# Parametric Analysis of Flexible Logic Control Model

Authors: Lihua Fu; Dan Wang; Jinyun Kuang
Journal: Discrete Dynamics in Nature and Society
 (2013)

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## Abstract

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 flexible logic control method to realize the effective control for complex system. In practical control application, how to determine the general correlation coefficient of flexible 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 [a,b]. And some important theorems are given and proved, which provide a foundation for the flexible 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. Then, this paper has carried out the parametric analysis of flexible 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.


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## Body

## 1. 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 H-infinity control [6]. To reduce dimensionality, hierarchical fuzzy logic controller separates the set of control rules into several sets based on different functions [7, 8]. The 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]. Though 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 flexible logic. It considers the continuous change of not only the truth value of proposition, which is called truth value flexibility, but also the relation between propositions, which is called relational flexibility. Based on fuzzy logic, it puts up two important coefficients: generalized correlation coefficient “h” and generalized self-correlation coefficient “k.” The flexible change of universal logic operations is based on “h” and “k.” 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 two-dimensional fuzzy controller, such as E and EC. 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 flexible logic control method was put forward.Through the previous analysis, it is clear that flexible ability of flexible logic control model is resulted from the following aspect. We use universal combinatorial operation to reflect 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 fixed operator, but a continuous cluster of combinatorial operators determined by thegeneral correlation coefficienth 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 thegeneral correlation coefficienth of flexible 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 [a,b]. And some important theorems are given and proved, which provide a foundation for the control application of universal combinatorial operation. Then, this paper carries out the parametric analysis of flexible 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.The rest of the paper is organized as follows. Section2 introduces necessary background on universal combinatorial operation model and flexible logic control method and gives and proves some important theorems of universal combinatorial operation model based on the interval [a,b]. Section 3 carries out the parametric analysis of flexible logic control model and gives some research results. Finally, concluding remarks are given in Section 4.


## 2. Universal Combinatorial Operation Model and Flexible Logic Control Method

### 2.1. Universal Combinatorial Operation Model

It is well known that there is complex relation between every factor of complex system, which may be conflictive or consistent. For dealing reasonably with the complex relations, various aggregation operators have been given which are mostlyT-norm, S-norm, or Mean operators. Nevertheless, T-norm, S-norm, and Mean operators have the following properties:
(1)0≤T(x,y)≤min(x,y)max(x,y)≤S(x,y)≤1min(x,y)≤M(x,y)≤max(x,y).As a result,T-norm or S-norm can only handle mutually conflictive relation. In contrast, Mean operators can only handle mutually consistent relation. Therefore, 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 thegeneralized correlation coefficient  h. So, zero-order universal combinatorial operation model is defined as follows.Definition 1 (see [10]).
Zero-order universal combinatorial operation model is the cluster(2)Ce(x,y,h)=ite{Γe[(xm+ym-em)1/m]∣x+y<2e;1-(Γ1-e[(1-x)m+(1-y)m-(1-e)m(1-y)m])1/m∣x+y>2e;e},
where m=(3-4h)/(4h(1-h)),h∈[0,1],m∈R,ande∈[0,1].Remark 2.
The conditional expressionite{β∣α;γ} represents that if α is true, then the result is β, otherwise γ. Similarly, ite{β1∣α1;β2∣α2;γ}=ite{β1∣α1;ite{β2∣α2;γ}}. And Γ1[x]=ite{1∣x>1;0∣x<0orxisanimaginarynumber;x}.
Universal combinatorial operation model is a continuous cluster of combinatorial operators, which can be continuously changeable withgeneralized correlation coefficienth 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.Table 1
Some special combinatorial operators of universal combinatorial operation model.
Value ofh
General correlationbetween propositions
Combinatorial operator
Name of combinatorial operator
1
Max-attracting
C
e(x, y, 1) = ite{min(x, y) ∣x+y<2e;     max(x, y) ∣x+y>2e;     e}
Zadeh combinatorial operatorC3e
0.75
Independent correlation
C
e(x, y, 0.75) = ite{xy/e∣x+y<2e;     (x+y-xy-e)/(1-e)∣x+y>2e;     e}
Probability combinatorial operatorC2e
0.5
Max-rejecting
C
e(x, y, 0.5) =Γ1[x+y-e]
Bounded combinatorial operatorC1e
0
Max-restraining
C
e(x, y, 0) = ite{0∣x, y<e;     1∣x, y>e;     e}
Drastic combinatorial operatorC0e


### 2.2. Universal Combinatorial Operation Model on Any Interval[a,b]

In practical control application, fuzzy domain of fuzzy variables,E and EC, 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 [a,b].Definition 3 (see [11]).
Normal universalNotoperation model in any interval [a,b] is defined as follows:
(3)GN(x)=b+a-x.
For the previous definition, some common properties of normal universal Not operation model are given:(1)
closure(4)GN(x)∈[a,b],(2)
two polar law(5)GN(a)=b,GN(b)=a,(3)
symmetric involution(6)GN(GN(x))=x.Definition 4 (see [11]).
Universal combinatorial operation model on any interval[a,b] is the cluster(7)GCe(x,y,h)=ite{min((x-a)m+(y-a)m-(e-a)m(b-a)me,(b-a)×[max(0,(x-a)m+(y-a)m-(e-a)m(b-a)m)]1/m+a(x-a)m+(y-a)m-(e-a)m(b-a)m)∣x+y<2e;b+a-min((x-a)m+(y-a)m-(e-a)m(b-a)me′,(b-a)×[max(0,(b-x)m+(b-y)m-(b-e)m(b-a)m)]1/m+a(x-a)m+(y-a)m-(e-a)m(b-a)m)∣x+y>2e;e},where m=(3-4h)/(4h(1-h)), h∈[0,1], m∈R, e,e′∈[a,b], and e′=GN(e).
Without loss of generality, we assume that the fuzzy domain of fuzzy variables,E and EC, is the interval [-1,1]. As a result, when a, b, and e are equal to −1, 1, and 0, respectively, the corresponding zero-order universal combinatorial operator is the cluster(8)GC0(x,y,h)=ite{min([max(0,(x+1)m+(y+1)m-12m)]1/m0,2×[max(0,(x+1)m+(y+1)m-12m)]1/m-1[max(0,(x+1)m+(y+1)m-12m)]1/m)∣x+y<0;-min(0,2[max(0,(1-x)m+(1-y)m-12m)]1/m-1[max(0,(x+1)m+(y+1)m-12m)]1/m)∣x+y>0;0}.
According to the definition ofUniversal Combinatorial Operation Model in any interval, the following characters [11] are attained:(1)
GCe(x,y,h) conforms to the combination axiom:
(a)
boundary conditionGC1
whenx,y<e,GCe(x,y,h)≤min(x,y);whenx,y>e,GCe(x,y,h)≥max(x,y);whenx+y=2e,GCe(x,y,h)=e;otherwise,min(x,y)≤GCe(x,y,h)≤max(x,y),(b)
monotonicityGC2
GCe(x,y,h) increases monotonously along with x and y,(c)
continuityGC3
whenh∈(0,1), GCe(x,y,h) is continuous for all x and y,(d)
commutative lawGC4(9)GCe(x,y,h)=GCe(y,x,h),(e)
law of identical elementGC5(10)GCe(x,e,h)=x,(2)
closure(11)GCe(x,y,h)∈[a,b],(3)
inverse law(12)GCe(x,2e-x,h)=e,(4)
renunciation law(13)GCe(e,e,h)=e.Theorem 5.
LetGN(GCGN(e)(GN(x),GN(y),h))=GCe(x,y,h), x,y∈[a,b], e∈[a,b].Proof.
x
,
y
∈
[
a
,
b
], e∈[a,b], according to the closure of central generalized negation operation and universal combinatorial operation:
GN
(
x
)
,
GN
(
y
)
∈
[
a
,
b
],GN(e)∈[a,b]  and GCGN(e)(GN(x),GN(y),h)∈[a,b], and according to the definition of universal combinatorial operation: (1)
whenx+y<2e,
(14)GN(x)+GN(y)=(b+a-x)+(b+a-y)=2(b+a)-(x+y)>2(b+a-e)=2GN(e).
Then, according to the definition ofGCe(x,y,h),
(15)GCGN(e)(GN(x),GN(y),h)=b+a-min(b+a-GN(e),(b-a)(b-GN(x))m+(b-GN(y))m-(b-GN(e))m(b-a)m)]1/m×[max(0,(b-GN(x))m(b-a)m(b-GN(x))m+(b-GN(y))m-(b-GN(e))m(b-a)m)]1/m+(b-GN(y))m(b-a)m-(b-GN(e))m(b-a)m)]1/m+a).
According to the definition of central generalized negation operation,(16)GN(x)=b+a-x(17)GN(y)=b+a-y(18)GN(e)=b+a-e.
Substituting (16), (17), and (18) separately into (15),
(19)GCGN(e)(GN(x),GN(y),h)=b+a-min([max(0,(x-a)m+(y-a)m-(e-a)m(b-a)m)]1/m-(e-a)m(b-a)m)]1/me,(b-a)×[max([max(0,(x-a)m+(y-a)m-(e-a)m(b-a)m)]1/m0,(x-a)m+(y-a)m(b-a)m[max(0,(x-a)m+(y-a)m-(e-a)m(b-a)m)]1/m-(e-a)m(b-a)m)]1/m+a),
and then(20)GN(GCGN(e)(GN(x),GN(y),h))=b+a-(+(y-a)m-(e-a)m(b-a)m)]1/m+a)b+a-min(+(y-a)m-(e-a)m(b-a)m)]1/me,(b-a)×[max(+(y-a)m-(e-a)m(b-a)m)0,(x-a)m(b-a)m+(y-a)m-(e-a)m(b-a)m)]1/m+a(y-a)m-(e-a)m(b-a)m))=min(e,(b-a)[max(0,(x-a)m+(y-a)m-(e-a)m(b-a)m)]1/m×[max(+(y-a)m-(e-a)m(b-a)m)]1/m0,(x-a)m(b-a)m+(y-a)m-(e-a)m(b-a)m[max(0,(x-a)m+(y-a)m-(e-a)m(b-a)m)]1/m)]1/m+a).=GCe(x,y,h).(2)
Whenx+y>2e,
(21)GN(x)+GN(y)=(b+a-x)+(b+a-y)=2(b+a)-(x+y)<2(b+a-e)=2GN(e).
So, according to the definition ofGCe(x,y,h),
(22)GCGN(e)(GN(x),GN(y),h)=min(-(GN(e)-a)m(b-a)m(GN(x)-a)m+(GN(y)-a)m(b-a)m)]1/mGN(e),(b-a)×[max(0,(GN(x)-a)m+(GN(y)-a)m(b-a)m-(GN(e)-a)m(b-a)m(GN(x)-a)m+(GN(y)-a)m(b-a)m)]1/m+a).
Substituting (16), (17), and (18) separately into (22),
(23)GCGN(e)(GN(x),GN(y),h)=min(+(b-y)m-(b-e)m(b-a)m)]1/mb+a-e,(b-a)×[max(+(b-y)m-(b-e)m(b-a)m)]1/m0,(b-x)m(b-a)m+(b-y)m-(b-e)m(b-a)m)+(b-y)m-(b-e)m(b-a)m)]1/m]1/m+a).
And then,
(24)GN(GCGN(e)(GN(x),GN(y),h))=b+a-min(+(b-y)m-(b-e)m(b-a)m)]1/mb+a-e,(b-a)×[max(0,(b-x)m(b-a)m+(b-y)m-(b-e)m(b-a)m)]1/m+(b-y)m-(b-e)m(b-a)m)]1/m+a)=GCe(x,y,h).(3)
Whenx+y=2e,
(25)GN(x)+GN(y)=(b+a-x)+(b+a-y)=2(b+a)-(x+y)=2(b+a-e)=2GN(e).
According to the definition of   GCe(x,y,h),
(26)GCGN(e)(GN(x),GN(y),h)=GN(e).
Then,
(27)GN(GCGN(e)(GN(x),GN(y),h))=GN(GN(e))=e.
From the aforementioned, the theorem is true.Lemma 6.
LetGCGN(e)(GN(x),GN(y),h)=GN(GCe(x,y,h)).Proof.
According to Theorem5 and involution law of central generalized negation operator in interval [a,b], the theorem can be proved simply.Lemma 7.
LetCe(x,y,h)=1-C1-e(1-x,1-y,h).Proof.
Setting the interval[a,b] of x,y as [0,1], the lemma can be proved simply.Lemma 8.
When interval[a,b] relates to e symmetry,  GCe(x*,y*,h)=(GCe(x,y,h))*, wherein x*represents the points of x relating to e symmetry, namely, x*=2e-x,y*, e*, (GCe(x,y,h))* is similar, e∈[a,b], h∈[0,1].Proof.
Since interval[a,b] relates to e symmetry, then a+b=2e, and GN(x)=a+b-x=2e-x. Thus,
(28)GN(x)=x*.
Similarly,
(29)GN(y)=y*,GN(e)=a+b-e=2e-e=e.
From (28) and (29), the following could be obtained:
(30)GCGN(e)(GN(x),GN(y),h)=GCe(x*,y*,h),GN(GCe(x,y,h))=(GCe(x,y,h))*.
And then from Lemma 6,
(31)GCe(x*,y*,h)=(GCe(x,y,h))*.
So, the theorem is true.Lemma 9.
LetC0.5(1-x,1-y,h)=1-C0.5(x,y,h), wherein h∈[0,1].Lemma 10.
When interval[a,b] relates to symmetry of original point, C0(-x,-y,h)=-GC0(x,y,h), h∈[0,1].This lemma indicates that when the interval[a,b] is symmetrical about the origin point and identity element e is 0, Universal Combinatorial OperationGCe(x,y,h) is also symmetrical about the origin point.As pointed out in the literature [3], the internal relation between fuzzy input variables, the deviation E, and the deviation change EC 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 flexible 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 [-5,5]. Obviously, it is the prerequisite of control model that the operation model relates to symmetry of original point. Therefore, Lemma 10 provides a basis for the Universal Combinatorial Operation’s application in control.


### 2.3. 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.


#### 2.3.1. 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. The weighted operator proposed byYager is one of the famous ones.The weighted operator proposed by Yager is defined as follows.Definition 11 (see [12]).
Assume that an operatorh(α,x) is a mapping from [0,1] to [0,1].
h
(
α
,
x
) is called a Yager weighted operatorif it satisfies the following properties.(I1)
Monotonicity with respect to the value,x. In particular if x>x′, then we require
(32)h(α,x)≥h(α,x′).(I2)
We desire that elements with weight zero have no effect on the aggregation process; thus, ife is the fixed identity of the aggregation to be used on the resulting bag, we must have
(33)h(0,x)=e.(I3)
A normalcy with respect to the weights(34)h(1,x)=x.(I4)
Finally, we desire that the transformation moves monotonically from its value forα=0 to α=1. That is, if x≥e, h(α,x) increases monotonically with respect to the value α; if x≤e, h(α,x)decreases monotonically with respect to the value α,where α is the weight associated with an argument, and x is the argument. Both α and x are drawn from [0,1].
And due to the definition ofYager weighted operator, we can draw the following theorems.Theorem 12.
Assume thath(α,x) is a Yager weighted operator. So, one has the following:
(35)ifx≥e,thenh(α,x)≥e,andifx≤e,thenh(α,x)≤e.Proof.
Due to the properties (I2) and (I4) of the definition, the theorem can be proved easily.Theorem 13.
Assume thath(α,x) is a Yager weighted operator. So, one has the following:
(36)ifx≥e,thenh(α,x)≥e,andifx≤e,thenh(α,x)≤e.Proof.
Due to the properties (I3) and (I4) of the definition, the theorem can be proved easily.
The chart ofYager weighted operatorh(α,x) changing along with the weight α is given in Figure 1.
One formulation that satisfies these conditions ish(α,x)=αx+(1-α)e. It is described by Figure 2.
According to the previous analyses, we discover thatYager weighted operator has some shortcomings as follows.(1)
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)
The weighted value changes in the interval[e,x] or [x,e] for any weight α∈[0,1] as shown in Figure 1. However, the weighted value is desired to change in the total interval [0,1] in some practical applications.(3)
Yager weighted operatoris limited in the interval [0,1]. But weighted operators are desired to change in the general interval [a,b] in some practical applications.
For solving the previous problems, the paper puts forward a kind of general weighted operatorsGh(α,x), which change in the general interval [a,b].Figure 1
Yager weighted operator
h
(
α
,
x
) varies with the weight α.Figure 2
Yager weighted operator
h
(
α
,
x
)
=
α
x
+
(
1
-
α
)
e with e is equal to 0.3.Definition 14.
Assume that an operatorGh(α,x) is a mapping from [a,b] to [a,b]. Gh(α,x) is called a general weighted operatorif it satisfies the following properties.(I1)
Monotonicity with respect to the value,x. In particular, if x>x′, then we require
(37)Gh(α,x)≥Gh(α,x′).(I2)
Gh(0,x)=ite{a∣x=a;b∣x=b;e}.(I3)
Gh(α,e)=e.(I4)
Gh(α,a)=a and Gh(α,b)=b.(I5)
Finally, we desire that the transformation moves monotonically for the weightα∈(0,1). That is, if x≥e, Gh(α,x)increases monotonically with respect to the value α; if x≤e, Gh(α,x) decreases monotonically with respect to the value α.(I6)
Gh(1,x)=ite{e∣x=e;a∣x<e;b∣x>e}.α is the weight associated with an argument x, and e is the fixed identity of the aggregation operator GCe. α is drawn from [0,1], and e is drawn from [a,b].
According to the previous definition, we can know that the absolute value of the argumentx decreases first and then increases with the weight α changing continuously from 0 to 1. The chart of general weighted operatorchanging along with the weight α is given in Figure 3.
The common general weighted operation model is polynomial model:(38)Gh(α,x)=ite{(x-e)(b-e)(1+n)1/2((b-x)+(1+n)1/2(x-e))+e∣x>e;e-(e-x)(e-a)(1+n)1/2((x-a)+(1+n)1/2(e-x))∣x<e;e(x-e)(b-e)(1+n)1/2((b-x)+(1+n)1/2(x-e))}.
Assume thatn=(2α-1)/(1-α)2, α∈[0,1], n≥-1, and Gh(α,x) is the limit as α=0 and α=1. So, (38) describes a general weighted operatoras shown in Figure 4.Figure 3
General weighted operator varies with the weight α.Figure 4
General weighted operator of polynomial model with a=-1, b=1, and e=-0.3.


#### 2.3.2. Universal Combinatorial Operation Model with Unequal Weights

According to the previous definition of general weighted operators, we can get the definition of universal combinatorial operation model with unequal weights as follows.Definition 15.
Assume that an operatorUGCe(x,y,αx,αy,h) is a mapping from [a,b] to [a,b]. UGCe(x,y,αx,αy,h)  is called universal combinatorial operation model with unequal weights:
(39)UGCe(x,y,αx,αy,h)=GCe(Gh(αx,x),Gh(αy,y),h),
where Gh(α,x) is general weighted operator, GCe(x,y,h)is universal combinatorial operator, e is the fixed identity of GCe(x,y,h), h is the generalized correlation coefficient, and αxandαy denote, respectively, the weights associated with the arguments x and y. x, y, and e are drawn from [a,b], and h, αx,andαy are drawn from [0,1].
The universal combinatorial operation model with unequal weights is given in Figure5.Figure 5
Universal combinatorial operation model with unequal weights.


### 2.4. 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 reflect synthetically the deviation between reference and the process output with a fuzzy logic system.There are four output variables in single inverted-pendulum, which arex,x′,θ, and θ′. In the four input variables of control system, it is θ and θ′ 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 define a fuzzy composed variable GEθ with θ and θ′. The fuzzy composed variable GEθ can describe synthetically the movement state of the rod. Similarly, a fuzzy composed variable GEx can be defined with x and x′ to describe synthetically the movement state of the cart. For multivariable system, one does not need to define, 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 quantification factors to obtain different fuzzy composed variables.Table 2
Fuzzy rules defining fuzzy composed variableGEθ.Remark 16.
In the paper, the input variables of the fuzzy controllers discussed aree and ec, which denote, respectively, the deviation and the deviation change. The output variable u is the control signal. The variables, e, ec, and u are crisp values from the practical process. The fuzzy variables, E, EC, and U 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 Table2 describe the essential relation between all deviation and the deviation change. As shown Table 2 can be divided approximately into four parts, which describe, respectively, fuzzy rules used to define composed variable as E and EC are both negative, E is negative but EC is positive, E and EC are both positive, and E is positive but EC is negative. E and EC both describe the deviation between the reference and the process output; so, we can define a composed variable, denoted as E′, based on the essential relation between them.
According to the physical meanings of fuzzy variables,E and EC, we can get the following conclusions.(1)
Suppose that bothE and EC are positive. That is to say, the deviation is positive and it will increase continuously. So, the value of the composed variable E′ should be positive and bigger than both of E and EC in this case. The combinatorial rules are shown by the right-bottom part of Table 2.(2)
Suppose thatE is positive and EC is negative. That is to say, the deviation is positive but it will decrease. So, the value of the composed variable E′ should be between E and EC in this case. The combinatorial rules are shown by the left-bottom part of Table 2.(3)
Similarly, we can obtain the value ofE′ in the two of other cases.
Based on the previous analysis, we get the conclusion that the essential relation amongE, EC, and the composed variable E′ is a kind of universal combinatorial one in Universal Logic. As a result, we have
(40)E′=GCe(E,EC,h).
So, we can obtain the relation amongE, EC, and the output variable U of control system as follows:
(41)U=-GCe(E,EC,h),
where fuzzy variables, E,EC,U∈[-1,1], e=0, h∈[0,1]. Fu and He [3] named the method Flexible Logic Control Method.
For effectively controlling different things, we can lead into a weighted factorα,α∈[0,1]. As the correlation coefficient h is equal to 0.5, we have
(42)U=-GCe(αE,(1-α)EC,0.5)=Γ-11[αE+(1-α)EC-e]=αE+(1-α)EC.
However, (42) is just the fuzzy control method proposed by Long et al. [14]. They used a linear equation, such as (42), to describe fuzzy control rules. But the relation among E, EC, and U is not only linear. So, (41) is a cluster of operators determined by correlation coefficient h, and (42) is only a special operator in the cluster as h is equal to 0.5. As a result, flexible logic control method can realize the accurate control for complex system.


### 2.5. Flexible Logic Control Model of Single Inverted-Pendulum System

The objective is to maintain the pole in an upright position and the cart in an appointed position in the rail. There are four output variables and one input variable in single inverted-pendulum, which arex, x′, θ, θ′, and u. In the four input variables of control system, both θ and θ′ describe the movement state of the pole, and both x and x′ 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 Ex and ECx. The other one is to maintain the pole in an upright position with two input variables Eθ and ECθ.We have designed the two subcontrollers with the flexible logic control method. And we led into weighted factorsαθ, αx, αx,αθ∈[0,1]. Two controllers are designed as follows:
(43)Uθ=-GC0(Gh(αθ,Eθ),Gh(1-αθ,ECθ),hθ),(44)αθ=(αs_θ-α0_θ)|Eθ|+α0_θ,(45)Ux=-GC0(Gh(αx,Ex),Gh(1-αx,ECx),hx),(46)αx=(αs_x-α0_x)|Ex|+α0_x.When the control signalU is combined, we lead into two weighted factors kθ,kx,kθ,kx∈[-1,1], for the two subcontrollers. According to (43) and (45), we can get the output of the controller as follows:
(47)U=kxUx+kθUθ=kx(-GC0(Gh(αx,Ex),Gh(1-αx,ECx),hx))+kθ(-GC0(Gh(αθ,Eθ),Gh(1-αθ,ECθ),hθ)),
where Ex,ECx,Ux,Eθ,ECθ,Uθ,U∈[-1,1], Gh(α,x) (38), UGC0(x,y,α1,α2,h)is (39), hx,hθ are general correlation coefficients, hx,hθ∈[0,1], αx,αθ∈[0,1], 0≤α0_x≤αs_x≤1, and 0≤α0_θ≤αs_θ≤1.By the previous analyses, we can obtain the control model of single inverted-pendulum as shown in Figure6.Figure 6
The flexible logic control model of single inverted-pendulum.


## 2.1. Universal Combinatorial Operation Model

It is well known that there is complex relation between every factor of complex system, which may be conflictive or consistent. For dealing reasonably with the complex relations, various aggregation operators have been given which are mostlyT-norm, S-norm, or Mean operators. Nevertheless, T-norm, S-norm, and Mean operators have the following properties:
(1)0≤T(x,y)≤min(x,y)max(x,y)≤S(x,y)≤1min(x,y)≤M(x,y)≤max(x,y).As a result,T-norm or S-norm can only handle mutually conflictive relation. In contrast, Mean operators can only handle mutually consistent relation. Therefore, 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 thegeneralized correlation coefficient  h. So, zero-order universal combinatorial operation model is defined as follows.Definition 1 (see [10]).
Zero-order universal combinatorial operation model is the cluster(2)Ce(x,y,h)=ite{Γe[(xm+ym-em)1/m]∣x+y<2e;1-(Γ1-e[(1-x)m+(1-y)m-(1-e)m(1-y)m])1/m∣x+y>2e;e},
where m=(3-4h)/(4h(1-h)),h∈[0,1],m∈R,ande∈[0,1].Remark 2.
The conditional expressionite{β∣α;γ} represents that if α is true, then the result is β, otherwise γ. Similarly, ite{β1∣α1;β2∣α2;γ}=ite{β1∣α1;ite{β2∣α2;γ}}. And Γ1[x]=ite{1∣x>1;0∣x<0orxisanimaginarynumber;x}.
Universal combinatorial operation model is a continuous cluster of combinatorial operators, which can be continuously changeable withgeneralized correlation coefficienth 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.Table 1
Some special combinatorial operators of universal combinatorial operation model.
Value ofh
General correlationbetween propositions
Combinatorial operator
Name of combinatorial operator
1
Max-attracting
C
e(x, y, 1) = ite{min(x, y) ∣x+y<2e;     max(x, y) ∣x+y>2e;     e}
Zadeh combinatorial operatorC3e
0.75
Independent correlation
C
e(x, y, 0.75) = ite{xy/e∣x+y<2e;     (x+y-xy-e)/(1-e)∣x+y>2e;     e}
Probability combinatorial operatorC2e
0.5
Max-rejecting
C
e(x, y, 0.5) =Γ1[x+y-e]
Bounded combinatorial operatorC1e
0
Max-restraining
C
e(x, y, 0) = ite{0∣x, y<e;     1∣x, y>e;     e}
Drastic combinatorial operatorC0e


## 2.2. Universal Combinatorial Operation Model on Any Interval[a,b]

In practical control application, fuzzy domain of fuzzy variables,E and EC, 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 [a,b].Definition 3 (see [11]).
Normal universalNotoperation model in any interval [a,b] is defined as follows:
(3)GN(x)=b+a-x.
For the previous definition, some common properties of normal universal Not operation model are given:(1)
closure(4)GN(x)∈[a,b],(2)
two polar law(5)GN(a)=b,GN(b)=a,(3)
symmetric involution(6)GN(GN(x))=x.Definition 4 (see [11]).
Universal combinatorial operation model on any interval[a,b] is the cluster(7)GCe(x,y,h)=ite{min((x-a)m+(y-a)m-(e-a)m(b-a)me,(b-a)×[max(0,(x-a)m+(y-a)m-(e-a)m(b-a)m)]1/m+a(x-a)m+(y-a)m-(e-a)m(b-a)m)∣x+y<2e;b+a-min((x-a)m+(y-a)m-(e-a)m(b-a)me′,(b-a)×[max(0,(b-x)m+(b-y)m-(b-e)m(b-a)m)]1/m+a(x-a)m+(y-a)m-(e-a)m(b-a)m)∣x+y>2e;e},where m=(3-4h)/(4h(1-h)), h∈[0,1], m∈R, e,e′∈[a,b], and e′=GN(e).
Without loss of generality, we assume that the fuzzy domain of fuzzy variables,E and EC, is the interval [-1,1]. As a result, when a, b, and e are equal to −1, 1, and 0, respectively, the corresponding zero-order universal combinatorial operator is the cluster(8)GC0(x,y,h)=ite{min([max(0,(x+1)m+(y+1)m-12m)]1/m0,2×[max(0,(x+1)m+(y+1)m-12m)]1/m-1[max(0,(x+1)m+(y+1)m-12m)]1/m)∣x+y<0;-min(0,2[max(0,(1-x)m+(1-y)m-12m)]1/m-1[max(0,(x+1)m+(y+1)m-12m)]1/m)∣x+y>0;0}.
According to the definition ofUniversal Combinatorial Operation Model in any interval, the following characters [11] are attained:(1)
GCe(x,y,h) conforms to the combination axiom:
(a)
boundary conditionGC1
whenx,y<e,GCe(x,y,h)≤min(x,y);whenx,y>e,GCe(x,y,h)≥max(x,y);whenx+y=2e,GCe(x,y,h)=e;otherwise,min(x,y)≤GCe(x,y,h)≤max(x,y),(b)
monotonicityGC2
GCe(x,y,h) increases monotonously along with x and y,(c)
continuityGC3
whenh∈(0,1), GCe(x,y,h) is continuous for all x and y,(d)
commutative lawGC4(9)GCe(x,y,h)=GCe(y,x,h),(e)
law of identical elementGC5(10)GCe(x,e,h)=x,(2)
closure(11)GCe(x,y,h)∈[a,b],(3)
inverse law(12)GCe(x,2e-x,h)=e,(4)
renunciation law(13)GCe(e,e,h)=e.Theorem 5.
LetGN(GCGN(e)(GN(x),GN(y),h))=GCe(x,y,h), x,y∈[a,b], e∈[a,b].Proof.
x
,
y
∈
[
a
,
b
], e∈[a,b], according to the closure of central generalized negation operation and universal combinatorial operation:
GN
(
x
)
,
GN
(
y
)
∈
[
a
,
b
],GN(e)∈[a,b]  and GCGN(e)(GN(x),GN(y),h)∈[a,b], and according to the definition of universal combinatorial operation: (1)
whenx+y<2e,
(14)GN(x)+GN(y)=(b+a-x)+(b+a-y)=2(b+a)-(x+y)>2(b+a-e)=2GN(e).
Then, according to the definition ofGCe(x,y,h),
(15)GCGN(e)(GN(x),GN(y),h)=b+a-min(b+a-GN(e),(b-a)(b-GN(x))m+(b-GN(y))m-(b-GN(e))m(b-a)m)]1/m×[max(0,(b-GN(x))m(b-a)m(b-GN(x))m+(b-GN(y))m-(b-GN(e))m(b-a)m)]1/m+(b-GN(y))m(b-a)m-(b-GN(e))m(b-a)m)]1/m+a).
According to the definition of central generalized negation operation,(16)GN(x)=b+a-x(17)GN(y)=b+a-y(18)GN(e)=b+a-e.
Substituting (16), (17), and (18) separately into (15),
(19)GCGN(e)(GN(x),GN(y),h)=b+a-min([max(0,(x-a)m+(y-a)m-(e-a)m(b-a)m)]1/m-(e-a)m(b-a)m)]1/me,(b-a)×[max([max(0,(x-a)m+(y-a)m-(e-a)m(b-a)m)]1/m0,(x-a)m+(y-a)m(b-a)m[max(0,(x-a)m+(y-a)m-(e-a)m(b-a)m)]1/m-(e-a)m(b-a)m)]1/m+a),
and then(20)GN(GCGN(e)(GN(x),GN(y),h))=b+a-(+(y-a)m-(e-a)m(b-a)m)]1/m+a)b+a-min(+(y-a)m-(e-a)m(b-a)m)]1/me,(b-a)×[max(+(y-a)m-(e-a)m(b-a)m)0,(x-a)m(b-a)m+(y-a)m-(e-a)m(b-a)m)]1/m+a(y-a)m-(e-a)m(b-a)m))=min(e,(b-a)[max(0,(x-a)m+(y-a)m-(e-a)m(b-a)m)]1/m×[max(+(y-a)m-(e-a)m(b-a)m)]1/m0,(x-a)m(b-a)m+(y-a)m-(e-a)m(b-a)m[max(0,(x-a)m+(y-a)m-(e-a)m(b-a)m)]1/m)]1/m+a).=GCe(x,y,h).(2)
Whenx+y>2e,
(21)GN(x)+GN(y)=(b+a-x)+(b+a-y)=2(b+a)-(x+y)<2(b+a-e)=2GN(e).
So, according to the definition ofGCe(x,y,h),
(22)GCGN(e)(GN(x),GN(y),h)=min(-(GN(e)-a)m(b-a)m(GN(x)-a)m+(GN(y)-a)m(b-a)m)]1/mGN(e),(b-a)×[max(0,(GN(x)-a)m+(GN(y)-a)m(b-a)m-(GN(e)-a)m(b-a)m(GN(x)-a)m+(GN(y)-a)m(b-a)m)]1/m+a).
Substituting (16), (17), and (18) separately into (22),
(23)GCGN(e)(GN(x),GN(y),h)=min(+(b-y)m-(b-e)m(b-a)m)]1/mb+a-e,(b-a)×[max(+(b-y)m-(b-e)m(b-a)m)]1/m0,(b-x)m(b-a)m+(b-y)m-(b-e)m(b-a)m)+(b-y)m-(b-e)m(b-a)m)]1/m]1/m+a).
And then,
(24)GN(GCGN(e)(GN(x),GN(y),h))=b+a-min(+(b-y)m-(b-e)m(b-a)m)]1/mb+a-e,(b-a)×[max(0,(b-x)m(b-a)m+(b-y)m-(b-e)m(b-a)m)]1/m+(b-y)m-(b-e)m(b-a)m)]1/m+a)=GCe(x,y,h).(3)
Whenx+y=2e,
(25)GN(x)+GN(y)=(b+a-x)+(b+a-y)=2(b+a)-(x+y)=2(b+a-e)=2GN(e).
According to the definition of   GCe(x,y,h),
(26)GCGN(e)(GN(x),GN(y),h)=GN(e).
Then,
(27)GN(GCGN(e)(GN(x),GN(y),h))=GN(GN(e))=e.
From the aforementioned, the theorem is true.Lemma 6.
LetGCGN(e)(GN(x),GN(y),h)=GN(GCe(x,y,h)).Proof.
According to Theorem5 and involution law of central generalized negation operator in interval [a,b], the theorem can be proved simply.Lemma 7.
LetCe(x,y,h)=1-C1-e(1-x,1-y,h).Proof.
Setting the interval[a,b] of x,y as [0,1], the lemma can be proved simply.Lemma 8.
When interval[a,b] relates to e symmetry,  GCe(x*,y*,h)=(GCe(x,y,h))*, wherein x*represents the points of x relating to e symmetry, namely, x*=2e-x,y*, e*, (GCe(x,y,h))* is similar, e∈[a,b], h∈[0,1].Proof.
Since interval[a,b] relates to e symmetry, then a+b=2e, and GN(x)=a+b-x=2e-x. Thus,
(28)GN(x)=x*.
Similarly,
(29)GN(y)=y*,GN(e)=a+b-e=2e-e=e.
From (28) and (29), the following could be obtained:
(30)GCGN(e)(GN(x),GN(y),h)=GCe(x*,y*,h),GN(GCe(x,y,h))=(GCe(x,y,h))*.
And then from Lemma 6,
(31)GCe(x*,y*,h)=(GCe(x,y,h))*.
So, the theorem is true.Lemma 9.
LetC0.5(1-x,1-y,h)=1-C0.5(x,y,h), wherein h∈[0,1].Lemma 10.
When interval[a,b] relates to symmetry of original point, C0(-x,-y,h)=-GC0(x,y,h), h∈[0,1].This lemma indicates that when the interval[a,b] is symmetrical about the origin point and identity element e is 0, Universal Combinatorial OperationGCe(x,y,h) is also symmetrical about the origin point.As pointed out in the literature [3], the internal relation between fuzzy input variables, the deviation E, and the deviation change EC 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 flexible 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 [-5,5]. Obviously, it is the prerequisite of control model that the operation model relates to symmetry of original point. Therefore, Lemma 10 provides a basis for the Universal Combinatorial Operation’s application in control.


## 2.3. 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.


### 2.3.1. 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. The weighted operator proposed byYager is one of the famous ones.The weighted operator proposed by Yager is defined as follows.Definition 11 (see [12]).
Assume that an operatorh(α,x) is a mapping from [0,1] to [0,1].
h
(
α
,
x
) is called a Yager weighted operatorif it satisfies the following properties.(I1)
Monotonicity with respect to the value,x. In particular if x>x′, then we require
(32)h(α,x)≥h(α,x′).(I2)
We desire that elements with weight zero have no effect on the aggregation process; thus, ife is the fixed identity of the aggregation to be used on the resulting bag, we must have
(33)h(0,x)=e.(I3)
A normalcy with respect to the weights(34)h(1,x)=x.(I4)
Finally, we desire that the transformation moves monotonically from its value forα=0 to α=1. That is, if x≥e, h(α,x) increases monotonically with respect to the value α; if x≤e, h(α,x)decreases monotonically with respect to the value α,where α is the weight associated with an argument, and x is the argument. Both α and x are drawn from [0,1].
And due to the definition ofYager weighted operator, we can draw the following theorems.Theorem 12.
Assume thath(α,x) is a Yager weighted operator. So, one has the following:
(35)ifx≥e,thenh(α,x)≥e,andifx≤e,thenh(α,x)≤e.Proof.
Due to the properties (I2) and (I4) of the definition, the theorem can be proved easily.Theorem 13.
Assume thath(α,x) is a Yager weighted operator. So, one has the following:
(36)ifx≥e,thenh(α,x)≥e,andifx≤e,thenh(α,x)≤e.Proof.
Due to the properties (I3) and (I4) of the definition, the theorem can be proved easily.
The chart ofYager weighted operatorh(α,x) changing along with the weight α is given in Figure 1.
One formulation that satisfies these conditions ish(α,x)=αx+(1-α)e. It is described by Figure 2.
According to the previous analyses, we discover thatYager weighted operator has some shortcomings as follows.(1)
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)
The weighted value changes in the interval[e,x] or [x,e] for any weight α∈[0,1] as shown in Figure 1. However, the weighted value is desired to change in the total interval [0,1] in some practical applications.(3)
Yager weighted operatoris limited in the interval [0,1]. But weighted operators are desired to change in the general interval [a,b] in some practical applications.
For solving the previous problems, the paper puts forward a kind of general weighted operatorsGh(α,x), which change in the general interval [a,b].Figure 1
Yager weighted operator
h
(
α
,
x
) varies with the weight α.Figure 2
Yager weighted operator
h
(
α
,
x
)
=
α
x
+
(
1
-
α
)
e with e is equal to 0.3.Definition 14.
Assume that an operatorGh(α,x) is a mapping from [a,b] to [a,b]. Gh(α,x) is called a general weighted operatorif it satisfies the following properties.(I1)
Monotonicity with respect to the value,x. In particular, if x>x′, then we require
(37)Gh(α,x)≥Gh(α,x′).(I2)
Gh(0,x)=ite{a∣x=a;b∣x=b;e}.(I3)
Gh(α,e)=e.(I4)
Gh(α,a)=a and Gh(α,b)=b.(I5)
Finally, we desire that the transformation moves monotonically for the weightα∈(0,1). That is, if x≥e, Gh(α,x)increases monotonically with respect to the value α; if x≤e, Gh(α,x) decreases monotonically with respect to the value α.(I6)
Gh(1,x)=ite{e∣x=e;a∣x<e;b∣x>e}.α is the weight associated with an argument x, and e is the fixed identity of the aggregation operator GCe. α is drawn from [0,1], and e is drawn from [a,b].
According to the previous definition, we can know that the absolute value of the argumentx decreases first and then increases with the weight α changing continuously from 0 to 1. The chart of general weighted operatorchanging along with the weight α is given in Figure 3.
The common general weighted operation model is polynomial model:(38)Gh(α,x)=ite{(x-e)(b-e)(1+n)1/2((b-x)+(1+n)1/2(x-e))+e∣x>e;e-(e-x)(e-a)(1+n)1/2((x-a)+(1+n)1/2(e-x))∣x<e;e(x-e)(b-e)(1+n)1/2((b-x)+(1+n)1/2(x-e))}.
Assume thatn=(2α-1)/(1-α)2, α∈[0,1], n≥-1, and Gh(α,x) is the limit as α=0 and α=1. So, (38) describes a general weighted operatoras shown in Figure 4.Figure 3
General weighted operator varies with the weight α.Figure 4
General weighted operator of polynomial model with a=-1, b=1, and e=-0.3.


### 2.3.2. Universal Combinatorial Operation Model with Unequal Weights

According to the previous definition of general weighted operators, we can get the definition of universal combinatorial operation model with unequal weights as follows.Definition 15.
Assume that an operatorUGCe(x,y,αx,αy,h) is a mapping from [a,b] to [a,b]. UGCe(x,y,αx,αy,h)  is called universal combinatorial operation model with unequal weights:
(39)UGCe(x,y,αx,αy,h)=GCe(Gh(αx,x),Gh(αy,y),h),
where Gh(α,x) is general weighted operator, GCe(x,y,h)is universal combinatorial operator, e is the fixed identity of GCe(x,y,h), h is the generalized correlation coefficient, and αxandαy denote, respectively, the weights associated with the arguments x and y. x, y, and e are drawn from [a,b], and h, αx,andαy are drawn from [0,1].
The universal combinatorial operation model with unequal weights is given in Figure5.Figure 5
Universal combinatorial operation model with unequal weights.


## 2.3.1. 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. The weighted operator proposed byYager is one of the famous ones.The weighted operator proposed by Yager is defined as follows.Definition 11 (see [12]).
Assume that an operatorh(α,x) is a mapping from [0,1] to [0,1].
h
(
α
,
x
) is called a Yager weighted operatorif it satisfies the following properties.(I1)
Monotonicity with respect to the value,x. In particular if x>x′, then we require
(32)h(α,x)≥h(α,x′).(I2)
We desire that elements with weight zero have no effect on the aggregation process; thus, ife is the fixed identity of the aggregation to be used on the resulting bag, we must have
(33)h(0,x)=e.(I3)
A normalcy with respect to the weights(34)h(1,x)=x.(I4)
Finally, we desire that the transformation moves monotonically from its value forα=0 to α=1. That is, if x≥e, h(α,x) increases monotonically with respect to the value α; if x≤e, h(α,x)decreases monotonically with respect to the value α,where α is the weight associated with an argument, and x is the argument. Both α and x are drawn from [0,1].
And due to the definition ofYager weighted operator, we can draw the following theorems.Theorem 12.
Assume thath(α,x) is a Yager weighted operator. So, one has the following:
(35)ifx≥e,thenh(α,x)≥e,andifx≤e,thenh(α,x)≤e.Proof.
Due to the properties (I2) and (I4) of the definition, the theorem can be proved easily.Theorem 13.
Assume thath(α,x) is a Yager weighted operator. So, one has the following:
(36)ifx≥e,thenh(α,x)≥e,andifx≤e,thenh(α,x)≤e.Proof.
Due to the properties (I3) and (I4) of the definition, the theorem can be proved easily.
The chart ofYager weighted operatorh(α,x) changing along with the weight α is given in Figure 1.
One formulation that satisfies these conditions ish(α,x)=αx+(1-α)e. It is described by Figure 2.
According to the previous analyses, we discover thatYager weighted operator has some shortcomings as follows.(1)
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)
The weighted value changes in the interval[e,x] or [x,e] for any weight α∈[0,1] as shown in Figure 1. However, the weighted value is desired to change in the total interval [0,1] in some practical applications.(3)
Yager weighted operatoris limited in the interval [0,1]. But weighted operators are desired to change in the general interval [a,b] in some practical applications.
For solving the previous problems, the paper puts forward a kind of general weighted operatorsGh(α,x), which change in the general interval [a,b].Figure 1
Yager weighted operator
h
(
α
,
x
) varies with the weight α.Figure 2
Yager weighted operator
h
(
α
,
x
)
=
α
x
+
(
1
-
α
)
e with e is equal to 0.3.Definition 14.
Assume that an operatorGh(α,x) is a mapping from [a,b] to [a,b]. Gh(α,x) is called a general weighted operatorif it satisfies the following properties.(I1)
Monotonicity with respect to the value,x. In particular, if x>x′, then we require
(37)Gh(α,x)≥Gh(α,x′).(I2)
Gh(0,x)=ite{a∣x=a;b∣x=b;e}.(I3)
Gh(α,e)=e.(I4)
Gh(α,a)=a and Gh(α,b)=b.(I5)
Finally, we desire that the transformation moves monotonically for the weightα∈(0,1). That is, if x≥e, Gh(α,x)increases monotonically with respect to the value α; if x≤e, Gh(α,x) decreases monotonically with respect to the value α.(I6)
Gh(1,x)=ite{e∣x=e;a∣x<e;b∣x>e}.α is the weight associated with an argument x, and e is the fixed identity of the aggregation operator GCe. α is drawn from [0,1], and e is drawn from [a,b].
According to the previous definition, we can know that the absolute value of the argumentx decreases first and then increases with the weight α changing continuously from 0 to 1. The chart of general weighted operatorchanging along with the weight α is given in Figure 3.
The common general weighted operation model is polynomial model:(38)Gh(α,x)=ite{(x-e)(b-e)(1+n)1/2((b-x)+(1+n)1/2(x-e))+e∣x>e;e-(e-x)(e-a)(1+n)1/2((x-a)+(1+n)1/2(e-x))∣x<e;e(x-e)(b-e)(1+n)1/2((b-x)+(1+n)1/2(x-e))}.
Assume thatn=(2α-1)/(1-α)2, α∈[0,1], n≥-1, and Gh(α,x) is the limit as α=0 and α=1. So, (38) describes a general weighted operatoras shown in Figure 4.Figure 3
General weighted operator varies with the weight α.Figure 4
General weighted operator of polynomial model with a=-1, b=1, and e=-0.3.


## 2.3.2. Universal Combinatorial Operation Model with Unequal Weights

According to the previous definition of general weighted operators, we can get the definition of universal combinatorial operation model with unequal weights as follows.Definition 15.
Assume that an operatorUGCe(x,y,αx,αy,h) is a mapping from [a,b] to [a,b]. UGCe(x,y,αx,αy,h)  is called universal combinatorial operation model with unequal weights:
(39)UGCe(x,y,αx,αy,h)=GCe(Gh(αx,x),Gh(αy,y),h),
where Gh(α,x) is general weighted operator, GCe(x,y,h)is universal combinatorial operator, e is the fixed identity of GCe(x,y,h), h is the generalized correlation coefficient, and αxandαy denote, respectively, the weights associated with the arguments x and y. x, y, and e are drawn from [a,b], and h, αx,andαy are drawn from [0,1].
The universal combinatorial operation model with unequal weights is given in Figure5.Figure 5
Universal combinatorial operation model with unequal weights.


## 2.4. 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 reflect synthetically the deviation between reference and the process output with a fuzzy logic system.There are four output variables in single inverted-pendulum, which arex,x′,θ, and θ′. In the four input variables of control system, it is θ and θ′ 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 define a fuzzy composed variable GEθ with θ and θ′. The fuzzy composed variable GEθ can describe synthetically the movement state of the rod. Similarly, a fuzzy composed variable GEx can be defined with x and x′ to describe synthetically the movement state of the cart. For multivariable system, one does not need to define, 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 quantification factors to obtain different fuzzy composed variables.Table 2
Fuzzy rules defining fuzzy composed variableGEθ.Remark 16.
In the paper, the input variables of the fuzzy controllers discussed aree and ec, which denote, respectively, the deviation and the deviation change. The output variable u is the control signal. The variables, e, ec, and u are crisp values from the practical process. The fuzzy variables, E, EC, and U 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 Table2 describe the essential relation between all deviation and the deviation change. As shown Table 2 can be divided approximately into four parts, which describe, respectively, fuzzy rules used to define composed variable as E and EC are both negative, E is negative but EC is positive, E and EC are both positive, and E is positive but EC is negative. E and EC both describe the deviation between the reference and the process output; so, we can define a composed variable, denoted as E′, based on the essential relation between them.
According to the physical meanings of fuzzy variables,E and EC, we can get the following conclusions.(1)
Suppose that bothE and EC are positive. That is to say, the deviation is positive and it will increase continuously. So, the value of the composed variable E′ should be positive and bigger than both of E and EC in this case. The combinatorial rules are shown by the right-bottom part of Table 2.(2)
Suppose thatE is positive and EC is negative. That is to say, the deviation is positive but it will decrease. So, the value of the composed variable E′ should be between E and EC in this case. The combinatorial rules are shown by the left-bottom part of Table 2.(3)
Similarly, we can obtain the value ofE′ in the two of other cases.
Based on the previous analysis, we get the conclusion that the essential relation amongE, EC, and the composed variable E′ is a kind of universal combinatorial one in Universal Logic. As a result, we have
(40)E′=GCe(E,EC,h).
So, we can obtain the relation amongE, EC, and the output variable U of control system as follows:
(41)U=-GCe(E,EC,h),
where fuzzy variables, E,EC,U∈[-1,1], e=0, h∈[0,1]. Fu and He [3] named the method Flexible Logic Control Method.
For effectively controlling different things, we can lead into a weighted factorα,α∈[0,1]. As the correlation coefficient h is equal to 0.5, we have
(42)U=-GCe(αE,(1-α)EC,0.5)=Γ-11[αE+(1-α)EC-e]=αE+(1-α)EC.
However, (42) is just the fuzzy control method proposed by Long et al. [14]. They used a linear equation, such as (42), to describe fuzzy control rules. But the relation among E, EC, and U is not only linear. So, (41) is a cluster of operators determined by correlation coefficient h, and (42) is only a special operator in the cluster as h is equal to 0.5. As a result, flexible logic control method can realize the accurate control for complex system.


## 2.5. Flexible Logic Control Model of Single Inverted-Pendulum System

The objective is to maintain the pole in an upright position and the cart in an appointed position in the rail. There are four output variables and one input variable in single inverted-pendulum, which arex, x′, θ, θ′, and u. In the four input variables of control system, both θ and θ′ describe the movement state of the pole, and both x and x′ 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 Ex and ECx. The other one is to maintain the pole in an upright position with two input variables Eθ and ECθ.We have designed the two subcontrollers with the flexible logic control method. And we led into weighted factorsαθ, αx, αx,αθ∈[0,1]. Two controllers are designed as follows:
(43)Uθ=-GC0(Gh(αθ,Eθ),Gh(1-αθ,ECθ),hθ),(44)αθ=(αs_θ-α0_θ)|Eθ|+α0_θ,(45)Ux=-GC0(Gh(αx,Ex),Gh(1-αx,ECx),hx),(46)αx=(αs_x-α0_x)|Ex|+α0_x.When the control signalU is combined, we lead into two weighted factors kθ,kx,kθ,kx∈[-1,1], for the two subcontrollers. According to (43) and (45), we can get the output of the controller as follows:
(47)U=kxUx+kθUθ=kx(-GC0(Gh(αx,Ex),Gh(1-αx,ECx),hx))+kθ(-GC0(Gh(αθ,Eθ),Gh(1-αθ,ECθ),hθ)),
where Ex,ECx,Ux,Eθ,ECθ,Uθ,U∈[-1,1], Gh(α,x) (38), UGC0(x,y,α1,α2,h)is (39), hx,hθ are general correlation coefficients, hx,hθ∈[0,1], αx,αθ∈[0,1], 0≤α0_x≤αs_x≤1, and 0≤α0_θ≤αs_θ≤1.By the previous analyses, we can obtain the control model of single inverted-pendulum as shown in Figure6.Figure 6
The flexible logic control model of single inverted-pendulum.


## 3. Parametric Analysis of Flexible Logic Control Model

Through the previous analysis, it is clear that flexible ability of flexible logic control model is resulted from the following aspect. Universal combinatorial operation model is not a single fixed operator, but a continuous cluster of combinatorial operators determined by thegeneral correlation coefficienth 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 thegeneral correlation coefficienth of flexible logic control model is a problem for further studies. In this section, we will analyze the general correlation coefficients, hx, hθ, of the flexible logic control model and give some research results.


### 3.1. Experimentation

We experiment the flexible logic control model in some single inverted-pendulum physical system. The physical parameters of the system are given in Table3.Table 3
Physical parameters of the quadruple inverted-pendulum.
Symbol
Value
Meaning
m
0
0.924 kg
Mass of the cart
m
1
See Table5
Mass of the pole
f
0
0.1 N·s/m
Dynamic friction coefficient between the cart and the track
f
1
0.007056 N·s/m
Dynamic friction coefficient for the pole
l
1
See Table5
Distance from the position sensor to the center of gravity of the poleBy simulating the inverted-pendulum and looking up the optimization with genetic algorithm, we can get the initial values of the control parameters. Then, by testing in real-time experimentations and repeatedly making some fine tuning, we can get the control parameters shown in Table4. And the variation of the length of the pole is shown in Table 5.Table 4
Control parameters of the control model.
Symbol
Value
Meaning
K
e
_
x
23.5294
Quantification factor forEx
K
c
_
x
9.4118
Quantification factor forECx
α
0
_
x
0.1725
Minimum value ofαx
α
s
_
x
0.4683
Maximum value ofαx
K
e
_
θ
50.5882
Quantification factor forEθ
K
c
_
θ
3.1765
Quantification factor forECθ
α
0
_
θ
0.2510
Minimum value ofαθ
α
s
_
θ
0.5953
Maximum value ofαθ
k
x
−0.2235
Weighted factor of the subcontroller for the cart
k
θ
0.4902
Weighted factor of the subcontroller for the pole
K
u
8.7843
Proportion factor forU
h
x
0.2118
General correlation coefficient betweenEx and ECx
h
θ
0.9490
General correlation coefficient betweenEθ and ECθTable 5
The variation of the length of the pole.
No.
m
1 (kg)
L
1 (m)
1
0.0149
0.1
2
0.0216
0.149
3
0.0284
0.199
4
0.0378
0.266
5
0.0493
0.354
6
0.0621
0.4436
7
0.0773
0.553
8
0.0966
0.691Genetic algorithm is used to optimize the general correlation coefficients,hx, hθ, and the fitness function is defined as follows:
(48)Dis=∑i=0Ni2×(x2(i)/10+x′2(i)/20+θ2(i)+θ′2(i))N,fitness=1(Dis/10).Through simulated optimization, we can get the corresponding values ofhx, hθ and the fitness shown in Table 6.Table 6
The optimization results ofhx, hθ.
No.
m
1 (kg)
L
1 (m)
h
x
h
θ
Fitness
1
0.0149
0.1
0.985
0.845
0.4310
2
0.0216
0.149
0.975
0.845
0.3310
3
0.0284
0.199
0.96
0.845
0.2713
4
0.0378
0.266
0.955
0.87
0.2240
5
0.0493
0.354
0.63
0.88
0.1838
6
0.0621
0.4436
0.37
0.875
0.1531
7
0.0773
0.553
0.32
0.865
0.1260
8
0.0966
0.691
0.37
0.85
0.1029Figure7 shows how the fitness varies with hx,hθ. And Figures 8 and 9 illustrate how the maximum fitness varies with hθ taking different values when hx, in turn, is equal to some value on standard interval [0,1]. Figure 8 shows how the maximum fitness varies with hx, and Figure 9 shows how hθ varies with hx when the fitness is the maximum.The fitness varies withhx,hθ.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)Figure 8
The maximum fitness varies withhx.Figure 9
h
θ varies with hx when the fitness is the maximum.Similarly, Figures10 and 11 illustrate how the maximum fitness varies with hx taking different values when hθ, in turn, is equal to some value on standard interval [0,1]. Figure 10 shows how the maximum fitness varies with hθ, and Figure 11 shows how hx varies with hθ when the fitness is the maximum.Figure 10
The maximum fitness varies withhθ.Figure 11
h
x varies with hθ when the fitness is the maximum.


### 3.2. Parametric Analysis of Flexible Logic Control Model

Firstly, calculate the correlation coefficient between fitness andhx when hθ, in turn, is equal to some value on standard interval [0,1]. The formula for calculating the correlation coefficient is defined as follows:
(49)r=∑m∑n(Amn-A-)(Bmn-B-)(∑m∑n(Amn-A-)2)(∑m∑n(Bmn-B-)2),
where A and B are m*n matrices, A- and B- are the means of the values in A and B, respectively, and r is the correlation coefficient between A and B.Figure12 illustrates how the correlation coefficient between fitness and hx varies with hθ taking different values on standard interval [0,1].Figure 12
The correlation coefficient between fitness andhx varies with hθ.Similarly, calculate the correlation coefficient between fitness andhθ when hx, in turn, is equal to some value on standard interval [0,1]. And Figure 13 illustrates how the correlation coefficient between fitness and hθ varies with hx taking different values on standard interval [0,1].Figure 13
The correlation coefficient between fitness andhθ varies with hx.After that, the correlation coefficients between the maximum fitness andhx, hθ are computed for poles of different lengths, as shown in Table 7.Table 7
The correlation coefficients between the maximum fitness andhx, hθ.
No.
m
1 (kg)
L
1 (m)
The correlation coefficient between the maximum fitness andhx
The correlation coefficient between the maximum fitness andhθ
1
0.0149
0.1
0.3801
0.9602
2
0.0216
0.149
0.3956
0.9503
3
0.0284
0.199
0.4104
0.9537
4
0.0378
0.266
0.4261
0.9604
5
0.0493
0.354
0.4285
0.9513
6
0.0621
0.4436
0.4294
0.9314
7
0.0773
0.553
0.4308
0.9125
8
0.0966
0.691
0.4327
0.8895By observing and analyzing the previous results, we can draw the following conclusions.(1)
The longer the pole, the worse the control effects. Figures7, 8, and 10 all indicate that the longer the pole, the smaller the fitness with the optimum control parameters, and vice versa. Through 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. That 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 ofhθ. This means that we can realize the effective control only when hθ is in some very short interval. And the interval is relatively fixed. That is, the interval of hθ does not change with the length of the pole. Figures 7, 9 and 10 all show that the fitness is relatively big only when hθ is in the range 0.8 to 0.9, and the interval of hθ remains unchanged for poles of different lengths.Figure12 also shows that the correlation coefficient between fitness and hx is relatively big only when hθ is around 0.2 or in the range 0.8 to 0.9, and the interval of hθ remains unchanged for poles of different lengths.(3)
For some given physical system, the control effect is not sensitive to the value ofhx. That is, we can realize the effective control when hx is in some very long interval. Figure 10 shows that the control effect does not change much with hx taking different values of the long interval.Figure13 also shows that the correlation coefficient between fitness and hθ is relatively big when hx is in the range 0.15 to 0.98.However, the width of the interval varies with the length of pole. The longer the pole, the narrower the interval. The paper calls the interval ofhx as hx  platform.(4)
For poles of different lengths, there is much difference ofhx and little one of hθ when we realize the most effective control. As shown in Table 7, the correlation coefficient between the maximum fitness and hx is small, but the one between the maximum fitness and hθ is big. Table 6 shows that the longer the pole, the smaller the optimum hx, and the optimum hθ is in the range 0.8 to 0.9 with little difference.From the third conclusion, we can know that the width ofhx  platform varies with the length of pole. That is, the longer the pole, the smaller the width. For poles of different lengths, the corresponding hx  platforms are shown in Table 8. And the relation between the width of hx  platform and the length of pole and the one between the starting value of hx  platform and the length of pole can be obtained through fitting method according to the experimental results shown in Table 8.Table 8
The correspondinghx  platforms for poles of different lengths.
No.
L
1 (m)
Ratio of the lengths of poles
The starting value ofhx  platform
The finishing value ofhx  platform
The width ofhx  platform
1
0.1
1
0.175
0.98
0.805
2
0.149
1.49
0.22
0.98
0.76
3
0.199
1.99
0.235
0.98
0.745
4
0.266
2.66
0.25
0.975
0.725
5
0.354
3.54
0.26
0.955
0.695
6
0.4436
4.436
0.27
0.94
0.67
7
0.553
5.53
0.275
0.94
0.665
8
0.691
6.91
0.285
0.94
0.655Firstly, suppose that the smallest effective length of pole, that is, 0.1 m, is 1 unit. Then, the ratios of the other lengths to the smallest effective length of pole are shown in Table8. The relation between the width of hx  platform and the ratio of the length of pole is depicted by means of the linear fit, which is as shown in (50) and Figure 14. Consider
(50)y=0.85314-0.062624x+0.0049723x2.Figure 14
The width ofhx  platform varies with the ratio of the length.Similarly, the relation between the starting value ofhx  platform and the ratio of the length of pole is depicted by means of the linear fit, which is as shown in (51) and Figure 15. Consider
(51)y=0.095132+0.10701x-0.022054x2+0.0015301x3.Figure 15
The starting value ofhx  platform varies with the ratio of the length.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 ofhx  platform by (50) and (51). Then, hx can be any of the interval and hθ should be any of the interval [0.8,0.9]. Therefore, 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.


## 3.1. Experimentation

We experiment the flexible logic control model in some single inverted-pendulum physical system. The physical parameters of the system are given in Table3.Table 3
Physical parameters of the quadruple inverted-pendulum.
Symbol
Value
Meaning
m
0
0.924 kg
Mass of the cart
m
1
See Table5
Mass of the pole
f
0
0.1 N·s/m
Dynamic friction coefficient between the cart and the track
f
1
0.007056 N·s/m
Dynamic friction coefficient for the pole
l
1
See Table5
Distance from the position sensor to the center of gravity of the poleBy simulating the inverted-pendulum and looking up the optimization with genetic algorithm, we can get the initial values of the control parameters. Then, by testing in real-time experimentations and repeatedly making some fine tuning, we can get the control parameters shown in Table4. And the variation of the length of the pole is shown in Table 5.Table 4
Control parameters of the control model.
Symbol
Value
Meaning
K
e
_
x
23.5294
Quantification factor forEx
K
c
_
x
9.4118
Quantification factor forECx
α
0
_
x
0.1725
Minimum value ofαx
α
s
_
x
0.4683
Maximum value ofαx
K
e
_
θ
50.5882
Quantification factor forEθ
K
c
_
θ
3.1765
Quantification factor forECθ
α
0
_
θ
0.2510
Minimum value ofαθ
α
s
_
θ
0.5953
Maximum value ofαθ
k
x
−0.2235
Weighted factor of the subcontroller for the cart
k
θ
0.4902
Weighted factor of the subcontroller for the pole
K
u
8.7843
Proportion factor forU
h
x
0.2118
General correlation coefficient betweenEx and ECx
h
θ
0.9490
General correlation coefficient betweenEθ and ECθTable 5
The variation of the length of the pole.
No.
m
1 (kg)
L
1 (m)
1
0.0149
0.1
2
0.0216
0.149
3
0.0284
0.199
4
0.0378
0.266
5
0.0493
0.354
6
0.0621
0.4436
7
0.0773
0.553
8
0.0966
0.691Genetic algorithm is used to optimize the general correlation coefficients,hx, hθ, and the fitness function is defined as follows:
(48)Dis=∑i=0Ni2×(x2(i)/10+x′2(i)/20+θ2(i)+θ′2(i))N,fitness=1(Dis/10).Through simulated optimization, we can get the corresponding values ofhx, hθ and the fitness shown in Table 6.Table 6
The optimization results ofhx, hθ.
No.
m
1 (kg)
L
1 (m)
h
x
h
θ
Fitness
1
0.0149
0.1
0.985
0.845
0.4310
2
0.0216
0.149
0.975
0.845
0.3310
3
0.0284
0.199
0.96
0.845
0.2713
4
0.0378
0.266
0.955
0.87
0.2240
5
0.0493
0.354
0.63
0.88
0.1838
6
0.0621
0.4436
0.37
0.875
0.1531
7
0.0773
0.553
0.32
0.865
0.1260
8
0.0966
0.691
0.37
0.85
0.1029Figure7 shows how the fitness varies with hx,hθ. And Figures 8 and 9 illustrate how the maximum fitness varies with hθ taking different values when hx, in turn, is equal to some value on standard interval [0,1]. Figure 8 shows how the maximum fitness varies with hx, and Figure 9 shows how hθ varies with hx when the fitness is the maximum.The fitness varies withhx,hθ.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)Figure 8
The maximum fitness varies withhx.Figure 9
h
θ varies with hx when the fitness is the maximum.Similarly, Figures10 and 11 illustrate how the maximum fitness varies with hx taking different values when hθ, in turn, is equal to some value on standard interval [0,1]. Figure 10 shows how the maximum fitness varies with hθ, and Figure 11 shows how hx varies with hθ when the fitness is the maximum.Figure 10
The maximum fitness varies withhθ.Figure 11
h
x varies with hθ when the fitness is the maximum.


## 3.2. Parametric Analysis of Flexible Logic Control Model

Firstly, calculate the correlation coefficient between fitness andhx when hθ, in turn, is equal to some value on standard interval [0,1]. The formula for calculating the correlation coefficient is defined as follows:
(49)r=∑m∑n(Amn-A-)(Bmn-B-)(∑m∑n(Amn-A-)2)(∑m∑n(Bmn-B-)2),
where A and B are m*n matrices, A- and B- are the means of the values in A and B, respectively, and r is the correlation coefficient between A and B.Figure12 illustrates how the correlation coefficient between fitness and hx varies with hθ taking different values on standard interval [0,1].Figure 12
The correlation coefficient between fitness andhx varies with hθ.Similarly, calculate the correlation coefficient between fitness andhθ when hx, in turn, is equal to some value on standard interval [0,1]. And Figure 13 illustrates how the correlation coefficient between fitness and hθ varies with hx taking different values on standard interval [0,1].Figure 13
The correlation coefficient between fitness andhθ varies with hx.After that, the correlation coefficients between the maximum fitness andhx, hθ are computed for poles of different lengths, as shown in Table 7.Table 7
The correlation coefficients between the maximum fitness andhx, hθ.
No.
m
1 (kg)
L
1 (m)
The correlation coefficient between the maximum fitness andhx
The correlation coefficient between the maximum fitness andhθ
1
0.0149
0.1
0.3801
0.9602
2
0.0216
0.149
0.3956
0.9503
3
0.0284
0.199
0.4104
0.9537
4
0.0378
0.266
0.4261
0.9604
5
0.0493
0.354
0.4285
0.9513
6
0.0621
0.4436
0.4294
0.9314
7
0.0773
0.553
0.4308
0.9125
8
0.0966
0.691
0.4327
0.8895By observing and analyzing the previous results, we can draw the following conclusions.(1)
The longer the pole, the worse the control effects. Figures7, 8, and 10 all indicate that the longer the pole, the smaller the fitness with the optimum control parameters, and vice versa. Through 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. That 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 ofhθ. This means that we can realize the effective control only when hθ is in some very short interval. And the interval is relatively fixed. That is, the interval of hθ does not change with the length of the pole. Figures 7, 9 and 10 all show that the fitness is relatively big only when hθ is in the range 0.8 to 0.9, and the interval of hθ remains unchanged for poles of different lengths.Figure12 also shows that the correlation coefficient between fitness and hx is relatively big only when hθ is around 0.2 or in the range 0.8 to 0.9, and the interval of hθ remains unchanged for poles of different lengths.(3)
For some given physical system, the control effect is not sensitive to the value ofhx. That is, we can realize the effective control when hx is in some very long interval. Figure 10 shows that the control effect does not change much with hx taking different values of the long interval.Figure13 also shows that the correlation coefficient between fitness and hθ is relatively big when hx is in the range 0.15 to 0.98.However, the width of the interval varies with the length of pole. The longer the pole, the narrower the interval. The paper calls the interval ofhx as hx  platform.(4)
For poles of different lengths, there is much difference ofhx and little one of hθ when we realize the most effective control. As shown in Table 7, the correlation coefficient between the maximum fitness and hx is small, but the one between the maximum fitness and hθ is big. Table 6 shows that the longer the pole, the smaller the optimum hx, and the optimum hθ is in the range 0.8 to 0.9 with little difference.From the third conclusion, we can know that the width ofhx  platform varies with the length of pole. That is, the longer the pole, the smaller the width. For poles of different lengths, the corresponding hx  platforms are shown in Table 8. And the relation between the width of hx  platform and the length of pole and the one between the starting value of hx  platform and the length of pole can be obtained through fitting method according to the experimental results shown in Table 8.Table 8
The correspondinghx  platforms for poles of different lengths.
No.
L
1 (m)
Ratio of the lengths of poles
The starting value ofhx  platform
The finishing value ofhx  platform
The width ofhx  platform
1
0.1
1
0.175
0.98
0.805
2
0.149
1.49
0.22
0.98
0.76
3
0.199
1.99
0.235
0.98
0.745
4
0.266
2.66
0.25
0.975
0.725
5
0.354
3.54
0.26
0.955
0.695
6
0.4436
4.436
0.27
0.94
0.67
7
0.553
5.53
0.275
0.94
0.665
8
0.691
6.91
0.285
0.94
0.655Firstly, suppose that the smallest effective length of pole, that is, 0.1 m, is 1 unit. Then, the ratios of the other lengths to the smallest effective length of pole are shown in Table8. The relation between the width of hx  platform and the ratio of the length of pole is depicted by means of the linear fit, which is as shown in (50) and Figure 14. Consider
(50)y=0.85314-0.062624x+0.0049723x2.Figure 14
The width ofhx  platform varies with the ratio of the length.Similarly, the relation between the starting value ofhx  platform and the ratio of the length of pole is depicted by means of the linear fit, which is as shown in (51) and Figure 15. Consider
(51)y=0.095132+0.10701x-0.022054x2+0.0015301x3.Figure 15
The starting value ofhx  platform varies with the ratio of the length.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 ofhx  platform by (50) and (51). Then, hx can be any of the interval and hθ should be any of the interval [0.8,0.9]. Therefore, 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.


## 4. Conclusion

Flexible logic method uses universal combinatorial operation model to describe the logic relation betweenE and EC, which are the fuzzy input variables of normal two-dimensional fuzzy controller. Universal combinatorial operation model is not a single fixed operator, but a continuous cluster of combinatorial operators determined by the general correlation coefficienth 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 thegeneral correlation coefficienth of flexible 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 any interval [a,b]. And some important theorems are given and proved, which provide a foundation for the flexible 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. Then, this paper has carried out the parametric analysis of flexible 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.



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