Microsyst Technol (2007) 13:831–835 DOI 10.1007/s00542-006-0284-y
TECHNICAL PAPER
Nonlinear controller design for a magnetic levitation device Ehsan Shameli Æ Mir Behrad Khamesee Æ Jan Paul Huissoon
Received: 30 June 2006 / Accepted: 2 October 2006 / Published online: 4 November 2006 Springer-Verlag 2006
Abstract Various applications of micro-robotic technology suggest the use of new actuator systems which allow motions to be realized with micrometer accuracy. Conventional actuation techniques such as hydraulic or pneumatic systems are no longer capable of fulfilling the demands of hi-tech micro-scale areas such as miniaturized biomedical devices and MEMS production equipment. These applications pose significantly different problems from actuation on a large scale. In particular, large scale manipulation systems typically deal with sizable friction, whereas micro manipulation systems must minimize friction to achieve submicron precision and avoid generation of static electric fields. Recently, the magnetic levitation technique has been shown to be a feasible actuation method for microscale applications. In this paper, a magnetic levitation device is recalled from the authors’ previous work and a control approach is presented to achieve precise motion control of a magnetically levitated object with sub-micron positioning accuracy. The stability of the controller is discussed through the Lyapunov method. Experiments are conducted and showed that the proposed control technique is capable of performing a positioning operation with rms accuracy of 16 lm over a travel range of 30 mm. The nonlinear control strategy
E. Shameli M. B. Khamesee (&) J. P. Huissoon Department of Mechanical and Mechatronics Engineering, University of Waterloo, Waterloo, ON, Canada e-mail:
[email protected] E. Shameli e-mail:
[email protected] J. P. Huissoon e-mail:
[email protected]
proposed in this paper showed a significant improvement in comparison with the conventional control strategies for large gap magnetic levitation systems. Keywords Mechatronics Magnetic levitation Micromanipulation Control
1 Introduction Certain applications of microrobotics technology require the use of new actuator systems, which allow motions to be realized with micrometre accuracy. Conventional actuation techniques such as hydraulic or pneumatic systems are not able to fulfill the demands of micromanipulation. In recent years, magnetic levitation actuation has become accepted as a feasible alternative when developing high performance actuators that deliver ultra fine position control. Several magnetic levitation systems (MLS) with small travel ranges have been proposed and studied in the literature (Shakir and Wonjong 2005). As the air gap of these systems either remained constant or its variation was relatively small, modeling and control strategies employed in these studies were often uncomplicated. In the case where DC electromagnets are employed and the air gap varies during the course of motion control, the highly unstable aspect of MLS and its inherent nonlinearities make the modeling and control problems very challenging. Several dynamic models of magnetic force have been proposed over the past years and different control strategies have been used with these and their performances compared. Slotine (1991) assumed that the magnetic force is directly proportional to the squared current in the coil and inversely proportional to the squared gap
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between the electromagnet and the levitated object. Jalili-Kharaajoo (2003, 2004), Alvarez-Sanchez et al. (2005), Jiunshian et al. (2005), and Fallaha et al. (2005) used the magnetic force model of Slotine and proposed sliding mode controllers (SMC) for magnetic levitation systems. Jalili-Kharaajoo (2003, 2004) compared the performance of sliding mode and feedback linearization techniques by simulation. The results were very conclusive concerning the effectiveness of the sliding mode technique over the feedback linearization. According to Jiunshian et al. (2005) a neural network estimator was implemented to calculate the equivalent control and reject chattering; as a result, chattering was successfully eliminated and the error performance of the SMC was improved. Khamesee et al. (2002, 2003) performed a comparative study of a model-reference adaptive controller and a conventional PID controller for 3D position control of a magnetically levitated microrobot. The experiments showed that the adaptive control technique has a better performance in rejecting the modeling uncertainties and variations in the payload with positioning accuracy of 0.1 mm over a 30 mm traveling range. Chen et al. (2004, 2001) proposed an adaptive sliding mode controller for a dual axis magnetic levitation system and reduced the positioning error to 100 lm over a levitation domain of 25 mm. Ximin et al. (2002a, b) and Shih-Kang et al. (2003) proposed an indirect adaptive control approach for a 6DOF ultra precise magnetic levitation stage. They used two different disturbance rejection algorithms to increase the positioning stiffness of the levitated stage. An internal modelbased control is used to reject the narrow band disturbances. For wide band disturbances, a chatter free sliding mode algorithm is proposed. The stage positioning error is reduced to 20 lm over a 800 lm travel range. The fuzzy control technique is also used in (Lepetic et al. 2001; Pen et al. 2002; Yen-Chen et al. 2003) for single axis levitation systems. In this paper a large gap magnetic levitation system is introduced based on the authors’ previous work in (Khamesee and Shameli 2005). A new model for the magnetic force is proposed and experimentally verified, and the maximum force estimation error of 3.2% has been measured. A nonlinear control technique was designed, and the accuracy of the position control system was significantly improved so that the rms error was reduced to 16 lm over a travel range of 30 mm.
2 Description of the experimental setup The experimental setup schematic is shown in Fig. 1. The levitation system consists of a set of electromag-
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Fig. 1 Schematic of the levitation system
nets, a disk pole piece and an iron yoke. The pole piece is made of soft magnetic iron and connects the electromagnet poles together. A laser line displacement sensor is used to measure the position of the levitated object in the vertical direction. The laser sensor has a resolution of 0.05 lm and an accuracy of 2 lm. The controller communicates with the current control amplifier and the laser sensor via a 16-bit A/D converter and a 16-bit D/A converter. The objective of the closed-loop control system is to levitate a small permanent magnet by adjusting the electrical currents in the electromagnets. Generating an appropriate control command needs a precise dynamic model of the levitation system which will be discussed in the next sect.
3 Dynamic modeling and controller design Assuming the movement of the magnet is in the vertical direction and neglecting the friction and drag force of the air, the dynamic equation of motion for the permanent magnet is expressed as: € ¼ fmag mg mZ
ð1Þ
where m is the permanent magnet’s mass in kg, g is the € is the magnet’s acceleration due to gravity (m/s2), Z acceleration (m/s2) and fmag is the magnetic force in the Z direction. The magnetic force is assumed to be of the form: fmag ði; ZÞ ¼ ða Z þ bÞ i
ð2Þ
where a and b are constants that should be identified based on the system properties. The force model in Eq. 2 was experimentally verified for the levitation system and successfully estimated the magnetic force over a range of 32 mm with maximum estimation error of 3.2%. Figure 2 provides the force estimation error over the operation range of the levitation system.
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the controller gains are designed for the linearized system, the nonlinear nature of the system might result in control instability. In the next section, the stability of the proposed controller will be investigated to guarantee the stability of the control system for a given K1 and K2
4 Stability analysis The equivalent mathematical model of the closed loop control system in Fig. 3 is expressed as:
Fig. 2 Percent of force estimation error
Combining Eqs. 1, 2, the dynamic equation of motion for a levitated permanent magnet is obtained as: € ¼ a iZþ b ig Z m m
ð3Þ
Where m = 0.00659, a = 0.899 and b = 0.1164. Equation 3 is used to design a controller for the levitation system. Figure 3 illustrates the schematic diagram of the proposed control technique. The control signal is composed of three parts: the signal from the state feedback controller, the weight compensator signal and the integrator signal. The state feedback controller is designed for the linearized system in the vicinity of the operating point. Coefficients K1 and K2 are obtained using the pole placement technique for the state space representation of the linearized system. The values K1 and K2 are set as 46.3 and 6.3 respectively. Signal ib is used to compensate the weight of the levitated permanent magnet at any point and is calculated by equating fmag in Eq. 2 with the weight. An integrator term was used to cancel out the steady state error. Gain k ranging from 0 to 15 · 10–6 can have different values and is scheduled based on the positioning error. For high error values, k is large while for small errors, k is small. The appropriate choice of k depends on the state feedback gains K1 and K2 and the error bound of the laser sensors. It is important to note that although
x_ 1 ¼ x2 a b x1 þ ug x_ 2 ¼ m m
ð4Þ ð5Þ
where u¼
mg K1 ðx1 xd Þ K2 ðx2 x_ d Þ ax1 þ b
ð6Þ
and x_ d is defined as: x_ d ¼
200ksgnðx1 xcmd Þ if jx1 xcmd j>0:000010 0 if jx1 xcmd j\0:000010
and xd ð0Þ ¼ xcmd
ð7Þ
If a positive definite Lyapunov function can be found such that its boundaries are negative along the state space equations of the system, then the stability of the controller is proven. As the first step, the equilibrium point of the system should be mapped to the origin through a change of variable: ^y ¼ ^x ^xd
ð8Þ
Hence, the resultant system will be: y_ 1 ¼ y2 a b y_ 2 ¼ ð y1 þ xd Þ þ ðK1 y1 K2 y2 Þ m m
ð9Þ ð10Þ
Fig. 3 Schematic diagram of the proposed controller
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If the Lyapunov function is defined as: P11 P12 y1 1 V ¼ ½ y1 y2 2 P12 P22 y2 1 ¼ P11 y21 þ 2P12 y1 y2 þ P22 y22 2 then V_ is calculated as: V_ ¼ ðP11 y1 þ P12 y2 Þy_ 1 þ ðP12 y1 þ P22 y2 Þy_ 2
Therefore: W ¼ y1 ðP11 K1 UP22 K2 UP12 Þy2 6
Also, based on algebraic inequalities, ð12Þ
Substituting y_ 1 and y_ 2 from Eqs. 9 and 10 in Eq. 12 and arranging the terms, V_ is obtained as: V_ ¼ W K1 UP12 y21 ðK2 UP22 P12 Þy22
ð13Þ
where W ¼ y1 ðP11 K1 UP22 K2 UP12 Þy2
ð14Þ
and U¼
a b ð y1 þ xd Þ þ m m
ðK1 P22 þ K2 P12 Þkyk2
ð11Þ
K1 UP12 y21 þ ðK2 UP22 P12 Þy22 >Hkyk2 where H ¼ min K1 UP12 ; K2 UP22 P12
max U min U _ V6 ðK1 P22 þ K2 P12 Þ H kyk2 2
ð15Þ
It should be noted that since (y1 + xd) is bounded to the working domain of the levitation system, the value of F is always positive to guarantee an attractive magnetic force to the permanent magnet. The positive definiteness of function V is guaranteed through conditions P11 > 0 and P11 P22 – P212 > 0. Assuming that:
P11 ¼
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ð20Þ
ð21Þ
max U min U ðK1 P22 þ K2 P12 Þ 2
ð22Þ
the system will be asymptotically stable. In the next section, the performance of the closed loop controller is shown experimentally. 5 Experimental results
ð16Þ
and
Fig. 4 Step response of the proposed controller
Therefore
H[
max U þ min U ðK1 P22 þ K2 P12 Þ 2 ¼ 7:3ðK1 P22 þ K2 P12 Þ
ð19Þ
If Q is selected such that
P12 [0
max U min U 2 ð18Þ
ð17Þ
The experimental setup shown in Fig. 1 was used to levitate a small cylindrical permanent magnet. The magnet has diameter of 1 cm and height of 1 cm and weights 6.59 g. Figure 4 illustrates the step response of the levitated permanent magnet. As is shown in this
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Fig. 4, the proposed control strategy provides precise and stable positioning over a 30 mm traveling range. The maximum steady state error measured was 41 lm. The rms of the steady state step response error is 16 lm and the approximate settling time of the controller is Ts = 0.38 s. 6 Conclusion A nonlinear model for the magnetic force of magnetic levitation device was presented. The magnetic force model was then used to propose a control technique for position control of a magnetically levitated permanent magnet. A Lyapunov based stability analysis was performed to prove the stability of the control technique. The proposed controller performed a precise positioning operation over an operation range of 30 mm with an rms positioning error of 16 lm, which is a significant improvement over available control strategies in the literature for large gap magnetic levitation systems. As a future step, an advanced control scheme for 3D-position control of a levitated object is under development to provide large gap three-dimensional levitation in practical applications.
References Alvarez-Sanchez E, Alvarez-Gallegos J, Castro-Linares R (2005) Modeling and controller design of a magnetic levitation system. Second International Conference on Electrical and Electronics Engineering 5:330–334 Chen MY, Wang CC, Fu LC (2001) Adaptive sliding mode controller design of a dual-axis maglev positioning system. Proceedings of the 2001 American Control Conference 5:3731–3736 Chen MY, Tsai CF, Fu LC (2004) Design and control of a 2Delectro-magnetic suspension actuator. Proceedings of the 2004 IEEE International Conference on Control Applications 1:93–98 Fallaha C, Kanaan H, Saad M (2005) Real time implementation of a sliding mode regulator for current-controlled magnetic levitation system. Proceedings of the 2005 IEEE International symposium on intelligent control 1:696–701
835 Jalili-Kharaajoo M (2003) Sliding mode control of voltagecontrolled magnetic levitation systems. Proceedings of 2003 IEEE Conference on Control Applications 1:83–86 Jalili-Kharaajoo M (2004) Robust variable structure control applied to voltage controlled magnetic levitation systems. 2nd IEEE International Conference on Industrial Informatics 1:335–338 Jiunshian P, Jianming L, Yasser M, Yahaghi T (2005) Neurosliding mode control for magnetic levitation systems. IEEE International Symposium on Circuits and Systems (ISCAS 2005) 5:5130–5133 Khamesee M, Kato N, Nomura Y (2002) Design and control of a microrobotic system using magnetic levitation. IEEE ASME Trans Mechatron 7(1):1–14 Khamesee M, Kato N, Nomura Y (2003) Performance improvement of a magnetically levitated microrobot using an adaptive control. Proceedings of the International Conference on MEMS NANO and Smart Systems 1:332–338 Khamesee MB, Shameli E (2005) Regulation technique for a large gap magnetic field for third non-contact manipulation. J Mechatron 15(9):1073–1087 Lepetic M, Skrjanc I, Chiacchiarini HG, Matko D (2001) Predictive control based on fuzzy model: a case study. The 10th IEEE International Conference on Fuzzy Systems 3:868–871 Peng Y, Qiang Z, Lianbing L (2002) Design of fuzzy weight controller in singleaxis magnetic suspension system. Proceedings of the fourth World Congress on Intelligent Control and Automation 4:3027–3030 Shakir H, Won-jong K (2005) Nanoscale path planning and motion control. Proc 2005 American Control Conf 5:3604– 3609 Shih-Kang K, Ximin S, Chia-Hsiang M (2003) Large travel ultra precision x– y– h motion control of a magnetic-suspension stage. IEEE ASME Trans Mechatron 8(3):334–341 Slotine JJ (1991) Applied nonlinear control. Englewood Cliffs, Prentice-Hall, NJ Ximin S, Shih-Kang K, Jihua Z, Chia-Hsiang M (2002a) Ultra precision motion control of a multiple degrees of freedom magnetic suspension stage. IEEE ASME Trans Mechatron 7(1):67–78 Ximin S, Chia-Hsiang M (2002b) Robust disturbance rejection for improved dynamic sti.ness of a magnetic suspension stage IEEE ASME Trans Mechatron 7(3):289–295 Yen-Chen C, Shinq-Jen W, Tsu-Tian L (2003) Minimum-energy neural-fuzzy approach for current/voltage-controlled electromagnetic suspension system, Proceedings of the 2003 IEEE International Symposium on Computational Intelligence in Robotics and Automation 3:1405–1410
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