Linear Parameter Varying Induction Motor Control with ... - IEEE Xplore

4th International Conference on Power Engineering, Energy and Electrical Drives

Istanbul, Turkey, 13-17 May 2013

Linear Parameter Varying Induction Motor Control with Two-degree-of freedom Controller Dalila. Khamari Abdessalem, Makouf and Drid Said, member IEEE Departement of Electrical Engineering LSPIE Laboratory Batna, Algeria [email protected], [email protected], [email protected]

Abstract— This paper proposes a self-scheduled control method for the induction motor. We design a controller with two degree of freedom for the induction motor using the linear matrix inequality (LMI) based approach to linear parameter varying (LPV) systems which takes into account the nonlinear dynamics of the system We propose a two hierarchical control structure. The inner-loop current controller which considers the mechanical speed, rotor resistance, and rotor current frequency as a variable parameter achieves robust tracking of the stator current reference signal. The outer loop mechanical speed controller aims to ensure reference speed tracking. In simulation the controller has been tested, stability and high performances have been achieved over the entire operating range of the induction motor.

includes a feed forward part and a feedback part which is potentially more powerful to achieve strong performance requirement. For the speed controller a proportional integrator (PI) is largely sufficient to ensure a good tracking. The paper is organized as follows. In the following section the LPV model of induction motor and polytopic representation taking into account parameter variation defined above is presented. In section 3 the stator current controller synthesis method and speed controller are given and simulations are carried out to confirm the synthesis. The last section contains concluding remarks. II.

I.

INTRODUCTION

The control of induction motors has attracted much attention from researchers and engineers; it is still a very active research area. The industrial interest in induction motor control is documented by more than 80.000 patents on this subject. The availability of low cost powerful digital signal processor and sgnificant advances in power electronics motivated the design of complex induction motor controls. The aim is to achieve the same, or even superior, performance on speed tracking and efficiency which are obtained by more sophisticated and expensive, but less reliable, electric motors such as direct or permanent magnet ones. Recently, there is a great interest to develop robust controllers for induction motor drives to achieve high performances and robustness. In this context, the gain scheduled LPV controller is designed for a nonlinear model of the induction motor [2],[3],[4]. Instead the motor dynamic model is brought to a LPV system via a state transformation. The current fed of the induction motor model has a particular structure that can be written as an LPV system because of affine dependence of rotational speed, rotor resistance and rotor current frequency. The variable parameter value θ (t ) can be estimated on line during control operation experimentally by a Kalman filter [5]. According to [6] and based on experimental results it is proved that the usual one degree of freedom controller structure [7] cannot meet simultaneously the performance and robustness objectives. To overcome this limitation a two degree of freedom controller structure is used. This structure

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INDUCTION MOTOR MODEL

A. LPV model The state space representation of the induction motor with the rotor field orientation [1] is as follows:

ª −1 θ « Lr 2 Φ ª rd º « « » = « −λ θ G (θ ) : isd « » «σ 2 2 «¬ isq »¼ « « −λ θ «¬ σ 1 ª º « 0 0 » « » « 1 » ªu º 0 « » + «σ L » u « s » ¬ sq ¼ « 1 » «0 σ L » ¬ s ¼ sd

M Lr −(

λ σ

2

θ2

θ2 + −θ 3

º » » ª Φ rd º »«i » θ3 » « sd » » «¬ isq »¼ λ τs −( 2 θ 2 + ) » σ σ » ¼ 0

τs σ

)

(1)

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Where θ1 ,

θ2

and

θ3

are time varying parameters. The other

parameters are as given below:

M

λ=

Ls

2

,τ s =

Ls Rs

,τ r =

Lr Rr

,σ = 1 −

M

2

(2)

Ls Lr


and A(θ (t )) = A0 + θ1 A1 + θ 2 A2 + θ3 A3 B, C , D , are fixed. Where θ (t ) is the time varying parameter vector and takes the following convex form for our case:

θ1 = ω , θ 2 = Rr , θ 3 = ω r

θ (t ) = α1θ1 + α 2θ 2 + α 3θ3 + α 4θ 4 + α 5θ5 + α 5θ5 + α 7θ 7 + α 8θ8 .

where ω is the rotational speed, ωr the rotor current frequency and

=1, α ≥ 0

(6)

where θi is the corner of the polytope θ. For each vertex values of θ the induction motor polytopic model can be given by:

The electromagnetic torque is given by

Te = p

r

r =1

τr

the rotor time constant. In this case is considered as time varying parameter and it can be accurately estimated on line [10].

8

¦α

M ( is ⊗ Φ ) Lr

(3)

B. Polytopic representation of induction motor The LPV model of induction motor in the ( d , q ) frame can be described in compact form as

° x = A (θ ( t ) ) x + B(θ (t )u G (θ ) : ® °¯ y = C (θ (t ) x + D(θ (t ))u

G (θ ) = α1G (θ1 ) + α 2 G (θ 2 ) + α 3G (θ 3 ) + α 4 G (θ 4 ) + α 5G (θ 5 ) + α 6G (θ 6 ) + α 7 G (θ 7 ) + α 8G (θ 8 )

where the

αi

(7)

is the barycentric coordinates of the parameter

vector in the polytope Θ for more details see [11] III.

GAIN SCHEDULED LPV CONTROLLER DESIGN

(4) A. LPV theory background (L2 Gain performance)

with

Consider an open loop LPV system P described by

ª −1 « L θ2 « r « −λ A(θ (t )) = « 2 θ 2 σ « « −λ « σ ¬ ªφrd º « » x = « isd » , « isq » ¬ ¼

C = [ 0 1 1]

º » » τs » λ −( 2 θ 2 + ) θ3 » σ σ » τs » −λ −θ3 −( 2 θ 2 + ) σ σ »¼ ª º « » « 0 0 » « 1 » ªusd º u = « », B=« 0» ¬ usq ¼ « σ Ls » « 1 » «0 » ¬ σ Ls ¼ ª0 0 º D=« » (5) ¬0 0 ¼ M θ2 Lr

0

x (t ) = A(θ (t )) x (t ) + B1 (θ (t )) w(t ) + B2 (θ (t ))u (t ), z (t ) = C1 (θ (t )) x(t ) + D11 (θ (t )) w(t ) + D12 (θ (t ))u (t ),

(9)

y (t ) = C2 (θ (t )) x(t ) + D21 (θ (t )) w(t ) Where y denote the measured output, z the controlled output, w the reference and disturbance inputs and u the control inputs. The LPV synthesis problem consists in finding a controller K (.) described by :

xK (t ) = AK (θ (t )) xK (t ) + BK (θ (t )) y (t ), u (t ) = CK (θ (t )) xK ,

(10)

Such that the closed- loop system is given by:

ªξ(t ) º ª Acl (θ (t )) Bcl (θ (t )) º ª ξ (t ) º P:« »=« »« » ¬ z (t ) ¼ ¬Ccl (θ (t )) Dcl (θ (t )) ¼ ¬ w(t ) ¼

(11)

where w and z represent respectively the input and output of the system.

We assume that the parameter dependence of the plant G is affine and Θ is a polytope with vertices θi ,i = 1, 2,..., r :

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The system is internally stable and the induced

L2 -norm of

w → z is bounded by a given number γ ; 0 for all possible The characterization of robust stability and performance for the closed-loop system (11) is proved by the following theorem: Theorem: The LPV system (10) has quadratic stability and L2 gain level γ ; 0 if there exists a matrix X ; 0 such that

ˆ , Bˆ , Cˆ ) and symmetric matrices step gives ( A Kj Kj Kj

AKj = N −1 ( Aˆ Kj − XA jY − Bˆ Kj C2 jY − XB2 j Cˆ Kj ) M −T , BKj = N −1 Bˆ Kj

CKj = Cˆ Kj M − T , where N and M are matrices such that

(12

Online computation: 1.

θ (t ) and compute its vertex decomposition

Measure

θ (t ) = α1θ1 + α 2θ 2 + .... + α rθ r

θ

(14)

belongs to a parameter

θi

polytope Θ and varies with vertices

of this polytope. where

L2 -norm): For each θ ∈ Θ , let

Gθ be the transfer function of the stable system obtained by taking θ (t ) = θ in (1). The induced L2 -norm of the family

G (θ ) = Gθ is defined by G = sup Gθ

∞

r

where

¦α

i

= 1, α i ≥ 0 . Note that α i ,..., α r can be

i =1

chosen (not in unique way) to be continuous functions of 2.

θ

Compute the state-space matrices of the controller

K (.) as a convex combination of the vertex controllers is given by:

.

B. Computation of self scheduled LPV controller Assuming that parameter dependence of the plant P is affine

Θ is polytopic with vertices θ i , i = 1, 2,..., r. , one

LPV controller

and

I − XY = NM T .

XBcl (θ ) Ccl (θ )T º » Dcl (θ )T » E 0 −γ I Dcl (θ ) −γ I »¼

The time varying parameter

Definition: (Induced

X and

Second : Compute AK j , BK j and CK j by

θ ∈Θ

for all

and

( ) denotes terms requiring the matrix to be self-adjoint. This

Y

parameter trajectories

ª AclT (θ ) X + XAcl (θ ) « Bcl (θ )T X « « Ccl (θ ) ¬


K (.) can be computed through the following

ª AK «C ¬ K

BK º

r

0¼

j =1

ª AK

» (θ ) = ¦ α j «

¬«CK

j

j

BK º j

».

0 ¼»

(15)

Note that the online computation (14) and (15) are not ime consuming.

steps: C. Induction motor stator controller synthesis

offline computation: compute the vertex controllers

The design procedure takes into account the customary H∞ and

K j = ( AK j , BK j , CK j , 0), (1 ≤ j ≤ r ) as follows:

based loop-shaping methods. A mixed sensitivity criterion is chosen and adapted to the control structure given by closed

First: Solve the set of LMIs (12) and (13)

ª XA + B C + * * * K 2 « j « T Aˆ K + A j Aj Y + B2 C K + * * « « T T ˆ B1 j −γ I «( XB1 + BK D21 ) « C1 C1 j Y + D12 Cˆ K D11 ¬ j

j

j

j

j

j

j

j

j

j

j

j

º » » * » E 0(12) » * » » −γ I ¼ *

through the sensitivity operator S(θ) while additive robustness is captured by the operator K(θ)S(θ). The self-scheduled H∞ control problem consists in finding an LPV controller such as: K(θ)=K1(θ)K2(θ). For all trajectories (variations) of θ(t), the closed loop system showed in Fig.2 and minimization of the L2-induced gain, the

and

ªX «I ¬

loop system in Fig.1 the performance objective is expressed

Iº ;0 Y »¼

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internal stability is satisfied.

(13)

1750


D. Speed controller design The PI speed controller is designed with pole placement method ( ξ = 0.707, ωn = 17.3rd / s )

Z

W G (θ) U


Y K (θ)

is

Ωref + −

θ w = ª¬isdref isqref º¼ are the reference

of stator currents. The controller output of stator voltage.

Fig: 3 general block of the proposed control of induction motor.

u = ª¬usd usq º¼ consists

The tracking errors are

y = ª¬esd , esq º¼ where esd = isdref − isd and esq = isqref − isq are the measured output.

K

− LPV Controller

Ω

Fig:1 Closed loop system The external input variables

Speed is _ref Controller +

PWM & Converter

Gp

IV.

SIMULATION RESULTS

. The proposed controller has been simulated on induction motor drive system. The drive system is subject to a benchmark test as shown on Fig. 4 In order to validate the controller design procedure. The simulations results were carried out for the speed and stator current. The motor is controlled to reach a speed as it is showed by the profile given in Fig.4. At 1.5s a load charge (10Nm) is applied. The time response of the rotor speed follows the specified reference with good accuracy and the current control error converge to zero after 5s in Fig.5 and Fig.6. The Fig.7 shows the stator voltage changes. We can note furthermore that the stator voltage peak stays within the limit. 150 mechanical speed reference speed

The weight functions choice is based on a time analysis of the LPV system and follows the same lines as classical H∞ synthesis [11]. W1 and W2 are used to shape the sensitivity function and control effort respectively. The following filters were adopted after several tests: § 10 , 10 · W1 = diag ¨ ¸ © s + 10 s + 10 ¹ (16) W = diag ( 0.8, 0.8 ) 2

The polytopic controller synthesized is described by 23 = 8 LTI controller at each corner for a performance level of γ=0.624. The resulting controller exploits all available on ω, Rr and ωr to adjust the current in induction motor as it is shown in figure3 where only one PI speed controller is used. This provides smooth and automatic gain scheduling with respect to the varying parameters.

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50

0

-50

0

1

2

3

4 Time (s)

5

6

7

8

6

7

8

Fig:4 rotor speed 2.5 C u rre n t e rror is d (A )

Fig:2 Two degree -of -freedom control structure

S p e e d (rd /s )

100

2 1.5 1 0.5 0 -0.50

1

2

3

4 Time (s)

Fig:5 stator current error

5

isd component

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[5]

S ta to r c u rre n t e rro r is q (A )

1.2 1

[6]

0.8 0.6

[7]

0.4 0.2

[8]

0 -0.20

1

2

3

4 Time (s)

5

6

7

8

[9]

isq component

Fig:6 stator current error

[10]

400 usq usd

S ta to r v o lta g e ( V )

300

[11] [12]

200

[13] 100

0

-100

0

1

2

3

4 Time (s)

5

6

7

8

Fig:7 stator voltage V.

CONCLUSION

A two degree of freedom controller for field oriented control of induction motor has been designed applying LPV H∞


D.Fodor and R.Toh "Speed Sensorless Linear Parameter Variant H∞ Control of the Induction motor" 43rd IEEE Conference on decision and Control December 14-17, 2004 Atlantis. J.C. Basilio J.A Silva and all, " H∞ design of rotor flux oriented currentcontrolled induction drives: speed control, noise attenuation and stability robustness" IET Control Theory App.,2010, Vol.4,Iss.11,pp.2491-2505. D.Khamari, A. Makouf and S. Drid ,"Control of Induction Motor Using Polytopic LPV Models,"IEEE-2011 International Conference on Communications, Computing and Control Application (CCCA'11) at Hammamet, Tunisia. E. Premapain I. Pstletwaite and A. Benchaib: “A Linear Parameter Variant H∞ Control Design for an Induction Motor”, Control Engineering Practice, No. 10, 2002, pp. 663-644. V.Uray Electrotechnics, in Hungarian, Muszaki konyvkiado, budapest:1970,pp.82-84. K. Wang J. Chaison M.Bodson "An on line rotor time constant estimator for the induction motor" IEEE Trans.Control Syst.Technol.,2007, 15,pp.330-348. S.Grenaille "Filter synthesis for diagnostic systems modeled as LPV system" Ph doctor Thesis Bordeaux university 2006. A. Apkarian and P.Gahinet,"A Convex arecterisation of gain scheduled H∞ controller IEEE.Trans. on Automatic Control, 40 (1995), pp.853-863. P. Apkarian, P. Gahinet, and G. Beker, "Self scheduled H∞ control of linear parameter varying a design exemple", Automatica, vol.31-9,1995, pp.1251-1261.

APPENDIX The machine parameters are as follows: Resistance e of the rotor; Rr = [4 Ω ,12] Resistance of the stator; Rs = 8 Ω Inductance of the rotor; Lr = 0.47H Inductance of the rotor; Ls = 0.47 H Mutual inductance; M= 0.44H Number of poles; p = 2 Inertia; J = 0.04 kg.m² ωr =[-50Hz, 50Hz] Rotor current frequency; ω = [0rd/s, 120rd/s] Mechanical speed

control theory. An LMI based approach has been proposed to design a two degree freedom controller to track the stator current. It is clearly turned out that with the use of the LPV techniques the performance and robustness and of the whole drive was demonstrate. The simulation results demonstrate clearly high performances of the induction motor control according to the profile defined above. REFERENCES [1]

[2]

[3]

[4]

F. Blaschke "The of Field orientation applied to the new Transvector closed loop Control System for rotating Field Machine",SiemensRev.,1972,39,pp.2209. B.Lu ,H. Choi, D.Gregory, K.Tammi "Linear parameter-varying technique for control of a magnetic bearing system" Control Eengineering Practice 16 (2009) pp 1161-1172 B. Paijmans, W. Symens, H. Van Brussel and J. Swevers, A gainscheduling-control technique for mechatronic systems with positiondependent dynamics, American Control Conference 2006. C.Wang " Control, Stability analysis and Grid integration of Wind Turbines" Doctor of Philosophy thesis March 2008

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