2008 American Control Conference Westin Seattle Hotel, Seattle, Washington, USA June 11-13, 2008
a and
ThBI02.3
Dynamic D c Output Feedbacck Controol of a Magnetic M Bearing Syystem via LMIs
M. Daavoodi, Studennt Member, IE EEE, P. Kohaan Sedgh, Student Memberr, IEEE, and R. R Amirifar, Mem mber, IEEE
Abstract— This T paper pressents a new dessign procedure for a robust stabiliity, robust ∞ control c and rob bust control via dynamic output feedback for f a class of uncertain lin near systems. The uncertainties u arre norm bounded type. The sttate space matricess of the contro ollers are the solutions of soome linear matrix inequalities i pro oblems. Finally,, these procedu ures are applied too an active rad dial magnetic bearing b system m to support a high-speed energy storage s flywheeel.
A
I. INTR RODUCTION
N active magnetic beaaring (AMB) is i a collectionn of electromaagnets used to suspend an obbject via feedbaack control. The principal ben nefits of AMB Bs, compared to mechanical and a hydrostatic bearings, are a dramaatic reduction in friction, whiich, in turn, allows efficiient operation at extremely higgh speeds, thee elimination of lubricants andd their associateed supply systeems, the abilityy to operate in a vacuum and at high tempperature, and the ntrolling the stiffness of the capability forr actively con bearing. Due to t these advan ntages, AMBs have h found usaage in many indusstrial applicatio ons, such as electric auxiliarries for aircraft, ennergy storage flywheels, f as well w as high-speeed turbines and compressors, ettc. [1]. c a bassic AMB systeem comprisingg an This paper considers electromagnett on each side of o a rigid rotor,, as shown in F Fig. 1. The modell on which thee controller deesign is basedd is described by a second-order linear interrval system w with unknown distturbances. Thee parameter uncertainty in the system is well described by y the given parrameter intervaals, modeled dynaamics may be included in the while the unm disturbance [2]. [ To elim minate the neeed for veloccity feedback, an output o feedbacck controller iss proposed whhich uses only the rotor r position signal. s Linear matrrix inequalitiees (LMIs) havve emerged ass a powerful form mulation and deesign techniquue for a varietyy of linear control problems. Sin nce solving LMIs L is a convvex s formulaations offer a optimization problem, such p that laack numerically trractable means of attacking problems an analytical solution. Connsequently, red ducing a conttrol m to an LMI can c be consideered as a practiical design problem solution to this problem. Manuscript recceived Sept. 21, 20 007. M. D. is with the Electrical Eng gineering Departm ment, Tarbiat Moddares University, Tehraan, Iran (e-mail:
[email protected]). P. K. is with the t Electrical Eng gineering Departm ment, Tarbiat Moddares University, Tehraan, Iran (e-mail: ko
[email protected]). R. A. is with the Electrical Enggineering Departm ment, Tarbiat Moddares University, Tehrran, Iran (corressponding author to provide phoone: +(98)(21)828843222; fax: +(98)(21)880005040;
[email protected]).
978-1-4244-2079-7/08/$25.00 ©2008 AACC.
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Fig. 1: Basic B magneticc bearing. y researchers have consid dered the prooblem of Many designinng robust sttability, ∞ performance and perform mance for nom m-bounded unceertain systems in recent years. In this paper, w we address robust stability, ∞ and perform mance via outpput feedback control c for thee class of uncertaiin linear systtems. We present the condditions of problem m in terms of a number of linnear matrix inequalities . Finally,, these probleems are applied to an actiive radial magnetiic bearing systtem. This paper is organnized as follow ws: Section III includes descripttion of system m and required lemmas. In Seection III, theorem ms for robust sstability, annd ∞ perform mance are presenteed. The appliccation of desiggn methods in magnetic bearing system and simulation reesults are presented in Sectionn IV. Finally, Concluding remarks are given in Sectionn V. III.
PRELIMINAR RIES
This paper p is conceerned with the class of uncerttain linear systemss that can be ddescribed by state-space s equuations of the form m: ∆
∆ 1
is thee state vector, is the disturrbance input veector, is the conttrol input vectoor, is the conntrolled outputt vector, is the measurement vector. , , , , , , , nstant matrices with approppriate dimensiions, and are con ∆ , ∆ m bounded parameter p reppresent norm wing form: uncertaiinties which arre in the follow ∆ , ∆ where , , , are known reaal constant mattrices, and , iss an unknown matrix that belongs to the following set:
,
|
Ω
Lemma 3 (Schur Complement): The Linear inequality:
By applying output feedback controller in the following form:
0 0 and S is an affine function of
with , is equivalent to:
2
,
0 X
0
to (1), the closed loop system will be: ∆
∆
3
where
Lemma 4: Let Σ, Ω, Γ be matrices with appropriate dimensions which Ω is a symmetric matrix Then for every matrix F with , Ω Γ Σ Γ Σ 0 is equivalent to the Ω ΓΓT Σ Σ 0 , if and only if there exist a constant 0 [6].
,
III. MAIN RESULTS ,
and the closed loop uncertainties ∆ ∆
and ∆
A. Robust Stability via Output Feedback The following theorem proposes an LMI for designing output feedback controller satisfying robust stability. Theorem 1: consider the change of controller variables as follows [3]:
are:
∆
,
where 0
0
0
,
0
0
,
0
7
Lemma 1: Suppose that system (4) is asymptotically stable.
where that:
and
are invertible and should be chosen such 8
4 let
denote its transfer function. if 0 then the following statements are equivalent: .
γ 0 and
.There exists
such that:
0
9
and
0,
0,
For system (1), there exists an output feedback controller in the form of system (2) such that closed loop system (3) for every admissible uncertainties, satisfies robust stability, if the following system of LMI's is feasible. Find , , , , , and scalar 0 such that:
0
0 5 Lemma 2 (Bounded Real Lemma): For system (4), ∞ performance, with 0 is equivalent to the existence of 0 satisfying: 0
6
2523
10
where: , , The state matrices of the controller ( be recoverd from (7). Note that symmetric terms.
,
, ) can is used to show
Proof: The system (3) is said to be stable for 0 such perturbations ∆ if there exists a matrix that: ∆
∆
0
where ,
11
18 ,
By separating uncertain part of inequality (11) and using lemma 4, the following inequality is obtained: 0
12
0
13
Since matrices and are multiplied in inequality (13), it is non convex. Therefore and are partitiond as follow:
It is readily verified that
Π ,Π =
,
are defined in (7).
∆
∆ 0
19
where denotes the Hermitian transpose. By separating uncertain part of inequality (19) and using lemma 4, the following inequality is obtained:
0
satisfies the identity
0
,Π =
(20) 15
0
with pre-and post multiplying inequality 0 by Π and Π respectively, inequality (9) is obtained. Similarly, the last LMI condition (10) is derived from (13) by pre- and post multiplication by diag(Π , , ) and diag(Π , , ) respectively.
B. Robust ∞ Control via Output Feedback The following theorem proposes an LMI for designing output feedback controller satisfying ∞ performance. Theorem 2: For system (1), there exists an output feedback controller in the form of system (2) such that closed loop system (3) for every admissible uncertainties, satisfies ∞ performance with 0, if the following system
of LMI's is feasible. Find
scalar
,
14
,
Π
,
Proof: By considering (6) for closed loop system (3), following inequality is obtained:
Using Schur complement for inequality (12), yields: 0
where
, 0 such that:
,
,
,
,
0
and
16
and
0
0 0 0
0
17
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Using Schur complement for inequality (20), yields:
0
0 0 0
0 21
and are multiplied in inequality (21), Since matrices it is non convex. Therefore and are partitioned the same as (14) and Π and Π are defined similar to (15). With pre-and post multiplying inequality 0 by Π and Π respectively, inequality (16) is obtained. Similarly, the last LMI condition (17) is derived from (21) by pre- and diag(Π , , , , ) and post multiplication by diag(Π , , , , ) respectively.
Remark: Since and are multiplied in (17), this inequality is non convex, but according to that is a scalar, this inequality can be easily solved by line search on . Suppose an arbitrary positive scalar, then if the problem was infeasible, change it with respect to this fact that the problem is feasible. C. Robust Control via Output Feedback The following theorem proposes an LMI for designing output feedback controller satisfying performance. Theorem 3: For system (1), there exists an output feedback controller in the form of system (2) such that closed loop system (3) for every admissible uncertainties,
satisfies
0 and
scalar ⎡R I ⎢I S ⎢ ⎢ψ ⎢⎣ C
, such that:
, matrices
,
,
⎤ ⎥ ⎥>0 Z ⎥ ⎥⎦
ψ
A. Problem Formulation A dynamical mathematical model for the AMB shown in Fig. 1, can be established as follows:
,
where
0
(23) ,
0
, , , are defined in (18) and are defined in (7).
Proof: By considering (5) for closed loop system (3), following inequalities is obtained: ∆
(27)
(22)
C
, ,
,
T
0 0
where , ,
IV. APPLICATION TO MAGNETIC BEARING
performance, if the following system of LMI's is
feasible. Find
∆
0,
0
Mass of the rotor (kg); Position displacement of the rotor (m); Nominal air gap (m); Permeability of free space H/m; Total pole-face area of each electromagnet (m ); Number of turns on each electromagnet coil; Electromagnet coil currents (A); An unknown disturbance (N); Some known force acting on the rotor (N).
When (1) is linearized at the equilibrium point, , 0 and augmented with the control structure shown in Fig. 2, the linearized model is obtained as the following secondorder system: (28) where (29)
24 By separating uncertain part of inequality (24) and using lemma 4, the following inequality is obtained: 0
0, 25 Using Schur complement for inequality (25), yields: Fig. 2. Diagram of the control system.
0 0
0
26
0, Since matrices and are multiplied in inequality (26), it is non convex. Therefore and are partitiond the same as (14) and Π and Π are defined similar to (15). With pre-and post multiplying inequality
0
and Π , respectively, inequality (22) is by Π , obtained. Similarly, LMI condition (23) is derived from (26) by pre- and post multiplication by diag(Π , , , ) and diag(Π , , , ) respectively.
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Due to inaccuracies in the measurement of some of the physical parameters and changing environmental conditions, the system parameters and are generally uncertain. However, without loss of generality, it can be assumed that their values lie within some known intervals:
where,
,
,
and
are known scalars satisfying: 0
0
In order to avoid the need for velocity feedback, while at the same time achieving satisfactory stability, this paper addresses the control of the system (28) using a output
feedback controller. Assuming that only the rotor displacement position is measured, and denoting:
the system (28) can be converted into the following equivalent state-space form:[2]
TABLE 1 Parameters and and lower and their upper bounds
Parameter
∆
∆
B. Simulation Results The values of and, and those of the interval and , 1,2, are given in Table 1. boundaries,
0
∆
0
1 , 0
0
∆
,
1 0
(30)
Value 359.6 ‐10.35 240 390 ‐22 ‐4.5
TABLE 2 Magnetic Parameters
Parameter value 6kg m 4 2.610 A 40 N 5 A .4mm
, and the parameters and satisfy 0 and 0. By defining nominal values , and scaled where,
, Δ
errors as Δ
, then they implies:
Δ with and by the class of uncertainties:
with
∆
∆
|
1
∆
Δ
1
, Note that the original parameters has been transformed into the new parameters , (−1, 1) by using a nominal values , and , as weights. Therefore: 0
1 ,∆ 0 0 ∆
∆
0 ∆
0 , 0
0
,
0
0
1 0,
,
and uncertain matrices are: 0
0 , 0
0
0 , 0
1 0 , 0 1
1 0
A general output feedback controller for the system (30) can be written in the following form: (31) where , , and are four scalar controller is introduced coefficients, to be designed, and the term to compensate for the effect of the force , the coefficient being given by [2]: 1
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Fig. 3. Closed-loop system response using robust stabilizer controller: =1.8, =6.5 (solid), =0.1, =1 (dotted), =0.4, =3 (dash).
The nominal parameters for the bearings are given in Table 2. The close loop system response with respect to robust stabilizer controller is shown in Fig. 3. The close loop system response with respect to robust controller is shown in Fig. 4 and its bode-magnitude diagram is shown in Fig. 5, Fig. 6 shows the ratio of regulated output energy to The disturbance energy of this system. The close loop system response with respect to robust controller is shown in Fig. 7 and its bode-magnitude diagram is shown in Fig. 8. The obtained value of from controller without uncertainty is 0.016, and the obtained value of from controller without uncertainty is 2.1538e006.Comparing Fig. 3, Fig. 4 and Fig. 7 shows that controller has a better robustness than the other controllers, but controller has a better transient response. V. CONCLUSION and In this paper, we addressed robust stability, performance via output feedback control for the class of uncertain linear systems. We presented the conditions of problem in terms of a number of linear matrix inequalities. Then these problems were applied to an active radial magnetic bearing system to support a high-speed energy storage flywheel. The effectiveness of the design was shown in simulation results.
Fig 4. Closed-loop system response using robust =6.5 (solid), =0.1, =1 (dotted), =0.4,
controller: =3(dash).
=1.8, Fig 8. Bode-magnitude diagram using robust controller: =6.5 (solid), =0.1, =1 (dotted), =0.4, =3(dash).
Fig 5. Bode-magnitude diagram using robust controller: =6.5 (solid), =0.1, =1 (dotted), =0.4, =3(dash).
=1.8,
=1.8, Fig 9. Closed-loop system response: Robust stabilizer (dotted), controller (dash), controller (solid).
REFERENCES [1]
M. Dussaux, “The industrial applications of active magnetic bearing technology,” in Proc. 2nd Int. Symp. Magnetic Bearings, Tokyo, Japan, 1990, pp. 33–38.
[2]
G. R. Duan, Z. Y. Wu, C. Bingham, and D. Howe, “Robust magnetic bearing control using stabilizing dynamical compensators,” in Proc. Int. Elect. Machines Drives Conf., Seattle, WA, 1999, pp. 493–495.
Fig 6. The ratio of regulated output energy to the disturbance energy of the system.
[3]
C. W. Scherer, P. Gahient, and M. Chilali, “Multi-objective output feedback control via LMI optimization”, IEEE Trans. Automat. Contr., vol. 42, no. 7, pp. 896-911, July 1997.
[4]
M. Chilali and P. Gahinet, “
∞
design with pole placement
constraints: An LMI approach”, IEEE Trans. Automat. Contr., vol. 41, no. 3, pp. 358-367, Mar. 1996. [5]
P. Gahinet, A. Nemirovski, A. J. Laub, and M. Chilali, “The LMI control toolbox”, in Proc. 33rd IEEE CDC, Florida, USA, 1994, pp. 2038-2041.
[6]
L. He and G. Duan, “Multi-objective control synthesis for a class of uncertain fuzzy systems”, in Proc. 4th International conference on mechanical learning and cybernetics, Guangzhou, China, 2005.
[7]
J. Lofberg, “YALMIP: A toolbox for modeling and optimization in MATLAB,” in Proc. CACSD Conf., Taipei, Taiwan, 2004.
Fig 7. Closed-loop system response using robust controller: =6.5 (solid), =0.1, =1 (dotted), =0.4, =3(dash).
=1.8,
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Available: http://control.ee.ethz.ch/˜joloef/yalmip.php.
