Abstract
Based on the basic physical meaning of error and error variety , this paper analyzes the logical relationship between them and uses Universal Combinatorial Operation Model in Universal Logic to describe it. Accordingly, a flexible logic control method is put forward to realize effective control on multivariable nonlinear system. In order to implement fusion control with artificial neural network, this paper proposes a new neuron model of Zero-level Universal Combinatorial Operation in Universal Logic. And the artificial neural network of flexible logic control model is implemented based on the proposed neuron model. Finally, stability control, anti-interference control of double inverted-pendulum system, and free walking of cart pendulum system on a level track are realized, showing experimentally the feasibility and validity of this method.
1. Introduction
In recent years, fuzzy control has made a rapid development, and it has found a considerable number of successful industrial applications [1–3]. But fuzzy control has two shortcomings in the process of controlling some practical complex systems. One is that the number of control rules increases exponentially with the increase of the number of inputs, and the other one is that the precision of control system is low [4].
To reduce the dimension of control model, hierarchical fuzzy logic control divides the collection of control rules into several collections based on different functions [5, 6]. Compound control combines fuzzy control and other relatively mature control methods to realize the effective control [7], such as Fuzzy-PID Compound Control [8], fuzzy predication control [9], adaptive fuzzy control [10], and so forth. The basic idea of adaptive fuzzy control based on variable universe [11, 12] is to keep the form of rules and varies universe of discourse according to the control error. 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 approximator. However, to date, there has been relatively little research conducted on the internal relations among input variables of fuzzy controllers.
Based on analysis of the logical relationship between the system’s error and error variety , this paper indicates that the relationship is just universal combinatorial relation in Universal Logic [13], and the simple Universal Combinatorial Operation can be used instead of complex fuzzy rule-based reasoning process. As a result, a flexible logic control method is proposed to realize effective control on multivariable nonlinear system.
Artificial neural network is widely used in modelling and controlling thanks to its properties of self-learning, self-organizing, and self-adapting [14]. In order to realize fusion control with artificial neural network, this paper attempts to study the neuron model of Zero-level Universal Combinatorial Operation in Universal Logic and propose a new neural model. Based on this neuron model, the artificial neural network of flexible logic control model is implemented. Finally, stability control, anti-interference control of double inverted-pendulum system, and free walking of cart pendulum system on a level track are realized to prove the feasibility and validity of this method.
The rest of the paper is organized as follows. Section 2 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 in the interval . Section 3 puts forward a new neuron model of Zero-level Universal Combinatorial Operation. Based on this neuron model, the artificial neural network of flexible logic control model is implemented in Section 4. The designed flexible logic control model is applied to treat the double inverted-pendulum system in Section 5. Finally, concluding remarks are given in Section 6.
2. Universal Combinatorial Operation Model
2.1. Universal Combinatorial Operation Model
In order to deal reasonably with the complex relation between factors in complex system, -norm, -norm or Mean operators are taken as Aggregation Operators.
However, -norm result is not bigger than the minimum value, and -norm result is not less than the maximum value. As a result, -norm, or -norm can only handle mutually conflictive relation. In contrast, Mean operators can vary only between the minimum and maximum values based on its “tradeoff” concept, so it can only handle mutually consistent relation [13].
Universal Logic [13], proposed by Professor He et al., is a kind of flexible logic. It considers the continuous change of not only the truth value of propositions, which is called truth value flexibility, but also the relation between propositions, which is called relation flexibility. Based on fuzzy logic, it puts forward two important coefficients: generalized correlation coefficient “” and generalized self-correlation coefficient “”. The flexible change of universal logic operations is based on “” and “”. So Universal Logic provides a new theoretical foundation to realize more effective control for complex systems.
Universal Combinatorial Operation Model is the combinatorial connective of Universal Logic. In this paper, we will only consider generalized correlation coefficient . So Zero-level Universal Combinatorial Operation Model is defined as follows.
Definition 1 (see [13]). Set mapping and
So is Zero-level Universal Combinatorial Operation Model, denoted by , where , , , .
Note 1. Conditional expression means if is true, then ; otherwise . Amplitude limiting function .
Universal Combinatorial Operation Model is a cluster of combinatorial operators, which is determined by general correlation coefficient between propositions. In practical application, according to general correlation between propositions, we can take the corresponding one from the cluster. As generalized correlation coefficient is equal to some special values, the corresponding combinatorial operators are given as follows.(1)When , it means two propositions attract each other to the maximum extent. And is Zadeh combination .(2)When , it means two propositions are independently correlated. And is probability combination .(3)When , it means two propositions reject each other to the maximum extent or restrain each other to the minimum extent. And is bounded combination .(4)When , it means two propositions restrain each other to the maximum extent. And is drastic combination .
2.2. Universal Combinatorial Operation Model in Any Interval
In practical control application, fuzzy domain of fuzzy variables, and , is mostly symmetrical, such as . However, the conventional Universal Combinatorial Operation Model has been limited in the interval . To this end, Chen, based on the basic idea of Universal Logic, sets up Fractal Logic in his doctoral dissertation [15], which can make inference in any interval .
The combinatorial operation model in Fractal Logic is described below.
Definition 2 (see [15]). Set mapping and Then is normal universal Not operation model in any interval , denoted by .
For the above definition, normal universal Not operation model has the following characters.(1)Closure:(2)Two polar law:(3)Symmetric involution:
Definition 3 (see [15]). Set mapping and
Then is Zero-level Universal Combinatorial Operation Model in any interval , denoted by , where , , , , and .
2.3. Demonstration of Attributes of Universal Combinatorial Operation Model in Any Interval
According to the definition of Universal Combinatorial Operation Model in any interval, the following characters [15] are attained.(1) conforms to the combination axiom:(i)Boundary condition : If , then . If , then . If , then . Otherwise, .(ii)Monotonicity : increases monotonously along with and .(iii)Continuity : When , is continuous for all and .(iv)Commutative law : (v)Law of identical element : (2)Closure: (3) Inverse law: (4)Renunciation law
Theorem 4. .
Proof. , according to the closure of normal universal Not operation and universal combinatorial operation: , and and according to the definition of universal combinatorial operation:(1)when Then, according to the definition of : According to the definition of central generalized negation operation: Substituting (14), (15), and (16) separately into (13): And then (2)when So, according to the definition of : Substituting (14), (15), and (16) separately into (20): And then (3)when According to the definition of : Then: From the above, the theorem is true.
Lemma 5. .
Proof. According to Theorem 4 and involution law of normal universal Not operation in any interval , the theorem can be proved simply.
Lemma 6. .
Proof. Setting the interval of as , the lemma can be proved simply.
Lemma 7. If the interval is symmetrical about , then , where and are symmetrical about , namely, , , and are similar, , .
Proof. Since the interval is symmetrical about , then and ; thus
Similarly
From (26) and (27) the following could be obtained:
And then from Lemma 5
So the theorem is true.
Lemma 8. , where .
Lemma 9. If the interval is symmetrical about the original point, then , .
This lemma indicates that, when the interval is symmetrical about the origin and identity element is 0, Universal Combinatorial Operation is also symmetrical about the origin.
As pointed out in the literature [4], the logical relationship between input variables, error and error variety of normal two-dimensional fuzzy controller, is universal combinatorial relation in Universal Logic. Consequently, the complex reasoning process based on fuzzy rule can be replaced by the simple universal combinatorial operation, and a flexible logic control model is presented accordingly. In fuzzy control, the domains of input variables and output variable are generally symmetric to the origin, such as . Obviously, it is the prerequisite of control model that the operation model be symmetric to the origin. Therefore, Lemma 9 provides a basis for the Universal Combinatorial Operation’s application in control.
2.4. Flexible Logic Control Method
Xiao et al. have put forward a concept of fuzzy composed variable to reduce effectively fuzzy control rules in multivariable nonlinear system [16]. According to the characteristics of controlled system and the internal relationship between input variables, the core is to construct a fuzzy composed variable by the fuzzy logic system to synthetically reflect the deviation between reference and the process output.
Four output variables in single inverted pendulum are considered, which are the displacement and speed of cart, and , and the angle and angle speed between pendulum bar and vertical line, and . Four input variables are involved for the fuzzy controller. In the input variables of control system, the angle and angle speed, and , directly reflected the motion of pendulum. Therefore, according to the angle and angle speed of pendulum and the language rules shown in Table 1, a fuzzy composed variable, the error of pendulum, can be defined to synthetically describe the motion of pendulum with the fuzzy logic system. Similarly, a fuzzy composed variable, the error of cart, can be defined according to the displacement and speed of cart, and , so as to synthetically describe the motion of cart. For multivariable system, it need not define, respectively, fuzzy logic system for every fuzzy composed variable. We can use a uniform fuzzy rule table, such as Table 1, and just select different quantization factors to obtain different fuzzy composed variables.
Note 2. In the paper, the fuzzy controller discussed has inputs, such as error and error variety , and output variable is the control signal. The variables, , , and , are crisp values from the practical process. The fuzzy language variables, , , and , are the corresponding fuzzy ones, and the fuzzy domains are unified as with fuzzy subsets, such as negative big (NB), negative middle (NM), negative small (NS), zero (ZE), positive small (PS), positive middle (PM), and positive big (PB).
Obviously, the language rules shown in Table 1 actually reflect the essential relationship between error and error variety . The contents in Table 1 can be approximately divided into four parts, which separately give the language rules used to define composed variables under four conditions, such as error and error variety are both negative, error is negative but error variety is positive, error is positive but error variety is negative, and error and error variety are both positive.
Both error and error variety reflect the deviation between the reference and the output; and then a composed variable can be defined to synthetically describe the control deviation of system, based on the essential relationship between them.
According to the physical meanings of error and error variety , we can get the following conclusions.(1)When error and error variety are both positive, the control deviation of system is positive and tends to further increase positively. So the value of composed variable should not be smaller than their maximum value and the direction is positive. The combination rules are shown by the lower right bold corner of Table 1.(2)When error is positive but error variety is negative, the control deviation of system is positive but tends to decrease. So the value of composed variable should be between and in this case. The combination rules are shown by the lower left bold corner of Table 1.(3)Similarly, the value of composed variable under two other conditions can also be obtained.
Through above analysis, it is easy to know that the essential relationship among error , error variety and the composed variable is just a kind of universal combinatorial one in Universal Logic. As a result, we have
Obviously, now the identity element is zero.
Moreover, according to the concept of negative feedback control, there is only a difference of a single sign between composed variable describing control deviation and the output variable of controller, namely;
Therefore, the relationship among error , error variety , and the output variable is where , , and . The control method is called Flexible Logic Control Method [4].
At the same time, a weighted factor is introduced to meet the requirements of different controlled objects. By adjusting the value of , it is possible to change the weighting degrees for error and error variety . When the general correlation coefficient is 0.5, there is
And the formula (33) is just the fuzzy control method proposed by Long and Wang [17]. He used a linear equation, such as (33), to describe fuzzy control rules. But the relationship among , , and is not only linear. So (32) is a cluster of operators determined by general correlation coefficient , and (33) is only a special operator in the cluster as is equal to 0.5. As a result, flexible logic control method can realize the effective control for complex system.
3. Neuron Model of Zero-Level Universal Combinatorial Operation
The uniform neuron model of generalized logic operators in any intervals [] is established in the literature [18]. Seven neuron models of logic operation are given separately, such as Not, And, Or, Implication, Equation, Average, and Combination.
But the neuron model of combination logic operation is too complicated for practical applications.
In this section, a new neuron model of zero-level universal combinatorial operation is constructed based on Lemma 6 in Section 2.3.
It is learnt from Lemma 6 in Section 2.3 that
Therefore, the model of zero-level universal combinatorial operation can be represented by the neuron shown in Figure 1.
The neuron model is composed of several subneurons which are interconnected. In the model, there are three input parameters: , , and . And the output , net denotes the weighted sum of all inputs, and all unmarked weights are 1. The transfer functions of the subneurons marked , , are with the parameter and . And the transfer function of the sub-neuron marked is . The sub-neurons marked , , , , and have the following transfer functions, respectively:
We will discuss if the neuron model shown in Figure 1 has realized zero-level universal combinatorial operation.
(1) . According to the definition of transfer functions , , and , it is easy to find that the outputs of the sub-neurons marked , , separately are , , and . According to the definition of transfer function , it is found that the outputs of the sub-neuron marked , , separately are , , and . Therefore, the output of the sub-neuron marked is . According to the definition of transfer function , the output of the sub-neuron marked is . According to the definition of transfer function , the output of the sub-neuron marked is . When , the output of the neuron is
(2) . It is similarly learnt that the output of the sub-neuron marked is , the output of the sub-neuron marked is , and the output of the sub-neuron marked is . When , the output of the neuron is
(3) . It is similarly learnt that the output of the sub-neuron marked is , the output of the sub-neuron marked is , and the output of the sub-neuron marked is . When , the output of neuron is .
In summary, the neuron model shown in Figure 1 has fully realized the model of zero-level universal combinatorial operation.
In practical control application, the fuzzy domains of system variables are generally expressed as , for example, . To simplify the neuron model, we can unify the fuzzy domains of system variables as . Obviously, the identity element is 0. Therefore, the model of zero-level universal combinatorial operation in the interval can be obtained by the definition of universal combinatorial operation model in any intervals in Section 2.2,
Therefore, it is easy to obtain the neuron model of zero-level universal combinatorial operation in the interval according to the neuron model given in Figure 1, as shown in Figure 2.
The neuron model is composed of interconnected multiple sub-neurons, and its internal structure is a small artificial neural network. In the model, and are the inputs, is the output, net1, net2, net3, and net4 are the weighted sum of all inputs, the values of connecting weights without specific marking are 1, and are marks of some sub-neurons for convenient discussion. The sub-neurons marked and have the transfer function with the parameter m. And the sub-neuron marked also has the transfer function with the parameter , where . The sub-neuron marked also has the transfer function with the parameter , and the sub-neurons marked , , , and have the following transfer functions, respectively:
4. Realization of Artificial Neural Network of Flexible Logic Control Model
In the practical complex system, the control objects are always not less than one, so multiple system variables are selected for feedback control. The basic idea of flexible logic control model presented in Section 2.4 is to design sub-goal controller based on the control object, and each subgoal controller is designed with the flexible logic control method. Then the output variable of control system is the weighted sum of the outputs of these sub-goal controllers.
Obviously, most of the operators involved in this model are universal combinatorial operation. Therefore, it is easy to implement the artificial neural network of the flexible logic control model by using the neuron model of zero-level universal combinatorial operation in Section 3. In this section, the artificial neural network structure of flexible logic control model is presented using the double inverted-pendulum system as the control object.
The objective is to maintain the rods in an upright position and the cart in an appointed position in the rail. There are six output variables and one input variable in double inverted-pendulum, which are , , , , , , and . The variables, , , , , , , and , are the corresponding fuzzy ones, and the fuzzy domains are unified as .
We can design three subcontrollers with the flexible logic control method. One is to maintain the cart in an appointed position with two input variables, and . The other ones are to maintain, respectively, the rods in an upright position with two input variables and . And we lead into weighted factors, such as , , and . The three sub-controllers are designed as follows:
When the control signal is combined, we lead into three weighted factors, such as , , and , for the three sub-controllers. According to (40) and (42), we can get the output of the controller as follows: where , , , , and , is (38), , , and are general correlation coefficients, , , and , , , and , , , and , , , and .
By the above analysis, we can obtain the control model of double inverted-pendulum, as shown in Figure 3. Based on the neuron model of zero-level universal combinatorial operation, the artificial neural network structure of the control model could be obtained, as shown in Figure 4.
The displacement and speed of cart, and , the angle and angle speed between the lower pendulum bar and vertical line, and , and the angle and angle speed between the upper pendulum bar and vertical line, and , are the input variables of the artificial neural network. The output variable is the control signal. , ,, are marks of some sub-neurons for convenient discussion. Net is the weighted sum of these inputs of the corresponding sub-neurons. The connection weight values of the unmarked sub-neurons are 1. The sub-neurons marked , , and are the neuron model of zero-level universal combinatorial operation in the interval , as given in Section 3. The other sub-neurons have the following transfer functions, respectively: where the transfer function has the parameters such as and , and the transfer functions of the sub-neurons marked , , and separately have the parameters , , , , , and .
From the Figure 4, we can see that the artificial neural network structure of the control model is a feed forward neural network. Genetic algorithms (GAs) are a robust and efficient optimization technique based on the mechanism of natural selection and natural genetics [19]. One of the important features of GAs is that they are a population-based search technique. Instead of moving from one single point to another like traditional mathematical programming techniques, GAs always maintain and manipulate a solution set. Therefore, GAs are used to train the artificial neural network.
5. Results of Experiments
This section takes a double inverted-pendulum physical system, for example, to show the feasibility and validity of the flexible logic control method based on artificial neural network.
Owing to the rapidity and the absolute instability of the inverted-pendulum system, it requires a high real-time processing frequency. Therefore, the sampling interval of this system is set as 5 ms. When the system is running for 20 s, the control effect comparison could be conducted. The physical parameters of the system are given in Table 2.
The flexible logic control method based on artificial neural network is applied into the above double inverted-pendulum physical system. The network parameters are optimized by genetic algorithms. The definition of the fitness function is shown as (46) and the control parameters of the system are shown in Table 3. Consider
Three experiments as the stability control, anti-interference control, and the free movement in a level track of cart pendulum system for the above double inverted-pendulum physical system are carried out by using the controller made of these control parameters.
5.1. Stability Control Experiment
Under the same initial state such as m, rad and rad, the stability control of the double inverted-pendulum physical system has been realized. The experimental results are shown in Figures 5, 6, 7, and 8 (running time: 20 s).
(a) The cart displacement curve
(b) The cart speed curve
(a) The lower rod angle curve
(b) The lower rod angular velocity curve
(a) The upper rod angle curve
(b) The upper rod angular velocity curve
From the experimental results, we can find when the system is in a stable state, the deviation of the cart displacement could be about 0.01 m with a good stability control effect.
5.2. Anti-Interference Control Experiment
After knocking the upper pendulum twice when the system reaches a steady state, it shows strong anti-interference ability. The experimental results are shown in Figures 9, 10, 11, and 12.
(a) The cart displacement curve
(b) The cart speed curve
(a) The lower rod angle curve
(b) The lower rod angular velocity curve
(a) The upper rod angle curve
(b) The upper rod angular velocity curve
From the experimental results, we can find the system becomes stable again after 1.9 s.
5.3. Free Movement Control Experiment of Cart Pendulum System
When the system reaches a steady state, the target position is changed on-line, such as from 0 m to −0.2 m. So the free movement control of cart pendulum on a level track could be realized. The experimental results are shown in Figures 13, 14, 15, and 16.
(a) The cart displacement curve
(b) The cart speed curve
(a) The lower rod angle curve
(b) The lower rod angular velocity curve
(a) The upper rod angle curve
(b) The upper rod angular velocity curve
From the experimental results, we can see the cart moves to the new target location after 2.7 s and the system remains stable.
5.4. Self-Adaptive Experiment of Control Model
Some physical parameters have been changed in another double inverted-pendulum physical system. The physical parameters of the system are shown in Table 4. The control model also can realize the stability control of the new double inverted-pendulum after changing some parameters. The experimental results are shown in Figures 17, 18, 19, and 20. The experimental results show that the control model based on artificial neural network has excellent self-adaptability and portability.
(a) The cart displacement curve
(b) The cart speed curve
(a) The lower rod angle curve
(b) The lower rod angular velocity curve
(a) The upper rod angle curve
(b) The upper rod angular velocity curve
5.5. Comparison of the Experimental Results
Cheng et al. [20] put forward a parameter fuzzy control method in 1996. The core idea of this method is to find the synthetical relationship among state variables by using modern control theory and form composed error and composed variety of error so as to construct a fuzzy controller.
The parameter fuzzy control method is used to stabilize the above double inverted-pendulum physical system. The control parameters of this system are shown in Table 5.
Real-time control is carried out in the above double inverted-pendulum physical system. When the initial state of the system is m, rad, and rad, the running time is 20 s. The comparison of the experimental results for two control methods is shown in Table 6.
From Table 6, the adjusting time and the steady-state errors of the displacement and the angle are compared with parameter fuzzy control method about the stability control in double inverted-pendulum physical system.
In summary, the flexible logic control model based on artificial neural networks has good control effects. And the controlling system has good stability and anti-interference ability.
6. Conclusion
Based on the basic physical meaning of error and error variety , this paper analyzes the logical relationship between them and uses universal combinational operation model to describe it. And a flexible logic control method is put forward to realize effective control on multivariable nonlinear system.
In order to implement the fusion control of Universal Logic and artificial neural networks, this paper puts forward a new neuron model of zero-level universal combinational operation. Based on this neuron model, artificial neural network is designed for flexible logic control model. Meanwhile, the stability control of the double inverted-pendulum physical system, anti-interference control, and free movement of the cart pendulum have been realized in a level track by using the proposed control model.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work is supported by the Beijing Municipal Natural Science Foundation of China (nos. 4113069 and 4122007) and the Beijing Municipal Education Commission Foundation (no. 007000546311501).