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.
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 [
Fuzzy systems can be constructed by splines [
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 [
The paper is organized as follows. Section
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 [
Let
The Frobenius norm on
Obviously,
In fuzzy inference, to acquire a group of inference rules and to acquire a group of I/O data are the same thing [
Let
The MISO first class of Bspline fuzzy system (1BFS) is
where
The MISO second class of Bspline fuzzy system (2BFS) is
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.
The uniform form of (
If we have
While, if we have
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 [
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 [
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
From the following two extreme cases, we can recognize the concrete significance of
When
Thus the faired BFS
When
For the cases in between, when
Let
Then, the objective function of the optimal problem (
From (
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.
Extract I/O data (
Given the initial weight
Calculate
If
Evaluate the obtained faired BFS,
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
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
A simplified model of the double inverted pendulum.
Let the clockwise angle and moment in Figure
For the control system of the double inverted pendulum, our control aim is to make the angles
The variable universe adaptive fuzzy controller for the double inverted pendulum is designed by the method in [
The universe of
Control rules for double inverted pendulum.










 

−0.8333  −0.8333  −0.6333  −0.5  −0.3333  −0.1667  0 

−0.8333  −0.6333  −0.5  −0.3333  −0.1667  0  0.1667 

−0.6333  −0.5  −0.3333  −0.1667  0  0.1667  0.3333 

−0.5  −0.3333  −0.1667  0  0.1667  0.3333  0.5 

−0.3333  −0.1667  0  0.1667  0.3333  0.5  0.6333 

−0.1667  0  0.1667  0.3333  0.5  0.6333  0.8333 

0  0.1667  0.3333  0.5  0.6333  0.8333  0.8333 
So the relatively exact I/O data for
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
Let
when the I/O data is
when the I/O data is
Response of controllers by 1BFS, 2BFS, faired 1BFS, and faired 2BFS with I/O data
Response of controllers by 1BFS, 2BFS, faired 1BFS, and faired 2BFS with I/O data
The control performance of control systems in terms of different weights is shown in Tables
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
For I/O data
For the relatively exact I/O data
For I/O data
From Tables
Performance index of control systems by 1BFS, faired 1BFSs with I/O data DISODS.
State  Index  1BFS  Faired 1BFS  


0.0002  0.3  2  
Diff  0  4.65%  1.19%  0.953%  
 

Steadystate error ( 
65.6  5.10  1.93  23.8 

150  16.4  0.461  52.7  

33.0  9.24  3.65  13.5  
 

Maximum overshoot  0.231  0.230  0.226  0.227 

0.118  0.111  0.113  0.115  

0.0570  0.0561  0.0554  0.0560  
 

3.36  3.44  3.33  3.34  

Settling  0.822  0.826  0.820  0.839 

0.713  0.319  0.313  0.315  
 

6.27  4.94  5.27  5.61 
Performance index of control systems by 2BFS, faired 2BFSs with I/O data DISODS.
State  Index  2BFS  Faired 2BFS  


3  5  10  
Diff  0.332%  1.20%  0.853%  0.610%  
 

Steadystate error ( 
13.4  2.45  3.66  0.587 

14.4  2.45  6.18  0.973  

17.5  3.31  3.34  0.549  
 

Maximum overshoot  0.229  0.228  0.227  0.227 

0.117  0.116  0.115  0.114  

0.0565  0.0563  0.0559  0.0557  
 

Settling time  3.35  3.35  3.34  3.35 

0.823  0.825  0.802  0.812  

0.31  0.308  0.315  0.314  
 

5.90  5.80  5.55  5.44 
Performance index of control systems by 1BFS, faired 1BFSs with I/O data DISODS^{1}.
State  Index  1BFS  Faired 1BFS  


0.001  0.5  4  
Diff  0  1.24%  1.21%  0.689%  
 

Steadystate error  1.77 

0.0899  0.277 












 

Maximum overshoot  2.01  0.228 

0.495 

0.219  0.115 

0.118  

0.149  0.0566 

0.0644  
 

Settling time  4.04  3.25 

3.59 

1.68  0.825 

0.846  

1.68  0.431 

0.796  
 

15.2  5.52 

6.84 
Performance index of control systems by 2BFS, faired 2BFSs with I/O data DISODS^{1}.
State  Index  2BFS  Faired 2BFS  


0.4  1.5  11  
Diff 

8.63%  1.35%  0.146%  
 

Steadystate error  1.71 

0.110  0.447 












 

Maximum overshoot  1.94 

0.322  0.664 

0.206  0.157 

0.1173  

0.143  0.0735 

0.0695  
 

Settling time  4.00 

3.45  3.66 

1.62  1.46 

0.881  

1.60  0.806 

0.866  
 

13.9  13.2 

6.64 
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.
Response of controllers by faired 1BFS and faired 2BFS with I/O data
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.
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).