposes, we consider the step size 0.001 second, the initial condition , the desired trajectory , and the control parameters , , , . The membership function of the input and the output of fuzzy system are shown in Figures 12. First, we adopt the general VSC method via fixing . The simulation results are shown in Figures 35. Next, we adopt the general VSC method via fixing . The simulation results are shown in Figures 68. Finally, we employ the given fuzzy VSC method. The simulation results are shown in Figures 911.

105074.fig.001
Figure 1: The membership function of the fuzzy input.
105074.fig.002
Figure 2: The membership function of the fuzzy output.
105074.fig.003
Figure 3: The time response of and in case 1.
105074.fig.004
Figure 4: The time response of the tracking error in case 1.
105074.fig.005
Figure 5: The time response of the control input in case 1.
105074.fig.006
Figure 6: Time response of and for case 2.
105074.fig.007
Figure 7: Time response of the tracking error for case 2.
105074.fig.008
Figure 8: Time response of for case 2.
105074.fig.009
Figure 9: The time response of and in case 3.
105074.fig.0010
Figure 10: The time response of the tracking error in case 3.
105074.fig.0011
Figure 11: The time response of the control input in case 3.

Remark 4.1. Figures 3, 6, and 9 show the time response of control disturbance and its estimate . Figures 4, 7, and 10 show the time response of the tracking error. Figures 5, 8 and 11 show the time response of the control input. In case 1, the control disturbance is fixed at 50, which is bigger than the upper bound of . From Figures 35 it can be seen, when , that there is an obvious chattering phenomenon in control input for the estimation error of . In case 2, the control disturbance is fixed at 20, which is less than the upper bound of . From Figures 68 it can be seen, when , that there exists a big tracking error because the VSC can not be guaranteed at this moment. In case 3, the control disturbance is estimated by the fuzzy auxiliary controller. From Figures 911 it can be seen that the control for the given system is realized within 1 second and the chattering phenomenon is reduced distinctly, which demonstrates the effectiveness of the presented fuzzy VSC method.

Remark 4.2. From the simulation results, we can see that the chattering phenomenon is reduced effectively by using the proposed fuzzy controller however there still exists the switching term in control signal although is a small constant. To further overcome the control chattering phenomenon, the switching term is recommended to be substituted for .

5. Conclusion

In this paper, the fuzzy variable structure control problem has been studied. The fuzzy control method and the switching control method have been employed to realize the control for uncertain system with disturbance, they possess the characteristic of simpleness in design and effectiveness in attenuating the control chattering, and aresuitable for the application in engineering. Some typical numerical examples have been included afterward to demonstrate the effectiveness of the given controller.

Acknowledgments

This work was partially supported by the Key Projects of Xihua University (Z1120946), the National Key Basic Research Program (973), China (no. 2012CB215202), the 111 Project (B12018), the National Natural Science Foundation of China (nos. 61174058, and 61170030), and the Engineering and Physical Sciences Research Council, UK (EP/F029195).

References

  1. V. I. Utkin, “Variable structure systems with sliding modes,” IEEE Transactions on Automatic Control, vol. 22, no. 2, pp. 212–222, 1977. View at Scopus
  2. H. H. Choi, “Adaptive controller design for uncertain fuzzy systems using variable structure control approach,” Automatica, vol. 45, no. 11, pp. 2646–2650, 2009. View at Publisher · View at Google Scholar · View at Scopus
  3. J. P. V. S. Cunha, R. R. Costa, F. Lizarralde, and L. Hsu, “Peaking free variable structure control of uncertain linear systems based on a high-gain observer,” Automatica, vol. 45, no. 5, pp. 1156–1164, 2009. View at Publisher · View at Google Scholar · View at Scopus
  4. H. Xing, C. C. Gao, and D. Li, “Sliding mode variable structure control for parameter uncertain stochastic systems with time-varying delay,” Journal of Mathematical Analysis and Applications, vol. 355, no. 2, pp. 689–699, 2009. View at Publisher · View at Google Scholar · View at Scopus
  5. X. Chen, T. Hisayama, and C. Y. Su, “Pseudo-inverse-based adaptive control for uncertain discrete time systems preceded by hysteresis,” Automatica, vol. 45, no. 2, pp. 469–476, 2009. View at Publisher · View at Google Scholar · View at Scopus
  6. L. Wu, D. W. C. Ho, and C. W. Li, “Sliding mode control of switched hybrid systems with stochastic perturbation,” Systems and Control Letters, vol. 60, no. 8, pp. 531–539, 2011. View at Publisher · View at Google Scholar · View at Scopus
  7. Z. Lin, Y. Xia, P. Shi, and H. Wu, “Robust sliding mode control for uncertain linear discrete systems independent of time-delay,” International Journal of Innovative Computing, Information and Control, vol. 7, no. 2, pp. 869–880, 2011. View at Scopus
  8. T. E. Lee, J. P. Su, K. W. Yu, and K. H. Hsia, “Multi-objective fuzzy optimal design of alpha-beta estimators for nonlinear variable structure control,” International Journal of Innovative Computing, Information and Control, vol. 7, no. 5, pp. 2123–2140, 2011. View at Scopus
  9. S. Qu, Z. Lei, Q. Zhu, and H. Nouri, “Stabilization for a class of uncertain multi-time delays system using sliding mode controller,” International Journal of Innovative Computing, Information and Control, vol. 7, no. 7, pp. 4195–4205, 2011. View at Scopus
  10. Y. Niu and D. W. C. Ho, “Stabilization of discrete-time stochastic systems via sliding mode technique,” Journal of the Franklin Institute, vol. 349, pp. 1497–1508, 2012. View at Publisher · View at Google Scholar · View at Scopus
  11. T. Jia, Y. Niu, and Y. Zou, “Sliding mode control for stochastic systems subject to packet losses,” Information Sciences, vol. 217, pp. 117–126, 2012.
  12. L. Wu and D. W. C. Ho, “Sliding mode control of singular stochastic hybrid systems,” Automatica, vol. 46, no. 4, pp. 779–783, 2010. View at Publisher · View at Google Scholar · View at Scopus
  13. H. R. Karimi, “A sliding mode approach to H synchronization of master-slave time-delay systems with Markovian jumping parameters and nonlinear uncertainties,” Journal of the Franklin Institute, vol. 349, no. 4, pp. 1480–1496, 2012.
  14. L. Wu, P. Shi, and H. Gao, “State estimation and sliding-mode control of markovian jump singular systems,” IEEE Transactions on Automatic Control, vol. 55, no. 5, pp. 1213–1219, 2010. View at Publisher · View at Google Scholar · View at Scopus
  15. B. Jiang, P. Shi, and Z. Mao, “Sliding mode observer-based fault estimation for nonlinear networked control systems,” Circuits, Systems, and Signal Processing, vol. 30, no. 1, pp. 1–16, 2011. View at Publisher · View at Google Scholar · View at Scopus
  16. M. Liu, P. Shi, L. Zhang, and X. Zhao, “Fault tolerant control for nonlinear Markovian jump systems via proportional and derivative sliding mode observer technique,” IEEE Transactions on Circuits and Systems, vol. 58, pp. 2755–2764, 2011. View at Publisher · View at Google Scholar · View at Scopus
  17. H. R. Karimi, “Robust synchronization and fault detection of uncertain master-slave systems with mixed time-varying delays and nonlinear perturbations,” International Journal of Control, Automation and Systems, vol. 9, no. 4, pp. 671–680, 2011.
  18. H. R. Karimi, “Adaptive H synchronization problem of uncertain master-slave systems with mixed time-varying delays and nonlinear perturbations: an LMI approach,” International Journal of Automation and Computing, vol. 8, no. 4, pp. 381–390, 2011.
  19. H. R. Karimi, M. Zapateiro, and N. Luo, “Stability analysis and control synthesis of neutral systems with time-varying delays and nonlinear uncertainties,” Chaos, Solitons and Fractals, vol. 42, no. 1, pp. 595–603, 2009. View at Publisher · View at Google Scholar · View at Scopus
  20. H. R. Karimi and P. Maass, “Delay-range-dependent exponential H synchronization of a class of delayed neural networks,” Chaos, Solitons and Fractals, vol. 41, no. 3, pp. 1125–1135, 2009. View at Publisher · View at Google Scholar · View at Scopus
  21. H. R. Karimi, B. Lohmann, B. Moshiri, and P. J. Maralani, “Wavelet-based identification and control design for a class of nonlinear systems,” International Journal of Wavelets, Multiresolution and Information Processing, vol. 4, no. 1, pp. 213–226, 2006. View at Publisher · View at Google Scholar · View at Scopus
  22. H. R. Karimi and H. Gao, “Mixed H2/H output-feedback control of second-order neutral systems with time-varying state and input delays,” ISA Transactions, vol. 47, no. 3, pp. 311–324, 2008. View at Publisher · View at Google Scholar · View at Scopus
  23. H. Yang, P. Shi, J. Zhang, and J. Qiu, “Robust H control for a class of discrete time fuzzy systems via delta operator approach,” Information Sciences, vol. 184, no. 1, pp. 230–245, 2012.
  24. Z. Wu, P. Shi, H. Su, and J. Chu, “Reliable H control for discrete-time fuzzy systems with infinite-distributed delay,” IEEE Transactions on Fuzzy Systems, vol. 20, no. 1, pp. 22–31, 2012.
  25. Q. Zhou, P. Shi, J. Lu, and S. Xu, “Adaptive output feedback fuzzy tracking control for a class of nonlinear systems,” IEEE Transactions on Fuzzy Systems, vol. 19, no. 5, pp. 972–982, 2011.
  26. X. Su, P. Shi, and L. Wu, “A novel approach to filter design for T-S fuzzy discrete-time systems with time-varying delay,” IEEE Transactions on Fuzzy Systems. In press.
  27. Z. Gao, B. Jiang, P. Shi, and Y. Xu, “Fault accommodation for near space vehicle attitude dynamics via T-S fuzzy models,” International Journal of Innovative Computing, Information and Control, vol. 6, no. 11, pp. 4843–4856, 2010. View at Scopus
  28. Z. Pei and P. Shi, “Fuzzy risk analysis based on linguistic aggregation operations,” International Journal of Innovative Computing Information and Control, vol. 7, no. 12, pp. 7105–7117, 2011.
  29. G. Wang, P. Shi, and C. Wen, “Fuzzy approximation relations on fuzzy n-cell number space and their applications in classification,” Information Sciences, vol. 181, no. 18, pp. 3846–3860, 2011. View at Publisher · View at Google Scholar · View at Scopus
  30. J. Zhang, P. Shi, and Y. Xia, “Robust adaptive sliding-mode control for fuzzy systems with mismatched uncertainties,” IEEE Transactions on Fuzzy Systems, vol. 18, no. 4, pp. 700–711, 2010. View at Publisher · View at Google Scholar · View at Scopus
  31. Q. Zhou, P. Shi, S. Xu, and H. Li, “Adaptive output feedback control for nonlinear time-delay systems by fuzzy approximation approach,” IEEE Transactions on Fuzzy Systems. In press.
  32. X. Zhao, Y. Xu, Z. Zhang, and P. Shi, “Design of PSO fuzzy neural network control for ball and plate system,” International Journal of Innovative Computing Information and Control, vol. 7, no. 12, pp. 7091–7103, 2011.
  33. K. Zhang, B. Jiang, and P. Shi, “Fault estimation observer design for discrete-time Takagi-Sugeno fuzzy systems based on piecewise Lyapunov functions,” IEEE Transactions on Fuzzy systems, vol. 20, no. 1, pp. 192–200, 2012.
  34. H. Khalil, Nonlinear Systems, Macmillan, New York, NY, USA, 1992.