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Project ppt 18 june.pptx
1. Linearization & Robust Control of
Active Magnetic Bearing System
Project presentation By
Amit Pandey
(BT16EE010)
Under supervision of
Mr. Sukanta Debnath
(Assistant Professor NIT MIZORAM)
Department of Electrical & Electronics
Engineering
National Institute of Technology Mizoram
1
2. Contents
Introduction
Objective
Active Magnetic Bearing Elements
Advantages & applications
Magnetic circuit analysis
Force analysis of 1-DOF AMB
Electrical circuit analysis
State space model
Linearization of non-linear AMB System
PID Controller & AMB System
Fractional Calculus
Transfer Function of Fractional Operator
Fractional Order PID Controller
Transfer Function of FOPID
Advantage of FOPID
FOPID optimization and tuning
Simulation & Results
Conclusion & Future Work 2
3. OBJECTIVE
• Mathematical modelling of 1-DOF AMB
• To linearize the non-linear system of AMB
• Analyse the displacement of rotor by PID Controller
• Robust FOPID Control of AMB System
3
4. INTRODUCTION
• An active magnetic bearing (AMB) system supports a rotating
shaft, without any physical contact by suspending the rotor in
the air ,with an electrically controlled (or/and permanent
magnet) magnetic force.
• It is a mechatronic product which involves different fields of
engineering such as Mechanical ,Electrical ,Control System and
Computer Sciences.
4
6. ADVANTAGES OF AMB
1. No mechanical contact so there is no friction.
2. No lubrication is necessary.
3.They can run in vacuum.
4.Very high rotation speed because of no friction.
5.Magnetic bearings have very small energy losses.
6. Dynamics of rotor can be controlled
7. Long life cycle, high reliability, and economic advantages.
6
7. APPLICATIONS OF MAGNETIC BEARING
1. In turbomolecular pumps,
2. In long-term energy storage flywheel systems,
3. In Magnetic levitated trains
4. Used in some centrifugal compressors for chillers
5. In watt-hour meters for electrical utilities
6. In Medical devices such as blood pump
7
9. IDEAL MAGNETIC CIRCUIT MODEL
• 𝐻. 𝑑𝑙 = 𝐽. 𝑛𝑑𝑎 (Ampere’s Law)
• 2𝐻𝑔 lg +𝐻𝑎𝑙𝑖 + 𝐻𝑠𝑙𝑠 = 𝑛ⅈ
• 𝐵 = 𝜇𝐻 𝑜𝑟 𝐻 =
𝐵
𝜇
• 2𝐵𝑔 lg +𝜇0
𝐵𝑎
𝜇𝑎
𝑙𝑎 +
𝐵𝑠
𝜇𝑠
𝑙𝑠 = 𝜇0𝑛ⅈ
• ,
Fig.4 MAGNETIC CIRCUIT 9
10. FORCE ANALYSIS OF 1-DOF AMB
• It can move only in Y-axis Direction.
• Notations
𝑦0 𝑚ⅈ𝑑 𝑝𝑜𝑛ⅈ𝑡
𝑚 𝑚𝑎𝑠𝑠 𝑜𝑓 𝑟𝑜𝑡𝑜𝑟
𝐹1 𝑎𝑡𝑡𝑟𝑎𝑐𝑡ⅈ𝑣𝑒 𝑓𝑜𝑟𝑐𝑒
𝐹2 𝑎𝑡𝑡𝑟𝑎𝑐𝑡ⅈ𝑣𝑒 𝑓𝑜𝑟𝑐𝑒
𝐹
𝑔 𝑔𝑟𝑎𝑣ⅈ𝑡𝑎𝑡ⅈ𝑜𝑛𝑎𝑙 𝑓𝑜𝑟𝑐𝑒
Table 1 Fig.5 1 –DOF AMB 10
11. FORCE ANALYSIS ON ROTOR
• Applying Newton’s 2nd Law,
• 𝑚𝜘 = 𝐹1 − 𝐹2 + 𝐹
𝑔 (1)
• Energy stored in a Magnetic Field in a given
volume ‘v’ is,
• 𝑤𝜙 =
1
2 𝑉
𝐻 ⋅ 𝐵 𝑑𝑉 =
1
2
𝐵2
𝜇0
𝑑𝑉 & dv=A.dg
(2)
Fig.6 Force components 11
24. EFFECT OF PID PARAMETERS ON SYSTEM
DYNAMICS
Response Rise Time Overshoot Settling Time SS Error
𝐾𝑝 Decrease Increase NT Decrease
𝐾𝑖 Decrease Increase Increase Eliminate
𝐾ⅆ NT (not fix) Decrease Decrease NT
24
25. FOPID CONTROLLER
• FOPID have been developed by A.Oustaloup through CRONE controller in
1991.
• It involves fraction order of “s” in the integration part and derivative part.
• It has form PIλDμ.
25
26. FRACTIONAL CALCULUS
• Differential operator denoted by: [{a𝐷𝑡
𝑞
}]
• It is defined by as:
Combined differentiation – integration
operator
Where ‘q’ is fractional order, which can
be complex number and a & t are limits
of operation.
𝑑𝑞
𝑑𝑡𝑞
𝑞 > 0
1 𝑞 = 0
𝑎
𝑡
𝑑𝜏−𝑞 𝑞 < 0
26
27. FRACTIONAL CALCULUS
• There are some definitions for fractional derivatives as:
• The Grunwald-Letnikov definition is given by as:
• Grunwald-Letnikov
• Riemann-Liouville
• Caputo
𝐷𝑡
𝑞
f t =
dqf t
d t − a
= lim
𝑁∞
𝑡 −
𝑎
𝑁
−𝑞
𝑗=0
𝑁−1
−1 𝑗
𝑞
𝑗 𝑓(𝑡 − 𝑗[𝑡 −
𝑎
𝑁
])
27
28. FRACTIONAL CALCULUS
• The Riemann-Liouville is given by:
• It is simplest and easiest definition to use.
𝐷𝑡
𝑞
f t =
dq
f t
d t − a
=
1
Γ 𝑛 − 𝑞
(
dn
dtn
)
0
𝑡
𝑡 − 𝜏 𝑛−𝑞−1𝑓 𝜏 𝑑𝜏
Where ‘n’ is the first integer
which is not less than ‘q’ i.e.
n-1 <q<1
&Γ ⅈ𝑠 𝑔𝑎𝑚𝑚𝑎 𝑓𝑢𝑛𝑐𝑡ⅈ𝑜𝑛
28
29. TRANSFER FUNCTION OF FRACTIONAL
OPERATOR
• A continuous time linear input / output system can be
described as –
• 𝑖=0
𝑛
𝑎𝑖 𝐷𝑡
𝛼𝑖
y t = 𝑘=0
𝑚
𝑏𝑘𝐷𝑡
𝛽𝑖
𝑢(𝑡)
• It’s transfer function--
𝑤ℎ𝑒𝑟𝑒 𝛼𝑛 > 𝛼𝑛−1 > ⋯ > 𝛼0 ≥ 0
𝛽𝑚 ≥ 𝛽𝑚−1 > ⋯ > 𝛽0
≥ 0 𝑎𝑟𝑒 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠.
𝐺 𝑠 =
𝑏𝑚𝑠𝛽𝑚 + 𝑏𝑚−1𝑠𝛽𝑚−1 + ⋯ + 𝑏0𝑠𝛽0
𝑎𝑛𝑠𝛼𝑛 + 𝑎𝑛−1𝑠𝛼𝑛−1 + ⋯ + 𝑎0𝑠𝛼0
29
30. FRACTIONAL ORDER PID CONTROLLER
• The most common form of a fractional order PID
CONTROLLER is 𝑃𝐼𝜆
𝐷𝜇
.
• Involving an integrator of order 𝜆 and a differentiator of order
𝜇 where 𝜆 and 𝜇 can be real number.
Fig.11 Block Diagram of FOPID 30
31. TRANSFER FUNCTION OF FOPID
• It has the form as:
𝐺𝑐 𝑠 =
𝑈 𝑠
𝐸 𝑠
= 𝐾𝑝 + 𝐾𝑖
1
𝑠𝜆
+ 𝐾𝐷𝑠𝜇
where Gc s is the transfer function of the controller
E s is the error & U s is controller output.
The
1
sλ
is integrator term on logarthimic table slope − 20db.
31
32. ADVANTAGE OF FOPID
• Enhance the systems control performance.
• Better control of dynamic system which are described by fractional
order mathematical model.
• FOPID controllers are less sensitive to change of parameters of a
controlled system.
• There is two extra degrees of freedom to better adjust the dynamical
properties of a fraction order control system.
32
34. FRACTIONAL ORDER PID CONTROLLER
TUNING
• FOPID controllers are tuned based on
1. Frequency domain specifications
2. Time domain based optimal control tuning
• Frequency domain Analysis
• “Monje – Vinagre” proposed it.
• Based on following specified values
…….continued
34
35. FREQUENCY DOMAIN ANALYSIS
1. No steady state error
2. Specified gain crossover frequency
1. 𝐶 𝑗𝑤𝑐𝑔 G j𝑤𝑐𝑔 dB = 0dB
3. Specified phase margin ∅𝑚 represented as—
1. −𝜋 + 𝜑𝑚 = arg(𝐶 𝑗𝑤𝑐𝑔 G j𝑤𝑐𝑔 )
4. For Robustness-- 𝑑(arg(𝐶 𝑗𝑤𝑐𝑔 G j𝑤𝑐𝑔 ))
𝑑𝑤
= 0
35
36. TIME DOMAIN ANALYSIS
• For designing controllers based on time domain ,controllers aim
at minimization of different integral performance indices as:
1. Integral square error ISE = 0
𝑡
𝑒2 𝑡 𝑑𝑡
2. Integral absolute error IAE= 0
𝑡
|𝑒 𝑡 |𝑑𝑡
3. Integral time square error ITSE = 0
𝑡
𝑡𝑒2 𝑡 𝑑𝑡
4. Integral time absolute error ITAE= 0
𝑡
𝑡|𝑒(𝑡)|𝑑𝑡
36
52. CONCLUSION
• Linearization of AMB System has been done.
• PID Controller & AMB system has been analysed.
• Fractional Calculus and Fractional Operator has been studied.
• Fractional Order PID controller and its modelling based on its
transfer function has been studied.
• Robustness of AMB has been analysed.
52
53. SCOPE OF FUTURE WORK
• Tuning and Optimization of parameters of AMB using
FOPID.
• Use of Fuzzy & Genetic Algorithm.
53
54. REFERENCES
1. Gao, Z. "Scaling and Bandwidth-Parameterization Based Controller Tuning," Proceedings of the IEEE
American Control Conference, pp. 4989 – 4
2. Gibbs, P. and Geim, A. “Is Magnetic Levitation Possible?” http://www.hfml.kun.nl/levitation-
possible.html, March, 1997.
3. Glover, K., and J.C. Doyle, "State-space formulae for all stabilizing controllers that satisfy an H-Infinity
norm bound and relations to risk sensitivity," Systems and Control Letters, Vol. 11, pp. 167 - 172, 1988.
4. Han, J. "Nonlinear Design Methods for Control Systems," Proceedings of the 14th IFAC World Congress,
Beijing, 1999.
5. Arijit iBiswas, iSwagatam iDas, iAjith iAbraham iand iSambarta iDasgupta, i“Design iof ifractional-order
icontrollers iwith ian iimproved idifferential ievolution”, iEngineering iApplications iof iArtificial
iIntelligence, iVolume i22, iIssue i2, ipp. i343-350, iMarch i2009.
6. Schweitzer, G. Active Magnetic Bearings,” ISBN 3728121320, Eidgenössische Technische Hochschule,
Zürich, 1994 . 54
