#### Abstract

We construct two classes of faired MISO B-spline fuzzy systems using the fairing method in computer-aided geometric design (CAGD) for reducing adverse effects of the inexact data. Towards this goal, we generalize the faring method to high-dimension cases so that the faring method only for SISO and DISO B-spline fuzzy systems is extended to fair the MISO ones. Then the problem to construct a faired MISO B-spline 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 B-spline fuzzy system to obtain a faired MISO B-spline fuzzy system. Furthermore, we design variable universe adaptive fuzzy controllers by B-spline fuzzy systems and faired B-spline fuzzy systems to stabilize the double inverted pendulum. The simulation results show that the controllers by faired B-spline fuzzy systems perform better than those by B-spline 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 input-output 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 B-spline fuzzy systems (B-FSs) [12, 13], which are linear combination of B-spline basis functions and rational B-spline basis functions, respectively. Therefore, the single input single output (SISO) and double input single output (DISO) of these two classes of B-FSs 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 B-FSs 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 B-spline fuzzy systems (faired B-FSs) 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 B-FSs, 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 B-FSs. So, we propose a regularization term taken as the energy function of the MISO B-FS. By using this energy function, the energy method in CAGD, which can only be applied to fairing SISO and DISO B-FSs, is extended to fair the MISO ones. Therefore, based on the above preparations, the problem to construct a faired MISO B-FS is transformed into solving an optimization problem with a strictly convex quadratic objective function. In our proposed method, the faired MISO B-FS is available by taking the unique optimal solution vector as linear combination coefficients of the corresponding MISO B-FS.

As we all know, fuzzy controllers are a type of closed-loop fuzzy systems, while adaptive fuzzy controllers are closed-loop 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 B-FSs, we design variable universe adaptive fuzzy controllers by B-FSs and faired B-FSs to stabilize the double inverted pendulum. The simulation results show that the controllers by faired B-FSs perform better than those by B-FSs, 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 B-FSs are constructed in Section 3. In Section 4, the variable universe adaptive fuzzy controllers by B-FSs and faired B-FSs 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 B-spline basis functions, the Frobenius norm, and briefly review the two classes of MISO B-FSs in [12, 13].

*Definition 1 (B-spline basis functions [24]). *Let be a nondecreasing sequence of real numbers, that is, , . The is called knots, and is the knot vector. The th B-spline 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 B-spline basis function with knots , , , , and . The two classes of MISO B-FSs we proposed in [12, 13] are listed below.(1) The MISO first class of B-spline fuzzy system (1-B-FS) is where , , are obtained by solving equations after extrapolating some points.(2) The MISO second class of B-spline fuzzy system (2-B-FS) is

#### 3. The Faired MISO B-FSs

In this section, two classes of the faired MISO B-FSs 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 B-FS, the energy function of MISO B-FS is proposed, which is a regularization term essentially. Consequently, the energy method is suitable for fairing the MISO B-FSs. Then, we transform the problem to construct a faired MISO B-FS 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 first-order optimality condition. Consequently, the faired MISO B-FS is obtained by taking the unique optimal solution vector as linear combination coefficients of the corresponding B-FS. In the following, we will describe the above procedure in details.

##### 3.1. The Unification SISO Form of Two Classes of B-FSs

The uniform form of (5) and (7) is, where , , and .

If we have then (8) will turn to the 1-B-FS (5).

While, if we have then (8) is exactly the 2-B-FS (7). Consequently, (8) is named as the unification SISO form of the two classes of B-FSs.

##### 3.2. Generalizing the Energy Method to High-Dimension 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 B-FS, 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 B-FSs with more than 2 input variables, we have to generalize the energy method to the high-dimension 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 B-FS. Therefore, we generalize the energy method to the high-dimension cases.

##### 3.3. The Faired MISO B-FSs

In general, the curves (surfaces) in CAGD are referred to the parametric B-spline curves (surfaces), while the B-FSs are the B-spline functions. When the parametric B-spline curves (surfaces) degenerate into B-spline functions, the control points will be the coefficients of B-spline basis functions. Write the original B-FS and the faired B-FS as respectively. Then, by (11), we can transform the problem to construct a faired MISO B-FS 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 B-FS turns to the original B-FS , which implies that the B-FS is not modified after the fairing process.(2) When , we immediately get that from . Therefore, the B-FS 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 B-FS 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 first-order 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 B-FS is available.

*Remark 3. *Since it is convenient to deal with a uniform cubic B-spline, when the knots of the B-spline basis functions of a B-FS are arbitrary (non-uniform), we can approximate to this B-FS by one fuzzy system with uniform cubic B-spline basis functions to fair it.

*Algorithm. Constructing a Faired MISO B-FS.*

*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 B-FS, , 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 B-FSs (hereinafter abbreviated as B-FCs) and faired B-FSs (hereinafter abbreviated as faired B-FCs) 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 B-FSs. Further, it may affect the control effect of the faired B-FCs, 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 B-FCs and faired B-FCs 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 B-FCs is nearly as good as that of the faired ones (Figure 2);(2)when the I/O data is , the faired B-FCs outperform that of the B-FCs (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 steady-state 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 B-FCs 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 B-FCs are slightly better than those by B-FCs. Moreover, for faired 1-B-FCs (faired 2-B-FCs), 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 B-FCs is much better than that of control systems with B-FCs (seen from Tables 4 and 5), only except the steady-state error of of in Table 4. Especially, we point out that the energy consumption of faired B-FCs is less than that of the B-FCs.(5)From Tables 4 and 5, we also found that, for I/O data , neither faired B-FCs 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 B-FSs 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 B-FCs is slightly better than that of the B-FCs, which means the faired B-FSs for the faired B-FCs improve the B-FSs for the B-FCs slightly. While the I/O data for fuzzy system is inexact, the control effect of the faired B-FCs outperforms that of the B-FCs, in this case, the corresponding faired B-FSs reduce adverse effects of the inexact I/O data on the corresponding B-FSs as well as improve them significantly.

*Remark 4. * To compare the control effect of faired B-FCs and B-FCs with inexact I/O data, the variance of the Gaussian white noise added to should be as small as . Otherwise, the corresponding B-FCs cannot stabilize the double invert pendulum. Actually, when the variance of Gaussian white noise is as large as , the corresponding faired B-FCs can also stabilize the double invert pendulum.

In order to investigate the control capability of the faired B-FCs, we choose and use denote the I/O data. In this case, the faired B-FCs can stabilize the double pendulum as well as locate the cart (Figure 4). However, the B-FCs cannot do these.

#### 5. Conclusion

In this paper, the energy method in CAGD was utilized to design the faired MISO B-FSs. Based on our generalized approach, the construction of a faired MISO B-FS 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 B-FS, a faired MISO B-FS is obtained. For the faired MISO B-FSs, 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 B-FSs to stabilize the double inverted pendulum by modifying the weights. It is concluded that the faired B-FCs outperform the B-FCs 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 B-FSs are fuzzy systems with robustness. In the future, we will try to fair the B-FSs 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).