Nonlinear controller design for a magnetic levitation device

Transcription

1 Microsyst Technol (007) 13: DOI /s y TECHNICAL PAPER Nonlinear controller design for a agnetic levitation device Ehsan Shaeli Æ Mir Behrad Khaesee Æ Jan Paul Huissoon Received: 30 June 006 / Accepted: October 006 / Published online: 4 Noveber 006 Ó Springer-Verlag 006 Abstract Various applications of icro-robotic technology suggest the use of new actuator systes which allow otions to be realized with icroeter accuracy. Conventional actuation techniques such as hydraulic or pneuatic systes are no longer capable of fulfilling the deands of hi-tech icro-scale areas such as iniaturized bioedical devices and MEMS production equipent. These applications pose significantly different probles fro actuation on a large scale. In particular, large scale anipulation systes typically deal with sizable friction, as icro anipulation systes ust iniize friction to achieve subicron precision and avoid generation of static electric fields. Recently, the agnetic levitation technique has been shown to be a feasible actuation ethod for icroscale applications. In this paper, a agnetic levitation device is recalled fro the authors previous work and a control approach is presented to achieve precise otion control of a agnetically levitated object with sub-icron positioning accuracy. The stability of the controller is discussed through the Lyapunov ethod. Experients are conducted and showed that the proposed control technique is capable of perforing a positioning operation with rs accuracy of 16 l over a travel range of 30. The nonlinear control strategy E. Shaeli M. B. Khaesee (&) J. P. Huissoon Departent of Mechanical and Mechatronics Engineering, University of Waterloo, Waterloo, ON, Canada e-ail: khaesee@uwaterloo.ca E. Shaeli e-ail: seshael@uwaterloo.ca J. P. Huissoon e-ail: jph@uwaterloo.ca proposed in this paper showed a significant iproveent in coparison with the conventional control strategies for large gap agnetic levitation systes. Keywords Mechatronics Magnetic levitation Microanipulation Control 1 Introduction Certain applications of icrorobotics technology require the use of new actuator systes, which allow otions to be realized with icroetre accuracy. Conventional actuation techniques such as hydraulic or pneuatic systes are not able to fulfill the deands of icroanipulation. In recent years, agnetic levitation actuation has becoe accepted as a feasible alternative when developing high perforance actuators that deliver ultra fine position control. Several agnetic levitation systes (MLS) with sall travel ranges have been proposed and studied in the literature (Shakir and Wonjong 005). As the air gap of these systes either reained constant or its variation was relatively sall, odeling and control strategies eployed in these studies were often uncoplicated. In the case DC electroagnets are eployed and the air gap varies during the course of otion control, the highly unstable aspect of MLS and its inherent nonlinearities ake the odeling and control probles very challenging. Several dynaic odels of agnetic force have been proposed over the past years and different control strategies have been used with these and their perforances copared. Slotine (1991) assued that the agnetic force is directly proportional to the squared current in the coil and inversely proportional to the squared gap

2 83 Microsyst Technol (007) 13: between the electroagnet and the levitated object. Jalili-Kharaajoo (003, 004), Alvarez-Sanchez et al. (005), Jiunshian et al. (005), and Fallaha et al. (005) used the agnetic force odel of Slotine and proposed sliding ode controllers (SMC) for agnetic levitation systes. Jalili-Kharaajoo (003, 004) copared the perforance of sliding ode and feedback linearization techniques by siulation. The results were very conclusive concerning the effectiveness of the sliding ode technique over the feedback linearization. According to Jiunshian et al. (005) a neural network estiator was ipleented to calculate the equivalent control and reject chattering; as a result, chattering was successfully eliinated and the error perforance of the SMC was iproved. Khaesee et al. (00, 003) perfored a coparative study of a odel-reference adaptive controller and a conventional PID controller for 3D position control of a agnetically levitated icrorobot. The experients showed that the adaptive control technique has a better perforance in rejecting the odeling uncertainties and variations in the payload with positioning accuracy of 0.1 over a 30 traveling range. Chen et al. (004, 001) proposed an adaptive sliding ode controller for a dual axis agnetic levitation syste and reduced the positioning error to 100 l over a levitation doain of 5. Xiin et al. (00a, b) and Shih-Kang et al. (003) proposed an indirect adaptive control approach for a 6DOF ultra precise agnetic levitation stage. They used two different disturbance rejection algoriths to increase the positioning stiffness of the levitated stage. An internal odelbased control is used to reject the narrow band disturbances. For wide band disturbances, a chatter free sliding ode algorith is proposed. The stage positioning error is reduced to 0 l over a 800 l travel range. The fuzzy control technique is also used in (Lepetic et al. 001; Pen et al. 00; Yen-Chen et al. 003) for single axis levitation systes. In this paper a large gap agnetic levitation syste is introduced based on the authors previous work in (Khaesee and Shaeli 005). A new odel for the agnetic force is proposed and experientally verified, and the axiu force estiation error of 3.% has been easured. A nonlinear control technique was designed, and the accuracy of the position control syste was significantly iproved so that the rs error was reduced to 16 l over a travel range of 30. Description of the experiental setup Fig. 1 Scheatic of the levitation syste The experiental setup scheatic is shown in Fig. 1. The levitation syste consists of a set of electroagnets, a disk pole piece and an iron yoke. The pole piece is ade of soft agnetic iron and connects the electroagnet poles together. A laser line displaceent sensor is used to easure the position of the levitated object in the vertical direction. The laser sensor has a resolution of 0.05 l and an accuracy of l. The controller counicates with the current control aplifier 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 syste is to levitate a sall peranent agnet by adjusting the electrical currents in the electroagnets. Generating an appropriate control coand needs a precise dynaic odel of the levitation syste which will be discussed in the next sect. 3 Dynaic odeling and controller design Assuing the oveent of the agnet is in the vertical direction and neglecting the friction and drag force of the air, the dynaic equation of otion for the peranent agnet is expressed as: Z ¼ f ag g ð1þ is the peranent agnet s ass in kg, g is the acceleration due to gravity (/s ), Z is the agnet s acceleration (/s ) and f ag is the agnetic force in the Z direction. The agnetic force is assued to be of the for: f ag ði; ZÞ ¼ ða Z þ bþi ðþ a and b are constants that should be identified based on the syste properties. The force odel in Eq. was experientally verified for the levitation syste and successfully estiated the agnetic force over a range of 3 with axiu estiation error of 3.%. Figure provides the force estiation error over the operation range of the levitation syste.

3 Microsyst Technol (007) 13: the controller gains are designed for the linearized syste, the nonlinear nature of the syste ight result in control instability. In the next section, the stability of the proposed controller will be investigated to guarantee the stability of the control syste for a given K 1 and K 4 Stability analysis The equivalent atheatical odel of the closed loop control syste in Fig. 3 is expressed as: Fig. Percent of force estiation error Cobining Eqs. 1,, the dynaic equation of otion for a levitated peranent agnet is obtained as: _x 1 ¼ x _x ¼ a x 1 þ b u g ð4þ ð5þ Z ¼ a i Z þ b i g ð3þ Where = , a = and b = Equation 3 is used to design a controller for the levitation syste. Figure 3 illustrates the scheatic diagra of the proposed control technique. The control signal is coposed of three parts: the signal fro the state feedback controller, the weight copensator signal and the integrator signal. The state feedback controller is designed for the linearized syste in the vicinity of the operating point. Coefficients K 1 and K are obtained using the pole placeent technique for the state space representation of the linearized syste. The values K 1 and K are set as 46.3 and 6.3 respectively. Signal i b is used to copensate the weight of the levitated peranent agnet at any point and is calculated by equating f ag in Eq. with the weight. An integrator ter was used to cancel out the steady state error. Gain k ranging fro 0 to can have different values and is scheduled based on the positioning error. For high error values, k is large while for sall errors, k is sall. The appropriate choice of k depends on the state feedback gains K 1 and K and the error bound of the laser sensors. It is iportant to note that although u ¼ g ax 1 þ b K 1ðx 1 x d Þ K ðx _x d Þ ð6þ and _x d is defined as: _x d ¼ 00ksgn ð x 1 x cd Þ if jx 1 x cd j>0: if jx 1 x cd j\0: and x d ð0þ¼x cd ð7þ If a positive definite Lyapunov function can be found such that its boundaries are negative along the state space equations of the syste, then the stability of the controller is proven. As the first step, the equilibriu point of the syste should be apped to the origin through a change of variable: ^y ¼ ^x ^x d Hence, the resultant syste will be: ð8þ _y 1 ¼ y ð9þ _y ¼ a ð y 1 þ x d Þþ b ð K 1 y 1 K y Þ ð10þ Fig. 3 Scheatic diagra of the proposed controller

4 834 Microsyst Technol (007) 13: If the Lyapunov function is defined as: V ¼ 1 ½ y P 11 P 1 y1 1 y Š P 1 P y ¼ 1 P 11y 1 þ P 1y 1 y þ P y ð11þ then V _ is calculated as: _V ¼ ðp 11 y 1 þ P 1 y Þ_y 1 þ ðp 1 y 1 þ P y Þ_y ð1þ Substituting _y 1 and _y fro Eqs. 9 and 10 in Eq. 1 and arranging the ters, _ V is obtained as: _V ¼ W K 1 UP 1 y 1 ð K UP P 1 Þy ð13þ W ¼ y 1 ðp 11 K 1 UP K UP 1 Þy ð14þ and U ¼ a ð y 1 þ x d Þþ b ð15þ It should be noted that since (y 1 + x d ) is bounded to the working doain of the levitation syste, the value of F is always positive to guarantee an attractive agnetic force to the peranent agnet. The positive definiteness of function V is guaranteed through conditions P 11 > 0 and P 11 P P 1 > 0. Assuing that: P 1 [0 ð16þ and axu þ inu P 11 ¼ ðk 1 P þ K P 1 Þ ¼ 7:3ðK 1 P þ K P 1 Þ ð17þ Therefore: ax U in U W ¼ y 1 ðp 11 K 1 UP K UP 1 Þy 6 ðk 1 P þ K P 1 Þkyk ð18þ Also, based on algebraic inequalities, K 1 UP 1 y 1 þ ð K UP P 1 Þy >Hkyk H ¼ in K 1 UP 1 ; K UP P 1 Therefore ax U in U _V6 ðk 1 P þ K P 1 Þ H kyk If Q is selected such that H[ ax U in U ð19þ ð0þ ð1þ ðk 1 P þ K P 1 Þ ðþ the syste will be asyptotically stable. In the next section, the perforance of the closed loop controller is shown experientally. 5 Experiental results The experiental setup shown in Fig. 1 was used to levitate a sall cylindrical peranent agnet. The agnet has diaeter of 1 c and height of 1 c and weights 6.59 g. Figure 4 illustrates the step response of the levitated peranent agnet. As is shown in this Fig. 4 Step response of the proposed controller

5 Microsyst Technol (007) 13: Fig. 4, the proposed control strategy provides precise and stable positioning over a 30 traveling range. The axiu steady state error easured was 41 l. The rs of the steady state step response error is 16 l and the approxiate settling tie of the controller is T s = 0.38 s. 6 Conclusion A nonlinear odel for the agnetic force of agnetic levitation device was presented. The agnetic force odel was then used to propose a control technique for position control of a agnetically levitated peranent agnet. A Lyapunov based stability analysis was perfored to prove the stability of the control technique. The proposed controller perfored a precise positioning operation over an operation range of 30 with an rs positioning error of 16 l, which is a significant iproveent over available control strategies in the literature for large gap agnetic levitation systes. As a future step, an advanced control schee for 3D-position control of a levitated object is under developent to provide large gap three-diensional levitation in practical applications. References Alvarez-Sanchez E, Alvarez-Gallegos J, Castro-Linares R (005) Modeling and controller design of a agnetic levitation syste. Second International Conference on Electrical and Electronics Engineering 5: Chen MY, Wang CC, Fu LC (001) Adaptive sliding ode controller design of a dual-axis aglev positioning syste. Proceedings of the 001 Aerican Control Conference 5: Chen MY, Tsai CF, Fu LC (004) Design and control of a Delectro-agnetic suspension actuator. Proceedings of the 004 IEEE International Conference on Control Applications 1:93 98 Fallaha C, Kanaan H, Saad M (005) Real tie ipleentation of a sliding ode regulator for current-controlled agnetic levitation syste. Proceedings of the 005 IEEE International syposiu on intelligent control 1: Jalili-Kharaajoo M (003) Sliding ode control of voltagecontrolled agnetic levitation systes. Proceedings of 003 IEEE Conference on Control Applications 1:83 86 Jalili-Kharaajoo M (004) Robust variable structure control applied to voltage controlled agnetic levitation systes. nd IEEE International Conference on Industrial Inforatics 1: Jiunshian P, Jianing L, Yasser M, Yahaghi T (005) Neurosliding ode control for agnetic levitation systes. IEEE International Syposiu on Circuits and Systes (ISCAS 005) 5: Khaesee M, Kato N, Noura Y (00) Design and control of a icrorobotic syste using agnetic levitation. IEEE ASME Trans Mechatron 7(1):1 14 Khaesee M, Kato N, Noura Y (003) Perforance iproveent of a agnetically levitated icrorobot using an adaptive control. Proceedings of the International Conference on MEMS NANO and Sart Systes 1: Khaesee MB, Shaeli E (005) Regulation technique for a large gap agnetic field for third non-contact anipulation. J Mechatron 15(9): Lepetic M, Skrjanc I, Chiacchiarini HG, Matko D (001) Predictive control based on fuzzy odel: a case study. The 10th IEEE International Conference on Fuzzy Systes 3: Peng Y, Qiang Z, Lianbing L (00) Design of fuzzy weight controller in singleaxis agnetic suspension syste. Proceedings of the fourth World Congress on Intelligent Control and Autoation 4: Shakir H, Won-jong K (005) Nanoscale path planning and otion control. Proc 005 Aerican Control Conf 5: Shih-Kang K, Xiin S, Chia-Hsiang M (003) Large travel ultra precision x y h otion control of a agnetic-suspension stage. IEEE ASME Trans Mechatron 8(3): Slotine JJ (1991) Applied nonlinear control. Englewood Cliffs, Prentice-Hall, NJ Xiin S, Shih-Kang K, Jihua Z, Chia-Hsiang M (00a) Ultra precision otion control of a ultiple degrees of freedo agnetic suspension stage. IEEE ASME Trans Mechatron 7(1):67 78 Xiin S, Chia-Hsiang M (00b) Robust disturbance rejection for iproved dynaic sti.ness of a agnetic suspension stage IEEE ASME Trans Mechatron 7(3):89 95 Yen-Chen C, Shinq-Jen W, Tsu-Tian L (003) Miniu-energy neural-fuzzy approach for current/voltage-controlled electroagnetic suspension syste, Proceedings of the 003 IEEE International Syposiu on Coputational Intelligence in Robotics and Autoation 3: