`Journal of Applied MathematicsVolume 2012, Article ID 158720, 15 pageshttp://dx.doi.org/10.1155/2012/158720`
Research Article

## The Hybrid Adaptive Control of T-S Fuzzy System Based on Niche

1Department of Mathematics, East China Normal University, Shanghai 200241, China
2Department of Mathematics, Nantong Shipping College, Jiangsu, Nantong 226010, China
3Department of Mathematics, Jiangsu College of Information Technology, Jiangsu, Wuxi 214400, China
4Faculty of Science, Jiangsu University, Jiangsu, Zhenjiang 212013, China

Received 31 March 2012; Revised 14 May 2012; Accepted 14 May 2012

Copyright © 2012 Tong Zhao 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

Based on the niche characteristics, a hybrid adaptive fuzzy control method with the function of continuous supervisory control is proposed in this paper. Considering the close degree of Niche as the consequent of adaptive T-S fuzzy control system, the hybrid control law is designed by tracking, continuous supervisory, and adaptive compensation. Adaptive compensator is used in the controller to compensate the approximation error of fuzzy logic system and the effect of the external disturbance. The adaptive law of consequent parameters, which is achieved in this paper, embodies system adaptability as biological individual. It is proved that all signals in the closed-loop system are bounded and tracking error converges to zero by Lyapunov stability theory. The effectiveness of the approach is demonstrated by the simulation results.

#### 1. Introduction

In recent years, fuzzy technique has gained rapid development in complex nonlinear plants [1]. Fuzzy logic offers human reasoning capabilities to capture nonlinearities and uncertainties, which cannot be described by precise mathematical models. Theoretical proofs of fuzzy models as universal approximators have been presented in the last decade [2, 3]. Fuzzy adaptive control methodologies have emerged in recent years as promising ways to approach nonlinear control problems. Fuzzy control, in particular, has had an impact in the control community because of the simple approach it provides to use heuristic control knowledge for nonlinear control problems. Recently, fuzzy systems have aroused a great resurgence of interest from part of the control community on the ground that they may be applied to model ill-defined complex systems. Direct and indirect schemes are two staple configurations for adaptive fuzzy controls. It has been established that indirect and direct controls (DAFC and IAFC) are able to incorporate plant knowledge and control knowledge, respectively, to yield stable and robust control systems [4]. The last two decades or so have witnessed a large quantity of results on indirect [5, 6] or direct [711] adaptive control (IAC/DAC) using fuzzy systems. However, the fuzzy adaptive controller proposed in [6, 7] ensure that the tracking error asymptotically convergence to zero (or a neighborhood of the origin) if the minimum approximation error is squared integrable along the state trajectory. And those algorithms are just confined to the linear and nonlinear systems whose state variables are assumed to be available for measurement. In many complicated cases, all state variables are not measurable such that output feedback or observer-based adaptive control techniques have to be applied. In [1215], observer-based IAC and DAC algorithms are proposed for nonlinear systems, respectively. But [6, 12, 13] proposed adaptive control gain is only applicable to nonlinear systems with unknown constant gain. So a hybrid adaptive fuzzy control is needed.

In order to exploit the relative advantages of indirect and direct adaptive configurations at the same time, some researchers have developed several hybrid IAC and DAC algorithms [1618] where a weighting factor, which can be adjusted by the tradeoff between knowledge of the plant and knowledge of the control, is adopted to sum together the control efforts from both the IAC and DAC. However, those schemes have their limitations. Above all, they take full-state feedback, which can be unsuitable for nonlinear systems without state variables available. Moreover, the conventional adaptive controller proposed in [19] combines indirect, direct, and variable structure methods; nonetheless its plants are assumed to be linear systems only. The author of [6] has developed a hybrid adaptive fuzzy control (HAFC) for nonlinear systems. However, input gains are required to be a constant 1 and control gain is an unknown constant. The HAFC algorithm of [20] can just be applied to robot manipulators with bounds estimation whereas that of [21] derives an unsupported HAFC scheme from a faulty Lyapunov derivative. On the other hand, a certain observer-based combined direct and indirect adaptive fuzzy neural controller is developed in [22]. Sensor fault may be in any form, even unbounded, which will make the system failure unavoidably. Paper [23] proposed a reliable observer-based controller, which makes the system work well no matter whether sensor faults occur or not.

Among various kinds of fuzzy models, there is a very important class of Takagi-Sugeno (T-S) fuzzy models [24] which have recently gained much popularity because of their special rule consequent structure and success in a functional approximation [6, 25]. In the recent two years, with the stability theory of T-S fuzzy system drawing lots of researchers attention, Yeh et al. [25] proposed stability analysis of interconnected and robustness design of time-delay fuzzy systems using fuzzy Lyapunov method. Moreover, T-S fuzzy model plays an important role in dealing with practical problems, such as oceanic structure [26], Vehicle occupant classification [27], and engineering systems [28], in which T-S fuzzy systems are applied to sensor fault estimation.

#### 2. Problem Formulation

Consider the th-order nonlinear system of the following form: where and are unknown but bounded functions, are the input, output of the system, respectively, and and is the external bounded disturbance. is the state vector where not all are assumed to be available for measurement. In order for the system (2.1) to be controllable, it is required that for in a certain controllability region . Since is continuous, we assume for . The control objective is to design a combined controller, tune the correlation parameters of adaptive law, and force the system output to follow a bounded reference signal under the constraint that all signals involved must be bounded. To begin with, the signal vector , the tracking error vector , and estimation error vector    are defined as ,  ,  , where and denote the estimates of and , respectively. If the functions and are known and the system is free of external disturbance , then we can choose the controller to cancel the nonlinearity and design controller. Let be chosen such that all roots of the polynomial are in the open left half-plane and control law of the certainty equivalent controller is obtained as Substituting (2.2) to (2.1), we get the main objective of the control is . However and are unknown, the ideal controller (2.2) cannot be implemented, and not all system states can be measured. So we have to design an observer to estimate the state vector in the following.

#### 3. T-S Fuzzy System Based on Niche

##### 3.1. Description of T-S Fuzzy System Based on Niche

Because and are unknown, T-S fuzzy systems are used to approximate them. The basic configuration of T-S [24] system is expressed as Here, are fuzzy sets, represents the real ecological factor of ecologic niche, and is the ideal ecological factor. Consequent indicates the difference between real state of ecologic niche and ideal one. we choose Gaussian membership function, which satisfies , where are the centers and is the variance of the functions. Then difference could be showed by approach functions as , where . From approach functions above, we could know consequent is a zero-order T-S mode. Here, we use as for convenience. Let be the number of systems with central average defuzzifier, and product inference and singleton fuzzifier can be expressed as where , , , ; and can be expressed similarly.

##### 3.2. Niche T-S Model Parameter Optimization Method

Niche fuzzy T-S model based on parameter optimization of backpropagation algorithm known input and output data , the task is to determine the form (3.2) Niche T-S model parameters, To fitting error the minimum. Assume is known, by adjusting , and let be the minimum. To facilitate the discussion, use to represent , and . , .

Using gradient descent to determines the parameters [6]    determines the steps.

Then Then, Equally, adjustable parameters with the same way: where and can be optimized directly.

#### 4. Hybrid Controller with Supervisory and Compensation Control Scheme

The overall control law is constructed as where is an indirect controller, is the output of the T-S controller, is the supervisory control to force the state within the constraint set, is a weighting factor, and is the compensate controller of adaptive control. Since cannot be available and and are unknown, we replace the functions , and error vector    by estimation functions , and . The certainty equivalent controller can be rewritten as The indirect control law is written as  . Applying (4.2), (4.1) to (2.1), the error dynamic equation is defining    as the observer vector, the observation errors are defined as from (4.2) and [24] and we get where    Let be strictly Hurwitz matrix; so there exists a positive definite symmetric   matrix , which satisfies the Lyapunov equation , ,   where and are arbitrary   definite symmetric matrix. Let . Since and are designer by the designer, we can choose and , such that . Hence, is a bounded function and there exists a constant value , such that .

#### 5. Hybrid Adaptive Control of Niche

We will develop the hybrid adaptive control such that the closed-loop output follows . Let us replace , and by , and , respectively. Therefore, the error dynamics (4.3) can be rewritten as Let then using (5.1), we have

In order to design so that , we need the following assumption.

Assumption 5.1. We could find three functions as and get and , in which and . This is due to the fact that we can choose to let . Also external disturbance is bounded. We design .

From Assumption 5.1, we choose the supervisory control as And we choose , and is a nonnegative constant. Considering the case and substitute (5.3) into (5.2), we obtain . Therefore, if the closed-loop system with the fuzzy controller    as works well in the sense that the error is not too large, if , then the supervisory control is zero. If the system tends to diverge, that is, , then be gins to force .

#### 6. Design of Adaptive Law

In order to adjust the parameters in the fuzzy logic system, we have to derive adaptive laws. Hence, the optimal parameters estimates , and are defined as where , and , are compact sets of suitable bounds on , and , respectively, and they are defined as where and , are positive constants. Define the minimum approximation errors as

Now consider the Lyapunov function where   and and is expressed similarly. We get the adaptive law as So    Furthermore, the adaptive law of niche factors is derived as follows. We derived the adaptive law of real ecologic factors, see Formula (6.7), (6.8), which represented the real niche always develop towards the ideal one. It reflected the compensation of the control system to the external disturbance: The same way we have

Applyingwe have .

Design parameters of, vector adaptive law of , stability, and performance analysis are similar to those of [18]; here we omit it.

#### 7. Simulation

Form combined fuzzy adaptive control of two-dimensional predation system [32] as where   means total number of food at the time of and denotes the total number of predators; are the regular numbering ecology, and is transforming factor,   represents death ratio of predators. is the function of Holling’s functional responses, and is the third kind of Holling’s functional responses. The demonstration of the two-dimensional predator system without the controller is shown in Figure 1.

Figure 1: Two-dimension predator system without the controller.

In order to reach an ideal ecologic balance in this two-dimension predators system, we get a way to control it, where .

Then In order to establish the direct relation between output    and controller  , we need the derivation of  . After derivating twice, we get Here, we command Then (7.2) and (7.3) could be shown as If hoping to apply adaptive fuzzy controlling system here, we have to firstly confirm the boundary of ,  and  . From [29], we know that   , then According to the above, we find out that the scope of   is larger than  . Therefore, we choose . Then we select , ,  ,  . The proposal method of this paper does not need to predefine the reference signal, but to reach the balance of each individual self-adaptively. Conveniently, we suppose (in this way, could be stable) and ,   we select , from , then we solve . as , and so we have ; when letting , we could get . In this way, we get the ideal . The adaptive law comes out as follows: We choose , then the trend of the prey and predator numbers with T-S controller show as in Figures 2 and 3. The trend of the T-S controller is shown in Figure 4.

Figure 2: The trend of the prey numbers with T-S controller.
Figure 3: The trend of the predator numbers with T-S controller.
Figure 4: The trend of the T-S controller.

From the simulation figures, we can see that the prey and predator numbers reach a stable status in a short period of time under the control of the proposed method in this paper. The creature individuals show a characteristic of self-adaptation according to outside changes.

From the figure of the trend of the T-S controller, it also reaches an ideal status to maintain the overall balance of the two-dimension predator system. The ecological system is optimized with the use of this control method.

#### 8. Conclusion

For the ecological niche, a hybrid adaptive fuzzy control method with the function of continuous supervisory control is proposed in this paper. Let the close degree of Niche which contains parameters as the consequent of adaptive T-S fuzzy control system, then designs the hybrid control law by tracking, continuous supervisory and adaptive compensation. Using gradient descent to optimize the parameters, we get the adaptive law of consequent parameters, embodying biological individual’s ability of adaptability. Based on Lyapunov stability theory, it is proved that all signals in the closed-loop system are bounded and tracking error converges to zero. This paper shows that the fuzzy methods provide good results in practical engineering problems. The performance of the developed approach is illustrated by simulation, on two-dimension predation system model.

#### Acknowledgment

Work supported by national science foundation (11072090).

#### References

1. C. W. Chen, K. Yeh, and K. F. R. Liu, “Adaptive fuzzy sliding mode control for seismically excited bridges with lead rubber bearing isolation,” International Journal of Uncertainty, Fuzziness and Knowlege-Based Systems, vol. 17, no. 5, pp. 705–727, 2009.
2. B. Kosko, “Fuzzy systems as universal approximators,” IEEE Transactions on Computers, vol. 43, no. 11, pp. 1329–1333, 1994.
3. R. Rovatti, “Fuzzy piecewise multilinear and piecewise linear systems as universal approximators in sobolev norms,” IEEE Transactions on Fuzzy Systems, vol. 6, no. 2, pp. 235–249, 1998.
4. L. X. Wang, A Course in Fuzzy Systems and Control, Prentice Hall, Englewood Cliffs, NJ, USA, 1997.
5. R. Shahnazi and M. R. Akbarzadeh-T, “PI adaptive fuzzy control with large and fast disturbance rejection for a class of uncertain nonlinear systems,” IEEE Transactions on Fuzzy Systems, vol. 16, no. 1, pp. 187–197, 2008.
6. L. X. Wang, “Stable adaptive fuzzy control of nonlinear systems,” IEEE Transactions on Fuzzy Systems, vol. 1, no. 2, pp. 146–155, 1993.
7. P. A. Phan and T. J. Gale, “Direct adaptive fuzzy control with a self-structuring algorithm,” Fuzzy Sets and Systems, vol. 159, no. 8, pp. 871–899, 2008.
8. S. Labiod and T. M. Guerra, “Adaptive fuzzy control of a class of SISO nonaffine nonlinear systems,” Fuzzy Sets and Systems, vol. 158, no. 10, pp. 1126–1137, 2007.
9. J. H. Park, S. H. Huh, S. H. Kim, S. J. Seo, and G. T. Park, “Direct adaptive controller for nonaffine nonlinear systems using self-structuring neural networks,” IEEE Transactions on Neural Networks, vol. 16, no. 2, pp. 414–422, 2005.
10. M. Wang, B. Chen, and S.-L. Dai, “Direct adaptive fuzzy tracking control for a class of perturbed strict-feedback nonlinear systems,” Fuzzy Sets and Systems, vol. 158, no. 24, pp. 2655–2670, 2007.
11. S. S. Ge and T. T. Han, “Semiglobal ISpS disturbance attenuation with output tracking via direct adaptive design,” IEEE Transactions on Neural Networks, vol. 18, no. 4, pp. 1129–1148, 2007.
12. R. Qi and M. A. Brdys, “Stable indirect adaptive control based on discrete-time T-S fuzzy model,” Fuzzy Sets and Systems, vol. 159, no. 8, pp. 900–925, 2008.
13. C.-H. Hyun, C.-W. Park, and S. Kim, “Takagi-Sugeno fuzzy model based indirect adaptive fuzzy observer and controller design,” Information Sciences, vol. 180, no. 11, pp. 2314–2327, 2010.
14. S. Tong, H.-X. Li, and W. Wang, “Observer-based adaptive fuzzy control for SISO nonlinear systems,” Fuzzy Sets and Systems, vol. 148, no. 3, pp. 355–376, 2004.
15. Y. G. Leu, W. Y. Wang, and T. T. Lee, “Observer-based direct adaptive fuzzy-neural control for nonaffine nonlinear systems,” IEEE Transactions on Neural Networks, vol. 16, no. 4, pp. 853–861, 2005.
16. M. K. Ciliz, “Combined direct and indirect adaptive control for a class of nonlinear systems,” IET Control Theory & Applications, vol. 3, no. 1, pp. 151–159, 2009.
17. Y. Q. Zheng, Y. J. Liu, S. C. Tong, and T. S. Li, “Combined adaptive fuzzy control for uncertain MIMO nonlinear systems,” in Proceedings of the American Control Conference (ACC '09), pp. 4266–4271, St. Louis, Mo, USA, June 2009.
18. Q. Ding, H. Chen, C. Jiang, and Z. Chen, “Combined indirect and direct method for adaptive fuzzy output feedback control of nonlinear system,” Journal of Systems Engineering and Electronics, vol. 18, no. 1, pp. 120–124, 2007.
19. X. Ye and J. Huang, “Decentralized adaptive output regulation for a class of large-scale nonlinear systems,” IEEE Transactions on Automatic Control, vol. 48, no. 2, pp. 276–281, 2003.
20. N. Hovakimyan, E. Lavretsky, B. J. Yang, and A. J. Calise, “Coordinated decentralized adaptive output feedback control of interconnected systems,” IEEE Transactions on Neural Networks, vol. 16, no. 1, pp. 185–194, 2005.
21. S. Tong, H. X. Li, and G. Chen, “Adaptive fuzzy decentralized control for a class of large-scale nonlinear systems,” IEEE Transactions on Systems, Man, and Cybernetics, Part B, vol. 34, no. 1, pp. 770–775, 2004.
22. S. S. Stanković, D. M. Stipanović, and D. D. Šiljak, “Decentralized dynamic output feedback for robust stabilization of a class of nonlinear interconnected systems,” Automatica, vol. 43, no. 5, pp. 861–867, 2007.
23. Z. Gao, T. Breikin, and H. Wang, “Reliable observer-based control against sensor failures for systems with time delays in both state and input,” IEEE Transactions on Systems, Man, and Cybernetics Part A, vol. 38, no. 5, pp. 1018–1029, 2008.
24. 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.
25. K. Yeh, C.-Y. Chen, and C.-W. Chen, “Robustness design of time-delay fuzzy systems using fuzzy Lyapunov method,” Applied Mathematics and Computation, vol. 205, no. 2, pp. 568–577, 2008.
26. C.-W. Chen, “The stability of an oceanic structure with T-S fuzzy models,” Mathematics and Computers in Simulation, vol. 80, no. 2, pp. 402–426, 2009.
27. Z. Gao and Y. Zhao, “Vehicle occupant classification algorithm based on T-S fuzzy model,” Procedia Engineering, vol. 24, pp. 500–504, 2011.
28. Z. Gao, X. Shi, and S. X. Ding, “Fuzzy state/disturbance observer design for T-S fuzzy systems with application to sensor fault estimation,” IEEE Transactions on Systems, Man, and Cybernetics, Part B, vol. 38, no. 3, pp. 875–880, 2008.
29. A. L. Goldberger, “Fractal mechanisms in the electrophysiology of the heart,” IEEE Engineering in Medicine and Biology, vol. 11, no. 2, pp. 47–52, 1992.
30. A. Babloyantz and A. Destexhe, “Is the normal heart a periodic oscillator?” Biological Cybernetics, vol. 58, no. 3, pp. 203–211, 1988.
31. X.-M. Wang and L. I. Yi-min, “Kind of new niche fuzzy control method,” Science Technology and Engineering, vol. 24, pp. 6318–6321, 2007.
32. S. P. Li, “Dynamical behavior analysis and control of two functional ecosystem,” Jiang Su University, pp. 33–43, 2005.