Parametric Analysis of Flexible Logic Control Model

Based on deep analysis about the essential relation between two input variables of normal two-dimensional fuzzy controller, we used universal combinatorial operation model to describe the logic relationship and gave a �exible logic control method to realize the effective control for complex system. In practical control application, how to determine the general correlation coefficient of �exible logic control model is a problem for further studies. First, the conventional universal combinatorial operation model has been limited in the interval [0, 1]. Consequently, this paper studies a kind of universal combinatorial operation model based on the interval [aa, aa]. And some important theorems are given and proved, which provide a foundation for the �exible logic control method. For dealing reasonably with the complex relations of every factor in complex system, a kind of universal combinatorial operation model with unequal weights is put forward. en, this paper has carried out the parametric analysis of �exible logic control model. And some research results have been given, which have important directive to determine the values of the general correlation coefficients in practical control application.


Introduction
Fuzzy control has made the rapid development, and it has found a considerable number of successful industrial applications in recent years [1].However, from the mathematical viewpoint, Professor Li revealed the interpolation mechanism of fuzzy control and proved that fuzzy controller is in essence an interpolator [2].So, there are two problems in controlling some practical complex systems.One is that control rules will grow exponentially with the growing of inputs, and the other one is that the precision of control system is not high [3].
Compound controllers combine fuzzy control and other relatively mature control methods to obtain effective control effect, such as Fuzzy-PID controllers [4], fuzzy prediction control [5], and adaptive fuzzy -in�nity control [6].To reduce dimensionality, hierarchical fuzzy logic controller separates the set of control rules into several sets based on different functions [7,8].e basic idea of adaptive fuzzy controllers based on variable universe is to keep the control rules unchanged and change the region bound of fuzzy variables with the values of input fuzzy variables in order to increase control rules indirectly [9].ough a great deal of research has been done to improve the performance of fuzzy control, most of these methods are based on the basic idea that fuzzy controller is a piecewise approximation.However, to date, there has been relatively little research conducted on the internal relations among input variables of fuzzy controllers.
Universal Logic [10], proposed by He et al., is a kind of �exible logic.It considers the continuous change of not only the truth value of proposition, which is called truth value �exibility, but also the relation between propositions, which is called relational �exibility.Based on fuzzy logic, it puts up two important coefficients: generalized correlation coefficient "ℎ" and generalized self-correlation coefficient "." e �exible change of universal logic operations is based on "ℎ" and "." So, Universal Logic provides a new theoretical foundation to realize more accurate control for complex systems.
In our previous work [3], we focused on the basic physical meanings of fuzzy input variables of the normal twodimensional fuzzy controller, such as  and .And we proved that the essential relation between them is just universal combinatorial relation in Universal Logic [10].So, the simple universal combinatorial operation can be used instead of complex fuzzy reasoning process.As a result, a �exible logic control method was put forward.
rough the previous analysis, it is clear that �exible ability of �exible logic control model is resulted from the following aspect.We use universal combinatorial operation to re�ect the essential relation between the deviation and the deviation change of control system, which considers the continuous change of the relation between things.And universal combinatorial operation model is not a single �xed operator, but a continuous cluster of combinatorial operators determined by the general correlation coefficient ℎ between propositions.In practical control application, according to the general correlation between propositions, we can take the corresponding one from the cluster to realize effective control for complex system.
However, in practical control application, how to determine the general correlation coefficient ℎ of �exible logic control model is a problem for further studies.First, the conventional universal combinatorial operation model has been limited in the interval [0, 1].Consequently, this paper studies a kind of universal combinatorial operation model based on the interval [, ].And some important theorems are given and proved, which provide a foundation for the control application of universal combinatorial operation.en, this paper carries out the parametric analysis of �exible logic control model.And some research results have been given, which have important directive to determine the values of the general correlation coefficients in practical control application.
e rest of the paper is organized as follows.Section 2 introduces necessary background on universal combinatorial operation model and �exible logic control method and gives and proves some important theorems of universal combinatorial operation model based on the interval [, ].Section 3 carries out the parametric analysis of �exible logic control model and gives some research results.Finally, concluding remarks are given in Section 4.

Universal Combinatorial Operation Model
and Flexible Logic Control Method As a result, -norm or -norm can only handle mutually con�ictive relation.In contrast, Mean operators can only handle mutually consistent relation.erefore, the operating regions of these aggregation operators are localized.Universal combinatorial operation model is the combinatorial connective of Universal Logic, whose operating region is the standard interval [0, 1].
In the paper, we will only consider the generalized correlation coefficient ℎ.So, zero-order universal combinatorial operation model is de�ned as follows.
�e�nition 1 (see [10]).Zero-order universal combinatorial operation model is the cluster where Universal combinatorial operation model is a continuous cluster of combinatorial operators, which can be continuously changeable with generalized correlation coefficient ℎ between propositions.In practical application, according to the general correlation between propositions, we can take the corresponding one from the cluster.As generalized correlation coefficient h is equal to some special values, the corresponding combinatorial operators are given in Table 1.

Universal Combinatorial Operation Model on Any Interval
[, ].In practical control application, fuzzy domain of fuzzy variables,  and , is mostly symmetrical, such as [−6, 6].However, the conventional universal combinatorial operation model has been limited in the interval [0, 1].So, Chen [11] put forward a kind of universal combinatorial operation model, which is on any interval [, ].�e�nition � (see [11]).Normal universal Not operation model in any interval [, ] is de�ned as follows: For the previous de�nition, some common properties of normal universal Not operation model are given: (2) two polar law (3) symmetric involution According to the de�nition of Universal Combinatorial Operation Model in any interval, the following characters [11] are attained: (1) GC  (  ℎ) conforms to the combination axiom: Substituting ( 16), (17), and (18) separately into (15), GC GN() From the aforementioned, the theorem is true.is lemma indicates that when the interval [  is symmetrical about the origin point and identity element  is 0, Universal Combinatorial Operation GC  (   is also symmetrical about the origin point.
As pointed out in the literature [3], the internal relation between fuzzy input variables, the deviation , and the deviation change  of normal two-dimensional fuzzy controller is the universal combination relation in universal logic.Consequently, the complex fuzzy rule inference process could be replaced by the simple universal combinatorial operation, and a �exible logic control model was presented accordingly.In fuzzy control, the domain of input variables and output variable is generally symmetrical about the original point, such as [− 5𝑥 5].Obviously, it is the prerequisite of control model that the operation model relates to symmetry of original point.erefore, Lemma 10 provides a basis for the Universal Combinatorial Operation's application in control.

Universal Combinatorial Operation Model with Unequal
Weights.In practical complex system, every factor is generally with unequal weight.But the existing universal combinatorial operation only discusses an ideal state that every factor is with equal weight.

Weighted
Operator.For dealing reasonably with the complex relations of every factor in complex system, various properties which weighted operators should have are put forward.e weighted operator proposed by Yager is one of the famous ones.
e weighted operator proposed by �ager is de�ned as follows.
�e�nition 11 (see [12]).Assume that an operator (  is a mapping from F 1: Yager weighted operator (  varies with the weight .Proof.Due to the properties (I3) and (I4) of the de�nition, the theorem can be proved easily.e chart of Yager weighted operator (  changing along with the weight  is given in Figure 1.
One formulation that satis�es these conditions is (  =   (1 − .It is described by Figure 2.
According to the previous analyses, we discover that Yager weighted operator has some shortcomings as follows.
F 3: General weighted operator varies with the weight .
( According to the previous de�nition, we can know that the absolute value of the argument  decreases �rst and then increases with the weight  changing continuously from 0 to 1. e chart of general weighted operator changing along with the weight  is given in Figure 3. Assume that   (2 − /( −  2 ,   [ ,  ≥ −, and Gh(  is the limit as    and   .So, (38) describes a general weighted operator as shown in Figure 4. e universal combinatorial operation model with unequal weights is given in Figure 5.

Flexible Logic Control Method.
In order to decrease effectively the number of fuzzy control rules for multivariable nonlinear system, Xiao et al. [13] gave a new concept of fuzzy composed variable.Its basic idea can be summarized as follows.According to characteristics of controlled system and internal relations between input variables, a fuzzy composed variable is constructed to re�ect synthetically the deviation between reference and the process output with a fuzzy logic system.
ere are four output variables in single inverted-pendulum, which are   ′  , and  ′ .In the four input variables of control system, it is  and  ′ that describe the movement state of the rod.So, a fuzzy logic system can be designed, which is described by the fuzzy rules in Table 2, to de�ne a fuzzy composed variable   with  and  ′ .e fuzzy composed variable   can describe synthetically the movement state of the rod.Similarly, a fuzzy composed variable   can be de�ned with  and  ′ to describe synthetically the movement state of the cart.For multivariable system, one does not need to de�ne, respectively, fuzzy logic system for every fuzzy composed variable.A uniform fuzzy rule table can be used, such as Table 2, but only select different quanti�cation factors to obtain different fuzzy composed variables.
Remark 16.In the paper, the input variables of the fuzzy controllers discussed are  and , which denote, respectively, the deviation and the deviation change.e output variable  is the control signal.e variables, , , and  are crisp values from the practical process.e fuzzy variables, , , and  are the corresponding fuzzy ones, and the fuzzy domains are uniformed to be [−1 1 with fuzzy subsets, such as NB, NM, NS, ZE, PS, PM, and PB.
Apparently, the fuzzy rules in According to the physical meanings of fuzzy variables,  and , we can get the following conclusions.
(1) Suppose that both  and  are positive.at is to say, the deviation is positive and it will increase continuously.So, the value of the composed variable  ′ should be positive and bigger than both of  and  in this case.e combinatorial rules are shown by the right-bottom part of Table 2.
(2) Suppose that  is positive and  is negative.at is to say, the deviation is positive but it will decrease.So, the value of the composed variable  ′ should be between  and  in this case.e combinatorial rules are shown by the le-bottom part of Table 2.
(3) Similarly, we can obtain the value of  ′ in the two of other cases.
Based on the previous analysis, we get the conclusion that the essential relation among , , and the composed variable  ′ is a kind of universal combinatorial one in Universal Logic.As a result, we have So, we can obtain the relation among , , and the output variable  of control system as follows: where fuzzy variables,     [−1 1,  = 0,   [0 1.Fu and He [3] named the method Flexible Logic Control Method.For effectively controlling different things, we can lead into a weighted factor     .As the correlation coefficient ℎ is equal to 0.  42) is only a special operator in the cluster as ℎ is equal to 0.5.As a result, �exible logic control method can realize the accurate control for complex system.

Flexible Logic Control Model of Single Inverted-Pendulum
System.e objective is to maintain the pole in an upright position and the cart in an appointed position in the rail.ere are four output variables and one input variable in single inverted-pendulum, which are ,  ′ , ,  ′ , and .In the four input variables of control system, both  and  ′ describe the movement state of the pole, and both  and  ′ describe the movement state of the cart.So, we have designed two subcontrollers.One is to maintain the cart in an appointed position with two input variables   and   .e other one is to maintain the pole in an upright position with two input variables   and   .
We have designed the two subcontrollers with the �exible logic control method.And we led into weighted factors   ,   ,        .Two controllers are designed as follows: By the previous analyses, we can obtain the control model of single inverted-pendulum as shown in Figure 6.

Parametric Analysis of Flexible Logic Control Model
rough the previous analysis, it is clear that �exible ability of �exible logic control model is resulted from the following aspect.Universal combinatorial operation model is not a single �xed operator, but a continuous cluster of combinatorial operators determined by the general correlation coefficient ℎ between propositions.In practical control application, according to the general correlation between propositions, we can take the corresponding one from the cluster to realize effective control for complex system.However, in practical control application, how to determine the general correlation coefficient ℎ of �exible logic control model is a problem for further studies.In this section, we will analyze the general correlation coefficients, ℎ  , ℎ  , of the �exible logic control model and give some research results.

Experimentation.
We experiment the �exible logic control model in some single inverted-pendulum physical system.e physical parameters of the system are given in Table 3.
By simulating the inverted-pendulum and looking up the optimization with genetic algorithm, we can get the initial values of the control parameters.en, by testing in real-time experimentations and repeatedly making some �ne tuning, T 3: Physical parameters of the quadruple inverted-pendulum.

Symbol
Value Meaning  0 0.924 kg Mass of the cart  1 See Table 5  rough simulated optimization, we can get the corresponding values of ℎ  , ℎ  and the �tness shown in Table 6.
Figure 7 shows how the �tness varies with ℎ  , ℎ  .And Figures 8 and 9 illustrate how the maximum �tness varies with ℎ  taking different values when ℎ  , in turn, is equal to some value on standard interval [0, 1]. Figure 8 shows how the maximum �tness varies with ℎ  , and Figure 9 shows how ℎ  varies with ℎ  when the �tness is the maximum.
Similarly, Figures 10 and 11 illustrate how the maximum �tness varies with ℎ  taking different values when ℎ  , in turn, is equal to some value on standard interval [0, 1]. Figure 10 shows how the maximum �tness varies with ℎ  , and Figure 11 shows how ℎ  varies with ℎ  when the �tness is the maximum.

Parametric Analysis of Flexible Logic Control Model.
Firstly, calculate the correlation coefficient between �tness and ℎ  when ℎ  , in turn, is equal to some value on standard interval [0, 1].e formula for calculating the correlation coefficient is de�ned as follows: where  and  are    matrices,  and  are the means of the values in  and , respectively, and  is the correlation coefficient between  and .Similarly, calculate the correlation coefficient between �tness and ℎ  when ℎ  , in turn, is equal to some value on standard interval [0, 1].And Figure 13 illustrates how the correlation coefficient between �tness and ℎ  varies with ℎ  taking different values on standard interval [0, 1].
Aer that, the correlation coefficients between the maximum �tness and ℎ  , ℎ  are computed for poles of different lengths, as shown in Table 7.
By observing and analyzing the previous results, we can draw the following conclusions.
(1) e longer the pole, the worse the control effects.
Figures 7, 8, and 10 all indicate that the longer the pole, the smaller the �tness with the optimum control parameters, and vice versa.rough experiment, we can know that the given single inverted-pendulum physical system can be controlled effectively when the pole is in the range 0.1 m to 0.691 m. at is to say, when the pole is longer than 0.691 m or shorter than 0.1 m, we cannot realize the stable control for the given physical system.
(2) For some given physical system, the control effect is sensitive to the value of ℎ  .is means that we can realize the effective control only when ℎ  is in some very short interval.And the interval is relatively ��ed.at is, the interval of ℎ  does not change with the length of the pole.Figures 7, 9 and 10 all show that the �tness is relatively big only when ℎ  is in the range 0.8 to 0.9, and the interval of ℎ  remains unchanged for poles of different lengths.Figure 12 also shows that the correlation coefficient between �tness and ℎ  is relatively big only when ℎ  is around 0.2 or in the range 0.8 to 0.9, and the interval of ℎ  remains unchanged for poles of different lengths.
(3) For some given physical system, the control effect is not sensitive to the value of ℎ  .at is, we can realize the effective control when ℎ  is in some very long interval.Figure 10 shows that the control effect does not change much with ℎ  taking different values of the long interval.
Figure 13 also shows that the correlation coefficient between �tness and ℎ  is relatively big when ℎ  is in the range 0.15 to 0.98.
However, the width of the interval varies with the length of pole.e longer the pole, the narrower the interval.e paper calls the interval of ℎ  as ℎ  platform.
(4) For poles of different lengths, there is much difference of ℎ  and little one of ℎ  when we realize the most effective control.As shown in Table 7, the correlation  coefficient between the maximum �tness and ℎ  is small, but the one between the maximum �tness and ℎ  is big.Table 6 shows that the longer the pole, the smaller the optimum ℎ  , and the optimum ℎ  is in the range 0.8 to 0.9 with little difference.
From the third conclusion, we can know that the width of ℎ  platform varies with the length of pole.at is, the longer the pole, the smaller the width.For poles of different lengths, the corresponding ℎ  platforms are shown in Table 8.And the relation between the width of ℎ  platform and the length of pole and the one between the starting value of ℎ  platform and the length of pole can be obtained through �tting method according to the experimental results shown in Table 8.
Firstly, suppose that the smallest effective length of pole, that is, 0.1 m, is 1 unit.en, the ratios of the other lengths to the smallest effective length of pole are shown in Table 8. e relation between the width of ℎ  platform and the ratio of the length of pole is depicted by means of the linear �t, which is as shown in (50) and Figure 14 Similarly, the relation between the starting value of ℎ  platform and the ratio of the length of pole is depicted by means of the linear �t, which is as shown in (51) and Figure 15.Consider   1  11     11   (51) Hence, for the given physical system, when the length of pole takes different values in the previous controllable interval, we can calculate the corresponding interval of ℎ  platform by (50) and (51).en, ℎ  can be any of the interval and ℎ  should be any of the interval [8, ].erefore, it does not need to optimize the control parameters again, and we can realize the effective control for the physical system with new length of pole.

Conclusion
Flexible logic method uses universal combinatorial operation model to describe the logic relation between  and EC, which are the fuzzy input variables of normal two-dimensional fuzzy controller.Universal combinatorial operation model is not a single �xed operator, but a continuous cluster of combinatorial operators determined by the general correlation coefficient ℎ between propositions.In practical control application, according to the general correlation between propositions, we can take the corresponding one from the cluster to realize effective control for complex system.
However, in practical control application, how to determine the general correlation coefficient ℎ of �exible logic control model is a problem for further studies.First, the conventional universal combinatorial operation model has been limited in the interval [, 1].Consequently, this paper studies a kind of universal combinatorial operation model based on any interval [, ].And some important theorems are given and proved, which provide a foundation for the �exible logic control method.For dealing reasonably with the complex relations of every factor in complex system, a kind of universal combinatorial operation model with unequal weights is put forward.en, this paper has carried out the parametric analysis of �exible logic control model.And some research results have been given, which have important directive to determine the values of the general correlation coefficients in practical control application.

F 6 :
e �exible logic control model of single inverted-pendulum.

Figure 12 illustrates
how the correlation coefficient between �tness and ℎ  varies with ℎ  taking different values on standard interval [0, 1].

F 9 :
ℎ  varies with ℎ  when the �tness is the ma�imum.
eorem 12. Assume that (  is a Yager weighted operator.So, one has the following: if  ≥  then  (  ≥  and if    then  (   .
Yager weighted operator is likely to transform entire True (False) proposition into partial True (False) proposition.However, due to the view of logic, entire True (False) proposition should still be transformed into entire True (False) proposition by weighted operator.(2)e weighted value changes in the interval [  or [  for any weight   [  as shown in Figure 1.However, the weighted value is desired to change in the total interval [  in some practical applications.(3)Yager weighted operator is limited in the interval [ .But weighted operators are desired to change in the general interval [  in some practical applications.For solving the previous problems, the paper puts forward a kind of general weighted operators Gh( , which change in the general interval [ .�e�nition��.Assume that an operator Gh(  is a mapping from [  to [ .Gh(  is called a general weighted operator if it satis�es the following properties.I5) Finally, we desire that the transformation moves monotonically for the weight   ( .atis, if  ≥ , Gh(  increases monotonically with respect to the value ; if   , Gh(  decreases monotonically with respect to the value .(I6)Gh(   ite{              . is the weight associated with an argument , and  is the ��ed identity of the aggregation operator GC (I1) Monotonicity with respect to the value, .In particular, if    ′ , then we require Gh (  ≥ Gh   ′  .(37) (I2) Gh(   ite{          .(I3) Gh(   .(I4) Gh(    and Gh(   .( . is drawn from [ , and  is drawn from [ .
Assume that an operator UGC  (        ℎ is a mapping from [  to [ .UGC  (        ℎ is called universal combinatorial operation model with unequal weights: UGC          ℎ  GC  Gh      Gh      ℎ  (39)   (, ,   ,   , ℎ)   is general weighted operator, GC  (   is universal combinatorial operator,  is the �xed identity of GC  (  ,  is the generalized correlation coefficient, and   and   denote, respectively, the weights associated with the arguments  and ., , and  are drawn from [ , and ,    and   are drawn from [0 1.
Table 2 describe the essential relation between all deviation and the deviation T 2: Fuzzy rules de�ning fuzzy composed variable   .As shown Table 2 can be divided approximately into four parts, which describe, respectively, fuzzy rules used to de�ne composed variable as  and  are both negative,  is negative but  is positive,  and  are both positive, and  is positive but  is negative. and  both describe the deviation between the reference and the process output; so, we can de�ne a composed variable, denoted as  ′ , based on the essential relation between them.

Table 4 .
Mass of the pole And the variation of the length of the pole is shown in Table5.Genetic algorithm is used to optimize the general correlation coefficients, ℎ  , ℎ  , and the �tness function is de�ned as follows: . Consider   1       (50) T 8: e corresponding ℎ  platforms for poles of different lengths.F 15: e starting value of ℎ  platform varies with the ratio of the length.