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.
|
| 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).
|
| 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:
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).
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:
where, B is the flux density and kf is expressed as:
|
| 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).
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:
The nominal flux density can be measured by the following equation (Mohamed
et al., 1994):
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:
By neglecting the second order term, which is valid if |δB|<<|B0|, then
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:
where,
On the other hand, after substituting equations in Eq. 5, it yields
where,
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:
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:
The state variables can be arranged in a vector form as:
Therefore, the dynamic model of the magnetic levitation system can be represented in a state space form as:
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
The system state space defined by Eq. 37 can be converted
to a transfer function using:
And the following transfer functions can be obtained:
where,
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 |
|
|
| 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):
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 h∞s.
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:
where Gp is the nominal plant and the multiplicative uncertainty
can be expressed as:
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 W
T
can be calculated using curve fitting commands in MATLAB such that |Δ
m
(jw)|# 1 and can be expressed as:
The H∞ controller has been designed so that the infinity norm from input
to output
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:
|
| Fig. 5: |
Overall block diagram of the system |
The lower linear fractional transformation of the generalized plant P and controller K∞ can be described by:
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):
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:
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:
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
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.
|
| 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.
|
| Fig. 7: |
The frequency response characteristics of the sensitivity
function S(s) compared with the inverse of the performance weighting function
Wp |
|
| Fig. 8: |
The frequency response characteristics of the KS (s) compared
with the inverse of the weighting function WP |
|
| Fig. 9: |
The frequency response characteristics of the complementary
sensitivity function T(s) compared with the inverse of the weighting function
Wt |
|
| Fig. 10: |
The frequency response characteristics of the system with
the controller |
|
| 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.
|
| Fig. 12: |
The uncertain system output response for a 0.1 N step input
with the controller |
|
| 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 |
|
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.