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									 Model-free Control Design for Hybrid Magnetic Levitation System
                           Rong-Jong Wai 1, Member, IEEE, Jeng-Dao Lee 2, and Chiung-Chou Liao 3
                      Department of Electrical Engineering, Yuan Ze University, Chung Li 320, Taiwan, R.O.C.
                     Department of Electronic Engineering, Ching Yun University, Chung Li 320, Taiwan, R.O.C.

AbstractThis study investigates three model-free control            magnetic force generated by the additional permanent magnet
strategies including a simple proportional-integral-differential     is used to alleviate the power consumption for levitation.
(PID) scheme, a fuzzy-neural-network (FNN) control and a                 Because the EMS system has unstable and nonlinear
robust control for a hybrid magnetic levitation (maglev) system.     behaviors, it is difficult to build a precision dynamic model.
In general, the lumped dynamic model of a hybrid maglev
                                                                     Some researches have derived various mathematical models
system can be derived by the transforming principle from
electrical energy to mechanical energy. In practice, this hybrid     for many kinds of maglev systems in numerical simulation
maglev system is inherently unstable in the direction of             [13], [14], but there still exist uncertainties in practical
levitation, and the relationships among airgap, current and          applications. In general, linearized-control strategies based on
electromagnetic force are highly nonlinear, therefore, the           a Taylor series expansion of the actual nonlinear dynamic
mathematical model can not be established precisely. In order to     model and force distribution at nominal operating points are
cope with the unavailable dynamics, model-free control design is     often employed. Nevertheless, the tracking performance of
always required to handle the system behaviors. In this study,       the linearized-control strategy [15][18] deteriorates rapidly
the experimental comparison of PID, FNN and robust control           with the increasing of deviations from nominal operating
systems for the hybrid maglev system is reported. From the
performance comparison, the robust control system yields
                                                                     points. Many approaches introduced to solve this problem for
superior control performance than PID and FNN control                ensuring consistent performance independent of operating
systems. Moreover, it not only has the learning ability similar to   points have been reported in opening literature. Backstepping
FNN control, but also the simple control structure to the PID        methods were incorporated into [12], [19] due to the
control.                                                             systematic design procedure. Huang et al. [12] addressed an
                                                                     adaptive backstepping controller to achieve a desired stiffness
                           I. INTRODUCTION                           for a repulsive maglev suspension system. In [19], a
                                                                     nonlinear model of a planer rotor disk, active magnetic
    In recent years, magnetic levitation (maglev) techniques
                                                                     bearing system was utilized to develop a nonlinear
have been respected for eliminating friction due to
                                                                     backstepping controller for the full-order electromechanical
mechanical contact, decreasing maintainable cost, and
                                                                     system. Unfortunately, some constrain conditions should be
achieving high-precision positioning. Therefore, they are
                                                                     satisfied for the precision positioning. Moreover, the
widely used in various fields, such as high-speed trains
                                                                     approach of gain scheduling [20], [21] can linearize the
[1][5], magnetic bearings [6], [7], vibration isolation             nonlinear relationship of the magnetic suspension at various
systems [8], wind tunnel levitation [9] and photolithography         operating points with a suitable controller designed for each
steppers [10]. In general, maglev systems can be classified          of these operating points. In order to achieve better control
into two categories: electrodynamic suspension (EDS) and             performance under the entire operation range, it needs to
electromagnetic suspension (EMS). EDS systems are                    subdivide the operating range into appropriate intervals. By
commonly known as “repulsive levitation”, and                        this way, the favorable control gains collected in the lookup
superconductivity magnets [11] or permanent magnets [12]             table will occupy a large memory to bring about the heavy
are always taken as the levitation source. However, the              computation burden. In addition, Sinha and Pechev [22]
repulsive magnetic poles of superconductivity magnets can            presented an adaptive controller to compensate for payload
not be reacted on low speed so that they are only suitable for       variations and external force disturbance using the criterion
long-distance and high-speed train systems. Basically, the           of stable maximum descent. Overall, the detailed or partial
magnetic levitation force of EDS is partially stable and it          mathematical models acquired by complicated modeling
allows a large clearance. Nevertheless, the productive process       processes are usually required to design a suitable control law
of magnetic materials is more complex and expensive. On the          for achieving the positioning demand. The aim of this study
other hand, EMS systems are commonly known as “attractive            is attempted to introduce model-free control strategies for a
levitation”, and the magnetic levitation force is inherently         hybrid maglev system and to compare their superiority or
unstable so that the control problem will become more                defect via experimental results.
difficult. Generally speaking, the manufacturing process and
cost of EMS are lower than EDS, but extra electric power is                          II. HYBRID MAGLEV SYSTEM
required to maintain a predestinate levitation height. To
merge the merits of these two kinds of levitation systems, a             The configuration of a hybrid maglev system is depicted
hybrid maglev system adopted in this study is combined with          in Fig. 1(a), which consists of a hybrid electromagnet, a
an electromagnetic magnet and a permanent magnet. The                ferrous plate, a load carrier and a gap sensor. Among these,
the hybrid electromagnet is composed of a permanent magnet                                        power produced by the magnetic field can be represented via
and an electromagnet. It forms two flux-loops in the E-type                                       the principle of the conservation of energy as
hybrid electromagnet, and the flux passes through a                                                   dWf      d      dx
permanent magnet, a ferrous plate, an air gap and a core in                                                 i     F                                      (5)
each loop. The magnetic equivalent circuit can be represented                                           dt     dt       dt
as Fig. 1(b). The magnetomotive force (mmf) of this hybrid                                        where F is the produced mechanical force, and x is the
electromagnet is the summation of the permanent magnet                                            displacement of levitation. Multiply dt on both sides of (5),
mmf ( FP ) and the electromagnet mmf ( Nmi ), where N m is                                        then
the coil turns and i is the coil current. Moreover, the total                                         dWf  i d  F dx                                    (6)
reluctance of the magnetic path is                                                                Since the magnetic energy W f is a function of  and x ,
    R  ( RP  RFe  Rx  Rc ) //( RP  RFe  Rx  Rc )   (1)                                     one can obtain
where RP , RFe , Rx and Rc are the reluctances of the                                                                  W f        W f
                                                                                                      dWf ( , x)            d          dx                       (7)
permanent magnet, ferrous plate, air gap and core in the                                                                           x
magnetic path, respectively. In addition, the flux (  )                                          To compare (6) with (7), the mechanical force F ( x, i) can
produced against the magnetic reluctance ( R ) by this hybrid                                     be expressed via (3) as
electromagnetic mmf can be denoted as
                                                                                                                    Wf ( , x)         1                P
         F  N mi                                                                                     F ( x, i)                   N m ( FP  N mi) i     G f i (8)
      P          .                                     (2)                                                             x             2                x
The energy in this magnetic field is                                                              where the term G f is related to the total magnetomotive
           1       1                                                                              force, coil turns and the permeance in the magnetic path.
    W f  Lt i 2  N m ( FP  N mi ) iP ,                (3)                                          According to the Newtonian law, the dynamic behavior of
           2       2
                                                                                                  the hybrid maglev system can be governed by the following
where P  1/ R is the permeance of the magnetic path; Lt
is the inductance of the hybrid electromagnet and is defined
                                                                                                               G              f    GG             f
as                                                                                                    (t )  f i  g  d  f i U  g  d
                                                                                                                m             m      m           m                  (9)
          N  N F  N mi
     Lt   m  m P                                      (4)                                                 G( x, t )U (t )  M ( x, t )
          i      i     i      R
                                                                                                  where m is the mass of total suspension object, g is the
in which  means the flux linkage.
                                                                                                  acceleration of gravity, f d is the external disturbance force,
                                                                        Ferrous Plate
                                                                                                   Gi is the function representation of a power amplifier and U
                                                                     F ( x, i )
                                                                                                  is the control voltage. Moreover, G( x, t )  G f Gi m
          U (t )
                                                                                                  expresses the control gain, and M ( x, t )   g  f d m . Due to
                                                                                                  the nonlinear and time-varying characteristics of the hybrid
                                                                                                  maglev system, the accurate dynamics model ( G( x, t ) and
                                                                                                   M ( x, t ) ) are assumed to be unknown in this study. Without
                                                                                                  loss of generality it is assumed that G( x, t ) is finite and
                                                                                                  bounded away from zero for all x.
                                                                Load Carrier

                                                                                                                  III. CONTROL SYSTEMS DESIGN
                                                                                                  A.   PID Control System
                                                                                                      In industrial application, a PID control system is the
                                                                                                  common usual due to its simple scheme. Define a tracking
                                        RFe                         RFe
                                                                                                  error as
                                                                                                      e  xm  x                                          (10)
                                              RP              RP       
                               Rx                                                   Rx            in which xm           represents the reference levitation
                                                          FP                                      displacement. The PID control law can be represented as
                                                      N mi                                                                                     de
                                              Rc                Rc                                                                    
                                                                                                      U  U P  U I  U D  K Pe  K I e  K D
Fig. 1.       Hybrid maglev system: (a) Configuration. (b) Equivalent circuit.                    where U P is a proportional controller; U I is an integral
                                                                                                  controller; U D is a differential controller; K P , K I and
    Assume that there is no loss in energy transmission, the
 K D are the corresponding control gains. Selection of the                 the energy function E is defined as
values for the gains in the PID control system has a                           E  ( xm  x)2 / 2  e2 / 2                          (15)
significant effect on the control performance. In general, they            In the output layer, the error term to be propagated is given
are determined according to desirable system responses, e.g.,              by
raising time, settling time, etc.                                                     E        E e        E e x
                                                                               o                                             (16)
                                                                                      yo       e yo       e x yo
B.   FNN Control System
                                                                           and the weight is updated by the amount
    In the FNN control system, a four-layer network structure
with the input (i layer), membership (j layer), rule (k layer)                                E         E  yo 
                                                                                Δwko   w         w      o    w ok
                                                                                                                                   (17)
and output (o layer) layers is adopted [23]. The membership                                   wk
                                                                                                         yo  wk 
layer acts as the membership functions. Moreover, all the
                                                                           where  w is the learning-rate parameter of the connecting
nodes in the rule layer form a fuzzy rule base. The signal
propagation and the basic function in each layer of the FNN                weights. The weights of the output layer are updated
are introduced in the following paragraph.                                 according to the following equation:
    For every node i in the input layer transmits the input                    wko ( N  1)  wko ( N )  Δwko                    (18)
variables xi (i  1,, n) to the next layer directly, and n is             where N denotes the number of iterations. Since the weights
the total number of the input nodes. Moreover, each node in                in the rule layer are unified, only the error term to be
the membership layer performs a membership function. In                    calculated and propagated.
this study, the membership layer represents the input values                            E
                                                                               k           o wko                             (19)
with the following Gaussian membership functions:                                      k
                   ( x  mij ) 2                                           In the membership layer, the error term is computed as
   n e jt( xi )   i j 2 , i j (net j ( xi ))  exp(net j ( xi )) (12)
                      ( i )                                               follows:
where      mij    and    i j (i  1,, n; j  1,, n p ) are,
                                                            i                  j  
                                                                                      net j
                                                                                               k k                        (20)
respectively, the mean and the standard deviation of the                                      k

Gaussian function in the jth term of the ith input variable x i            The update laws of mij and  i j can be denoted as
to the node of this layer, and n p is the total number of the                   mij ( N  1)  mij ( N )  Δmij                     (21a)

linguistic variables with respect to the input nodes. In                                       E                  2( xi  mi )

                                                                                mij  m                m  j                   (21b)
addition, each node k in the rule layer is denoted by  ,                                      mi     j
                                                                                                                      ( i j ) 2
which multiplies the input signals and outputs the result of
                                                                               i j ( N  1)   i j ( N )  Δ i j                 (22a)
the product. The output of this layer is given as
                                                                                              E                  2( xi  mij ) 2
                                                                               i j  s                s  j

     k   wkji i j (net j ( xi ))                               (13)                        i   j
                                                                                                                     ( i j )3
            i 1

where k (k  1,, ny ) represents the kth output of the rule              where  m and  s are the learning-rate parameters       of the
                                                                           mean and the standard deviation of the Gaussian function.
layer; w kji , the weights between the membership layer and                The exact calculation of the Jacobian of the actual plant,
the rule layer, are assumed to be unity; n y is the total                  x yo in (16), cannot be determined due to the
number of rules. Furthermore, the node o in the output layer               uncertainties of the plant dynamics. Similar to [23], the delta
is labeled with  ; each node yo (o  1, , no ) computes                  adaptation law  o  e  es is adopted in this study. Moreover,
the overall output as the summation of all input signals, and              varied learning rates derived in [23], which guarantee
 n o is the total number of output nodes.                                  convergence of the tracking error based on the analyses of a
            ny                                                             discrete-type Lyapunov function, are also used in this study.
     yo   wkok                                                  (14)
            k 1                                                           C.   Robust Control System
where the connecting weight wko is the output action                           In order to control the levitation displacement of the
strength of the oth output associated with the kth rule. In this           hybrid maglev system more effectively, a robust control
study, the inputs of the FNN control system are the tracking               system [24] is implemented and the state variables are
error ( x1  e ) and its derivative ( x2  es ), and the single            defined as follows:
output is the control effort for the hybrid maglev system, i.e.,                X1  x                                           (23)
U  yo .                                                                         v X
                                                                                X 1           2
    To describe the on-line learning algorithm of this FNN                 where v represents the levitation velocity of the hybrid
control system via supervised gradient decent method, first                maglev system. Rewrite (9), then the hybrid maglev system
can be represented in the following state space form:                    *  B  ( D  L)                                                 (33)
     X 1  0 1   X 1   0 
                                            0                                    
              X   G ( x, t ) U   M ( x, t ) (25)      where B is the left penrose pseudo inverse of B , i.e.,
     X 2  0 0   2                                         B   ( B T B )-1 B T . Since the dynamic model and the
The above equation can be expressed as                              uncertainties of the controlled system may be unknown or
 X P  AX P  BU  D  AX P  ( B  B)U  ( D  D)                perturbed, the ideal control gains shown in (31)(33) can not
      AX P  B U  D  L                                           be implemented in practice. Reformulate (30), then
                                                                        E  A E  B (E X  E R  E )                         (34)
                                     0 1                                 C            M    C            P     k       
where X P  [ X 1     X 2 ]T ; A         ; B  [0 G( x, t )]T ;
                                     0 0
                                                                    in which the control parameter errors E , Ek and E are
                                                                    defined as
 D  [0  M ( x, t )]T ; B and D are the nominal
                                                                        E   *                                      (35)
parametric matrixes of B and D ; B and D denote
the uncertainties introduced by parameter variation and                  Ek  K *  K                                                      (36)
external disturbance; L  BU  D is the lumped                         E     *
    In the robust control system design, the desired behavior       THEOREM 1: Consider the hybrid maglev system represented
of the hybrid maglev system is expressed through the use of a       by (25), if the robust control law is designed as (29), in which
reference model driven by a reference input. Typically, a           the adaptation laws of the control gains are designed as
linear model is used. A reference model of the following state      (38)(40), then the stability of the robust control system can
variable form is selected:                                          be guaranteed.
            0      1          0                                      
                                                                          ET BX T                                            (38)
     XM              X M   B  R  AM X M  BM R
                                                                                1       C        P
            Am1 Am 2          m1                                     
                                                                        K   2 B T EC R                                                   (39)
where X M  [ xm vm ]T represents the reference levitation               
                                                                          B T E                                                         (40)
                                                                               3             C
displacement and velocity; R is a reference input; AM and           where  1 ,  2 and  3 are positive tuning gains.
 BM are given constant matrices. AM  0 is assumed to be a              From Theorem 1, it follows that the tracking error will
stable matrix.                                                      tend to zero under the level of slowly varying uncertainties.
    The control problem is to find a control law so that the        However, the control gains will not necessarily converge to
state X P can track the reference trajectory X M in the             their ideal values in (31)(33); it is shown only that they are
presence of the uncertainties. Let the control error vector be      bounded. To have parameter convergence, it is necessary to
                                                                    impose the persistent excitation condition on the system.
     EC  X M  X P  [ xm  x vm  v]T  [e es ]T          (28)
                                                                    Moreover, according to the unavailable system parameters,
To make the control error vector tend to zero with time, the
robust control law U is assumed to take the following form          the nominal parameter G ( x, t ) in the tuning algorithms is
[24]:                                                               reorganized as G ( x, t ) sgn( G ( x, t )) in practical applications.
    U  U s  U f  U d  X P  KR                       (29)
                                                                    Therefore, the adaptation laws of the robust control system
where U s  X P is a state feedback controller; U f  KR           shown in (38)(40) can be reorganized as follows:
is a feedforward controller; U d   is an uncertainty                  
                                                                         1 es X P sgn( G ( x, t ))
controller. The control gains (  , K and  ) are adjusted             K   e R sgn( G ( x, t ) )                         (42)
                                                                                    2   s
according to dynamic adaptation laws introduced later. After             
some straightforward manipulation, the control error equation             3 es sgn( G ( x, t ))                                         (43)
governing the closed-loop system can be obtained from (25)          where sgn() is a sign function; the terms  1 G ( x, t ) ,
through (29) as follow:
                                                                    2 G ( x, t )          and  3 G ( x, t )       are absorbed by the tuning
     E C  AM E C  ( AM  A  B  ) X P
           ( BM  B K ) R  ( D  B   L)                         gain,  1 ,  2 and  3 individually. Consequently, only the
    If the precise model dynamics and the uncertainties in          sign of G ( x, t ) is required in the design procedure, and it
practical applications are available, there exist ideal control     can be easily obtained from the physical characteristic of the
gains Θ* , K * and  * in the following equations such that         hybrid maglev system.
the control error vector tend to zero with time:
                                                                                                 IV. EXPERIMENTAL RESULTS
    Θ*  B  ( AM  A)                                      (31)
                                                                        The block diagram of a computer-based control system
    K *  B  BM                                            (32)
                                                                    for the hybrid maglev system is depicted in Fig. 2. In the
hybrid maglev system, it divides into two parts: A ferrous                                                                2mm-step          command          are        designed        as
frame and a levitation table. A hybrid electromagnet is fixed                                                                 K P  25 , K I  10 , K D  0.5                         (45)
on the levitation table, and the attracting levitation force is                                                           In Fig. 3(b), there still have similar results as Fig. 3(a), and
produced by the magnetization of the electromagnetic coil. A
                                                                                                                          the MSE value is 6.902  103 mm2 . Consequently, the
servo control card is installed in the control computer, which
                                                                                                                          control coefficients of the PID control system should be
includes multi-channels of D/A, A/D, PIO and encoder
                                                                                                                          redesigned for various demands to satisfy the desirable
interface circuits. The moving displacement of the levitation
                                                                                                                          dynamic behavior.
table is fed back using a gap sensor. The control systems in
this study are realized in the Pentium PC via “Turbo C”                                                                          1.428mm    Unloading                                   2.01mm     Unloading     Table
                                                                                                                                                         Table Position
language to manipulate the coil current (i) in the
electromagnetic coil by way of voltage control (U), and the                                                                                 Position
                                                                                                                                                                                                  Command          Loading
                                                                                                                           0mm             Command           Loading              0mm
control intervals are all set at 6ms.                                                                                                                            Tracking Error                                            Tracking Error

                                                                                                                           0mm         MSE=1.108×10-3 mm2                                     MSE=1.492×10-3 mm2
                                                                                                                                                                  Coil Current                                             Coil Current

                                                                                                                                 10A                                                    10A
                                                                                                                            0A                                                    0A
                                                                                                                                                   (a)                                                    (b)
                                                                                                                          Fig. 4. Experimental results of FNN control system at load-variation
                                                                                                                          condition: (a) 1mm-step command. (b) 2mm-step command.
                                                                                                                              For comparison, the FNN control system in Section III-B
                                                                                                                          is also applied to control the hybrid maglev system. To show
                                                                                                      Digital             the effectiveness of the FNN with small rule set, the FNN has
        Servo                       x                        U
                                                                                                                          two, six, nine and one neuron at the input, membership, rule
        Control                                                                     xm
                                                                                                                          and output layer, respectively. It can be regarded that the
                                          A/D                       D/A                                                   associated fuzzy sets with Gaussian function for each input
                                        Converter                 Converter
                                                                                                                          signal are divided into N (negative), Z (zero) and P (positive),
                                                                                                                          and the number of rules with complete rule connection is nine.
                                         Pentium                      Memory                                              Moreover, some heuristics can be used to roughly initialize
                                                                                                                          the parameters of the FNN for practical applications. The
                                               Control Computer
                                                                                                                          effect due to the inaccurate selection of the initialized
Fig. 2. Computer-based control system.
                                                                                                                          parameters can be retrieved by the on-line learning
       1.428mm      Unloading Table Position                           2.01mm       Unloading Table Position              methodology. Therefore, for simplicity, the means of the
                                                                                                                          Gaussian functions are set at -1, 0, 1 for the N, Z, P neurons
                                                                                                                          and the standard deviations of the Gaussian functions are set
                                        Loading                                   Command
 0mm                                                             0mm                                                      at one. In addition, to test the learning ability of the FNN
                                            Tracking Error                                               Tracking Error
                                                                                                                          control system, all the initial connecting weight between the
             MSE=6.041×10-3 mm2                                                MSE=6.902×10-3 mm2
                                             Coil Current
                                                                                                          Coil Current    output layer and the rule layer are set to zero in the
       10A                                                             10A
                                                                                                                          experimentation. The responses of the table position, tracking
  0A                                                             0A
                              (a)                                                             (b)
                                                                                                                          error and coil current using the FNN control system due to
Fig. 3. Experimental results of PID control system at load-variation                                                      1mm-step and 2mm-step commands are depicted in Fig. 4(a)
condition: (a) 1mm-step command. (b) 2mm-step command.                                                                    and (b), where the respective MSE values are
    Some experimental results are provided here to                                                                        1.108  103 mm2 and 1.492  103 mm2 . From the
demonstrate the effectiveness of the PID, FNN and robust                                                                  experimental results, the overshoot responses at the transient
control systems. In the experimentation, the initial condition                                                            state are caused by the rough initialization of the network
of this hybrid maglev system is loaded by two pieces of iron                                                              parameters. After this, the tracking errors reduce to zero
disk with 3.7kg weight. The experimental results of the PID                                                               quickly even under the load variations. Although favorable
control system due to step commands are depicted in Fig. 3.                                                               tracking performance can be obtained, this control scheme
In Fig. 3(a), a 1mm-step command is set, and the gains of the                                                             seems to be too complex in practical applications.
PID control system are given as follows:                                                                                      In the end, the experimentation of the robust control
     K P  32 , KI  10 , K D  0.6                          (44)                                                         system is implemented. The gains of the robust control
Then, unloads one iron disk at 6s and reloads it at 12s, it is                                                            system are given as follows:
obvious that the position drift of the levitation is almost 1mm                                                                1  1000 , 2  65 , 3  65                         (46)
when unloading. The mean-square-error (MSE) value is                                                                      The selection of the positive tuning gains ( 1 , 2 , 3 ) is
 6.041103 mm2 . Because the gains in (44) are selected at                                                               concerned with the tracking speed. The tracking response
1mm-step command, these control gains may not keep the                                                                    converges slowly with small tuning gains, and the tracking
levitation height at 1mm during the unloading duration.                                                                   speed increases with large tuning gains. Due to the
Moreover, the gains of the PID control system for a
unavailable system dynamics, they are chosen via a trial and                                                   [3] M. Y. Chen, M. J. Wang, and L. C. Fu, “Modeling and controller design
                                                                                                                    of a maglev guiding system for application in precision positioning,”
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 6.150  104 mm2 and 8.061 104 mm2 . From the                                                                    control of a magnetic bearing system,” IEEE Trans. Syst., Man, and
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Fig. 5 with Figs. 3 and 4, it is obvious that the robust control                                                    vol. 50, no. 2, pp. 374–378, April 2003.
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                                                                                                                    levitation table with electromagnetic control and its vibration isolation
performance than the PID and FNN control systems.                                                                   method using direct disturbance cancellation combining optimal
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 0mm              Command                               0mm
                                       Tracking Error                                         Tracking Error        Jan. 2004.
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                                                                                               Coil Current
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  0A                                                    0A
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Fig. 5. Experimental results of robust control system at load-variation                                             915, March 2002.
condition: (a) 1mm-step command. (b) 2mm-step command.                                                         [12] C. M. Huang, J. Y. Yen, and M. S. Chen, “Adaptive nonlinear control of
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                                      V. CONCLUSIONS                                                                no. 12, pp.13571367, Dec. 2000.
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                                                                                                                    system,” IEEE Proc. Computational Intelligence Robotics and
FNN and robust control systems for a hybrid maglev system
                                                                                                                    Automation, vol. 3, pp. 14051410, 2003.
to demonstrate the performance comparison of these three                                                       [14] X. Jie and B. T. Kulakowski, “Sliding mode control of active
model-free control strategies. The PID control system                                                               suspension for transit buses based on a novel air-spring model,” IEEE
belongs to an event-based linear controller. There are larger                                                       Proc. Amer. Cont. Conf., vol. 5, pp. 3768–3773, 2003.
MSE values under the occurrence of load variations;                                                            [15] W. G. Hurley, M. Hynes, and W. H. Wolfle, “PWM control of a
                                                                                                                    magnetic suspension system,” IEEE Trans. Educ., vol. 47, no. 2, pp.
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                                   ACKNOWLEDGMENTS                                                                  Application of gain scheduled H∞ robust controllers to a magnetic
                                                                                                                    bearing,” IEEE Trans. Contr. Syst. Technol., vol. 4, no. 5, pp. 484–493,
    The authors would like to acknowledge the financial
                                                                                                                    Sept. 1996.
support of the National Science Council of Taiwan, R.O.C.                                                      [21] C. Y. Kim and K. H. Kim, “Gain scheduled control of magnetic
through grant number NSC 93-2213-E-155-014.                                                                         suspension systems,” IEEE Proc. Amer. Contr. Conf., pp. 3127–3131,
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