Tran Xuan Minh Tap chi KHOA HOC & CONG NGHE 122(08): 155- 160
AN OPTIIVIAL STATE FEEDBACK CONTROL METHOD FOR 4 DEGREES OF FREEDOIVI - RIGID ROTOR ACTIVE IVIAGNETIC BEARING SYSTEM
T r a n Xuan Minh Thai Nguyen University ofTeclinolog}.
S U M M A R Y
Based on mechanical - electrical - magnetic principles, the paper presents detailed analyses to build a completed mathematical model for 4 degree of freedom - rigid rotor active magnetic bearing (AMB) system. Gyroscopic effect, one of significant reasons affecting to performances of system is mentioned in this research. By using the centralized approach, a state-space model for multi-input multi-output (MIMO) active magnetic bearing system is buih. An optimal state feedback controller is then designed in order to directly formulate the performance objectives of the control system and provides the best possible control system for a given set of performance objectives. Zero steady-state error of system outputs is also given by the means of integrators which are added into the system. As a result, MIMO system's responses achieve quick stabilization and good performances.
Keywords: Active Magnetic Bearing (AMB): gyroscopic. MIMO; stole-space. Linear Quadratic Regulator (LQR)
I N T R O D U C T I O N
Active Magnetic Bearing ( A M B ) comprises a set of electromagnetic m e c h a n i s m s to provide bearing forces which suspend rotor shaft freely in space. T h e s e systems utilize feedback control m e t h o d s to stabilize the rotating motion of them. This advanced bearing technology offers m a n y significant advantages, c o m p a r e d to conventional bearings, since mechanical non-contact between rotor shaft and static parts is generated by electromagnets. With a suitable active control approach, d a m p i n g and bearing stiffness characteristics of A M B can be adjusted [ 1 , 2] Control methods contribute an important role in designing an A M B system.
In many applications, however. the performance of a controller is highly influenced by the coupled impact in motion of the system which should not be neglected.
Many different control methods have been applied successfully for A M B , with or without the mention of the gyroscopic effect [4-9]. T h e s e include conventional decentralized a p p r o a c h e s such as PD. P I D . . . and nonlinear control methods such as
Tei 0913.^54975
feedback linearization, backstepping., [4, 5], [7], [9]. A new trend for modern control methods is also attracted many interests.
These centralized methods consisting of Pole- placement, LQR, LQG, H,,, n-synthesis... [6, 7], [9] increase quickly due to the rapid development of the sensor technology and digital signal processing recently. As a result.
measurement and computation tasks of various physical signals can be implemented easily f o r t h e purpose of feedback control.
In this research work, a fully mathematical model of 4 - D O F A M B is described, in which the gyroscopic effect is also included in the system d y n a m i c s . A modern centralized control method is designed for a M I M O radial suspension system. By using this approach, the optimal controller is then proposed in order to yield high performance for the system in terms of control energy and control error. Obtained results show that an improvement in dynamic performance of the system can be achieved
This paper is structured in four parts. Part 2 dedicates to modeling of the s \ s t e m in terms of dynamics and clectromagneltc issues The control design is described in part 3 Part 4 is the computation and simulation results
155
Tran Xuan Minh Tap chi KHOA HOC & CONG NGHE 122(08): 155- 160 MODELING OF 4-DOF AMB SYSTEM
Dynamic equations of rotor
The object of this research work is a drive system including 2 active magnetic bearings arranged in both ends of the shaft. The AMBl and AMB2 provide radial forces in the directions of XA, yA and Xg, ye respectively.
Figure 1. 4-DOF rigid rotor AMB system [5]
Based on dynamic principles of a rotating object in literature [1, 2, 3, 4, 5, 6], equations of motion for rotating shaft can be recognized by Newton's second law and Euler's second law of motion. This arrangement is assigned 4 physical degrees of freedom, describing translations in the x- and y- directions and angular displacements about the xz (P) and yz (a) planes. This contributes to elements for structural matrices of the mechanical system which enable the Lagrange representation of the second order dynamical system.
Mq + Gq = BUj,=F where, q — [/3 AJ:^
Figure 2. The rigid rotor provided with magnetic bearings andsensors[2]
0 ^L^r,
M = m 0
0 J,
G = 0 0The presentation of the generalized force F:
F^
a b 1 1 0 0 0 0 0 0 a 1 0 0
*
1
;C = c d 0 0
1 0 1 0
0 c 1 0 d 1 The mass matrix is symmetric and positive definite, IVI=M^>0; J^-J, because the rotor is assumed to be symmetric; the gyroscopic matrix is skew-symmetric, G=-G^ containing the rotor speed ^„„ as a linear factor;
Radial bearing forces are represented by four controlled forces, which act within the bearing planes in the x- and y- directions.
"/=[/« /,» f,A f,J\
Each bearing force is be described as a linearized function of the rotor displacement in the rotor and the coil current. Hence, the following relationship results for the force vector u^ can be shown as follows [2]:
- K , q ^ - l - K , i (2)
wh£^e,
K, =diag(K^
n ^ , . « ) ;K, is bearing stiffness vector; i: is vector of control current of four bearing coils;
K,=diag{K,^,K,^,K.^,K,i,y.\ = [i^^ Z,^ (,,^ ;;,,]' K,: is force/current vector; q^ expresses the rotor displacements within magnetic bearings.
For the purpose of simple differential
equation description, the center of gravity
(COG) coordinates, combined inq, have
been introduced. However, rotor
displacements involve the bearing
coordinates(\i,. It therefore is necessary to
transfer the bearing coordinates q^ into the
COG coordinates q through a linear
transformation matrix ^T^:
Tran Xuan Minh Tap chl KHOA HOC SL CONG NGHE 122(08): 1 5 5 - 1 6 0
o 1 0 0 6 1 0 0 0 0 <7 1 0 0 i 1
H
X
-a y q(=[-»w ^hh yitA yhi,\ ^
By inserting o f ( 2 ) a n d ( 3 ) into ( 1 ) , the complete equation o f motion c a n b e obtained as follows:
Mq + Gq + K ^ q - B K , i
where, K,^ = - B K , |,Ts is t h e negative bearing stiffness matrix converted into C O G coordinates.
In general, d i s p l a c e m e n t m o t i o n s Aj:^,Ay^ and angular motions o,/3 will b e coupled d u e t o the presence o f t h e g y r o s c o p i c effect in matrix G if rotor speedw„, ^ 0 ,
E l e c t r o m a g n e t i c e q u a t i o n s of m a g n e t i c bearing model
Assuming that in t h e air gaps and iron paths, the magnetic flux and flux density are constant. Moreover, t h e iron will be treated as operating below saturation then stored magnetic energy W^, and magnetic flux in the paths 0u can be calculated by [4]:
'.!-• I
w . •• t — f i - i / F - ^ ^ 25H 2,1 2pA !':„.
- + -
/'/.
where, K„ = 2.^/^,,; A,, is cross-section area o f flux path. It is assumed that flux leakage is not existed, <P = 0^ = <P,,.
Ni (J^A^Ni
[ft HiH™, MiC/il-i
When the magnetic flux in the air gaps is unity then malting: B = B„= '1>/At,:
[f rotor is displaced b y an a m o u n t ofzl^ then a magnetic force F is generated [ 1 , 2 ] , [4]:
S.' [ I I . ] ' F = - -
2i + Cv.
The relationship between magnetic force and current in (6) is represented by square term which indicates the nonlinearity.
—7T\H is neglected since t h e equation lA.J l^fi)
above does not take into account magnetization effect o f ferromagnetic materials. A s a result, (6) becomes:
aw^ fipA^N-
(7) (4)
Positive side ' Negative side Figure 3, An differential arrangement for
electromagnets in x-axis Two functions for magnetic forces will be given by:
The total magnetic force F will be the difference between F+ and F.; K - " "
4 Under the assumption that A5<g:,N„and A/<g:igthe linear function of F can be approximated by the first order of Taylor- expansion as follows [1,2]:
f = l ^ A s + ^ A ; ' = /:,As + ^,A/' (8)
State-space descnption
The rotor dynamic equations and the linearized electromagnetic equations constitute a set of equations describing dynamic behaviour of the system. We introduce a state vector presenting for bearing displacements and their derivatives:
\=\d Ax -a Ay 0 Ax - a Ay|
the(6)put vector: ^—[^^ '^ ^yA '>B| and, the output vector at the sensors o f bearings:
Then, the state-space description of the 4 - D O F A M B system yields:
157
Tran Xuan Minh Tap chi KHOA HOC & CONG NGHE 122(08): 155-160
B
y = Cx-|-Du
KA
1 M^'BK,
O 4 , . ^4X4
;C=[C,,, 04.,];D-[0,„,]
Controller design
Linear time-invariant state equations of proposed system can be expressed in generalized form:
[x = Ax+Bu
|y =Cx with
X e R";a,y €R'";A€ R'""';B e R""";C € R"""' This controlling design seek to minimize the quadratic performance index:
J = j(\''QTi + u^Ruyt
where, Q is state weighting matrix and positive semi-definite; R is control weighting matrix and positive definite; x is the state vector.
fl-"
Figure 4. LOR closed-loop block diagram In the LQR design approach, the only design parameters are the weighting matrices, Q and R. A LQR controller is constructed under the assumption that all states of the controlled system are available for feedback. Once the LQG controller is obtained, the dynamic behaviour of each controlled variable can be checked and the closed-loop poles can be evaluated [10].
Optimal control law denoting by u„^„ can be represented as: u,,^„ =-R"'B^Pxor u„^,, =-Kx where, K = R"'B^P is optima! feedback gain matrix.
Should /, in (11) be infinite, the matrix Riccati equations reduce to a set of 158
simultaneous equations, where P is unique positive-definite solution to the following eqi^dt^on:
P A - H A ^ P - H Q - P B R ' B ^ P = O (13)
Therefore, if we want to modify the system behaviour, the weighting matrices Q and R can be adjusted to obtain different optimal feedback gain matrix K. A reasonable simple choice for the matrices Q and R is given by Bryson 's rule [9]. Then Q and R are selected diagonally with:
Qr- max acceptable value of 37
",,= max acceptable value of u^
,/e(i,2,..-,y};/e{L2,...,/}
;ri corresponds to the following criterion:
whic]
• ' = / Ea^.w'+PT, "J,"/')' k (14) In this paper, we introduce an integral action based on the tracking error in order to achieve zero steady-state error under any changes in references. A number of integrators are added in the system model then the state-space model of the new augmented system cab be written as:
- B K , (
- C
BK,
(15)
= |c o|
where the additional auxiliary state variables e act as "accumulated errors".
SIMULATION RESULTS
In this section, AMB's simulating parameters are taken from [7] in order to prove the effectiveness of the method:
Rotor mass: w=12.4(kg); Distance between r n ^ . center and bearings/sensors: /„= 4=
0.2 Km); /, - /</= 0.215(m); Momen of inertia in /, 7 and k axes: J, = Jj = 2.22xl0"'(kg.m^) and ^t=6.88x10'^ (kg.m"). Speed of rotor corm
15000(RPM). Ratio of magnetic
force/current: K, = 102.325 ( \ A); Bearing
Tran Xuan Minh Tap chl KHOA HOC & CONG NGHE 122(08). 155-160 stiffiiess coefficient: K, = -4.65xlO'(N/m);
Earth's gravity acceleration: g=9.81(kg.m/s').
The computation results of the optimal feedback gain matrix K in equation (12) can be obtained with given weighting matrices Q and R and Riccati equation solution P.
K"^- - 0 , 0 2 6 00055 0 0089 1 85x10'"
-7.071 7.071 - 5 . 9 5 x 1 0 -
1 28x10"'"
0 026 0.0055 - 0 0089 - 1 . 6 6 x 1 0 " "
7.071 7.07]
4 8 2 x 1 0 " "
1 52x10"'"
- 0 0089 8 7 x ! 0 - "
- 0 026 0 0055 - 5 . 2 2 x 1 0 "
1,99x10-"
-7.071 7071
0 0089 - 2 , 7 x 1 0 " '
0,026 0.0055 5 5 6 x 1 0 " "
6,66x10""
7 071 7 071
With the optimal feedback gain matrix K above, eigenvalues of closed-loop system can be achieved:
-1242 1 + 5 1 7 7i - 1 2 6 . 7 5 - 5 2 . 8 1 . -58 5 + 2 6 7 6i
- 1 2 4 2 1 + 5 1 7 7 i , - 1 2 6 . 7 5 + 5 2 , 8 1 ; - 5 8 3 5 + 2 6 7 . 6 1
- 5 8 3 5 - 2 6 7 6i - 5 8 3 5 - 2 6 7 . 6 ) ;
h is can be seen that all the closed-loop eigenvalues have negative real parts
indicating that the closed-loop is asymptotically stable. We see in the Figure 5 and 6 that the LQR design exhibits excellent regulation performance. The zeros response for LQG design decays to zeros quickly with not much oscillation. The results indicate that the proposed optimal controller is superior in terms of both regulation performance and control efforts.
The figure 7 shows that the performance of LQR method is much better than pole- placement method when the same step inputs are applied to. With the addition of integrators in the LQR model, steady-state errors are improved significantly. Moreover, in contrast to the pole-placement method, where the desired performance is indirectly achieved through location of closed-loop poles, the optimal control system directly addresses the desired performance objectives while minimizing the control energy. Figures 8 and 9 indicate the significant difference in terms of control efforts.
^ .-..,
Figures. State responses of 4 DOF AMB Figure 6. Output responses of4DOFAMB using LQR + integrators using LQR + integrators
Figure 7. Comparisons of output responses between different methods
Figure 8. Control efforts of 4 DOF AMB using Figure 9. Control efforts of 4 DOF AMB
Pole-placement method using LQ controller + integrators
TrSn Xuan Minh Tap chi KHOA HOC & CONG N G H E 122(08): 1 5 5 - 1 6 0
C O N C L U S I O N
In this paper, a detailed m a t h e m a t i c a l m o d e l of a 4 D O F A M B system h a s b e e n constituted in which the g y r o s c o p i c effect is also mentioned. T h e p r o p o s e d L Q regulator h a s performs excellent qualities o f both regulation performance and control effort. This centralized control approach is successfully constructed under the assumption that all states can be measured. Moreover, as integrator parts have been added to L Q R model, steady-state errors are reduced significantly in dynamic output responses of the system. Therefore, this design can be a good reference for an alternative realistic designs, such as optimal output feedback design and observer design, since the issues of unmeasurable physical quantities are considered. Those unsolved problems are wished to be investigated in other research works.
Modeling and simulation with full-state feedback", IEEE Transactions on Magnetics, Vol.
3 1 , No. 2, 1995.
4. M. S- de Queiroz and D, M. Dawson,
"Nonlinear control of Active Magnetic Bearing: A backstepping approach", IEEE Transactions on Control Systems Technology, Vol. 4, No. 5, March 1996.
5. Abdul R. Husain, Mohamad N. Ahmad and Abdul H. M. Yatim, "Deterministic models of a Active Magnetic Bearing System", Joumal of Computers, Vol. 2, N o . 8, October 2007, 6 Tian Ye, Sun Yanhua, Yu Lie, "LQG Control of Hybrid Foil-Magnetic Bearing", I2th International Symposium on Magnetic Bearings, 2010.
7. Chunsheng Wei, Dirk Soffker, "MIMO-control of a Flexible Rotor with Active Magnetic Bearing", 12th International Symposium on Magnetic Bearings, 2010.
8. Quang Dich Nguyen, Nobukazu Shimai, Satoshi Ueno, "Control of 6 Degrees of Freedom Salient Axial-Gap Self-Bearing Motor", 12th International Symposium on Magnedc Bearings, August, 2010.
9. Luc Quan Tran, Xuan Minh Tran, "Design a state feedback controller with Luenberger observer for 4 degree of fi-eedom - rigid rotor active magnetic bearing system", Proceedings of the First Vietnam Conference on Control and Automation, November, 2011 (in Vietnamese).
10. Robert L. Williams 11. Douglas A. Lawrence, Linear State-Space Control Systems, John Wiley
& Sons. 2007 R E F E R E N C E S
1. Akira Chiba, Tadashi Fukao, Osamu Ichikawa, Masahide Oshima, Masatsugu Takemoto and David G. Dorrell, Magnetic Bearings and Bearingless Drives Newnes, 2005.
2. Gerhard Schweitzer and Eric H Maslen, Magnetic Bearings: Theory, Design, and Application to Rotating Machinery. Springer- Verlag, 2009.
3. R, D. Smith and W, F. Weldon. "Nonlinear control of a rigid rotor magnetic bearing system:
T O M T A T
M O T P H U ' O N G P H A P D I E U K H I E N T O I U U P H A N H O I T R A N G T H A I C H O H E T H O N G O BO T l T C H U D O N G 4 B A C TU" D O R O T O R C l T N G
T r 3 n Xuan M i n h ' Trudng Dgi hoc Ky ihual Cong nghiep - DH Tiidi Nguyen Dya tren cac nguyen ly ve ca- dien- tir, bai bao trinh bay nhijng phan tich chi tiet de xay dung nen mpt mo hinh toan hgc day du cho he thong 6 da tir chu dgng (Active Magnetic Bearing-AMB), rotor cirng, 4 bac tu do. Anh huong hoi chuyen la mgt trong nhirng nguyen nhan chinh anh hudng den chat lugng lam viec ciia he thong cdng dugc de cap den trong nghien ciru nay. Bang each su- dung giai phap dieu khien tap trung, mgt mo hinh khong gian trang thai cho he nhieu dau vao, nhieu dau ra (MIMO) dugc xay dung cho he thong 6 dd tir chu dpng. Mgt bg dilu khi^n phan h6i trang thai duoc thiet ke nham true tiep dua ra cac tieu chi chat lirong cila he thong dieu khien va tao thanh mpt he dieu khien tot nhat co the ung vdi cac tieu chi ch^t tugng lam viec cho trudc. Sai lech tTnh cua tin hieu dau ra cung dugc cai thien thong qua cac bg tich phan duoc bo sung vao mo hinh di^u khien cua he thong. Cac ket qua mo phong cho irng cua he MIMO dai dugc dp on dinh hoa nhanh chong va chat lugng lam viec tot.
Til- kh6a: D do- tir chu dong (AMB): dnh hir&ng hoi chuyen, M!M0;kh6ng gian Irang ihdi; LQR.
Ngciy nhgn bai 09/6/2014; Ngdy phdn hiin. 11/8/2014. Ngdy duyit ddng 25/8/2014 Phdn biin khoa hoc: PGS TS Lai Khdc Lai - Dai hoc Thai Nguyen
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