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H_infinity Controller Design to Control the Single Axis Magnetic Levitation System with Parametric Uncertainty



Basheer Noaman Hussein, Nasri Sulaiman, R.K. Raja Ahmad, Mohammad Hamiruce Marhaban and Hazem I. Ali
 
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ABSTRACT

In this study the force control design of single axis magnetic levitation system using H_infinity controller is presented. First, the system dynamics are linearized and described in transfer function form. Second, the magnetic force is regulated using H_infinity controller to achieve robust stability, disturbance/noise rejection and asymptotic tracking. A multiplicative unstructured model extracted from the parametric uncertainty is used for H_infinity control design. The obtained results showed that robust stability and performance have been achieved. On the other hand, an improved and more reliable time response compared with previous work also has been achieved in this study.

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Basheer Noaman Hussein, Nasri Sulaiman, R.K. Raja Ahmad, Mohammad Hamiruce Marhaban and Hazem I. Ali, 2011. H_infinity Controller Design to Control the Single Axis Magnetic Levitation System with Parametric Uncertainty. Journal of Applied Sciences, 11: 66-75.

DOI: 10.3923/jas.2011.66.75

URL: https://scialert.net/abstract/?doi=jas.2011.66.75
 
Received: April 29, 2010; Accepted: October 23, 2010; Published: November 10, 2010



INTRODUCTION

The potential uses of magnetic levitation systems range from standstill applications which use light current, such as active magnetic bearings, vibrating tables, wafer conveyors, micro machines, etc., to the levitation and guidance of electrical vehicles which need high current. The magnetic levitation has no contact between the moving object and fixed part. It is one of the good tools for a micro-machine because mechanical friction disappears, which increases the resolution and accuracy of the positioning device. Another advantage of the magnetic levitation system is that the manipulator can operate as a rigid body rather than using jointed parts such as robots, which means that position errors do not compound and the dynamic behavior is simple to model. The major disadvantage of levitation is that the system is inherently unstable and hence feedback control is usually required for stabilization (Yi et al., 1996; Chao-Lin et al., 2005). Due to the features of the open-loop instability and highly inherent nonlinearities in electromechanical dynamics of the magnetic levitation systems, the development of a high performance control design for the position control of the levitated magnets is very important. Since, it is very difficult to acquire an exact mathematical model to describe the electromechanical dynamics of the magnetic levitation systems, in general, only the approximated dynamic model are used. Therefore, many studies have been reported and discussed based on the approximated dynamic models in the recent years (Ono et al., 2002; Chen et al., 2003; Kaloust et al., 2004; Grochmal and Lynch, 2007). Magnetic levitation systems have been successfully and widely implemented for many engineering applications, such as vibration isolation, frictionless bearings, high-speed maglev passenger trains and fast-tool servo systems (Chiang et al., 2006; El-Hajjaji and Ouladsine, 2001; Trumper et al., 1997; Hurley et al., 1997).

The development of magnetically levitated micro machines has been reported in the literature. Tsuda et al. (1987) have designed a Magnetically Supported Intelligent Hand (MSIH) which uses active DC-type magnetic bearings. Hollis et al. (1987) have designed a hexagonal shaped magnetically levitated wrist using permanent magnets and air-core electromagnets. Park et al. (1996) have designed a six degree-of-freedom high precision positioner employing an antagonistic property which is generated by using permanent magnets and air-core electromagnets. Using precise control, the above mechanisms can carry out various automatic assembly tasks like active/advanced Remote Center Compliance (RCC) device. Banerjee et al. (2007) have designed and implemented a single axis magnetic levitation system for the suspension of a platform by using cascade lead compensator designed using the classical synthesis method. An attemption has been done to examine various micro-machine technologies, including piezoelectric, hydraulic, magnetic, shape memory alloy, and electrostatic approaches.


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Fig. 1: Conceptual design for the shear force tester

Of particular note and the attempts to build electrostatic micro-motors using silicon-based technology (Trimmer and Gabriel, 1987; Bart et al., 1988; Tai et al., 1989). The motivations for using a silicon-based technology are the ability to have good process control and the potential savings in cost obtained with batch fabrication, particularly if the electronics are integrated with the mechanical devices (Busch-Vishniac et al., 1990).

In some applications, micro-machines require a strategy of force control. For example, in semiconductor fabrication, force control is needed for mechanical quality control testing for bonds between circuits on devices. One possible conceptual design in the field of Micro-machines is the shear force tester shown in Fig. 1. The tester consists of U-shaped electromagnets and a manipulator (a rectangular piece of metal), both are made of iron. It is assumed that the xyz coordinate system is fixed at the centre of the manipulator and ψ, φ and θ are the Eulerian angles of the xyz coordinate system. Due to the magnetic suspension, spring forces serve as stabilizing forces on the manipulator and, hence, making the manipulator stable in both ψ and φ directions and unstable in è direction. A known force can be applied to the bond in the z direction (Yi et al., 1996).

In this application, only one degree-of-freedom motion in the z direction will be examined, with the assumption that all other motions are stabilized. The force is determined from measurement of the flux density.

The goal of robust systems design is to retain assurance of system performance in the presence of model uncertainty. A system is robust when its performance performs adequately under model uncertainty. A control system is robust when (1) it has low sensitivity, (2) it is stable over the range of parameter variations and (3) the performance continues to meet the specification in the presence of uncertainty (Ali et al., 2009).


Image for - H_infinity Controller Design to Control the Single Axis Magnetic Levitation System with Parametric Uncertainty
Fig. 2: A single axis magnetic levitation system

H_infinity is one of the most known techniques available nowadays to design a robust control (It is an optimization method that takes into consideration a strong definition of the mathematical way to express the ability to include both classical and robust control concepts within a single design framework). With this method, model uncertainties and performance requirements can be incorporated into a single framework of H_infinity controller achieve very robust stability and good performance in theory (Zhou and Doyle, 1998).

System dynamics A single axis force controlled Magnetic Levitation System (MLS), shown in Fig. 2, is considered to derive a mathematical formulation of system motion. In addition to the electromagnet and the manipulator, a spring and a linear bearing are used for providing a reaction force and constraining the manipulator motion to the vertical direction, respectively. The U-shaped iron core magnet can produce a strong magnetic force, which is a good advantage (Yi et al., 1996).

The fundamental equation of motion of the system is derived from Newton’s second law of motion as:

Image for - H_infinity Controller Design to Control the Single Axis Magnetic Levitation System with Parametric Uncertainty
(1)

Four physically different forces are acting on the manipulator mass. They are the gravitational force, the electromagnetic force (attraction or repulsion), the spring force and any possible internal or external disturbance. It is important to note here that the disturbance force is also introduced to compensate for any neglecting of specific physical modes, which may be considered in the mathematical formulation as will be shown below. Therefore, the equation of motion of the system is (Yi et al., 1996).

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(2)

where, m is the manipulator mass, g is the gravity of earth (acceleration), z (t) is the air gap length, zn is the air gap length when no spring force exists (assumed neglected), f (t) is the time function electromagnetic force, ks is the spring constant and fdis(t) is a time function of disturbance force or model uncertainty.

Figure 3 shows the free body diagram of the single axis magnetic levitation system to describe all the forces that may have an effect.

The magnetic force can be expressed in terms of the flux density B by assuming the magnetic flux is uniform in the air gap and the permeability of iron is high enough to be considered. The equation of magnetic force f in terms of flux density is as follows:

Image for - H_infinity Controller Design to Control the Single Axis Magnetic Levitation System with Parametric Uncertainty
(3)

where, B is the flux density and kf is expressed as:

Image for - H_infinity Controller Design to Control the Single Axis Magnetic Levitation System with Parametric Uncertainty
(4)

Image for - H_infinity Controller Design to Control the Single Axis Magnetic Levitation System with Parametric Uncertainty
Fig. 3: Free body diagram of single axis magnetic levitation system

where, A is the magnetic pole face area and μ0 is the permeability of free space.

The instantaneous flux density B(t) and, hence, the instantaneous electromagnetic force f (t) are produced by the electrical coil circuit. The equation relating the coil voltage e, flux density B and the air gap length z is expressed as (Theraja, 1984).

Image for - H_infinity Controller Design to Control the Single Axis Magnetic Levitation System with Parametric Uncertainty
(5)

where N is the number of coil turns, R is the coil resistance and edis is the voltage noise. Again this noise voltage is added not only to compensate for changes in the voltage source but also for any mathematical simplification that may be carried in Eq. 5.

Since the MLS consists of both electrical and mechanical elements, then the generalized Lagrange method of modelling can be used to obtain the complete mathematical modelling of the system.

Equations 2 and 5 represent the two basic differential equations of the considered MLS. These equations depict nonlinear elements, the square of flux density B2 and the product Bz. Moreover, if one assumes that fdis and/or edis are random quantities then the solution would not be found analytically. One way to solve the problem is to apply the theory of small perturbation around nominal steady state parameter values.

Let F0, B0, Z0 and E0 be the nominal values of the magnet force, flux density, air gap length and coil voltage, respectively and let δf, δB, δz, δe be the deviation of these quantities from the nominal values. To apply the theory of deviation to the time function variables, it yields:

Image for - H_infinity Controller Design to Control the Single Axis Magnetic Levitation System with Parametric Uncertainty
(6)

Image for - H_infinity Controller Design to Control the Single Axis Magnetic Levitation System with Parametric Uncertainty
(7)

Image for - H_infinity Controller Design to Control the Single Axis Magnetic Levitation System with Parametric Uncertainty
(8)

Image for - H_infinity Controller Design to Control the Single Axis Magnetic Levitation System with Parametric Uncertainty
(9)

The nominal flux density can be measured by the following equation (Mohamed et al., 1994):

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(10)

Next, the independent variable (t) will be dropped for simplicity of writing. The magnetic force can be related to the flux density by substituting Eq. 6 and 7 in Eq. 3:

Image for - H_infinity Controller Design to Control the Single Axis Magnetic Levitation System with Parametric Uncertainty
(11)

Image for - H_infinity Controller Design to Control the Single Axis Magnetic Levitation System with Parametric Uncertainty
(12)

Image for - H_infinity Controller Design to Control the Single Axis Magnetic Levitation System with Parametric Uncertainty
(13)

By neglecting the second order term, which is valid if |δB|<<|B0|, then

Image for - H_infinity Controller Design to Control the Single Axis Magnetic Levitation System with Parametric Uncertainty
(14)

Image for - H_infinity Controller Design to Control the Single Axis Magnetic Levitation System with Parametric Uncertainty
(15)

The last two equations show that the force can be determined from flux density measurement. By substituting these two equations and Eq. 8 in 2, the system dynamics can be written as follows:

Image for - H_infinity Controller Design to Control the Single Axis Magnetic Levitation System with Parametric Uncertainty
(16)

Image for - H_infinity Controller Design to Control the Single Axis Magnetic Levitation System with Parametric Uncertainty
(17)

Image for - H_infinity Controller Design to Control the Single Axis Magnetic Levitation System with Parametric Uncertainty
(18)

where,

Image for - H_infinity Controller Design to Control the Single Axis Magnetic Levitation System with Parametric Uncertainty
(19)

On the other hand, after substituting equations in Eq. 5, it yields

Image for - H_infinity Controller Design to Control the Single Axis Magnetic Levitation System with Parametric Uncertainty
(20)

Image for - H_infinity Controller Design to Control the Single Axis Magnetic Levitation System with Parametric Uncertainty
(21)

Image for - H_infinity Controller Design to Control the Single Axis Magnetic Levitation System with Parametric Uncertainty
(22)

where,

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(23)

Therefore, Eq. 18 and 22 represent a linear version of the original system equations and all nonlinear modes are put in the model uncertainties fd and ed.

There are three physical quantities in the system to be assigned as system states. They are the air gap length (z), the flux density (B) and the electric coil voltage (e).

Let the system states be defined as follows:

Image for - H_infinity Controller Design to Control the Single Axis Magnetic Levitation System with Parametric Uncertainty
(24)

Image for - H_infinity Controller Design to Control the Single Axis Magnetic Levitation System with Parametric Uncertainty
(25)

Image for - H_infinity Controller Design to Control the Single Axis Magnetic Levitation System with Parametric Uncertainty
(26)

Image for - H_infinity Controller Design to Control the Single Axis Magnetic Levitation System with Parametric Uncertainty
(27)

where u (t) is the system input signal.

By substituting Eq. 18 in Eq. 25 and 22 in 26 the state space representation of the system dynamics can be expressed as:

Image for - H_infinity Controller Design to Control the Single Axis Magnetic Levitation System with Parametric Uncertainty
(28)

Image for - H_infinity Controller Design to Control the Single Axis Magnetic Levitation System with Parametric Uncertainty
(29)

Image for - H_infinity Controller Design to Control the Single Axis Magnetic Levitation System with Parametric Uncertainty
(30)

The state variables can be arranged in a vector form as:

Image for - H_infinity Controller Design to Control the Single Axis Magnetic Levitation System with Parametric Uncertainty
(31)

Image for - H_infinity Controller Design to Control the Single Axis Magnetic Levitation System with Parametric Uncertainty
(32)

Image for - H_infinity Controller Design to Control the Single Axis Magnetic Levitation System with Parametric Uncertainty
(33)

Image for - H_infinity Controller Design to Control the Single Axis Magnetic Levitation System with Parametric Uncertainty
(34)

Therefore, the dynamic model of the magnetic levitation system can be represented in a state space form as:

Image for - H_infinity Controller Design to Control the Single Axis Magnetic Levitation System with Parametric Uncertainty
(35)

Image for - H_infinity Controller Design to Control the Single Axis Magnetic Levitation System with Parametric Uncertainty
(36)

where, n (t) is the sensor noise (random disturbance), d (t) is the input disturbance that represents the model uncertainty and voltage noise signal and

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Image for - H_infinity Controller Design to Control the Single Axis Magnetic Levitation System with Parametric Uncertainty
(37)

The system state space defined by Eq. 37 can be converted to a transfer function using:

Image for - H_infinity Controller Design to Control the Single Axis Magnetic Levitation System with Parametric Uncertainty
(38)

Image for - H_infinity Controller Design to Control the Single Axis Magnetic Levitation System with Parametric Uncertainty
(39)

And the following transfer functions can be obtained:

Image for - H_infinity Controller Design to Control the Single Axis Magnetic Levitation System with Parametric Uncertainty
(40)

Image for - H_infinity Controller Design to Control the Single Axis Magnetic Levitation System with Parametric Uncertainty
(41)

Image for - H_infinity Controller Design to Control the Single Axis Magnetic Levitation System with Parametric Uncertainty
(42)

where,

Image for - H_infinity Controller Design to Control the Single Axis Magnetic Levitation System with Parametric Uncertainty

Image for - H_infinity Controller Design to Control the Single Axis Magnetic Levitation System with Parametric Uncertainty
(43)

The parameters for the system are listed in Table 1. The system dynamics can be described by linear control system as shown in Fig. 4.

The H_infinity synthesis has been used to design a force controller for the magnetic levitation system such that the following requirements are achieved: robust stability against various model uncertainties, disturbance/noise attenuation, asymptotic tracking to command signals, less control energy and limiting closed loop bandwidth to achieve good robustness and noise rejection (Zhou et al., 1998; Mohamed et al., 1994).


Table 1: The parameters of the magnetically levitated system
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Fig. 4: Block diagram of the system

One of the important parts in the design of the H_infinity controller is the selection of weighting functions for specific design problems. This is not an easy procedure and often needs many iterations and fine-tuning and it is hard to find a general formula for the weighting functions that will work in every case (Anselmo and Moura, 1998). Therefore, to obtain a good control design, it is necessary to select suitable weighting functions. The performance and control weighting functions formula used in this work are (Zhou et al., 1998):

Image for - H_infinity Controller Design to Control the Single Axis Magnetic Levitation System with Parametric Uncertainty
(44)

Image for - H_infinity Controller Design to Control the Single Axis Magnetic Levitation System with Parametric Uncertainty
(45)

where wb is the minimum acceptable bandwidth (for disturbance rejection), Ms is the maximum peak magnitude of |s (jw)|, Ess is the steady state error (allowed), K1 and K2 are gains of the weighting functions, wbc is the controller bandwidth and Mu is the magnitude of hs.

The weighting functions parameters are selected by trial and error to be wb = 360, Ms = 1.5, Ess = 0.0001, K1 = 0.1, K2 = 0.00008, wbc= 150 and Mu = 10.

The H control design deals with both structured and unstructured uncertainty. However, since a design scheme involving unstructured uncertainty gives more control over the system (as it can cover not modelled dynamics at high frequencies (Nudehi and Farooq, 2007), the plant with structured uncertainty can be expressed in terms of unstructured multiplicative uncertainty. By selecting a set of nominal plants to evaluate the disk of uncertainty, the uncertainty plant is:

Image for - H_infinity Controller Design to Control the Single Axis Magnetic Levitation System with Parametric Uncertainty
(46)

where Gp is the nominal plant and the multiplicative uncertainty Image for - H_infinity Controller Design to Control the Single Axis Magnetic Levitation System with Parametric Uncertainty can be expressed as:

Image for - H_infinity Controller Design to Control the Single Axis Magnetic Levitation System with Parametric Uncertainty
(47)

The uncertain parameters of the system are R and m, which vary from 2 to 7Ω and 0.100 to 0.300 kg, respectively.

From Eq. 47, multiplicative uncertainty weight WT can be calculated using curve fitting commands in MATLAB such that |Δm (jw)|# 1 and can be expressed as:
Image for - H_infinity Controller Design to Control the Single Axis Magnetic Levitation System with Parametric Uncertainty
(48)

The H controller has been designed so that the infinity norm from input

Image for - H_infinity Controller Design to Control the Single Axis Magnetic Levitation System with Parametric Uncertainty

to output

Image for - H_infinity Controller Design to Control the Single Axis Magnetic Levitation System with Parametric Uncertainty

is minimized. Where w represents all exogenous input signals such as command signals (R), voltage noise (ed) and disturbances (fd); ef, uf are the weighted error and weighted control signals; u represents the actuator input; e is the error signal. Figure 5 shows the standard feedback diagram of the system with weights. The generalized plant P is expressed by:

Image for - H_infinity Controller Design to Control the Single Axis Magnetic Levitation System with Parametric Uncertainty
Fig. 5: Overall block diagram of the system

Image for - H_infinity Controller Design to Control the Single Axis Magnetic Levitation System with Parametric Uncertainty
(49)

The lower linear fractional transformation of the generalized plant P and controller K can be described by:

Image for - H_infinity Controller Design to Control the Single Axis Magnetic Levitation System with Parametric Uncertainty
(50)

Image for - H_infinity Controller Design to Control the Single Axis Magnetic Levitation System with Parametric Uncertainty
(51)

where, S is the sensitivity function of the nominal plant and T is the complementary sensitivity function, they can be defined as (Zhou et al., 1998):

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(52)

Image for - H_infinity Controller Design to Control the Single Axis Magnetic Levitation System with Parametric Uncertainty
(53)

The design goal is to find the controller I that internally stabilizes the system such that the maximum singular value of N is minimized.

In order to achieve robust stability, less control energy and robust performance, the following constraints must be achieved:

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(54)

The H_infinity controller is obtained by using the H_infinity robust controller command (hinf) in MATLAB. After some iteration the H_infinity controller which achieves the robustness and performance goals was found to be:

Image for - H_infinity Controller Design to Control the Single Axis Magnetic Levitation System with Parametric Uncertainty
(55)

SIMULATION RESULTS

Figure 6a and b show the system output response characteristics. It is clear that the system becomes unstable when (a) a disturbance of 0.1 N step or (b) noise signals of 0.1 N step are applied. One of the important achievements of applying Image for - H_infinity Controller Design to Control the Single Axis Magnetic Levitation System with Parametric Uncertainty controller is to ensure the robust stability of the system. Figure 7 shows the frequency characteristics of the sensitivity function compared with the inverse of the performance weighting function.

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Fig. 6: The system output response for a 0.1 N step input (a) with 0.1 N step disturbance force applied without H_infinity controller and (b) with 0.1 N step voltage noise applied without H_infinity controller

It is clear that the first performance criterion in Eq. 54 has been satisfied that means the sensitivity function is always less than the inverse of the performance weighting function wp . The second performance criterion in Eq. 54 also has been achieved as shown in Fig. 8 that means K (s) S (s) is always less than the inverse of the performance weighting function wp. Figure 9 shows the frequency characteristics of the complementary sensitivity function compared with the inverse of the multiplicative uncertainty weighting function. It is clear that the third performance criterion in Eq. 54 has been satisfied that means the complementary sensitivity function is always less than the inverse of the multiplicative uncertainty weighting function wT. Figure 10 shows the frequency response of the controlled system with all parameters uncertainty.

Image for - H_infinity Controller Design to Control the Single Axis Magnetic Levitation System with Parametric Uncertainty
Fig. 7: The frequency response characteristics of the sensitivity function S(s) compared with the inverse of the performance weighting function Wp

Image for - H_infinity Controller Design to Control the Single Axis Magnetic Levitation System with Parametric Uncertainty
Fig. 8: The frequency response characteristics of the KS (s) compared with the inverse of the weighting function WP

Image for - H_infinity Controller Design to Control the Single Axis Magnetic Levitation System with Parametric Uncertainty
Fig. 9: The frequency response characteristics of the complementary sensitivity function T(s) compared with the inverse of the weighting function Wt

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Fig. 10: The frequency response characteristics of the system with the controller

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Fig. 11: The system output response for a 0.1 N step input with the controller

It is shown that the system is stable with all parameters uncertainty; this means the robust stability of the system has been achieved. Figure 11 and 12 show the step response characteristics of the nominal system and uncertain system.

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Fig. 12: The uncertain system output response for a 0.1 N step input with the controller

Image for - H_infinity Controller Design to Control the Single Axis Magnetic Levitation System with Parametric Uncertainty
Fig. 13: The system output response for (a) 0.1 N step disturbance force with the controller and (b) for 0.1 N step voltage noise with the controller

Figure 13a and b show the time response of the system when (a) a disturbance of 0.1 N step or (b) noise signals of 0.1 N step are applied respectively. It can be seen that a settling time of 0.1 sec has been achieved. An improved response has been obtained in comparison with results obtained from previous work. The obtained time response specifications are better in term of speed and error steady state than those obtained by previous theoretical and experimental work (Yi et al., 1996) as shown in Table 2.

Table 2: Time response specifications of the controller
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CONCLUSION

A robust H_infinity controller for a single axis magnetic levitation system with model uncertainty has been presented. First the linear mathematical model of the system was derived. The unmodelled dynamic was taken into account as a disturbance to design the controller. Also the parameters uncertainty was considered. Adjusting the performance of the simulated closed loop response carried out the selection of the weighting functions. The appropriate selection of the weighting functions led to obtain a robust controller that achieves the force control in magnetic levitation system. Finally, the objectives of the controller were verified by simulation. An improved time response compared with previous work has been achieved.

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