Adaptive Fuzzy-Neural-Network Control for Maglev Transportation System

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Adaptive Fuzzy-Neural-Network Control for Maglev Transportation System
Rong-Jong Wai, Senior Member, IEEE, and Jeng-Dao Lee
Abstract—A magnetic-levitation (maglev) transportation system including levitation and propulsion control is a subject of considerable scienti?c interest because of highly nonlinear and unstable behaviors. In this paper, the dynamic model of a maglev transportation system including levitated electromagnets and a propulsive linear induction motor (LIM) based on the concepts of mechanical geometry and motion dynamics is developed ?rst. Then, a model-based sliding-mode control (SMC) strategy is introduced. In order to alleviate chattering phenomena caused by the inappropriate selection of uncertainty bound, a simple bound estimation algorithm is embedded in the SMC strategy to form an adaptive sliding-mode control (ASMC) scheme. However, this estimation algorithm is always a positive value so that tracking errors introduced by any uncertainty will cause the estimated bound increase even to in?nity with time. Therefore, it further designs an adaptive fuzzy-neural-network control (AFNNC) scheme by imitating the SMC strategy for the maglev transportation system. In the model-free AFNNC, online learning algorithms are designed to cope with the problem of chattering phenomena caused by the sign action in SMC design, and to ensure the stability of the controlled system without the requirement of auxiliary compensated controllers despite the existence of uncertainties. The outputs of the AFNNC scheme can be directly supplied to the electromagnets and LIM without complicated control transformations for relaxing strict constrains in conventional model-based control methodologies. The effectiveness of the proposed control schemes for the maglev transportation system is veri?ed by numerical simulations, and the superiority of the AFNNC scheme is indicated in comparison with the SMC and ASMC strategies. Index Terms—Fuzzy neural network (FNN), linear induction motor (LIM), maglev transportation system, magnetic levitation (maglev), sliding-mode control (SMC).
I. INTRODUCTION LIDING-MODE CONTROL (SMC) is one of the effective nonlinear robust control approaches since it provides fast system dynamic responses with an invariance property to uncertainties once the system dynamics are controlled in the sliding mode [1]. In the past decade, the SMC system has been widely used in various practical applications [2], [3]. He and Luo [2] introduced a new sliding-mode approach for the control of direct current (dc–dc) converters. Metin et al. [3] improved overall ef?ciency in series hybrid-electric vehicles by restricting the operation of the engine to the optimal ef?ciency region via
S
Manuscript received June 2, 2006; revised December 7, 2006 and April 19, 2007; accepted April 30, 2007. This work was supported by the National Science Council of Taiwan, R.O.C. under Grant NSC 95-2221-E-155-085. R.-J. Wai is with the Department of Electrical Engineering, Yuan Ze University, Chung Li 32003, Taiwan, R.O.C. (e-mail: rjwai@https://www.sodocs.net/doc/5c1833435.html,.tw). J.-D. Lee was with the Department of Electrical Engineering, Yuan Ze University, Chung Li 32003, Taiwan, R.O.C. He is now with the Army, Ministry of National Defense, Taiwan, R.O.C. (e-mail: jengdaolee@https://www.sodocs.net/doc/5c1833435.html,). Digital Object Identi?er 10.1109/TNN.2007.900814
a control strategy based on two chattering-free sliding-mode controllers. Unfortunately, the information of system uncertainties is usually required in conventional SMC strategies. In previous research literature [4], [5], various adaptive estimation mechanisms for uncertainty information have been embedded in SMC to be capable of keeping the robustness properties with respect to system uncertainties. Li et al. [4] proposed an adaptive sliding-mode ?ux observer for the sensorless speed control of induction motors. Wai and Chang [5] implemented an adaptive SMC system to control a dual-axis inverted-pendulum mechanism that is driven by permanent magnet synchronous motors. Though these control strategies had good control performances and were insensitive to uncertainties, adaptive estimation algorithms for system uncertainties are always monotonous functions, and tracking errors introduced by any uncertainty, such as sensor error or accumulation of numerical error, will cause the estimated values to increase even to in?nity with time. It results that actuators will eventually be saturated and the controlled system may be unstable. In recent years, many magnetic levitation (maglev) transportation systems have been constructed, tested, and improved. In general, a maglev transportation system can be divided into propulsion and levitation mechanisms. Linear induction motors (LIMs) have many excellent performance features, such as high-starting thrust force, alleviation of gear between motor and the motion devices, reduction of mechanical losses and the size of motion devices, high-speed operation, silence, and so on [6]. Owing to these advantages, the LIM has been used widely in the ?eld of industrial processes and transportation applications [7]. Because the LIM can be visualized to unroll a rotary induction motor (RIM), the driving principles of the LIM are similar to the RIM and the dynamic model of the LIM can be modi?ed from the one of the RIM. Thus, many decoupled control techniques in RIM, such as ?eld-orientated control [8] and nonlinear state feedback techniques [9], can be adopted to decouple the dynamics of the thrust and the ?ux amplitude of the LIM. In this paper, the LIM is used as the propulsive mechanism, and a simple decoupled dynamic of the LIM with the possible occurrence of uncertainties is considered in the entire dynamic model of the maglev transportation system. Today, maglev techniques have been manipulated for eliminating friction due to mechanical contact, decreasing maintaining cost, and achieving high-precision positioning. Therefore, they have been widely used in the maglev transportation systems [10]–[13]. In general, maglev techniques can be classi?ed into two categories: electrodynamic suspension (EDS) and electromagnetic suspension (EMS). EDS systems are commonly known as “repulsive levitation,” and the corresponding
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levitation sources are from superconductivity magnets [12] or permanent magnets [13]. However, the repulsive magnetic poles of superconductivity magnets cannot be activated at low speed so that they are only suitable for long-distance and high-speed train systems. Basically, the maglev force of EDS is partially stable and allows a large clearance. Nevertheless, the production process of magnetic materials is more complex and expensive. On the other hand, EMS systems are commonly known as “attractive levitation,” and the maglev force is inherently unstable so that the control problem becomes more dif?cult. Generally speaking, the manufacturing process and cost of EMS are lower than that of EDS, but extra electric power is required to maintain levitation height. In this paper, the EMS strategy is utilized for the fundamental levitation force of a maglev transportation system, and the corresponding levitated positioning and stabilizing control of the maglev system is one of the major control objectives to be manipulated. Because the EMS system has unstable and nonlinear behaviors, it is dif?cult to build a precise dynamic model. Some research has derived various mathematical models for many kinds of maglev systems [14], [15], but there still exists unmodeled dynamics or unpredictable uncertainties in practical applications. In general, linearized control strategies based on a Taylor-series expansion of the actual nonlinear dynamic model and force distribution at nominal operating points are often employed. Nevertheless, the tracking performances of linearized control strategies [16], [17] deteriorate rapidly with increasing deviation from nominal operating points. Many approaches introduced to solve this problem for ensuring consistent performances independent of operating points have been reported in previous literature. Recursive backstepping methods were reported in [11] and [18] due to the systematic design procedure. Kaloust et al. [11] proposed a recursive control in the sense of nonlinear state transformation and Lyapunov’s direct method to guarantee the global stability for a nonlinear maglev system. However, this nonlinear robust control for the levitation and propulsion of a maglev system was individually designed. Queiroz and Dawson [18] utilized a nonlinear model of an active magnetic bearing system for developing a nonlinear backstepping controller. Unfortunately, partial constrained conditions should be satis?ed for precise positioning. Moreover, the approach of gain scheduling [19], [20] can linearize the nonlinear relationships of the magnetic suspension at various operating points with a suitable controller design for each of these operating points. In order to further achieve better control performance over the entire operational range, it needs to subdivide the operating range into appropriate intervals. By this way, favorable control gains collected in the lookup table will occupy a large memory to bring about heavy computation burden. In addition, Sinha and Pechev [21] presented an adaptive controller to compensate for payload variations and external force disturbance using the criterion of stable maximum descent. On the other hand, the complementary control strategies with mechanical devices [22], [23] are utilized in the maglev system. Gutierrez and Ro [22] presented an SMC design for a magnetic servo–levitation system to possess the robustness with respect to parametric uncertainties and unmodeled dynamics. Bonivento et al. [23] proposed a balanced control design without relying upon complementarity or relaxed complementarity conditions
for a maglev system in presence of physical uncertainties. However, the information of system dynamics is required, and the installed position of mechanical devices may be limited via complementary or balanced control strategies. Moreover, the attractive levitation in the EMS system belongs to a nonnegative input system. Some researches focused on nonnegative input systems are addressed in [24] and [25]. Overall, partial mathematical models via complicated modeling processes or speci?c mechatronic device are usually required to design a suitable control law for achieving positioning demand. Recently, intelligent control methods have attracted more attention to deal with the complex nonlinear problem of the maglev [26]–[30]. Shiakolas et al. [26] discussed the use of a real-time digital control environment with a hardwarein-the-loop maglev device for modeling and controls education, with emphasis on neural network (NN) feedforward control. Phuah et al. [27] proposed a synergistic combination of NN with SMC methodology to carry out a maglev system. Buckner [28] utilized intelligent uncertainty bound estimation with traditional SMC to reveal excellent tracking performance without excessive control activity in a maglev system. However, the network weights in [26]–[28] were adjusted via modi?ed online error backpropagation algorithms so that the entire system stability is a challenging problem to be solved and the detailed control action behavior in NN is dif?cult to handle clearly. On the other hand, Yang et al. [29] improved the stability and reliability of a maglev train based on a composite fuzzy-proportional-integral differential (PID) controller. Santisteban et al. [30] used the fuzzy logic approach in the magnetic bearing system whose dynamic behavior was not well known, and these rules were based on experimental results of a proportional derivative (PD) controller previously implemented via computer simulations. Although these fuzzy logic techniques [29], [30] allow the constructing of a control system based in a group of rules in a similar way as the human thought does, how to build appropriate rules and how to ensure system stability are the major problems. Therefore, the concept of incorporating fuzzy logic into an NN has grown into a popular research topic [31]. The integrated fuzzy-neural-network (FNN) system possesses the merits of both fuzzy systems [32] (e.g., human-like IF-THEN rules thinking and ease of incorporating expert knowledge) and NNs [33] (e.g., learning and optimization abilities and connectionist structures). By this way, one can bring the low-level learning and computational power of NNs into fuzzy systems and also high-level, humanlike IF-THEN rule thinking and reasoning of fuzzy systems into NNs. In this paper, the main topic is to design actual control currents in electromagnets and LIM simultaneously via a model-free adaptive fuzzy-neural-network control (AFNNC) scheme without the requirement of auxiliary compensated controllers, strict constrains, and control transformations, and all network parameters in the FNN are autotuning according to suitable adaptation laws derived from the stability analyses. As far as our knowledge goes, no intelligent control design addressing the problems of levitation and propulsion simultaneously has been reported. In the network structure, the major difference between the proposed AFNNC scheme and similar network framework in existing literature [34]–[36] is the adoption of a sigmoid func-

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Fig. 1. (a) Entire framework of maglev transportation system. (b) Sketch diagram of maglev system.
tion in the output layer of the FNN. It is helpful to produce a nonnegative input for a maglev system without complicated control transformation. However, it will cause the challenging problem of developing the corresponding online learning algorithms for network parameters including connective weights inside layers and important coef?cients in activation/membership functions to further ensure the system stability. In the previous works, Lin et al. [34] demonstrated the application of a sliding-mode controller and an FNN controller to control the position of a slider of the motor-toggle servomechanism. However, this control strategy lacks stability analysis. Though the system stability of this FNN-based control scheme can be guaranteed in Lin et al. [35], supervisory control design and prior system knowledge were required. Wai et al. [36] presented a robust FNN control system for a linear ceramic motor driven by a unipolar switching full-bridge voltage source inverter using inductance–capacitance resonant technique. The requirement of prior system knowledge in [35] was relaxed in [36] because the FNN controller was designed to learn an ideal feedback linearization control law. But an extra robust controller was still
required to compensate the shortcoming of the FNN controller. Besides, the convergence of network parameters is a common problem to be solved in [34]–[36]. According to the survey of previous works with similar control frameworks, the superiority of the proposed AFNNC strategy over AFNN schemes from existing literature is summarized as follows. 1) The outputs of the AFNNC scheme can be directly supplied to the electromagnets and LIM without complicated control transformations for relaxing strict constrains in conventional model-based control methodologies. 2) Online learning algorithms via Taylor series expansion are designed to ensure the stability of the controlled system without the requirement of auxiliary control design. 3) The convergence of network parameters in the FNN is ensured by the use of projection algorithm. II. DYNAMIC ANALYSES OF MAGLEV TRANSPORTATION SYSTEM The entire framework of a maglev transportation system is depicted in Fig. 1(a), which is a linear motor via electromagnetic suspension technique to reduce friction forces during linear

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movement for promoting the electric machinery ef?ciency and simplifying the mechanical design procedure. The bottom of this mechanism is a motion basal platform equipped with an I-shaped steel bridge as the rail body. The motion basal platform is a granite foundation to keep the mechanism horizontal. By appropriately exciting the stator winding of an LIM ?xed on the overhead I-shaped steel bridge, it will produce the thrust force to propel the moving platform in horizontal displacement, and the corresponding displacement can be measured by a linear encoder. Moreover, the fenders placed at the ends of the LIM are designed to avoid the problem of collision, and vertically auxiliary wheels are used to decrease friction forces before the moving platform has been suspended completely. In order to achieve the maglev object, four electromagnets under the bottom of the moving platform are adopted in this paper. When the electromagnets are excited to attract the upper plate of the rail, the moving platform will be levitated, and the corresponding levitation height is acquired by a gap sensor. In addition, perpendicularly ?rm cubes are installed for holding the vertical angle between two conductor plates of the moving platform to reduce measure errors. When the transversal displacement is varied during linear movement, transversal auxiliary wheels are employed to guide the moving platform on the march. In general, six degrees-of-freedom (DOF) could be considered in a fully suspended substance. For decreasing the hardware cost, the transversal electromagnets and gap sensors are replaced by transversal auxiliary wheels in this paper. According to Fig. 1(a), rotating along -axis and moving in -direction are constricted by transversal auxiliary wheels, which will result in little friction forces under the rolling situation. Therefore, three kinds of motions including levitation ( , moving in -direction), rolling ( , rotating along -axis), and pitching ( , rotating along -axis) should be considered to analyze the dynamic model of the maglev system. In other words, the mechanical behavior is constrained on the side-to-side movements, and it only takes the reliance on the levitation, roll, and pitch into account in the dynamic derivation. To regard the moving platform as a 3-D coordinate of the levitation stage, the sketch diagram of the maare the glev system is illustrated in Fig. 1(b), where and length and width of the moving platform. , which is located in the 3-D coordinate, denotes the at -axis and are the cenmass center of the moving platform. tral levitation height and its maximum limitation. By dividing the moving platform equally into four subplatforms labeled as , and areas, , and represent the mass centers of four subplatforms, and the corresponding levitation , and are measured via gap senheights labeled as sors installed in the location of individual mass center. and are the horizontal distances from the mass centers of four subplatforms to -axis and -axis, respectively. The applied con, trol forces vector is represented as in which , and of four subplatforms are acquired through levitation forces produced by electromagnets, and is the propulsive force produced by the LIM. In general, a maglev force can be expressed as [25] (1)
and are the air–gap length and permeance of the where and are the turn number and magnetizing electromagnet; current of the electromagnetic coil. According to (1), the levitation forces produced by four electromagnets in this maglev system can be represented as follows:
(2) , and indicate the where the symbols with suf?xes variables for four electromagnets in different areas. According to Lagrangian method, the dynamic model of the maglev system is derived in Appendix I. By considering the mechanical equation of the propulsive LIM in Appendix II, the entire dynamic model of a maglev transportation system can be organized by the following matrix form: (3) and and vector where the detailed elements of matrices are given in Appendix III; and is the is the system state vector; control effort vector acquired by the levitation forces produced by electromagnets, and the propulsive force produced by the LIM. Note that the bending angle is regarded as a zero state in this paper because the moving platform is assumed to be a rigid body. The relation between the control effort and actual control current can be represented as (4) where is a control current vector, in and are the corresponding magnetizing curwhich rents for four electromagnets, and is the LIM control current
is an invertible constant matrix
in which is the LIM force constant, of the LIM,
is the mover mass
, and
,

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in which the de?nitions of , and are given in is the control effort (2). In (3), vector acquired by the levitation forces produced by electromagnets and the propulsive force produced by the LIM. Moreover, the relation between the control effort and actual control curin (4). As can be seen rent can be represented as in (4), the forces will be affected by different from (2) and air–gap lengths and magnetizing currents in electromagnets so that the nonlinear characteristics of the electromagnets make a challenging problem with controlling a maglev system. Despite the aforementioned problem of the maglev transportation system, it has several inherent technical challenges as follows: 1) maglev system is open-loop unstable, 2) since only a single moving platform generates all the motions, its dynamics is coupled, 3) owing to the absence of major damping or restricting force on the moving platform, the overshoots to the command steps are large, and 4) nonlinear relationship between the current and displacement may not allow large travel ranges. After mathematical manipulations, the dynamic model of the maglev transportation system with current control input can be represented as follows: (5) where , and . Note that, the property of a positive–de?nite matrix will be veri?ed in Appendix III. Since the actual measurable variables are the central levitation heights of four subplatforms, the relation of the and system states levitation heights can be transferred by
be a suf?ciently smooth function by a reference model. Let be bounded for . The control problem is of and to ?nd a suitable control law so that the system states can track desired reference commands. To achieve this control objective, and its derivative de?ne an error state vector . The detailed derivations of sliding-mode-based control strategies are described as follows. A. Sliding-Mode Control By viewing as a virtual control effort vector (i.e., ) and reformulating (7) as (9) in the conventional SMC design, a sliding surface vector is chosen as follows: (10) where , in which , and are all positive constants. Take the derivative of the sliding surface vector with respect to time and use (9), then (11) In this paper, the SMC law can be designed as (12) where a control gain is concerned with the upper bound of unis a sign function, and is a diagonal poscertainties, itive–de?nite matrix. According to Lyapunov theorem [1], the . stability of the SMC strategy can be guaranteed with However, the parameter variations of the system are dif?cult to measure, and the exact values of the external disturbance and unmodeled dynamics are also unknown in advance for practical applications. Selection of the upper bound of the lumped uncertainty vector has a signi?cant effect on the control performance. If the bound is selected too large, the sign function in (12) will result in serious chattering phenomena in the control efforts. The undesired chattering control efforts will wear the mechanical structure and might excite unstable system dynamics. On the other hand, if the bound is selected too small, the stability conditions may not be satis?ed. It will cause the controlled system to be unstable. B. Adaptive Sliding-Mode Control
(6) , and in (5) are nominal values and deSuppose that note unknown actual values of , and by , and , the actual dynamic model with an unknown external disturbance can be represented realistically as (7) where denotes the lumped uncertainty vector and its bound is assumed to be given by (8) in which constant. is the Euclidean norm, and is a given positive
III. SLIDING-MODE-BASED CONTROL STRATEGY The block diagram of sliding-mode-based control strategies for the maglev transportation system is depicted in Fig. 2(a), is the command vector of system states and where is the reference trajectory vector speci?ed
Although the conventional SMC strategy with a ?xed uncertainty bound results in a simple implementation, the compromise between robust control performance and chattering control force is usually tradeoff. Thus, a simple bound estimation embedded in the SMC strategy is organized to form an adaptive sliding-mode control (ASMC) scheme in this section. In the ASMC strategy, the adaptation law for the bound of the lumped uncertainty vector is designed as (13)

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Fig. 2. Control block diagram: (a) sliding-mode-based control strategy and (b) AFNNC scheme.
and the estimated error
is de?ned as (14)
the realistic control current vectors derived from the SMC and ASMC laws as shown in (12) and (15) have to be transferred by (16a) (16b) To ensure the stability of the controlled system despite the existence of uncertainties, alleviate chattering phenomena in control efforts, and avoid redundant transformation steps, an AFNNC scheme without auxiliary controller design is further investigated in Section IV. IV. ADAPTIVE FUZZY-NEURAL-NETWORK CONTROL In order to control the states of the maglev transportation system more effectively, an AFNNC scheme as shown in Fig. 2(b) is constructed in this section. Moreover, a four-layer FNN, which comprises the input, membership, rule, and output layers, is adopted to implement the AFNNC scheme in this paper [34]–[36]. The basic con?guration of the FNN includes a fuzzy rule base that is composed of a collection of fuzzy IF-THEN rules. Such an FNN implementing the procedures of
where is a positive constant. Replacing ASMC law can be represented as
by
in (12), the
(15) According to Lyapunov theorem and Barbalat’s lemma [1], it can be shown that the vector goes to zero as time tends to in?nite. Unfortunately, the adaptation law for the bound of the lumped uncertainty vector shown in (13) is always positive and tracking errors introduced by any uncertainty will cause the growth. It implies that the ASMC law (15) will result in large chattering with time gradually. In other words, the electromagnets and the LIM will eventually be saturated and the controlled maglev transportation system may be unstable. Besides, it is worth mentioning that the control currents to electromagnets belong , to positive inputs. According to the relation of

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fuzzi?cation, fuzzy inference, and defuzzi?cation performs the mappings from an input vector to an output variable. The input and membership layers in the FNN belong to the premise part of the fuzzy system, and the rule and output layers belong to the consequent part of the fuzzy system. The inputs of the FNN are the elements in the sliding surface vector relative to central levitation height, rolling angle, and pitching angle, respectively. Note that the error terms concerned with bending angle are not considered because the moving platform is assumed to be a rigid body. In addition, the output of the FNN is the control current vector . The signal propagation and the basic function in each layer of the FNN are introduced in the following. 1) Input layer transmits the input linguistic variables to the next layer, where denotes the elements except for the term conin the sliding surface vector cerned with bending angle. 2) Each node performs a membership function in the membership layer that can be referred to as the fuzzi?cation procedure. In this paper, the following Gaussian function is adopted as the membership function: (17) is the exponential function and and , respectively, are the mean and standard deviation of the Gaussian function in to the the th term of the th input linguistic variable node of this layer. In order to represent the general case including different clusters with respect to the network is utilized to denote the individual inputs, the symbol number of membership functions. For ease of notation, de?ne parameter vectors and collecting all mean and standard deviation of Gaussian membership functions as and , where denotes the total number of membership functions. 3) The output of each node in this layer is determined by fuzzy AND operation. Each node in the rule layer is denoted by , which multiplies the input signals and outputs the result of the product. The product operation is utilized to determine the ?ring strength. It can be referred to as the fuzzy inference mechanism. The output of this layer is given as (18) where represents the th output of the rule layer and all values can be collected by a parameter , the weights between the vector membership layer and the rule layer are assumed to be is the total number of rules. unity, and 4) Layer four is the output layer. The output node together with links connecting it act as a defuzzi?cation procedure. computes the output as the Each node summation of all input signals with the following type: (19) where
is the active function; , the adjustable where weights between the rule layer and the output layer, can be gathered by the following matrix:
. . .
. . .
..
.
. . . (20)
in which . Moreover, the output of the FNN can be rewritten in the following vector form: (21) The proposed AFNNC scheme comprises an FNN control and its associated network parameters tuning algorithms. The FNN control is designed to replace the SMC law in (16a) to maintain the robust control performance without the requirement of system information and auxiliary compensated control. Moreover, the network parameters tuning laws are derived in the sense of projection algorithm [32] and Lyapunov stability theorem [1] to ensure the network convergence as well as stable control performance. According to the powerful approximation to learn ability [31], there exists an optimal FNN control the SMC law, , such that (22) where is a minimum reconstructed error vector and and are the optimal parameters of , and in the FNN. Design the control current vector of the AFNNC scheme as (23) where and are the estimates of and , as provided by tuning algorithm to be introduced later. Subtracting (23) from (22), an approximation error is de?ned as (24) In this paper, the linearization technique is employed to transform the active functions into partially linear form so that the expansion of in Taylor series obtains
. . .
(25)

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where , , and are the optimal parameters of and , and are the and , are vectors of higher estimates of with order terms, ; and . Moreover, the linearization technique is employed again to transform the membership functions into partially linear form so that the expansion of in Taylor series obtains
Proof:
Consider
a
Lyapunov
function
candidate
. . .
. . .
. . .
(26)
where and a vector of higher order terms,
, , and
is
To substitute (26) into (25), the approximation error reformulated as
. can be
and take the derivative of with respect to time; there exists an AFNNC law in (23) and adaptation laws for FNN parameters in (28)–(30) such that . According to Lyapunov stability theorem and Barbalat’s lemma [1], it can be implied that the vector will converge to zero as . Moreover, parameter , and can be guaranteed to be bounded estimation errors in the sense of projection algorithm [32]. As a result, the stable control behavior can be ensured without the requirement of system information and the compensation of auxiliary control design. Simultaneously, the control currents can be obtained from the outputs of the proposed AFNNC scheme without any control transformation. Remark 1: Without loss of generality, the moving platform’s is assumed to be located at the center of the recmass center , and in this paper. But, it may tangle formed by affect the validness of the dynamics derived in the Appendices I and III if the masses of the four subplatforms are different. Fortunately, the imprecise system dynamics incurred by ideal structural assumptions could be involved in the lumped uncertainty vector, and the related control design still holds. Moreover, the role of the system dynamic model given in Section II is just taken as a controlled object for simulating plant behaviors in order to verify the effectiveness of the AFNNC system. In the proposed control scheme, the detailed dynamic model is not required. V. NUMERICAL SIMULATIONS In the levitation mechanism, EI-type (EI-96) electromagnets are adopted, and the coil turns for four electromagnets are all chosen as 230. The LIM used in the propulsive mechanism is a three-phase Y-connected two-pole 1.5-kW 60-Hz 110-V/10.2-A type. The detail parameters of the maglev transportation system are listed as follows: mm mm mm kg kg/s N/A (31)
(27) . In (25) and (26), the linearizawhere tions of the activation and membership functions are helpful for representing the approximation error (27) being linear in the parameters and it is convenient for the stability analyses. Theorem 1: Consider the dynamic model of a maglev transportation system including levitated electromagnets and a propulsive LIM represented by (9), if the AFNNC law is designed as (23) and the adaptation laws of the FNN parameters are designed as (28)–(30), then the convergence of network parameters and tracking error of the proposed AFNNC scheme can be assured, as shown in (28)–(30), at the bottom of the page, , and are positive learning rates and , and where are given positive parameter bounds.
if
or if if if if if or and
and and or and and and
(28a) (28b) (29a) (29b) (30a) (30b)

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where is the viscosity and iron-loss coef?cient of the LIM. Numerical simulations of the SMC, ASMC, and AFNNC systems are implemented via the MATLAB software based on the schemes as shown in Fig. 2. Moreover, a second-order transfer function of the following form with rise time 0.2 s is chosen as the reference model, which is used to specify the reference trajectories for the levitation height of the maglev transportation system, for step commands:
Moreover, the control parameters of the SMC, ASMC, and AFNNC systems are given as
(34) are chosen to achieve the best transient Note that and control performance in numerical simulations by considering the requirement of stability and the possible operating conditions. The choice of is a compromise between the superiority of control performance and the chattering of control effort. is related to the The selection of the learning rates convergence rate of the error state vector and the parameter are decided by the range of reasonable bounds network coef?cients. In this paper, the control objective is to make the central levitation height and moving position of the platform follow the reference trajectories, and to keep the corresponding rolling and pitching angles horizontal (i.e., ) under the possible occurrence of uncertainties. All the illustrations of the SMC, ASMC, and AFNNC systems are depicted in Figs. 3–10. Figs. 3(a)–(c)–10(a)–(c) are the tracking responses of the central levitation height, rolling, and pitching angles. Figs. 3(d)–10(d) show the tracking response of the platform position. Figs. 3(e)–10(e) show the related attitude control force. Figs. 3(f)–10(f) show the LIM control current. In the SMC strategy, the ?xed bound of the lumped uncertainty vector is determined roughly owing to the limitation of control effort and the possible perturbed range of parameter variations and external disturbance. The simulated results of the SMC strategy with a small uncertainty bound at cases 1 and 2 are depicted in Figs. 3 and 4, respectively. In Figs. 3(a) and 4(a), the central levitation heights follow the reference trajectories well before 2 s, but the performance becomes poor because of the additional weight loading and unloading as 2 and 5 s and external force disturbance occurring at 7 s. Similarly, the degenerate rolling and pitching angles in Figs. 3(b) and (c) and 4(b) and (c) caused by the inappropriate selection of also occurred after 2 s. In addition, the tracking responses of the platform position in Figs. 3(d) and 4(d) also possess degenerate performance under the occurrence of uncertainties. Figs. 5 and 6show the simulated results of the SMC strategy with a large uncertainty bound at cases 1 and 2, respectively. Though a larger bound of the lumped uncertainty vector can solve the problem of degenerate responses, it will result in more serious chattering and excessive control currents. Besides, the undesired chattering control efforts will wear the mechanical structure and might excite unstable system dynamics. Figs. 7 and 8 illustrate the simulated results of the ASMC strategy at cases 1 and 2, respectively. As can be seen from Figs. 7(a)–(d) and 8(a)–(d), acceptable tracking responses and robust characteristics can be obtained according to the
(32)
are the damping where is the Laplace operator and and ratio (set at one for critical damping) and undamped natural frequency. In addition, the mean-square-error (MSE) measures of position and angle responses are de?ned as
MSE
(33)
indicate the elements of the system state vector and the corresponding error state vector and is total sampling instants. According to (33), the normalized-mean-square-error (NMSE) values of the levitation-height response, angle response, and platform-position response using per-unit values with a -mm, -degree, and -cm bases are used for examining the control performance in this paper. To investigate the robustness of the proposed control systems, the following two cases with parameter variations and time-varying external force disturbance are considered. Case 1) A step command for the central levitation height is 0.5 mm initially, and a sinusoidal comset at mand for the platform position is chosen as cm after 1 s. An additional weight at 2 s and is unloaded at of 0.5 kg is loaded on 5 s. Case 2) A step command for the central levitation height is set at 0.5 mm initially, and is changed from 0.5 to 2 mm at 4 s and from 2 to 1 mm at 6 s. A sinusoidal command for the platform position is cm. An additional chosen as weight of 1 kg is loaded on at 2 s and is unloaded at 5 s. Other identical conditions in these two cases are represented as follows: 1) initial system state: mm and cm; 2) external force disturbance occurring at 7 s:
where
and
; 3) parameter variations: .
and

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Fig. 3. Numerical simulations of SMC strategy with small uncertainty bound ( = 15) at case 1. (a) Tracking response of central levitation height. (b) Rolling angle response. (c) Pitching angle response. (d) Platform position response. (e) Attitude control force. (f) LIM control current.
Fig. 4. Numerical simulations of SMC strategy with small uncertainty bound ( = 15) at case 2. (a) Tracking response of central levitation height. (b) Rolling angle response. (c) Pitching angle response. (d) Platform position response. (e) Attitude control force. (f) LIM control current.

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Fig. 5. Numerical simulations of SMC strategy with large uncertainty bound ( = 50) at case 1. (a) Tracking response of central levitation height. (b) Rolling angle response. (c) Pitching angle response. (d) Platform position response. (e) Attitude control force. (f) LIM control current.
Fig. 6. Numerical simulations of SMC strategy with large uncertainty bound ( = 50) at case 2. (a) Tracking response of central levitation height. (b) Rolling angle response. (c) Pitching angle response. (d) Platform position response. (e) Attitude control force. (f) LIM control current.

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Fig. 7. Numerical simulations of ASMC strategy at case 1. (a) Tracking response of central levitation height. (b) Rolling angle response. (c) Pitching angle response. (d) Platform position response. (e) Attitude control force. (f) LIM control current.
Fig. 8. Numerical simulations of ASMC strategy at case 2. (a) Tracking response of central levitation height. (b) Rolling angle response. (c) Pitching angle response. (d) Platform position response. (e) Attitude control force. (f) LIM control current.

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Fig. 9. Numerical simulations of AFNNC scheme at case 1. (a) Tracking response of central levitation height. (b) Rolling angle response. (c) Pitching angle response. (d) Platform position response. (e) Attitude control force. (f) LIM control current.
online adjustment of the bound of the lumped uncertainty vector. However, the estimated bound is gradually increased with time so that the attitude control forces and LIM control currents tend to diverge. These situations can be veri?ed by observing Figs. 7(e) and (f) and 8(e) and (f). Though the chattering amplitude at the beginning is smaller than the one in the SMC strategy, the magnitude and chattering phenomena of the control efforts raise gradually with the increasing of the estimated bound of the lumped uncertainty vector. As a result, the growth estimated bound will not only cause more serious chattering in control efforts but also may make the controlled system unstable. For comparison, the proposed AFNNC scheme is also applied to control the maglev transportation system. To show the effectiveness of the AFNNC scheme, the FNN has , and neurons at the input, membership, rule, and output layer, respectively. It can be regarded that the associated fuzzy sets with Gaussian (negative), function for each input signal are divided into (zero), and (positive), and the number of rules with . The complete rule connection is active functions in the output nodes are chosen as sigmoid functions (i.e., with a positive constant ) because of nonnegative control current inputs for four electromagnets, and the active function in the output node
is chosen as a bypass function (i.e., ) for the LIM control curcan be implemented as rent. According to (25), , in which and can be computed by (18). Moreover, some heuristics can be used to roughly initialize the parameters of the AFNNC scheme for practical applications; e.g., the means and the standard deviations of the Gaussian functions can be determined according to the maximum variation of elements in the sliding surface vector . By considering the same simulated cases as the SMC and ASMC strategies, the numerical simulation of the proposed AFNNC scheme at cases 1 and 2 are depicted in Figs. 9 and 10, respectively. Not loading the additional weights at the center of mass of the platform affects not only the levitation height and the propulsion force of the LIM, but also simultaneously affects the rolling and the pitching angles. As can be seen from Figs. 9(a)–(d) and Fig. 10(a)–(d), the error state vector converges quickly and the robust control characteristics under the occurrence of varied reference trajectories, parameter variations, and external force disturbance can be clearly observed. Because all parameters in the AFNNC are roughly initialized, the tracking errors are gradually reduced through online training process whether system uncertainties exist or not. By observing Figs. 9(e) and (f) and 10(e) and (f), the degenerate performance or undesirable

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Fig. 10. Numerical simulations of AFNNC scheme at case 2. (a) Tracking response of central levitation height. (b) Rolling angle response. (c) Pitching angle response. (d) Platform position response. (e) Attitude control force. (f) LIM control current.
chattering phenomena in the SMC strategy and the gradually increasing chattering phenomena in the ASMC scheme can be all removed. Consequently, the proposed AFNNC scheme indeed yields superior performance than the SMC and ASMC strategies. VI. CONCLUSION This paper has successfully investigated the SMC, ASMC, and AFNNC schemes for the maglev transportation system. The performance comparisons of the SMC, ASMC, and AFNNC systems are summarized in Table I. According to the NMSE measures, the proposed AFNNC scheme has over 99.34% and 98.59% height-tracking improvements, 99.66% and 97.87% angle-stabilizing improvements, and 80.3% and 68.44% position-tracking improvements than the SMC and ASMC strategies, respectively. The AFNNC scheme has the salient merits of model-free control design, favorable robust characteristic, and control effort without chattering in comparison with the SMC and ASMC strategies. Thus, the proposed AFNNC scheme is more suitable for the maglev transportation system than the SMC and ASMC strategies. The major contributions of this paper are recited as follows: 1) the successful construction of the mathematical model of a maglev transportation system including levitated electromagnets and a propulsive LIM based on the concepts of mechanical geometry and motion dynamics, 2) the successful development of a model-free AFNNC scheme without the requirement
of auxiliary compensated controllers, strict constrains, and control transformations, 3) all parameters in FNN are autotuning according to suitable adaptation laws derived from the stability analyses, and 4) the successful application and comparison of three different controllers (SMC, ASMC, and AFNNC) for a maglev transportation system to demonstrate the value of using the AFNNC approach over SMC.
APPENDIX I In the 3-D coordinate system as shown in Fig. 1(b), three DOFs of the moving platform including and are considered to analysis the dynamic model in this paper. After mathof the moving ematical manipulations, the kinetic energy platform can be represented as (A1) is the total mass of the moving platform, and are the inertia with respect to - and -axis, respectively [37]. Moreover, the of the moving platform can be expressed as potential energy (A2) where

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TABLE I PERFORMANCE COMPARISONS OF SMC, ASMC, AND AFNNC SYSTEMS
where is the gravity acceleration. Thus, the Lagrange equations of motion are
where vector be obtained by
. In addition, the attitude control force of the moving platform can
(A9) where (A3) where is Lagrangian and is de?ned as the total kinetic energy minus the potential energy (A4) APPENDIX II According to (A1)–(A4), the dynamic equations of the maglev system can be expressed as (A5) (A6) (A7) The dynamic motion in (A5)–(A7) is similar to the results derived from the Newton’s second law of motion [38]. Assume that the moving platform is a rigid body; then, the following condition always holds: In general, the mechanical equation of an LIM can be denoted as [8], [9] (B1) where is the mover mass of the LIM, is the visis the platform position, cosity and iron-loss coef?cient, denotes the propulsive force, and represents the external force disturbance, unstructured mutual force reaction between electromagnets and the LIM, and unpredictable uncertainties. With suitable impressed current or ?eld-oriented control [8], the propulsive force can be simpli?ed as (B2)
(A8)

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(C2)
is the LIM force constant and is the LIM conwhere trol current. According (B1) and (B2), the dynamic model of the propulsive LIM in the maglev transportation system can be reformulated as
(B3)
APPENDIX III The matrix and is de?ned as in (3) is a positive–de?nite matrix
(C1)
According to (4), the inverse matrix of can be denoted as (C2), shown at the top of the page. Because the diagonal elein (C2) are always positive, the property of a posiments of can be ensured. According to the positive–de?nite matrix in (C1) and in (C2), the property tive–de?nite matrices can be guaranteed. of a positive–de?nite matrix in (3) is de?ned as Moreover, the matrix
(C3)
In addition, the vector
in (3) is given as (C4)
ACKNOWLEDGMENT The authors would like to thank the referees and the associate editor for their valuable comments and helpful suggestions. REFERENCES
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Rong-Jong Wai (M’99–A’00–M’02–SM’05) was born in Tainan, Taiwan, R.O.C., in 1974. He received the B.S. degree in electrical engineering and the Ph.D. degree in electronic engineering from Chung Yuan Christian University, Chung Li, Taiwan, R.O.C., in 1996 and 1999, respectively. Since 1999, he has been with the Department of Electrical Engineering, Yuan Ze University, Chung Li, Taiwan, R.O.C., where he is currently a Professor. He is also the Director of the Electric Control and System Engineering Laboratory at Yuan Ze University, and the Energy Conversion and Power Conditioning Laboratory at the Fuel Cell Center. He is an author of a chapter in Intelligent Adaptive Control: Industrial Applications in the Applied Computational Intelligence Set (Boca Raton, FL: CRC Press, 1998) and the coauthor of Drive and Intelligent Control of Ultrasonic Motor (Tai-chung, Taiwan, R.O.C.: Tsang-Hai, 1999), Electric Control (Tai-chung, Taiwan, R.O.C.: Tsang-Hai, 2002), and Fuel Cell: New Generation Energy (Tai-chung, Taiwan, R.O.C.: Tsang-Hai, 2004). He has authored numerous published journal papers in the area of control system applications. His biography was listed in Who’s Who in Science and Engineering (Marquis Who’s Who) in 2004–2007, Who’s Who (Marquis Who’s Who) in 2004–2007, and Leading Scientists of the World (International Biographical Centre) in 2005, Who’s Who in Asia (Marquis Who’s Who), Who’s Who of Emerging Leaders (Marquis Who’s Who) in 2006–2007, and Asia/Paci?c Who’s Who (Rifacimento International) in Vol. VII. His research interests include power electronics, motor servo drives, mechatronics, energy technology, and control theory applications. Dr. Wai received the Excellent Research Award in 2000, and the Wu Ta-You Medal and Young Researcher Award in 2003 from the National Science Council, R.O.C. In addition, he was the recipient of the Outstanding Research Award in 2003 from the Yuan Ze University; the Excellent Young Electrical Engineering Award in 2004 from the Chinese Electrical Engineering Society, R.O.C; the Outstanding Professor Award in 2004 from the Far Eastern Y. Z. Hsu-Science and Technology Memorial Foundation, R.O.C.; the International Professional of the Year Award in 2005 from the International Biographical Centre, U.K., the Young Automatic Control Engineering Award in 2005 from the Chinese Automatic Control Society, R.O.C., and the Yuan-Ze Lecture Award in 2007 from the Far Eastern Y. Z. Hsu—Science and Technology Memorial Foundation, R.O.C.
Jeng-Dao Lee was born in Taipei, Taiwan, R.O.C., in 1980. He received the B.S. and Ph.D. degrees in electrical engineering from Yuan Ze University, Chung Li, Taiwan, R.O.C., in 2002 and 2007, respectively. Currently, he serves in the Army, Ministry of National Defense, Taiwan, R.O.C. His research interests include motor servo drives, power electronics, mechatronics, and intelligent control.