An Intelligent Coupling 3-Grade Fuzzy Comprehensive Evaluation Approach With AHP for Selection of Levitation Controller of Maglev Trains

During recent years, maglev transportation has made great progress, and as a result, many intelligent levitation control algorithms have emerged. However, enterprises often find it difficult to make a choice when faced with the selection of a controller. The main reason is that the performance evaluation of control algorithms is a complex, multiple-criteria, multifactor coupling problem that cannot be represented by a precise mathematic model. In this paper, a novel artificial intelligent evaluation method for the selection of a levitation controller is developed based on a 3-grade fuzzy method and analytic hierarchy process (AHP). Three kinds of intelligent levitation control algorithms are applied to a full-size test maglev train to collect experimental results with real data. The proposed artificial intelligence method to develop a 3-grade fuzzy multicriteria approach is used to select the best levitation controller for the maglev train. This method can then provide information consultation services to maglev train firms. To the best of our knowledge, for maglev trains, this is the first intelligent evaluation approach with real experimental data. The proposed method can also be applied to other information consultation and decision making systems with appropriate modifications.


I. INTRODUCTION
With the rapid improvement of the worldwide economic situation and, in particular, urbanization, urban traffic has many difficult problems Examples include traffic accidents, and more so, latterly, exhaust pollution. Although the use of subways can minimize these problems, the noise and, in particular, subway vibration not only affects passengers but, depending on foundation quality, can also affect the state of surrounding buildings and consequently their residents. In such circumstances, an environmentally friendly, comfortable, safe and intelligent transportation method is urgently The associate editor coordinating the review of this manuscript and approving it for publication was Mengchu Zhou . needed. The maglev train, as shown in Fig. 1, is a new type of urban transportation method [1]- [3]. It can travel faster than 500 km/h and has such advantages as riding comfort, safety, low maintenance relative to other transportation methods and also contributes to environmental protection. In the light of the above, maglev transportation is further developing and spreading vigorously worldwide [4]. The levitation control system, which determines the performance of a maglev train, is the core element. The characteristics of this system include such as strong nonlinearity, open loop instability, time-varying parameters and external disturbances, all of which challenge the control design. Currently, the traditional control algorithm is classic linear control, such as the PID controller. Increasingly, VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ alongside, the development of artificial intelligence technology many intelligent control algorithms have been proposed. An active levitation controller with a virtual energy harvester designed by Li et al. [5] is used to suppress vehicle-guideway coupling vibration. Sun et al. [6] proposed an adaptive sliding mode control of the maglev system, based on the radial basis function (RBF) neural network and an adaptive learning law for network weights, which can approximate unknown parameters effectively. Zhou et al. [7] designed an active control method with a finite impulse response (FIR) filter for a magnetic levitation system, which can suppress the vibration caused by the track irregularity. Qian and Fan [8] utilized RBF neural network to approximate the uncertainties, which can solve the load frequency control problem for the renewable power system. Fuzzy control approaches had, already, proved to be successful in various applications [9]- [15]. Sun et al. [9] established a Takagi-Sugeno (T-S) fuzzy model of a maglev vehicle-guideway with global nonlinearity. This greatly greatly facilitates stability analysis and control law design. Ding et al. [10] proposed a novel method to analyze the stability of the hybrid systems with fuzziness. The effectiveness of the method presented in this Paper is verified by the successful application to a differential-drive two-wheeled mobile robot. Precup et al. [11] presented a fuzzy logic control algorithm to stabilize the Rössler chaotic dynamic system with sufficient satisfactory simulation results. Sun et al. [12] designed a fuzzy PID controller to address vehiclerail coupling vibration problems. Radgolchin and Moeenfard [13] developed an adaptive supervised multi-level fuzzy controller to control the deflection of an electrostatically actuated microplate. The simulation results show that the proposed controller can effectively stabilize the microplate beyond the pull-in instability limit. Sun et al. [15] proposed a nonlinear robust control law with an adaptive fuzzy logic approximator. Although these levitation control algorithms have their own advantages and disadvantages, there is still no method for evaluating them, despite the fact that many maglev train companies want to choose new intelligent control algorithms, but in the face of the variety of control algorithms, it is difficult to determine which one is the most suitable. Currently there is an urgent need to comprehensively evaluate control effects, based on artificial intelligence algorithms and to hence be able to advise enterprises regarding suitable decisions. The purpose of a levitation intelligent control algorithm evaluation system is the selection and evaluation of the control scheme using artificial intelligence (AI) algorithms. In expert systems and AI, many rules and criteria cannot be accurately described [16]- [18], hence, fuzzy mathematic methods are used to do so. If the systems contain a relatively complicated and large amount of knowledge and associated experience, the results obtained by the fuzzy methods are more realistic. The evaluation of levitation control is a typical example. The values of the evaluation factors, which cannot be accurately described by mathematics, are based on multiple-criteria decision making. In recent years, AI analysis methods including, such as Bayesian networks, accident trees, fuzzy comprehensive evaluation, fatality analysis, gray theory and other evaluation methods, have been widely applied to consultations and decision-making and to find information,. Yao et al. [19] proposed a constrained parameter evolutionary learning (CPEL) algorithm for Bayesian network parameter learning to analyze the decision-making related to UAV autonomous missions. Lakehal and Harouz [20] presented a novel method, based on a fault tree and a BN to enable simpler information processing. The method has been successfully applied to turbo compressor analysis. Dong et al. [21] utilized a 2-tuple fuzzy linguistic approach in the analytic hierarchy process to improve the selection of the individual numerical scale and prioritization. Wang et al. [22] proposed a fuzzy case-based reasoning method based on a design thinking process and extracted the key form features by utilizing a fuzzy analytic hierarchy process. Abdel-Basset et al. [23] proposed a new method based on a neutrosophic analytical hierarchy process to evaluate risks in the supply chain. Mouronte-López et al. [24] utilized an analytic hierarchy process, neural networks, and software agents to improve the spare parts management process in a telecommunications' operation. Xu and Xu [25] developed a new method for the probability-hesitant analytic hierarchy process. Han et al. [26] proposed a fuzzy comprehensive evaluation method, which is effective and applicable for power grid enterprises in the assessment of the efficiency of a power plant program, The inconsistent elements can be rapidly and accurately detected with the proposed method.
Thus, it appears that the factors that affect controllers are numerous and coupled. However, traditional methods, such as the equal weight method, statistical experiment method, variable weight method and set-valued statistical iteration method, often yield little differences in those evaluation values, which cause decision-making difficulties, or require a deep understanding of the problems in applied mathematics. Thus, to evaluate and compare the performance of different levitation controllers in a one-dimensional space, it is important to scientifically and objectively synthesize a multi index problem into a single index form and subsequently be used by maglev trains companies to select new intelligent control algorithms.
The evaluation and analysis methods for maglev traffic have not yet been reported, however, the application of these intelligent analysis methods, in other aspects, provides a suggested method to evaluate the controllers. However, the evaluation is complicated, in that it involves such as control accuracy, dynamic performance, anti-disturbance ability, response speed. For example, some algorithms have quick response speed, but are sensitive to disturbances, causing overshoot. Additionally, there is an analysis deficiency if only a single mathematical analysis method is used. When fault tree analysis alone, (FTA) is used, it is difficult to determine the importance of each basic event from the top event, in the cases, in which the probability of each, cannot be accurately counted. When the analytic hierarchy process (AHP) alone, is used, there is some subjectivity, because the judgment matrix is constructed by an experts' understanding of the whole system. In addition, only when the random consistency index CR<0.1 is the consistency of the judgment matrix considered to meet the requirement. Otherwise, it must be readjusted. Therefore, to analyze and evaluate the levitation control algorithm, which can provide consulting services for enterprises to select the control algorithms, a new 3-grade fuzzy comprehensive evaluation approach with AHP method is proposed. The main contributions are summarized as follows: 1. In comparison with traditional methods, the designed 3-grade fuzzy comprehensive evaluation approach, with AHP, is able to provide a more comprehensive and effective evaluation.
2. The proposed method, to evaluate the performance of different levitation controllers in one dimension space, is able to synthesize a multi index problem into a single index form.
3. As far as we know, this is the first intelligent evaluation approach for maglev trains using real experimental data.
The rest of this paper is organized as follows: in Section 2, the preliminary knowledge is given. In Section 3, three intelligent levitation control algorithms are introduced. In Sec. 4, intelligent comprehensive levitation control algorithm evaluations are given. The paper concludes with conclusions and the future outlook.

II. PRELIMINARY KNOWLEDGE A. ANALYTIC HIERARCHY PROCESS (AHP)
The analytic hierarchical process takes the target of the research as a problem that requires systematic analysis, and decomposes the complex problems associated with the target layer by layer [23]- [27]. The factors in the same layer are compared, discriminated and calculated.
The numerical scale of AHP consists of 17 values and can be described as follows: 9) corresponds to the ith grade of the AHP linguistic scale. By choosing different values for f i (i = 1, 2, . . . , 9), different numerical scales can be obtained. Let A = a ij n×n , where a ij > 0 and a ij × a ji = 1, be a reciprocal numerical pairwise comparison matrix. The priority vector can be derived from A. The AHP linguistic scale has nine gradations [23]- [27], which are listed in Table 1 as follows.
The additive normalization method is a prioritization method; it can be expressed as follows: The principal eigenvector of A as the desired priority vector W can be obtained by solving the linear system where, λ is the principal eigenvalue of matrix A. λ max is the maximum eigenvalue of the judgment matrix. The consistency criterion of the judgment matrix can be written as follows: R is the average random consistency index value and η R can be calculated as follows: where, η R < 0.1 and the pairwise comparison matrix is generally considered to have complete consistency; otherwise, the matrix needs to be readjusted until it has satisfactory consistency.

B. COMBINATION WITH FUZZY COMPREHENSIVE EVAIUATION
The method combined with fuzzy comprehensive evaluation approach [21], [22], [28] can be described as follow.
2) Construct a priority vector according to the improvement AHP. The priority vector corresponding to U can be written as follows: The priority vector corresponding to U i can be obtained as follows: 3) To determine the rating level as V = (v 1 , v 2 , · · · , v n ), we use the comment set to rank items into 3 levels (i.e., n = 3), excellent, average, and poor.
4) The single factor fuzzy evaluation is carried out and the single factor evaluation matrix can be obtained as follows.
5) Let the fuzzy synthetic decision model be (U, V, R), the priority vector beW, and the corresponding comprehensive evaluation can be B =W • R. The multi-grade fuzzy evaluation model can be designed, in accordance with the complexity of the model. At present, most are two-grade fuzzy comprehensive evaluation models.

III. INTELLIGENT LEVITATION CONTROL ALGORITHMS
Firstly, it is necessary to determine which controllers will participate in the evaluation. The fuzzy PID controller [11], the adaptive neural-fuzzy sliding mode controller (ANF-SMC) [15], and the RBF neural network sliding mode controller with the minimum parameter learning method [6] were subsequently chosen as examples. For the design of the fuzzy, neuro-fuzzy and sliding mode controllers, refer to [6], [12] and [15]. The three kinds of levitation controllers are to be comprehensively evaluated and compared based on the proposed intelligent evaluation approach.

1) FUZZY PID CONTROL ALGORITHM
The fuzzy PID controller can be expressed as follows: The meaning of the symbols is found in [12]. The control schematic diagram of the fuzzy PID is illustrated in Fig. 2.
The maglev system fuzzy control rules tables are listed in Tables. 2-4. More detailed information about the fuzzy PID controller is given [12].

2) ADAPTIVE NEURAL-FUZZY SLIDING MODE CONTROL ALGORITHM
The adaptive neural-fuzzy sliding mode controller (ANF-SMC) is described below.
The meanings of the symbols are given in [15]. The structure of the neural network fuzzy system used in ANFSMC is shown in Fig. 3.
The detail of how to train, test and validate the neural network and the architecture is given in [15].

3) RBF NEURAL NETWORK SLIDING MODE CONTROL ALGORITHM
The RBF sliding mode controller [30][31] is described as follows: The meaning of these symbols and RBF neural network details can be found in [6].
These control algorithms are programmed and tested on a full-size maglev test vehicle, as shown in Fig. 4, and experimental data can be collected for later analysis and evaluation.

IV. INTELLIGENT COMPREHENSIVE EVALUATION OF THE LEVITATION CONTROL ALGORITHMS
To evaluate the levitation controller, is a complicated system engineering problem. The factors that affect controllers are numerous and complicated. The influence of each factor is correspondingly different, and there are particular relationships between them. The boundary between a good and bad performance is also quite vague, and hence difficult to describe by using classical mathematics. Fuzzy mathematic is a better choice when solving such complex large-scale problems. To comprehensively evaluate more factors and thereby overcome the difficulty of the weight distribution caused by the interlinkage of factors, to evaluate the controllers, an intelligent comprehensive evaluation approach, based on a 3-grade fuzzy method and AHP is proposed. The main steps of the proposed algorithm are given in Tab. 5.

A. SELECTION OF THE EVALUATION INDEX AND EVALUATION SET
Firstly, the selection of an evaluation index and evaluation set was made. The control performance is evaluated in terms of the carrying capacity and anti-disturbance capacity. The carrying capacity involves no load, full load and overload. The anti-disturbance capability includes high-frequency square wave input, low-frequency square wave input, highfrequency harmonic input and low-frequency harmonic input. The evaluation indices include two first-grade indexes, seven second-grade indexes and twenty third-grade indexes, as shown in Fig. 5. The first-grade indexes are the carrying capacity and anti-disturbance capacity. The second-grade indexes are the no-load, full-load, overload, high-frequency square wave input, low-frequency square wave input, high-frequency harmonic input and low-frequency harmonic input. The third-grade indexes are different indicators of the control performance. The evaluation set of a maglev train can be denoted as U = {u 1 , u 2 , · · · , u 20 }.
During the experiment, the different disturbance signals are artificially added at the sensor inputs of. The high-frequency square wave input signal refers to a square wave signal with a period of 0.5 s and amplitude of 1.5 mm. The low-frequency square wave input signal is a square wave signal with a period VOLUME 8, 2020   Fig. 6.

B. EVALUATION INDEX VALUES OF REAL DATA IN EXPERIMENTS
After selecting the evaluation index, experimental data for candidate controllers should be provided. Based on the data collected in the experiments, a trusted database is built according to the Apriori algorithm [29]. The relevant data are analyzed and extracted to form the required evaluation index values. The evaluation index values of the three controllers are listed in Tables 6-7.

C. DETERMINATION OF INTELLIGENT PRIORITY
The intelligent priority should be determined following the steps below: First, the hierarchical structure of various control indices of the maglev train is established as shown in Fig. 5.
Second, the pairwise comparison matrix of the levitation system for the maglev train is constructed. According to the pairwise comparison of factors of the same grade with respect to the importance of a factor of the previous grade, the pairwise comparison matrix is obtained as follows.
1) The importance pairwise comparison matrix in the criterion layer 1 relative to the target layer.
Based on Table 1 and expert experience, the pairwise comparison matrix of U 1 and U 2 to U is A = 1 4 1/4 1 .
2) The importance pairwise comparison matrix of criterion layer 2 relative to criterion layer 1.
The pairwise comparison of U 11 , U 12 , and U 13 to U 1 is The pairwise comparison matrix of U 21 , U 22 , U 23 and U 24 to U 2 is 3) The importance pairwise comparison matrix of scheme layers relative to criterion layer 2.
The pairwise comparison matrixes of U 111 , U 112 , U 113 and U 114 to U 11 , and U 121 , U 122 , U 123 and U 124 to U 12 , and U 131 , U 132 , U 133 and U 134 to U 13 are: Similarly, the pairwise comparison matrixes of U 211 and U 212 to U 11 , U 221 and U 222 to U 22 , U 231 and U 232 to U 23 , U 241 and U 242 to U 24 are: Third, the weight of each layer is calculated, and a consistency check of the weights is conducted. Because the same calculated method is used, only matrix A 2 is developed in detail here.
The consistency indicator is calculated as follows: The random consistency ratio is as below.
It is learned that η R < 0.1, so the priority vector of U 21 , U 22 , U 23

D. FUZZY COMPREHENSIVE EVALUATION OF LEVITATION CONTROL ALGORITHMS
The fuzzy comprehensive evaluation approach can be implemented as follows: First, the membership function between the 3-grade index and the evaluation set is determined. To express the fuzzy mapping of the factors set to the evaluations set, the trapezoidal distribution is used as the membership function of ''excellent'', ''average'' and ''poor''.
The membership function of the ''excellent'' controller can be expressed as follows:   The membership function of the ''average'' controller can be obtained as follows: The membership function of the ''poor'' controller is: where, ζ denotes the control performance index value of the maglev levitation control system. γ 1 , γ 2 , γ 3 and γ 4 represent the membership function reference points. The membership function is described in Fig. 7.
The reference point values of the levitation control system are reported in Table 9.
Second, the original data are standardized, and a single factor evaluation of the third-grade index is conducted to obtain a single factor evaluation matrix.
Let the fuzzy comprehensive decision model be (U, V, R) and the priority vector beW. The corresponding comprehensive evaluation is B =W • R, whereW = a 1 a 2 · · · a n T and R = (r ij ) n×m (i.e., comprehensive judgment). The principal factor determinant mode b j = n ∨ i=1 (a i · r ij ) (j = 1, 2, · · · , m) is utilized to obtain the comprehensive evaluation matrix as The results of the third-grade comprehensive evaluation are listed as follows: From the principle of maximum membership, it is concluded that the fuzzy PID is a ''poor'' controller, and the membership of ''poor'' is 0.287.
Additionally, the evaluation results for the ANFSMC and RBF neural network sliding mode controllers can be obtained in the same manner and are as follows: The results suggest that the ANFSMC is the best controller among the three controllers. This is consistent with the long-term experimental results from a national maglev transportation engineering R&D center.
Utilizing the classic AHP method, we can also obtain the weight vector of the three controllers for the total target is [0.214, 0.425, 0.361]. The greater value in the weight vector indicates better control effect. However, the results show that the difference between the evaluations is very small. Besides, once a new control algorithm is introduced for evaluation, the results of all control algorithms need to be recalculated. The proposed method in this work not only produces more comprehensive evaluation, but also only a single calculation is needed when a new control algorithm is introduced for evaluation. Therefore, the new method also can save a lot of computation time.

V. CONCLUSIONS
Proposed In the study, presented in this paper, is the provision of information consultation services for maglev train companies, and an intelligent comprehensive evaluation method for the selection of levitation controllers. To the best of our knowledge, the proposed method, herein, is the first artificial intelligence evaluation method enabling the selection of maglev levitation controllers capable of utilizing a 3-grade fuzzy multicriteria approach. The experimental results of three kinds of levitation controllers are provided for comprehensive evaluation, based on the proposed intelligent coupling 3-grade fuzzy comprehensive evaluation approach with AHP. The results show that the membership of ''good'' for the fuzzy PID, is 0.051. The membership of ''good'' for the RBF neural network sliding mode controller, is 0.056 and that for the ANFSMC, is 0.287. The ANFSMC is the best controller among the proposed three controllers. The evaluation results are consistent with the long-term experimental results. It should be noted that this result only relates to the evaluation of three controllers and under specific control parameters. Thus it is possible that different controller parameter values may produce different results. Additionally, the different importance levels selection (the pairwise comparison control indices matrix) also obtained different evaluation results. This can be determined by the companies according to the market rating. Focus, next, must be on the rules related to the selection of a numerical scale and the selection of a prioritization method for the proposed method, such that it can be extended to other systems for convenient evaluation and consultation.