In this chapter we will develop acontrol system that is able to keep a ball levitated under a coil against the gravitational force. In earlier chapters we analyzed the system 's dynamics and realized both a sensor and a current amplifier that, together with the controller, should stabilize the levitated object. While designing the controller and the control system we had to deal with the restrictions of our sensor and current amplifier. Successively we will explain why we separated the sensor and the current amplifier, describe the new system that thus appeared, design a controller for this system and finally descrihe the practical and simulated results of the created system.
5.1 Sensing and actuating on separate coils
Although it should theoretically he possihle to sense and actuate with the same coil, the practical application of this option appeared to he very difficult. As explained in chapter 4 the sensing sinusoidal current was inftuenced by the actuator, temperature, or the non-linear behaviour of the FET. Furthermore the sensor's reliability was effected negatively, when high actuating currents were supplied to the coil, as explained in chapter 3. All this made us decide to do the measurement and actuating on separate coils for this simple magnetic levitation system. However, this is not possible for the magnetic hearing, we finally want ta realize.
Thus, the sensing prohlem remains yet unsolved. Sensing and actuating on separate coils means electric and magnetic circuits different from those described in chapter 3 occur. The electric circuit, that exists, is presented in figure 5.1. In this figure we can see an actuating part and a sensing part. These parts are coupled by the mutual inductanee coefficient M[H]
as described in the fallowing equatians:
. dit di2
Ut
=
Rl*
tt+
Ll*
(ft+
M* di
. di2 dit
U2
=
R2*
t2+
L2*
(ft+
M* dt
We expect M to be a function of the balI position x and the frequency w. Ta complete our model of the levitation system it would be more exact ta determine the inftuence of M on the system 's dynamics. However, we ex peet M ta he very small, and sa we decided ta neglect it's influence while designing the controller. In figure 5.1 aresistanee R3 of 60
n
is placed para.llel with the actuating coil. This resistance limits the currents that will flow through the30
CHAPTER 5. THE CONTROL SYSTEM 31
i2 i1
R3
u1L2 M
R2
î
c: u2.~
sensing part actuating part
Figure 5.1: Eleetrie circuit of the sensor and actuator
coil to a eertain frequency. This ean he demonstrated by deriving the following equation:
(5.3)
i1(s) = iaetuator(s)
*
R3+ ::+
sL1 ::::: iactuator(s)* O.OOl~s +
1We presented this transfer in figure 5.2. It is very important to limit the frequency of the i1
t
(s)iact(s)
freq.
-20 db
I
decadeFigure 5.2: Bandlimiting the aetuating current
currents sllpplied to the actIlating eoil, as high-frequency currents wil! induee high voltages in
the coils. Furthermore it is important that na crosstalk occursbetween the actuating current i1 and the sensing voltage 11,2 at the sensing frequency, as this would severely influence the quality of our sensor output.
In this section we changed our electric and magnetic circuits in such a way that both actuating and sensing were made possible. Although this solution is not an elegant one, separating the sensing and actuating coil created the possibility to design a system that was able to levitate a ba.ll. However, further study must be done to analyze the influence of the sensing coil on the system 's dynamics to atta.in an appropriate model of the levitation system. In the next section we will present the system as des cribed in chapter 2 (figure 2.4), and we will design an appropriate controller, that can stabilize the cant rol system.
5.2 The control system
Up till now, we have developed a position sensor and a current amplifier and we analyzed the dynamic's of the magnetic levitation system. Together with the controller these components lay the foundation of our control system, that is shown in figure 5.3. The blocks presented in
controller 11. actuator in~1 plant XI~..~ sensor
x.~,.,
sum
Xref
Figure 5.3: The control system this figure perform the following transfer functions:
• The current amplifier block transforms the voltage ca1culated by the controller into the actuating current, according to the following equation (see (5.3) and figure 4.6):
iactuating 1
--~=
Ueont (0.0016s+ 1)-(0.00018+ 1) (5.4)
• The nonlinear plant is the magnetic levitation system that "transforms" the actuating current (iactuating) into the position of the balI (x), according to (2.16). This transfer fllnction is highly nonlinear with respect to the output x and the input iactuating. This will make it hard to design an appropriate controller. Therefore we have linearized
CHAPTER 5. THE CONTROL SYSTEM
(2.16) in some point (xo,ia) as explained in chapter 2, resulting in (2.21):
V
2.Q'9Xt(s)
m-
-11it(s) =-s2·(X+xa)-2g ~ (s+38)(s-38) with
it iactuating - ia[A]
Xt
=
x - Xa[mJXo desired position = 0.005[mJ ia = J2'
~.
9 . (X+
xa) = 1.72[A]33
(5.5)
This approach irnplies that the linearization of the system will only hoid for values ofx very close to Xoand currentsiactuating very close to ia and the controller, to he designed, will prohably only stahilize the system for a very small range of positions .
• The sensor measures the position of the hall and translates x to the sensor output U sen
according to (3.11):
Usen(S) ](
=---,--,-Xtrue(s) (2.2.10-3 .s
+
1)4 (5.6)• The controller must he designed to stabilize the system. Because the magnetic levitation system contains a poie that causes instahility, this will not he easy. We call the transfer function of the controller K(s)
=
~t~, with Ut = Ucon+
Ua. Ua is the part of the controller output that takes care of the equilibrium current io.If we take a look at the open-loop transfer function of the linearized control system we find the following open-Ioop-gain (o.l.g.):
1 -11
K
o.l.g.
=
]((8) . (0.00168+
1)(O.OOOls+
1) (s+
38)(8 _ 38) . (2.2.10-3 .8+
1)4 (5.7) If we make a pole-zero-plot of this o.l.g. we obtain figure 5.4. In this figure the poles and zeros of the controller are not taken into account. The poles, zeros and gain of the controller must he chosen in slich a way that the poies of the closed loop function (c.I.f.) are situated in the Ieft-half of the s·plane (c.I.f.:::o.lt ).
This assures a stabie system. Ta get this done,t+o..g.
root-loci are used. In the next section a few controllers will he designed hy using root-loci.
Figure 5.4 also shows us that a proportional controller will never he able to stabilize the system.
5.3 Controller design
In this section we will design a few controllers that stabilize the system. After that we will discuss their practical implementation. We will start with a simple PO-controller.
1000
Figure 5.4: Pole-zero-plot of the open-Ioop-gain.
Root loci with proportional control 5.3.1 Design of a PD-controller
A Proportional Derivative controller realises the following transfer function:
](s)=P+D.s=P.(p ·s+l)D (5.8)
As a matter of fact it introduces a zero at s
=
-~. Ifwe choose ~=
30 we obtain the root-loci plot of figure 5.5. We can see that, by choosing an appropriate gain P, we can stabilize the system. In the next section we will test this controller using SIMULINK. In practice, introducing a zero at -30 means you have to realize a pole at -300 to avoid local instabilities. In this case this pole only slightly influences the control system. Due to this extra pole the maximum P will decrease. It should be possible to use this controller to stabilize the actual system. In figure 5.5 the resulting pole positions are shown.5.3.2 Design of a PID-controller
A Proportional Integrating Derivative controller realises the following transfer function:
I:"() P I D D .92 and YP~D4DI
=
5 , results in the root loci of figure 5.6. In this figllre we can see that again the system ca.n be stabilized by choosing an appropriate gain. The final error of a PID-controller will be zero. This is a a big advantage. However, the PlO-controller will react very heavily to disturbances. This means the actual nonlinear system will be difficult to stabilize by th is controller. This will be shown in the next section.CHAPTER 5. THE CDNTRDL SYSTEM 35
Figure 5.5: Root Loci with PD-control and actual closed loop poles
50.
Figure 5.6: Root Loci with PID-control and actual closed loop poles
5.3.3 Another Design
While designing the controllers rnentioned before, we noticed that the gain, necessary to move the pole at s == 38 into the left-half plane, al most moves the pole at s
=
-38 intothe "unstable area" This means the range of gains stabilizing the system is very smaU.
Furthermore, disturbances will not be damped very quickly, due to tbe P91e!:> m9ving to the imaginary axis. This made us decide to design a controller that would also move the pole at
s
=
-38 to the left, resulting in the following controller:K(s)
=]( .
(s+
20)(s+
200)(s
+
1000)2 (5.10)The root loci occurring with this controller are shown in figure 5.7. In this figure we can see
o
-3~00 -250 -200 -150 -100
Real Axls
-60 o 50
Figure 5.7: Root Loci with new controller and actual closed loop poles
it is possibIe to reach a higher gain, without forcing some poIes into the right half pIane. In practice, the zeros will cause very high locaI noise gains. This may cause locaI instabilities.
It is a design rule to accompany every zero with apoIe, according to p
=
lOz, where p isthe pole and z the zero. The pole that's necessary for the zero at z = -20 must be situated at p
=
-200. This pole wouId cancel the effect of the second zero. In this way again a PD-controller occurs. In the next section we will show same step responses of the linearized system and the actual nonlinear system for each of the controllers, we designed. After that we will choose one of the controllers and implement it for the real magnetic Ievitation system.5.4 Results of the control system
In th is section we will analyze robustness and performance of the controllers suggested in the earlier section. We will do this by llsing a nonlinear SIMlJLINK model of the control system.
The model used is presented in appendix B. We will analyze the performance by looking at same step responses. We wil! a.nalyze robustness by looking at the stability of the system when the parameters (like La, R, Q and X) are changed. In practice this may o'ccur due to temperature changes or the presence of interfering electromagnetic fields.
CHAPTER 5. THE CONTROL SYSTEM 37
2
5.4.1 Results using a PD-controller
The results of the PD-controller are shown in figure 5.8. Originally the reference position was set at 0.005 [m] and at t=l [sJ it was changed to 0.006 rml. Furthermore the actuating current and voltage are shown. In this figure we can see that the position is only correct
:::t :' : 1
0.4 0.6 0.8 1 1.2 1.4 1.6 _ 1.8
0.2 0.4 0.6 0.8 1
timers)
Figure 5.8: Simulated step response of the PD-controller
in the linearization point. Otherwise a final error occurs. However, we must he satisfied with these results, whereas the curve, leading to the final position is very smooth and not too large or very fast changing currents occur. Moreover, this controller can he implemented in practice very easily. We also tested the system with PD-control on it's rohustness. We did is hy changing the parameters of the actual system. In the end we tested this control configuration, when an amount of white measurement noise was added. The results of these tests are presented in appendix C. The results teIl us the system remains stahle, hut very large final errors might oCCUT. Summarizing, we may state that the PD-controller should he ahle to levitate the hall, especially in the linearization point, provided that we know our system parameters (section 2.2) pretty weIl.
5.4.2 Results using a PID-controller
In earlier sections we spoke out some expectations about the PID-controller. We expected it to be very accurate, but not very robust, as due to the integrating action, very heavy reactions to disturbances were foreseen. The simulated step response results of the PID-controller are shown in figure 5.9. As expected we see the final error is zero, whereas the overshoot is pretty large. Furthermore, we see the current is changing very strong!y, which may introduce loca!
instabilities. Simulations testing the robustness of this controller are presented in appendix C.
They show, this controller is more robust than we expected. Therefore it deserves attention to try and realise th is controller in practice.
2 1.8 1.6 1.4 1.2 0.8
0.6 0.2 0.4
Ir<: , : :
o 0.2 0.4 0.6 0.8E: :I
1 1.2 1.4 1.6 1.8 2:':: E::: j
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
, : : : ë: : : : j
1
Figure 5.9: Simulated step response of the PID-controller 5.4.3 Results using the third controller
In subsection 5.3.3 we tried to design a. controller that would he as robust as possible even for high gains. However, this controller will not function in practice, as we explained before, it seemed interesting to test if our remarks in the earlier section were true. The step response results ofthe simulations with this controller are shown in figure 5.10. In this figure we can see
X10~
1:[
o==r-~ ~: j
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
o:<
o 0.2 0.4 0.6OM:
0.8 1f: : : : ]
1.2 1.4 1.6 1.8 2'J_"" ~~' ".. .;-'----~''--'---~~~~ "--,--t-.J-"' : '---,-'" ~---,'.~,--' ",
.L--:'"l
o 0.2 0.4 0.6 0.8 1 1.2 l.4 1.6 1.8 2
IImelsJ
Figure 5.10: Simulated step response of the third controller
very high currents and voltages, which will eause the loeal instahilities we discussed before.
CHAPTER 5. THE CONTROL SYSTEM 39
However, the sirnulations presented in appendix C show the controller to he very rohust for large disturbances.
5.4.4 Implementation of the PD-controller in practice
After we developed our controllers we decided to try out the PD-controller in practice. We realised an e1ectronic circuit perforrning the transfer function of 5.8. As explained, this realisation needed to be extended with a pole at s = -300, to avoid local instabilities. The circuit designed is shown in appendix A. All this resulted in a stable system, that, however, was very sensitive for external disturbances. Although the hall was levitated, it was not in sus pension at the equilibrium position xo. Furtherrnore, the ball was not steady, but kept rnoving with a certain frequency. These observations do not match the simulations. A track of the ballposition is not available. However, in appendix D a few tracks are given, realized with the DSP, that show the results of the system using a PID-controller. In these figures we can see the same movement as described above.
This rnay be caused by a number of reasons:
1. In our simulation model and controller design the influence of M (5.1) was not taken into account.
2. It is very hard to exactly realise the equilibrium current ia.
3. The sensor output may be disturbed hy noise and nonlinear effects.
4. Saturation effects of the ferroxcube.
To improve the performance of the control system, all effects mentioned above should he taken into account while designing our controller. Furthermore, to realize the control system designed exactly, the controller should be realized by a computer (Digital Signal Processor (DSP)). Furthermore it deserves attention to verify and stamp the results ofthe sensor. Con-cludendi, we can say, it is very hopeful the ball can be levitated, but still a lot of improvements are needed and possible to attain our control goal of being able to control the position as accurate as possible.
5.5 Exact linearization
5.5.1 Introduction
In this seetion we will show, that it is possible to exactly linearize our magnetic levitation system [6], provided we exactly know the system parameters Q and X and we can exaetly measure the ball's position. In this way a linear system exists, according to figure5.11. It goes without saying that a linear system is more easy to control, than a Ilonlinear one. In this figure, we must determine the linearizing feedback, to realize the following Linear system, with two poles in the origin:
x(s) =
ilin(S)
1 (5.11)
J
L
,i..-linearizing nonlinear
feedback plant
linear system
ilin controller x
1
XreFigure 5.11: Principle of exact linearization
If we look at the dynamic equation of our nonlinear system (2.16),
~x
Q
~m· - =m .9 - 1/2· .~
dt2 (X
+
x)2 (5.12)we can derive the fol1owing transfer function for our linearizing feedback:
.-J
2(m .9 - m .'u.) (X )~-
. +x
Q (5.13)
These derivations imply that between input ilin and output x the transfer function of a linear system appears, according to5.11. For this system we can easily design a PD-controller. This will be done in the next section. However, in practice the situation will not beas ideal as we suggested in this section. This is caused by a number of reasons:
1. The parameters of the system are not exactly known.
2. Between the actual position x and the position we can use to calculate the linearizing feedback is the transfer of the sensor.
3. Between the calculated current and the current we can supply to the coil is the transfer of the current amplifier.
These restrictions will cause the system to he nonlinear, especially when the ballposition is changing rapidly. In the simulations and derivation in the following section we did not take these restrictions ioto account.
5.5.2 Design of a PD-controller
In figure 5.12 a root locus plot is made of our exact linearized system with PD-control. The PD-controller performs the following transfer function: ~
=
K*
sstt~Oo We can see it isCHAPTER 5. THE CONTROL SYSTEM 41
-1400 -1200 -1000 -800 -600 -400 -200 0 200 400 600 Real Axis
50 , - - . , . . - - . . , . . . - - - , - " . - , . - -...-...,..--,r---.,..--...- - - ,
-30 -20 -10 0 10
Real Alels
20 30 40 50
Figure 5.12: Root loci with PD-control and actual c10sed loop poles
easy to get all our poles into the left-half plane. After this we tested the resulting system by simulating a step response from x = O.005(m] to x
=
O.007[m]. The result is shown in figure 5.13. In this simulation we can dearly notice that the control action consists of threeaX1O-S
Figure 5.13: Step response with exact linearization
parts. First the eurrent is deereased to make the ball move downward. We eall th is the falling phase. In the second phase, the stabilizing phase, "the fall is broken" and the movement is
stahilized. In the last part the hall is levitated in a new equilibrium point, where a 1;>igger equilibrium current is needed to compensate for the gravitational force. The result shown is very satisfying and absolutely better than the ot her control systems, we tested. Moreover, in appendix C we will show this control "algorithm" to he more robust as well. It deserves attention to try and realize this way of control.
In this chapter we have developed some control systems levitating the bali, of which we tested one in practice. However, as the model of our system was not completed and we didn't have time to test all algorithms in practice, we absolutely did not develop an optimal system in practice. Therefore, we mentioned same suggestions of how our system can be improved. In the next chapter we will make some conclusions about the system we developed and realized.
After that we will surnmarize some suggestions of how to improve the existing system.