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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 1,2 Department of Electrical Engineering, Yuan Ze University, Chung Li 320, Taiwan, R.O.C. 3 Department of Electronic Engineering, Ching Yun University, Chung Li 320, Taiwan, R.O.C. AbstractThis 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 R 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 equation: 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 x 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 Permanent Magnet F ( x, i ) x 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 Power Amplifier maglev system, the accurate dynamics model ( G( x, t ) and M ( x, t ) ) are assumed to be unknown in this study. Without Electromagnet loss of generality it is assumed that G( x, t ) is finite and Gap Sensor bounded away from zero for all x. Load Carrier III. CONTROL SYSTEMS DESIGN A. PID Control System mg In industrial application, a PID control system is the (a) 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 dt (11) (b) 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 ok (17) and output (o layer) layers is adopted [23]. The membership wk o 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: E 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) i linguistic variables with respect to the input nodes. In E 2( xi mi ) j 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 n k wkji i j (net j ( xi )) (13) i j ( i j )3 (22b) 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 wkok (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 (24) 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 (26) 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 * (37) uncertainty. 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 (27) 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 )) T (41) 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 (30) ( 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 6.902 103 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 Position Table Position language to manipulate the coil current (i) in the electromagnetic coil by way of voltage control (U), and the Position 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 0mm 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 x Oscilloscope two, six, nine and one neuron at the input, membership, rule Control xm and output layer, respectively. It can be regarded that the Card 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 Position Position Loading and the standard deviations of the Gaussian functions are set Loading Command 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 0mm Coil Current 0mm 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 1.108 103 mm2 and 1.492 103 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 6.041103 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,” error process to achieve the superior transient response in the IEEE Trans. Ind. Electron., vol. 50, no. 3, pp. 493–506, June 2003. experimentation considering the requirement of stability, the [4] J. Kaloust, C. Ham, J. Siehling, E. Jongekryg, and Q. 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