approach from problem studying to the finale solution with all necessary steps, beside that the translation of performance demands to the H, controller synthesis.
Robust controller design for position control of the metal ball in magnetic field Amor Chowdhury, Marjan Golob, Peter Cafuta, Rajko SveEko University of Maribor Faculty of Electrical Engineering and Computer Science Smetanova 17, SI-2000 Maribor, Slovenia amor.chowdhurv~uni-mb.si Article is describing a modeling and controller synthesis for the position control of the metal hall in magnetic field. The focus was paid to implementation of modern control theory for the controller design. The position control of the metal hall in magnetic field is known unstable non-linear control problem that was solved with different methods and with variety of the control synthesis tools 171. Our aim in this article is to point out the use of modern control theory such as H, control theory [6j,[9l and related difficulties by the implementation of the theory when solving real control problems. Controller design is a process composed with many steps and with even more obstacles and limitations 151. The article is presenting a systematic approach from problem studying to the finale solution with all necessary steps, beside that the translation of performance demands to the H, controller synthesis tools is described as a case study. For a use of modern control theory a system robustness was the primarily reason. H, control theory is enabling a controller synthesis with assurance of the system robustness hut there is also a possibility that the system will perform robust to the object perturbation and external disturbances regardless to the controller synthesis method. In article one such modified approach of the controller synthesis is presented. With the performance measurement results on the real system and comparison of those for two different controllers designed different ways we have cam to two important conclusions. First, that H, controller synthesis although mathematically very elegant can he a good deal of a problem when implemented on a real system and second, that is possible to achieve system robustness although using a different method of controller synthesis, important is only that the robustness is vaulted.
1 Introduction Active magnetic suspension systems are allowing contactfree suspension. They do not suffer from neither friction nor wear, and this is important advantage of these systems. Magnetic suspension technology is now used for various industrial purposes, and has already been applied to magnetically levitated trains, magnetic bearings, etc. Since an active controlled-type magnetic suspension system is inherently unstable, feedback control is indispensable to stabilize the system. In most cases the control system and energy supply requirements to levitate the object have a higher level of complexity. The non0-7803-7852-0/03/$17.00 02003 IEEE
linear nature of the system dynamics coupled with the nonlinear characteristics of the actuators complicate the controller design. The exact description of this behaviour is almost hopeless, and even if it should be achieved by infinite dimensional non-linear differential equations, the resulting model is only effective for the simulationslanalysis, hut can not he used for a control system design as it is. Then some approximations and assumptions must be employed, and consequently the gaps between the real physical system and the design model cannot be avoided. The classical controller design approach relies on a linearization of the model dynamics and on the application of a PID controller as a feedback compensator, to compensate the effects of non-modeled dynamics and exterior disturbances. This approach often yields enough stability and performance, but owing to model uncertainties and changes of the operating points, the entire system sometimes becomes unstable. Problems could exist in those cases where the set point is changeable within wide operating range. These devices employ linear control strategies that are based an approximate linear model of the actual non-linear force distribution at the nominal operating point. The performance of the classical linear control strategies continuously degrades with increasing deviation from the nominal operating point. To avoid these problems, the variety of possible controller design strategies can be used like adaptive control, soft computing-fuzzy control or robust control. Approach shown in this paper is an application of robust control methodologies based on H, control theory.
2 Modeling of the magnetic suspension dynamics A simple electromagnet and a steel hall in levitated position realize the magnetic suspension system. The magnetic force applied by the electromagnet is opposite to gravity force and maintains the suspended steel ball levitated as shown in Fig. 1. The purpose of this laboratory electromagnetic suspension system is a hall position control. The control problem is to levitate the iron hall stabile utilizing the electromagnetic force. The basic principle of a simple electromagnetic suspension system is shown in Fig. 2. The magnetic force F, depends on the electromagnetic current i(t), electromagnet characteristics and the air gap x between the steel ball and the electromagnet. For obtaining the linear mathematical model of the system dynamics a certain measurements of
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electromagnet characteristics have been done and also some simplifications have been accepted.
Under these assumptions, we have derive the following three equations, which describes a system dynamic as an equation of the motion of the iron ball, magnetic flux and equation of an electric circuit of ,the electromagnet, respectively.
(2)
V ( t )= L(x,i)i(f) W t ) dr ’
U(:) = i ( t ) R + -
Fig. I: Laboratory electromagndic suspensionsystem
(3)
For the system dynamic description a connection between magnetic force F, and magnetic flux f i t ) is needed. We derive this with the differential of the magnetic energy Wma8,which is a function of magnetic flux f i t ) and a ball position x(t): ‘--
.
Differential of the magnetic energy is composed from electrical and mechanical part.
Because electrical current i is independent from the ball positionx its follows
Fig. 2 The basic principle of a simple electromagnetic suspension system
Nominal physical system parameters: Steel ball weight m = 147 g ,
With integration of (6) by x and (7) by i we get the function of the magnetic energy Li‘ Wmw= f ( x ) + IiLdi = f ( x ) +-.
Maximal position height D = 0,025 m ,
(8)
2
0
Coil electrical resistance R = 2,8 R ,
With the partial differentiation of the function W,
Electromagnet inductance Lo = 0,377 U .
n we get
aw,, In order to derive a nominal linear model of this system by physical laws, we introduce the following assumptions:
Eddy currents in the electromagnet can he neglected. Inductance around the operating point is constant.
dx
i’dl, +--.
2dx
(9)
Comparing equations (7) and (9) lead us to the function of the magnetic force
Magnetic flux density and magnetic field do not have any hysteresis, and they are not saturated. Magnetic permeability of the electromagnet is infinite.
- df(x)
ax
2.1 Mathematical model
(8) by
F
=--
df(x) dx
i2dL +--.
2dx
(10)
Considering that magnetic force F, = O when the electrical current is i = 0 regardless to the position value x.
we see that the first art of eauation (10) . , is
and final equation of magnetic force is:
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dfb) =0 m
i2dL
F, =-.
2ak
The inductance L is combined from electromagnet inductance Lo and inductance contributed by the steel ball L,. The contributed part of the inductance L, is depending on the air gap x. With the measurements of the inductance by different values of air gap x we have obtained the static characteristic of the inductance in dependence fromx.
Fig. 4 Measured static characteristicofthe magnetic force
F ,( i , x )
Fig. 3: Characteristic afthe object inductance in relation with the ball position - air gap x
The behavior of inductance characteristic Fig. 3 was approximation with exponential equation:
L = Lo t L,e?,
(12)
Where L, (0,02875 H) an inductance is contributed by the steel ball and ~(64,3775m-I) is a coefficient of the inductance reduction. To verify approximation of object inductance we have also measured a static characteristic of the magnetic force in dependence from current i and air gap x (Fig. 4) and then compared this characteristic with the approximated one (Fig. 5). The difference between the measured and approximated static characteristic of the magnetic force F,(i,x) shown in Fig. 6 is big in nearness of the electromagnet (x = 0- 3mm), and is relatively small in the
Fig. 5: Approximated Static characteristic ofthe magnetic force
F,(i,x)
... .: ..,...., ........... . .' . ..
rest part of the air gap (x = 4-25mm) AF,,(i,x) < i 0 , 5 N . Considering equation (12) the object dynamics is described with two non-linear differential equations:
...
~
'..
0
I (nl
With the linearization of equations (13) and (14) in the operating point, the linear mathematical model of the object dynamics was obtained.
O X
x
Imml
Fig. 6: The differencebetween the measured and approximated static characteristicofthe magnetic force Mm(i,x)
By the linearization it was considered that all forces are in equilibrium. O=V
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(15)
Parameter uncertainties are consequence of physical changes in the system. For example electrical resistance R is changing with the temperature, the mass of the steel hall is not necessary constant also the electromagnet inductance is chancing with the ball position. Linearization error is unavoidable, because linear model is needed as base for linear control design. Considering the fact that mathematical description is just an approximation of the real data we have to deal also with influence of this error. Even by relatively simple plants the nominal model is not including all dynamics behaviour, especially for higher frequencies. Usually to avoid model complexity we are neglecting stabile poles with small influence on plant dynamics and this is so-called unmodeled dynamics. For better description of system dynamics we make a new extended model, which is a set of plant models, that is constructed with the nominal model and model uncertainties. With this set of models we can cover the relatively wider behaviours of the real plant, hut still not globally. This set is an extension of the nominal model. By model extension we can use many different approaches [5],[8],[9]. In OUT case we have chosen a model with multiplicative perturbation shown in Fig. 7, which is suitable for description of unstructured perturbation.
Next step of the mathematical model derivation is a partial differentiation of equations (16) and (17):
Fig. 7: Model with multiplicativepemrbation
Finally with the Laplace kansfonnation and on basis of equations (16-23) we have obtained the model transfer function: P(s) = s'
k,k, - k,s' - (k,+ k,k,)s + k2k,
AW(s) = P,(s)"(P(s)-P,(s))
(24)
With the choice of the operating point at x = 15mm the nominal model suitable for the controller design was calculated =
28,9
(s+8,063)(~+23>65)(~-24,257)
(25)
2.2 Model dynamics analysis
The derived model (25) with nominal model parameters works fairly well around the steady state operational point. However, if the state of the system deviates fiom the nominal operating point, the model no longer suitably describes the physical system. This gap or difference between nominal model and actual dynamics we treat as a model uncertainty. Tbere are three well known causes for model uncertainties and they are [51S81,[91: * Parameter uncertainty Linearization error * Unmodeled dynamics
(26)
By determination of perturbation function AW(s) we have followed the recommendations known kom the literature [1],[9]. Major problem was how to incorporate the influence of changing unstable pole, because the AW(s) must be stabile and proper function, beside this the extended model must have the same unstable poles as the nominal one. With the graphical analyses of the frequency characteristics (Fig. 8) of electrical part, which is stabile and with the consideration of recommendations [9] we have designed the perturbation function:
Considering (26) our extended plant model for controller synthesis is:
-
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P(s)= P,(s)(l+AW(s))
(28)
weight functions have been selected with the help of recommendation [9]: Nominal model
S
-+lo rq(s)=- 5,s s+ 1,5
-60
Fqs) = 10”
(30)
W,(s) = A W ( s )
(31)
The inverse function of ?(s) is upper magnitude frequency bound for the system sensitivity function S(s) , with which the performance criteria’s are selected. The goal was to find such controller that the system performance will satisfy next conditions: Overshot on step response lesser then 30%.
Fig. 8: Magoihlde frequency charactcristie ofthe nominal model and the model with included uncertainties
Settling time shorter then 0,5 second, Steady state enor must be lesser then 5%,
3 Robust controller design and experimental results
System must perform robustly, regardless to the model uncertainties and the influence of exogenous disturbance signals. Mentioned performance demands have been translated to the frequency domain with the performance weight W, (s) [5]. With the inverse of the weight function W,(s)the amplitude of controller output signal was limited. In cases when the model uncertainties are included in extended plant model with the multiplicative perturbation form (26), the weight function W,(s) is representing model perturbation. Because our system is a SISO, all input weights functions have been set to value one. With the H, controller synthesis we have searched for the controller that is minimizing the infinity norm of the transfer functions matrices:
In previous section we have derived the non-linear equations that are representing the motion of the mass. These equations were further linearized around an operating point to obtain linear model of the plant. Beside this an extended linear plant model was introduced with which we have included also the description of model uncertainties. Because the description of the model uncertainties is given in non-structured form the most appropriate controller design method is H, loop shaping [~l,Pl.
[:1 JLS
-4s -s -s
T,=
0 W, 0
-T
-Cl
v,
0
0
v,l
-U 0 V, 0
s -Jo
0
(32)
The idea is to find suchlike controller K ( s ) that the infmity norm of (32) will he at least less then one(l]Tm(s)ll~< I ) , Fulfilment of the condition
Fig. 9 System configuration for H, loop shaping
In Fig. 9 the system configuration for 11, loop shaping is shown. 4 ( s ) , W,(s),W,(s) are output weight functions with which we are modeling system performance, controller signal output boundary and model uncertainties. V,(s), V,(s), V,(s) are the input weights and are describing characteristics of disturbance signals. For SISO systems is possible to use only the weights W,(s),W,(s), W,(s) in which are the characteristics of exogenous disturbance signals already included [1],[9]. Because in this case we are dealing with the SISO system it was needed to determinate only y(s), W,(s),W,(s). For the model with multiplicative perturbation following
( ~ ~ T m ( s
