Research Article  Open Access
Faired MISO BSpline Fuzzy Systems and Its Applications
Abstract
We construct two classes of faired MISO Bspline fuzzy systems using the fairing method in computeraided geometric design (CAGD) for reducing adverse effects of the inexact data. Towards this goal, we generalize the faring method to highdimension cases so that the faring method only for SISO and DISO Bspline fuzzy systems is extended to fair the MISO ones. Then the problem to construct a faired MISO Bspline fuzzy systems is transformed into solving an optimization problem with a strictly convex quadratic objective function and the unique optimal solution vector is taken as linear combination coefficients of the basis functions for a certain Bspline fuzzy system to obtain a faired MISO Bspline fuzzy system. Furthermore, we design variable universe adaptive fuzzy controllers by Bspline fuzzy systems and faired Bspline fuzzy systems to stabilize the double inverted pendulum. The simulation results show that the controllers by faired Bspline fuzzy systems perform better than those by Bspline fuzzy systems, especially when the data for fuzzy systems are inexact.
1. Introduction
Since Zadeh introduced fuzzy theory in 1965, fuzzy systems have been utilized successfully in many areas, such as fuzzy control, classification, expert systems, and others. It is known that a fuzzy system is usually established by inputoutput data (I/O data) which can be obtained by experiments, expert knowledge, or observation records. However, the accuracy of these I/O data may be affected by hardware/software limitations, unavoidable round off, truncation error of a system, and some uncertainties [1]. This means that we cannot establish fuzzy systems on the exact I/O data. Therefore, it is important to construct appropriate fuzzy systems when the I/O data is inexact. Reference [2] gives the upper bounds of the output errors of two kinds of fuzzy systems affected by the perturbation of I/O data. However, the problem of performance improvement for fuzzy systems is seldom considered in the case of the inexact I/O data.
Fuzzy systems can be constructed by splines [3–6] because the design of a fuzzy system can be regarded as a function approximation problem [7–10] and spline functions have many nice structural properties and excellent approximation powers [11]. By investigating the relation between fuzzy systems and splines, we proposed two classes of Bspline fuzzy systems (BFSs) [12, 13], which are linear combination of Bspline basis functions and rational Bspline basis functions, respectively. Therefore, the single input single output (SISO) and double input single output (DISO) of these two classes of BFSs can be regarded as curves and surfaces in CAGD. As curves and surfaces in CAGD, fairness is necessary. Though fairness is the property about geometric shapes, a fair curve can seek through the digitizing errors in its design process [14]. Hence, it is necessary to reduce adverse effects of the inexact I/O data on a fair fuzzy system. Since BFSs can be regarded as curves and surfaces in CAGD, we can fair them and obtain good performance.
In this paper, we construct two classes of faired Bspline fuzzy systems (faired BFSs) to reduce adverse effects of the inexact I/O data on fuzzy systems as well as improve their performance. For faring these two classes of BFSs, the energy extremum principle (energy method) based faring method in CAGD is utilized for its overall modification nature. However, we note that the energy method in CAGD is only used to fair curves and surfaces, which means it can only fair the SISO and DISO BFSs. So, we propose a regularization term taken as the energy function of the MISO BFS. By using this energy function, the energy method in CAGD, which can only be applied to fairing SISO and DISO BFSs, is extended to fair the MISO ones. Therefore, based on the above preparations, the problem to construct a faired MISO BFS is transformed into solving an optimization problem with a strictly convex quadratic objective function. In our proposed method, the faired MISO BFS is available by taking the unique optimal solution vector as linear combination coefficients of the corresponding MISO BFS.
As we all know, fuzzy controllers are a type of closedloop fuzzy systems, while adaptive fuzzy controllers are closedloop fuzzy systems with adaptive or training algorithms [15–19]. Especially, Professor Li advances the variable universe method [20–22], and this method succeeded in the experiment of controlling the simulation model and physical model of quadruple inverted pendulum with variable universe fuzzy controllers in 2001 [23] and 2002. In order to verify the ability of the faired BFSs, we design variable universe adaptive fuzzy controllers by BFSs and faired BFSs to stabilize the double inverted pendulum. The simulation results show that the controllers by faired BFSs perform better than those by BFSs, especially when the I/O data for fuzzy systems are inexact.
The paper is organized as follows. Section 2 provides some preliminaries. The faired MISO BFSs are constructed in Section 3. In Section 4, the variable universe adaptive fuzzy controllers by BFSs and faired BFSs are designed to demonstrate their ability. The final section contains some conclusions and prospects of our research.
2. Preliminaries
In this section, we will introduce the definition of Bspline basis functions, the Frobenius norm, and briefly review the two classes of MISO BFSs in [12, 13].
Definition 1 (Bspline basis functions [24]). Let be a nondecreasing sequence of real numbers, that is, , . The is called knots, and is the knot vector. The th Bspline basis function of degree (order ), denoted by , is defined as
Definition 2 (Frobenius norm [25]). The Frobenius norm on is defined as
Obviously,
In fuzzy inference, to acquire a group of inference rules and to acquire a group of I/O data are the same thing [10]. In this paper, we always denote the I/O data for MISO fuzzy systems as where , , and , .
Let , where is the cubic Bspline basis function with knots , , , , and . The two classes of MISO BFSs we proposed in [12, 13] are listed below.(1) The MISO first class of Bspline fuzzy system (1BFS) is where , , are obtained by solving equations after extrapolating some points.(2) The MISO second class of Bspline fuzzy system (2BFS) is
3. The Faired MISO BFSs
In this section, two classes of the faired MISO BFSs are constructed. In order to fair them together, we write them in the unification SISO form. By analyzing the energy functions of SISO and DISO BFS, the energy function of MISO BFS is proposed, which is a regularization term essentially. Consequently, the energy method is suitable for fairing the MISO BFSs. Then, we transform the problem to construct a faired MISO BFS into solving an unconstrained optimization problem. Based on the unification SISO form, the objective function of the unconstrained optimization problem can be reduced to a quadratic function which is turned to a strictly convex quadratic function via using a proper weight. Therefore, the unique optimal solution is available through solving linear equations of the firstorder optimality condition. Consequently, the faired MISO BFS is obtained by taking the unique optimal solution vector as linear combination coefficients of the corresponding BFS. In the following, we will describe the above procedure in details.
3.1. The Unification SISO Form of Two Classes of BFSs
The uniform form of (5) and (7) is, where , , and .
If we have then (8) will turn to the 1BFS (5).
While, if we have then (8) is exactly the 2BFS (7). Consequently, (8) is named as the unification SISO form of the two classes of BFSs.
3.2. Generalizing the Energy Method to HighDimension Cases
The objects studied in CAGD are curves and surfaces which are parametric equations with single parameter and double parameters, respectively. The problem of fairing a curve (surface) by the energy method can be transformed into the following optimal one [26]: where the variable is the vector of control points, is the energy function, is the difference between the faired data points and the original ones, and is the weight which is assigned in advance. In CAGD, an approximated or simplified strain energy is used as energy function [26]. For SISO and DISO BFS, which can be regarded as curve and surface in CAGD, the energy functions can be written as By (2) and (3), the integrand of in (12) or (13) is the square of the Frobenius norm of (the Hessian matrix of ). That is,
When the number of input variable is more than 2, it is difficult to get an energy function with specific geometric meaning or physical meaning. In order to fair the BFSs with more than 2 input variables, we have to generalize the energy method to the highdimension cases. In fact, the energy method is a kind of regularization method with the energy function as its regularization term. Especially, when the observation data is inexact, the regularization method can identify a meaningful and stable solution [27]. This coincides with our motive to fair. It is noted that (14) can be viewed as a regularization term. Consequently, we call the regularization method with (14) as its regularization term the generalized energy method. It is known that (14) is the energy function for MIMO BFS. Therefore, we generalize the energy method to the highdimension cases.
3.3. The Faired MISO BFSs
In general, the curves (surfaces) in CAGD are referred to the parametric Bspline curves (surfaces), while the BFSs are the Bspline functions. When the parametric Bspline curves (surfaces) degenerate into Bspline functions, the control points will be the coefficients of Bspline basis functions. Write the original BFS and the faired BFS as respectively. Then, by (11), we can transform the problem to construct a faired MISO BFS into solving the following unconstrained optimization problem as where , the positive weight, is assigned in advance, and is defined as in (14).
From the following two extreme cases, we can recognize the concrete significance of .(1) When , since , we have Thus the faired BFS turns to the original BFS , which implies that the BFS is not modified after the fairing process.(2) When , we immediately get that from . Therefore, the BFS turns to be the fairest one in this case.
For the cases in between, when is set to be a small number, becomes small and the fuzzy system (16) is fair at the cost of much difference between and as well as much difference between and . On the other hand, the larger is, the less the difference is between and . Thus there is less difference between and . And the fairness of the fuzzy system (16) may be poor. So when the I/O data is inexact, a smaller is needed to make a larger difference between and , and then the original BFS may be improved. On the contrary, a larger might be appropriate.
Let , and substitute (8) into (14), we have,
Then, the objective function of the optimal problem (17) is where
From (20), we know that is a quadratic function. Moreover, by (21), is a symmetric matrix. Obviously, there always exists weight to make a symmetric positive definite matrix. Thus, becomes a strictly convex quadratic function. Consequently, through solving the linear equations of the firstorder optimality condition as shown in the following: we can get the unique optimal solution of (17). Let serve as the linear combination coefficients of (16), the faired MISO BFS is available.
Remark 3. Since it is convenient to deal with a uniform cubic Bspline, when the knots of the Bspline basis functions of a BFS are arbitrary (nonuniform), we can approximate to this BFS by one fuzzy system with uniform cubic Bspline basis functions to fair it.
Algorithm. Constructing a Faired MISO BFS.
Step 1. Extract I/O data (4) from the fuzzy inference rules;
Step 2. Given the initial weight ;
Step 3. Calculate by (21);
Step 4. If is positive definite, the optimal solution is available by solving linear equations (22). Otherwise, let , go to Step 4;
Step 5. Evaluate the obtained faired BFS, , if it works, end, if not, tuning the weight , go to Step 4.
4. Simulation Results
In this section, we design variable universe adaptive fuzzy controllers by BFSs (hereinafter abbreviated as BFCs) and faired BFSs (hereinafter abbreviated as faired BFCs) to stabilize the double inverted pendulum. Moreover, the control effect between them is compared. As the analysis in Section 3.3, the weight can affect the fairness and difference of the faired BFSs. Further, it may affect the control effect of the faired BFCs, too. Therefore, the control effect incited by different weights is demonstrated in the following simulations.
The double inverted pendulum is mainly made up of a cart, two rods which are freely linked together. The case where they are put in a coordinate system is shown in Figure 1.
Let the clockwise angle and moment in Figure 1 be in positive direction. And we assume that is the outer force of system, the displacement of the cart, the angle between rod and vertical direction, and the linking point and centroid of rod , the mass of the cart, the mass of rod , the moment of inertia of rod around , the distance from to , the length of rod , the fricative coefficient between the cart and its orbit, and the fricative coefficient of rod around . Then the differential equations to describe the locomotion of the double inverted pendulum are where ,,,,, and, and .
For the control system of the double inverted pendulum, our control aim is to make the angles , , respectively, converge to and, at the same time, drive the cart to the point which is pointed out by us in advance. In this simulation experiment, the parameters in the double inverted pendulum are taken as , (unit: kg), , , , (unit: m), , (unit: ), , (unit: ), and (unit: ).
The variable universe adaptive fuzzy controller for the double inverted pendulum is designed by the method in [23]. Firstly, we linearize (23) at the equilibrium point . Then, let , , and solve LQR by Matlab. And then we can get a state feedback matrix and . Let be the coefficient vector to reduce the dimensions and be a kind of integrated error and integrated error change rate. The variable universe adaptive fuzzy controller is where is a fuzzy system of two input variables.
The universe of and are both taken as . The fuzzy partition of is , , , , , , , and the fuzzy partition of is , , , , , , . The control rules are given in Table 1 [23].

So the relatively exact I/O data for is that where , , and matrix is formed by the data in Table 1.
When the I/O data is inexact, we consider the data with noise only which is obtained by adding the Gaussian white noise. In this simulation, the inexact I/O data is obtained by adding the Gaussian white noise with mean and variance to I/O data , and the obtained inexact I/O data is denoted as .
Let , , , and . Figures 2 and 3 show the control effect of BFCs and faired BFCs respectively, where Figure 3 is the average result of 100 independent runs. Obviously,(1)when the I/O data is , the control effect of the BFCs is nearly as good as that of the faired ones (Figure 2);(2)when the I/O data is , the faired BFCs outperform that of the BFCs (Figure 3).
The control performance of control systems in terms of different weights is shown in Tables 2, 3, 4, and 5, where the control performance includes dynamical performance such as maximum overshoot and settling time, and steady performance such as steadystate error; s; is defined as the relatively adjustment between fuzzy system and its corresponding I/O data; the setting time is defined as the time required for the system to settle within 5% of the steady value, and shows the consumption of energy. In particular, the results shown in Tables 4 and 5 are the average result of 100 independent runs. The following results can be seen from the above tables.(1)The smaller (larger) the weight is, the larger (smaller) the relatively adjustment between the fuzzy system and its corresponding I/O data is (seen from Tables 2, 3, 4 and 5). This agrees with our analysis in Section 3.3.(2)For I/O data , only small adjustment can make the faired BFCs have good control effect (seen from Tables 2 and 3). Since the variance of Gaussian white noise of is a small value, , the good control effect is also available by small adjustment (seen from Tables 4 and 5).(3)For the relatively exact I/O data , we can see from Tables 2 and 3 that, almost all the performance index of control systems by faired BFCs are slightly better than those by BFCs. Moreover, for faired 1BFCs (faired 2BFCs), we should note that the energy consumption gets higher (lower) as the weight increasing.(4)For I/O data , the control performance of control systems with faired BFCs is much better than that of control systems with BFCs (seen from Tables 4 and 5), only except the steadystate error of of in Table 4. Especially, we point out that the energy consumption of faired BFCs is less than that of the BFCs.(5)From Tables 4 and 5, we also found that, for I/O data , neither faired BFCs with larger weights nor those with smaller ones can stabilize the double inverted pendulum (figures not shown). In Table 4, when the weight is 0.5, we can obtain almost the best control performance, especially the energy consumption is the least, while the weight is 1.5 in Table 5 for the same goal. Therefore, one can conclude that the larger (smaller) weights lead to smaller (larger) adjustment to the inexact I/O data , and both cases are not suitable for the faired BFSs which are used to construct controllers.




In summary, when the I/O data for fuzzy system is relatively exact, the control effect of the faired BFCs is slightly better than that of the BFCs, which means the faired BFSs for the faired BFCs improve the BFSs for the BFCs slightly. While the I/O data for fuzzy system is inexact, the control effect of the faired BFCs outperforms that of the BFCs, in this case, the corresponding faired BFSs reduce adverse effects of the inexact I/O data on the corresponding BFSs as well as improve them significantly.
Remark 4. To compare the control effect of faired BFCs and BFCs with inexact I/O data, the variance of the Gaussian white noise added to should be as small as . Otherwise, the corresponding BFCs cannot stabilize the double invert pendulum. Actually, when the variance of Gaussian white noise is as large as , the corresponding faired BFCs can also stabilize the double invert pendulum.
In order to investigate the control capability of the faired BFCs, we choose and use denote the I/O data. In this case, the faired BFCs can stabilize the double pendulum as well as locate the cart (Figure 4). However, the BFCs cannot do these.
5. Conclusion
In this paper, the energy method in CAGD was utilized to design the faired MISO BFSs. Based on our generalized approach, the construction of a faired MISO BFS is equivalent to solve an optimization problem with a strictly convex quadratic objective function. By taking the unique optimal solution vector as the linear combination coefficients of a certain BFS, a faired MISO BFS is obtained. For the faired MISO BFSs, the fairness and difference can be adjusted by modifying the weights in the objective function. This gives us the opportunity to improve the performance of fuzzy systems and fuzzy controllers. Moreover, we use the obtained faired MISO BFSs to stabilize the double inverted pendulum by modifying the weights. It is concluded that the faired BFCs outperform the BFCs in the case of exact and inexact I/O data. In fact, there are many fairing methods in CAGD. We only choose the energy approach. Moreover, the faired MISO BFSs are fuzzy systems with robustness. In the future, we will try to fair the BFSs by other fairing methods and investigate their robustness.
Acknowledgments
This paper is supported by the National Natural ScienceFoundation of China (no. 61074044, no. 61104038, no. 60834004), and the National 973 Basic Research Program of China (no. 2009CB320602).
References
 M. Biglarbegian, W. Melek, and J. Mendel, “On the robustness of type1 and interval type2 fuzzy logic systems in modeling,” Information Sciences, vol. 181, no. 7, pp. 1325–1347, 2011. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 W. Y. Song, D. G. Wang, and W. G. Wang, “Analysis for perturbation of fuzzy systems modeling,” ICIC Express Letters, vol. 3, no. 3, pp. 573–578, 2009. View at: Google Scholar
 M. Brown and C. J. Harris, “A nonlinear adaptive controller: a comparison between fuzzy logic control and neurocontrol,” IMA Journal of Mathematical Control and Information, vol. 8, no. 3, pp. 239–265, 1991. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 J. Zhang and A. Knoll, “Constructing fuzzy controllers with Bspline models: principles and applications,” International Journal of Intelligent Systems, vol. 13, no. 23, pp. 257–285, 1998. View at: Google Scholar
 J. Zhang, K. Van Le, and A. Knoll, “Unsupervised learning of control surfaces based on Bspline models,” in Proceedings of the 6th IEEE International Conference on Fussy Systems, pp. 1725–1730, July 1997. View at: Google Scholar
 X. J. Zeng and M. G. Singh, “Approximation accuracy analysis of fuzzy systems as function approximators,” IEEE Transactions on Fuzzy Systems, vol. 4, no. 1, pp. 44–63, 1996. View at: Google Scholar
 A. Mencattini, M. Salmeri, and A. Salsano, “Sufficient conditions to impose derivative constraints on MISO TakagiSugeno fuzzy logic systems,” IEEE Transactions on Fuzzy Systems, vol. 13, no. 4, pp. 454–467, 2005. View at: Publisher Site  Google Scholar
 Q. Luo, W. Yang, and D. Yi, “Kernel shapes of fuzzy sets in fuzzy systems for function approximation,” Information Sciences, vol. 178, no. 3, pp. 836–857, 2008. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 Z. Long, X. Liang, and L. Yang, “Some approximation properties of adaptive fuzzy systems with variable universe of discourse,” Information Sciences, vol. 180, no. 16, pp. 2991–3005, 2010. View at: Publisher Site  Google Scholar
 L. Hongxing, “Interpolation mechanism of fuzzy control,” Science in China E, vol. 41, no. 3, pp. 312–320, 1998. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 L. L. Schumaker, Spline Functions: Basic Theory, Cambridge University Press, New York, NY, USA, 2007. View at: Publisher Site  MathSciNet
 T. Yanhua, L. Hongxing, and X. Jixiang, “Two classes of Bspline fuzzy systems and their applications,” Journal of Control Theory and Applications, vol. 28, no. 11, pp. 1651–1657, 2011 (Chinese). View at: Google Scholar
 T. Yanhua and L. Hongxing, “The MISO Bspline fuzzy systems and its applications in double inverted pendulum”. View at: Google Scholar
 G. Farin, Curves and Surfaces For CAGD, Morgan Kaufmann Publishers, San Francisco, Calif, USA, 5th edition, 2001.
 L. X. Wang, “Stable adaptive fuzzy control of nonlinear systems,” IEEE Transactions on Fuzzy Systems, vol. 1, no. 2, pp. 146–155, 1993. View at: Google Scholar
 S. Tong and Y. Li, “Observerbased fuzzy adaptive control for strictfeedback nonlinear systems,” Fuzzy Sets and Systems, vol. 160, no. 12, pp. 1749–1764, 2009. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 T. Wang, S. Tong, and Y. Li, “Robust adaptive fuzzy control for nonlinear system with dynamic uncertainties based on backstepping,” International Journal of Innovative Computing, Information and Control, vol. 5, no. 9, pp. 2675–2688, 2009. View at: Google Scholar
 S. Tong, C. Liu, and Y. Li, “Fuzzyadaptive decentralized outputfeedback control for largescale nonlinear systems with dynamical uncertainties,” IEEE Transactions on Fuzzy Systems, vol. 18, no. 5, pp. 845–861, 2010. View at: Publisher Site  Google Scholar
 S. C. Tong, Y. M. Li, G. Feng, and T. S. Li, “Observerbased adaptive fuzzy backstepping dynamic surface control for a class of MIMO nonlinear systems,” IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, vol. 41, no. 4, pp. 1124–1135, 2011. View at: Publisher Site  Google Scholar
 Li Hongxing, “To see the success of fuzzy logic from mathematical essence of fuzzy controlon the paradoxical success of fuzzy logic,” Fuzzy Systems and Mathematics, vol. 9, no. 4, pp. 1–14, 1995 (Chinese). View at: Google Scholar
 H.X. Li, Z.H. Miao, and E. S. Lee, “Variable universe stable adaptive fuzzy control of a nonlinear system,” Computers & Mathematics with Applications, vol. 44, no. 56, pp. 799–815, 2002. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 H. Li, “Adaptive fuzzy controllers based on variable universe,” Science in China E, vol. 42, no. 1, pp. 10–20, 1999. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 L. Hongxing, M. Zhihong, and W. Jiayin, “Variable universe adaptive fuzzy control on the quadruple inverted pendulum,” Science in China E, vol. 45, no. 2, pp. 213–224, 2002. View at: Publisher Site  Google Scholar  MathSciNet
 L. Piegl and W. Tiller, The NURBS Book, Springer, New York, NY, USA, 1996.
 X. Shufang, The Theory and Technique in Matrix Computation, Peking University Press, Beijing, China, 1995 (Chinese).
 Z. XinXiong, Curve and Surface Modeling Technology, Science Press, Beijing, China, 2000 (Chinese).
 W. Zaiwen, Least squares method and its application [M.S. thesis], Chinese Academy of Sciences, Academy of Mathematics and Systems Science of Chinese Academy of Sciences, 2004.
Copyright
Copyright © 2013 Tan Yanhua and Li Hongxing. 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.