Mode-switching control schemes for a Maglev platform

IFAC Workshop on Adaptation and Learning in Control and Signal Processing, and IFAC Workshop on Periodic Control Systems, Yokohama, Japan, August 30 – September 1, 2004

MODE-SWITCHING CONTROL SCHEMES FOR A MAGLEV PLATFORM Liang-Chun Yao and Jian-Shiang Chen Department of Power Mechanical Engineering National Tsing-Hua University Hsinchu, Taiwan, 30043, ROC Tel: +886-3-5715153 ext. 3792 E-mail: [email protected]

Abstract: This paper proposed a mode-switching control scheme (MSC) that combines different sliding mode control (SMC) schemes to alleviate adverse effect while achieve precise positioning of a maglev platform. Firstly, the MSC scheme: utilized a slidingmode control scheme and an integral sliding-mode controller, mode switching occurs while entering the vicinity of a preset operating point. Secondly, the MSC scheme utilized two integral SMC with added gain margin that can eliminate wind-up effort due to the use of integral SMC effectively. The feasibility of the proposed MSC schemes are demonstrated and compared through simulation studies. Preliminary simulation results on a maglev system have further validated its feasibility and applicability. Copyright © 2004 IFAC Keywords: Maglev Platform, Sliding Mode, Mode Switching

Iwasaki, et al. (1998), applied the MSC in a two degree-of-freedom position control system to achieve both fast response and high accuracy. Yamguchi, et al. (1998), proposed the MSC with initial value compensation to determine the optimal switching conditions for the disk drives. This paper investigates an inherently unstable maglev platform with six-degree-of-freedom that exhibits nonlinearity, disturbance and dynamic coupling. Here, we utilize MSC design with SMC schemes for a maglev platform and the simulation results further demonstrate that the controller can achieve precise positioning.

1. INTRODUCTION Ultra precision machining or manufacturing, e.g. in semiconductor industries, requires precise positioning control. Both air bearing and magnetic levitation have been proposed to reduce the friction effect motion systems to further enhance its positioning accuracy. Maglev system is always considered as a better choice over the air bearing due to its advantages in performance, construction, maintenance and applicability in space. The maglev platform is known to be a highly nonlinear system in nature. In general, the sliding mode control was applied to the control of maglev system mostly due to its robustness to system uncertainties. Lu (1995) proposed a sliding mode controller with a robust pole-assignment to place the poles at the desired locations to reject parameter variations and external disturbance. Hung (1998) simplified the coupling of each degree of freedom and used sliding mode control to achieve precise positioning in a maglev system. Shan and Menq (2002) added an adaptive filter between the sliding mode controller and the plant to attenuate the measurement noisy. Aimed at the nonlinear system, the Mode Switching Control (MSC) is usually adopted to improve the accuracy and robustness in the controller design.

2. DYNAMIC MODEL OF THE MAGLEV SYSTEM UNDER STUDY Fig. 1 shows the maglev platform under study, which is consisting of twelve electromagnets. Two magnets Mz1a and Mz2a pull up the table, while another two corresponding magnets Mz1b and Mz2b locate on the other side of the frame, respectively, that pull down the table. Four magnets My1b , My2b , My3b and My4b provide attraction forces in the direction of y-axis, while the other four magnets My1a, My2a, My3a and My4a exert forces on the table in the opposition direction. The geometric arrangement of these eight 403

where α is a constant, I denotes the controlled current(unit: A), g is the air gap length(unit: meter). Hence, each attraction force in vertical modes and lateral modes can be described as

magnets in the y-direction was proposed by Hung (1998) to decouple rolling and yawing motions when the table is at its neutral position. The table is driven by an embedded linear motor in the transverse (x-) direction with its displacement measured from a noncontact linear optical encoder. The geometric relationships between position of the movable table and magnets are shown in Fig. 2, where L and B denote half of the horizontal and vertical distances between the centers of magnets position My1b, My2b and My1b, My2b, respectively. Assuming that the table’ s center of mass and its center of geometry coincide with each other, and that the table operates with small angular motions, a linearized equations of motion is written as H&q&  Q

M 0  0 H 0 0  0 q  [ zc



0

0 0 M 0 0 0

I yy 0 0 0 0

θ

φ δ

yc

Q  Fz τ yy

0 0 0 I xx 0 0

0 0 0 0 I zz 0

2  i2 imb f m  α y  ma2  β y y w y  ym  m



0 0  0  0 0  M

Fx

vertical modes , lateral modes

wy

transverse modes



I yyθ&&  τ yy  d θ (t)  L( f r  fl )  xc ( f r  f l )  dθ (t) I zzδ&&  τ yy  d δ (t )  L( f 2  f1)  x c ( f1  f2  f 3  f4 )  d δ (t)

(6) 3. MODE SWITCHING CONTROL SCHEME USING SMC The SMC design provides an effective approach in maintaining both stability and robust performance to modeling imprecision. However, the performance of controller will be deteriorated with augmented boundary-layer approach, and the steady-state error will be inevitable. The integral SMC can reduce the steady-state error and chattering as compared to boundary-layer approach, but the additional integral term could cause the actuator’ s windup. This paper proposed two mode-switching schemes: In the first one, mode switching is performed between a slidingmode control scheme and an integral sliding-mode control at preset conditions while entering the vicinity of an operating point. In the second one, both two modes utilize integral SMC with gain margin that can reduce control effort effectively. 3.1 Mode-Switching Between SMC and ISMC

with y1  y2  y3  y4 . In general, the attraction force can be modeled as

g2

  , m  1, 2,3,4 (5)  

I xxφ&&  τ xx  dφ (t)  B( f3  f4 )  dφ (t )

1 y1 ( L  xc )  y2 ( L  x c ) 2 2L 1 zl ( L  xc )  zr ( L  xc )  w z (2) zc  2L 2 y  y3 z  zr y  y2 φ 4 θ l δ 1 2B 2L 2L

f 

(4)

Mz&&c  Fz  d z (t )  Mg   fl  fr   d z (t )

T



αI 2



2

n  l, r

Mx&&c  Fx  d x (t ) My&&c  Fy  d y (t )  f 1  f 2  f 3  f 4  d y (t )

where M is the mass of the levitated table, Ixx, Iyy and Izz denote the moment of the table around the x-, y-, zaxes, respectively. xc, yc and zc are the center of mass of the table in the x-, y-, z-directions, respectively. φ , θ and δ denote the table’ s rotation in the x-, y-, zdirections, respectively. Fx, Fy, Fz and τ xx , τ yy , τ zz are the control forces and torques exerting on the table in the x-, y-, z-directions. The control forces and torques generated by the twelve magnets can be expressed as follows. Each sensor located on the center of the six magnets (Mz1a, Mz2a, My1a , My2a, My3a , My4a) that can measure the air gap between table and magnets. Let the variables of air gaps of each degree of freedom are zl, zr, y1, y2, y3 and y4, the linear relationship between the displacement of the center of mass (yc, zc) and rotational angles( φ , θ , δ ) can be described as yc 

 ,  

where fl and fr are the total force between Mz1a, Mz1b and Mz2a, Mz2b; f1, f2, f3 and f4 denote the net forces between My1a , My2a , My3a , My4a and My1b, My2b, My3b , My4b. ila and ira are the controlled currents in Mz1a and Mz2a. Similarly in ilb, ira; α z and β z are constant parameters of the corresponding magnet. Because magnetic force and gravity force are the external forces, we can have the equations of motion expressed as

(1)

x c ]T ,

F y τ xx τ zz

2  i2 inb f n  α z  na2  β z z w z  zn 2  n

Combining the two schemes to yield  s  q~& i  λ ~ qi   2 ~& ~  s I  qi  2λ I q i  λ I 

(3)

404



t

ts

s  i q~i dt

sI  i

(7)

where s is the sliding variable, q~i  qi  qid , qi is the generalized coordinate, q id is the desire position,  i is the preset thickness, λ and λ I are the parameters of the sliding mode control and integral sliding mode control, respectively. t s is the timing for mode switching. However, if λ  2λ I , then sliding variable had produced discontinuous problem while switching. Moreover, the sliding variable inside the boundary layer, if λ  2 λ I , will switch to sI then sI will be greater than  . In this mode-switching condition, there exists a compatibility problem between s  

s qi



t

ts

~ qi dt

0  a ii min  a ii  a ii max

(9)

The estimate aˆ ii of gain a ii is the geometric mean of the above bounds

aˆii  aii min aii max

(10)

Eq.(9) can be rewritten as

γ ii1 

aˆ ii  γ ii aii

where γ ii  (a ii max / a ii min )1/ 2 . The bounds of geometry are depicted in Table 1 that assuming physical errors are all less than 10% in the Maglev platform under this study. The controller can be designed to perform robustness to the bounded multiplicative uncertainty, here γ is called the gain margin of this design (Slotine 1991). Assuming the desired positions are q d  0 , the control forces can be written as

Assuming the desired positions are q d  0 , the control forces can be written as ,

q i   qi

i  1, ..., 5 (9)

   s  aˆ ii 1   2λq 1q& i  λq2 1q i  K q 1 sat  qi 1   i i i   q 1     i  Qi    s qi 2    1  2  aˆ ii   2λqi 2 q&i  λ qi 2 qi  K qi 2 sat     2 q   i  

where Q i is generalized force,  qi 1 and  qi 2 are the boundary layer under two sliding-modes. Remarks: The mode-switching between SMC and ISMC has a compatibility problem in nominal control part. If the switching point satisfy q~i q~& i  0 and q~& i2  λ qi 1 ~ qi2 , then λ qi 2 

i

~ qi   qi ~ qi   qi

1 where the control gain aii  H ii is unknown but of known bounds

where  qi is the layer thickness of qi .

   

ts

i

&&  H 1Q q

q~i   qi , q~i   qi

q i   qi

0 t

Eq.(1) can be written as

i  1, ..., 5 (8)

  sq 1   λ qi1 H ii q& i  K qi 1 sat  i    q1    i  Qi    sq i 2  2   2λ qi 2 H ii q& i  λ qi 2 H ii qi  K qi 2 sat     qi 2

 q~ dt  q~ dt

where  qi is the switching region thickness of q~i .

Hence, this paper proposed a mode-switching condition to replace the original one. The physical variables q~i will replace the sliding variable to be the mode-switching condition. That will solve the compatibility problem and ensure the performance of MSC. Under this condition, the sliding variables and mode-switching condition can be rewritten as

s qi

ts

i  1, ..., 5 (8)

and s I   .

q~&i  λq ~ q i1 i   ~& ~ 2 q i  2λ qi 2 qi  λ qi 2 

 ~& 2 ~ q i  2λ qi 1 q i  λ qi1  q~& i  2λ q q~i  λ2 qi 2 i2 

q i   qi

,

q i   qi

i  1, ..., 5 (9)

where K qi 1 and K qi 2 can be expressed as

qi  ~ q& i  ~ qi q~i λ qi 1 ~ . We can solve λ qi 2 ~ q

k qi j  γ ii ( F j  η j )  (γ ii  1)  2λ qi j q& i  λ 2qi j q i , j=1, 2

i

to straighten out the compatibility in nominal control part.

The switching layer thickness  qi can decide the switching timing to combine two sliding mode flawlessly. If the switching timing is too early, then the switching layer thickness is too large that will create the integral redundancy problem. On the other hand, if the switching timing is too late, then the SMC is not robust to reject the uncertainty and disturbance. The detailed choice of switching layer thickness  qi

3.2 Mode-Switching Between Two ISMCs Another mode-switching is proposed to reduce control effort which using two ISMCs with gain margin. The compatibility problem is the same as the previous one, sliding variables and mode-switching condition can be rewritten as

had been determined in Yao, et al. (2003). 405

τ yy  L  xc Fz   f l  2L 2L  τ L  x yy f  c Fz   r 2L 2L

3.3 Stability of Mode-Switching Control Schemes Lemma 1 Itkis(1976), If the system dqi  f (t , qi , u ) dt

( i  1,..., n)

In the lateral modes, the relationship between the control forces( F y , τ xx , τ zz ) and total

satisfy the recahability condition,

electromagnetic forces( f1 , f 2 , f 3 , f 4 ) on the fourpair of magnets(My1a, My1b; My2a , My2b ; My3a, My3b ; My4a, My4b) can be written as

ds ds lim  0  lim s  0 dt s 0 dt

or, in a more concise way,

Fy  f 1  f 2  f 3  f 4

τ xx  B  f3  f 4 

d (s 2 ) 0 s 0 dt

lim

To control three output variables with four unconnected inputs, there exists redundancy in specifying these four electromagnetic forces. This paper proposed an optimal distribution of electromagnetic forces in the lateral modes. To yield optimal force distribution, the following performance index is minimized subject to three constraints described by (1):

Theorem 1 If the sliding surface S (t) in the statespace R (n ) by the scalar equation s (q; t ) =0 satisfy Lemma 1, the sliding motion is globally asymptotically stable of the proposed mode-switching control schemes in this paper.

J  f12  f 22  f32  f 42

which means to find out four desired forces that will minimize their own squared-sum value. The optimal distribution forces can be written as Liu (2003)

Proof. If the sliding surfaces can satisfy the conditions in Lemma 1, the Lyapunov function can be defined as

( L  2 xc )Fy τ zz    f1  4L 2L  ( L  2 x c ) Fy τ zz    f2  4L 2L  F τ  f  y  xx  3 4 2B   f  Fy  τ xx  4 4 2 B

1 2 (s  s2 2  L  sn 2 )  0 2 1

where n is sliding surfaces, (n-1) is switching times and V& 

n

 i 1

s i s&i  

n

η

(11)

τ zz  L  f 2  f1   x c  f1  f2  f3  f 4 

then it will drive the trajectories to switching surface and maintains it on this surface once it has been reached. The inequality suggests a necessary condition for system to have a singular Lyapunov function of the form V(q)=s(q)2.

V 

(10)

i

si  0

(12)

i 1

4. RESULTS AND DISCUSSIONS

where η is a strictly positive constant, V& is negative definite because V&  0 is only exist while of s i  0 . Moreover, we can choose the suitable sliding surfaces for proposed method and ensure V   as q   , the equilibrium is globally asymptotically stable. Consequently, the sliding motion is globally asymptotically stable of bi-direction switching of proposed mode -switching control schemes.

Simulations are performed in this section, which utilize the SMC, integral SMC, MSC I and MSC II without gain margin, MSC II. Moreover, the results have demonstrated the MSC I providing the good position control better than the other two and MSC II reducing control effort efficiently. CASE1: Mode-Switching Between SMC and ISMC The vertical modes including the displacement zc of zdirection and pitch angle θ , and the initial condition of the transverse displacement is defined at xc=0. Fig. 3 and Fig. 4 show the simulation results of the positioning control. When only the SMC is used, we can obtain good settling time, but if could cause larger steady-state error. When the SMC with integral control is used, the steady state error can be reduced, but this method requires longer settling time.

3.4 Distribution from electromagnetic forces Moreover, the geometric arrangement of these eight magnets must utilize the linear relationship to obtain the electromagnet force in each pair of magnets. In the vertical modes, the total electromagnetic forces f l and f r on the Mz1a, Mz1b and Mz2a, Mz2b can be written as

406

X. Shan and C. H. Menq (2002), Robust Disturbance Rejection for Improved Dynamic Stiffness of a Magnetic Suspension Stage, IEEE/ASME Trans. Mechatron., Vol. 7, No.3, pp.289-295, Sep. M. Iwasaki, K. Sakai and N. Matsui (1998), HighSpeed and High-Precision Table Positioning System by Using Mode Switching Control, Proceedings of the 24th Annual Conference of the IEEE, Vol.3, pp. 1727 –1732. T. Yamaguchi, H. Numasato(1998), H. Hirai, A mode-switching control for motion control and its application to disk drives: design of optimal mode-switching conditions, IEEE/ASME Transactions on Mechatronics, Vol.3, pp. 202209. Jean-Jacques E. Slotine and Weiping Li (1991), Applied Nonlinear Control, pp. 276-310, Prentice Hall, New Jersey. U. Itkis (1976), Control Systems of Variable Structure. pp. 16-20, Willey, New York. C. C. Liu (2003), The Design and Implementation of A Maglev Platform, Master Thesis, Department of Power Mechanical Engineering, National Tsing-Hua University, Taiwan. L. C. Yao, C. C. Liu, and J. S. Chen (2003), The Design and Implementation of a Maglev Platform Using Mode Switching, CSME Conference, B0125, pp. 161.

Furthermore, the MSC with a mode-switching condition proposed in this paper not only can achieve better settling time but also reduce the steady-state error. The lateral modes including the displacement yc of ydirection, yaw angle δ , roll angle φ , and the initial condition of the transverse displacement is defined at xc=0. From the Fig. 5 to Fig. 7, the MSC I can be divided into two parts, the SMC is used during the rising time and then switch to SMC with integral control near the operating point. Hence, this method can obtain precise positioning in this five -degree-offreedom efficiently. CASE2: Mode-Switching Between Two ISMCs In the Maglev platform, the magnitude of control effort is very important. Because electromagnets can’ t work under saturation region long time that cause damage of magnet winding. In this simulation, the modeling uncertainties are given in the system dynamics as described in Table 2. There choose the vertical displacement zc to demonstrate feasibility of proposed controller. This simulation results using MSC II are shown in Figs. 8-9 for regulator and tracking problem. From the simulation results, it can be seen that the control effort can be reduced effort efficiency by using MSC II. Moreover, the results show the capability of the MSC II to reduce the tracking error under parameter uncertainties.

Table 1 Bounds of the system parameters

5. CONCLUSIONS This paper compares two different Mode-Switching Control schemes and successfully applied to a Maglev platform to yield good positioning control performance. The simulation results have shown that MSC can provide better positioning performance than SMC and ISMC alone. However, parameter uncertainty will lead to the significant steady-state error. This problem can be solved by the adding of a gain margin to alleviate the integral windup. The second method not only can command the Maglev system to track the reference input, but also can reduce steady-state error due to parameters uncertainty and the control effort can yield better efficiency. The future wo rk would be to implement the proposed schemes on a Maglev platform to validate its efficacy further.

System parameter M Ixx

Bound 12.15  M  14.85 0.008838 I xx  0.010802

Iyy

0.14337  I yy  0.17523

Izz

0.06336  I zz  0.07744

Table 2 The system nominal parameters and actual parameters for simulate studies System parameter M Ixx Iyy Izz

Actual parameters 14 kg 0.011 kgm2 0.152 kgm2 0.067 kgm2

Nominal parameter 13.5 kg 0.00982 kgm2 0.15930 kgm2 0.07040 kgm2

REFERENCES Y. S. Lu (1995), Applications of Sliding Mode Control to Nonlinear Servo Systems, Ph.D. Dissertation, Department of Power Mechanical Engineering, National Tsing-Hua University, Taiwan. L. S. Hung (1998), A Multivariable Sliding Mode Control for Magnetic Suspension Systems and Its Applications, Ph.D. Dissertation, Department of Power Mechanical Engineering, National TsingHua University, Taiwan.

Fig. 1. Schematic drawing of a maglev platform. 407

Fig. 6. Regulate control of yaw angle using MSC I. Fig. 2. Geometric relationships between the movable table and electromagnets

Fig. 7. Regulate control of roll angle using MSC I.

Fig. 3. Regulate control of vertical direction using MSC I.

Fig. 8. Regulating control of vertical direction using MSC II.

Fig. 4. Regulate control of pitch angle using MSC I.

Fig. 5. Regulate control of lateral direction using MSC I. Fig. 9. Tracking control of the vertical direction using MSC II 408