Control of Induction Motor using Poly topic LPV Models

Dalila KRAMARI, Abdesslem MAKOUF and Said DRID LSPIE Laboratory, Department of Electrical Engineering, University ofBatna, Email:

a makouf@yahoo.fr and s drid@yahoo.fr

Abstract-A Gain scheduled control design for the stator current is presented, the approach is novel in that the gain scheduled design does not involve linearization about operating points. Instead the motor dynamics are brought to linear parameter varying form via state transformation. A linear parameter varying system is defined as a linear system whose dynamics depends on unknown but measurable exogeneous parameter .The current equations in the ( a,ll) frame have a particular structure, allowing to be written as an LPV system because of affine dependence of rotational speed which is taken as time varying parameter. This varying parameter values can be measured on line during control operations. The LMI based gain scheduled controller using H-infinity synthesis and polytopic representation is designed such that the robust quadratic stability and robust quadratic performance can be assured along the reference trajectory of the varying parameter.

Keywords - LPV, Induction Motor, H-infinity, LMI, Gain­scheduling, polytopic representation.

I. INTRODUCTION Induction motor modeling and control have always

been one of the most challenging problems. Because

the dynamics of the induction motor are highly

nonlinear and quite uncertain. The trends in induction

motor control system design is to use effective robust

controller design approaches such as Hoocontrol [4]

and other robust control approaches one of the recent

topics in control related fields is linear matrix

inequality (LMI) based control system design

because of some advantages . For LPV systems a

traditional control method is to design L TI controllers

for several points, and then using interpolating

technique to obtain the control law over the entire

operating range. The main drawback of this is a lack

of high performance of robustness, even of stability

[8]. In the framework of LPV systems proposed in

[3], [4] [9],[ 10], and [ 1 1] the controller synthesis

problem is formulated as convex optimization

problem. After solving some linear matrix inequalities

(LMI) the so called self-scheduled controller is

obtained by a simple interpolation and then stability

and certain performance bounds are guaranteed along

all possible trajectories of OCt). A self scheduled

LPV controller update its self on line using parameter

measurement, so that the changing plants dynamics

are taken into account.

In this paper, a gain scheduled current control system

in the (a,�) frame is designed for the whole

operating region using Linear Matrix Inequality

optimization and the Linear Parameter Varying (LPV) systems framework. The system non linearity's

inherent to the rotation speed are taken into account

by an affine LPV model with polytopic

representation. An optimal time-varying controller for

field -oriented (see[5] and [6]) control calculated

through a LMI problem formulation to minimize the

L2 gain criterion and assure the stability of the closed

loop composed by the different control objectives.

II. LINEAR PARAMETER VARYING PLANT LPV plants are described by state space equations of the form

x = A(O(t»)x + B(O(t»)u y = C(O(t»)x + D(O(t»)u

(1)

Where x, y and u denote state vector, measured output vector and control input, respectively. 0 is a vector of time varying plant parameters and plant matrices are fixed function of the O. in practice o. Can be the time varying physical parameter such as velocity, damping, stiffuess and etc, and be given by

(2)

The time varying parameter 0 belongs to a parameter polytope 8and varies with vertices 0, Olf'''' Or of this polytope

(3)

When OCt) undergoes large variations during control operation, it is ofen impossible to achieve high performance with a single robust L TI controller if the measurement of the OCt) are available in real time during control operation , the designed controllers have the same parameter dependence as the plant . The controller form is :

XK = AK(O(t»)x + BK(O(t»)y u = CK(O(t»)x + DAO(t»)y

(4)

Where y is the output measured vector and u is the control input. According with the parameter variation this controller is continuously adjusted in the plant dynamics and maintains stability and good performances.

III. LMI BASED GAIN SCHEDULED CONTROL FORMULATION

LMI based gain scheduled HXJ control formulation given here is aimed for practical use. For more detail, consider state- space representation of the plant and th[;e] c:n[t'EMg:,] [�]

(5)

Y C2 D21 0 U

(6)

Where x E Rn and XK E Rk are the plant and the controller state vector, respectively. x and y denote the controlled output and the measured output vectors. u is the control input and w is the distubance input vector. Combining the two systems, the closed-loop system can be obtained by

[X�l] = [Ael BCI] [Xel] Z Ccl Dcl w (7)

Closed loop matrices ACII Bcll Ccll Del are :

[ACI BCI] [

AD + AoE Bo + BOD21 ]

Ccl DCI = Co + D210E Dll + D120D21

(8)

where

A 0 B1 0 B2 [A' Bo �,]

= 0 0 0 Ik 0

Co Dll C1 0 Dll 0 D12 (9)

E D21 0 Ik 0 AK BK

C2 0 D21 CK DK

Note that controllers matrices are collected into a single matrix o.

The Lyapunov function V(x) = XT px > 0 etablishes global asymptotic stability for the closed- loop system (7) . The L2 induced norm from w to z for L TI system is bounded as

IIzlb < IIwlb ( 10)

Finally, there exists a positive definite Lyapunov function V(x) = xT px, P > 0 that satisfies

d -V(x) + ZTZ - y2wTw < 0 dt ( 1 1)

The validity of equality ( 1 1) is proved in [5]. The Hoo suboptimal control problem is equivalent to the existence of a solution to the following inequality for XcI> 0 [A�IXCI

T+ XclAcl XclBcl

BclXcl -yI Ccl Dcl

cIz ] Dcl < 0 -yI

Solution of the LMI ( 12) requires symmetric matrices R and S such that [AR + RAT RC[ B1 ] N"£NR C1R -yI Dll < 0

B[ D[l -yI

(12)

to find two

(l3)

[ATS + SA SB1 C[ ] NINs B[S -yI DI1 < 0

C1 Dll -yI ( 14)

Where NR and Ns denote basis of the null spaces of

(BI, DI2) and (C2, D12) , respectively. The above Hoo control problem is valid only for LTI system and can be extended for LPV sytems. Let's consider state-space representation of LPV plant [X] [ A(e) z = C1(e) y C2(e)

(15)

e is a vector of time varying system parmeters matrices A (. ) , B1 (. ), C1 (. ) and Dll fixed functions of the . B2, C2, D12f D21 matrix are indepenent of the parameter e because of tactability reasons. Finally , the solution of Hoo control problem for LPV system has the same form of LTI system as follows : [AiR + RAf RcI; B1 . ] N"£ CliR -yI D1:i NR < 0

B[i D[li -yI [Af S + SAi SB1i cI; ] NI B[iS -yI D[li Ns < 0

Cli D11i -yI [R I] > 0 I S -

( 17)

(18)

(19)

Where Au Bli , Cli , and Dlli denote the parameter values of A(e), B1 (e), C1 (e) and Dll (e) at the vertices e = ei of the parameter polytope. The solution of inequalities ( 17), ( 18), and ( 19) is possible using advanced software such as convex optimization algorithms. The construction of the controller matrix 0 from R and S matrices can be done by the same convex programs.

IV. MODELING OF INDUCTION MOTOR SYSTEM The state space model of stator current is in the (a, fJ) fram s as follows :

r:::j =

[} �y C/Jsa Ir.. 0 cPsp 0 T MITT w 11a * Ls 0 o ;Iao* Ls [�;] (20) o 0

Where (isa, isp) are the two component of the stator

current and (C/Jsa, C/J sp) are two component of the

magnetic flux, (Vsa,Vsp)is the stator voltage, and Wr is the rotational speed of the shaft acting as a disturbance.

The electromagnetic torque is given by:

M _

Te = P Lr (Is ® ¢;)

and its associated equation of motion is: dD.

Te - Tl = J Tt Where T, is the load torque

V. CONTROL STRATEGY A. Vector control by flux orientation This consists in orienting the rotor flux [6]. Thus, it results the constraints given below in (24). The rotor flux is oriented on the d axis.

¢rd = ¢r ¢rq = 0

B. Speed controller design

(24)

The PI speed controller is designed with pole

placement method (�= 0.707, ffin= 17.3 rd/s).

C. Stator current controller design The inner current controllers are designed in the (a,�) plant to avoid the discretization of the difeomorphism related to the Park transformation and using the LMI Gain scheduled Hoo approach. This control is designed independently of the speed controller.

VI. GAIN SCHEDULED Hoo CONTROLLER DESIGN The state space model of the system can be obtained by

x = A(w) + Bu y = Cx (25)

Parameter dependence range of the plant due to rotational speed is:

w E [0, wmaxl (26)

Specifically for this problem, the parameter vector O(t) has the following convex decomposition

a 2::0 (27)

Where 0i gives the comer of polytopic parameter range. The comer values of parameter range for this problem are:

0'1= (0,0), 0'1 = (0, wmax) At the comer values of 0 , the plant matrix is:

G(O) = a1G(01) + a2G(02)

Augmented plant can be given by [A(O) B1 B2 ] G (0) = C1 Dll D12

C2 D21 D22

(28)

(29)

(30)

The desired performance specification on control system is defined in terms of frequency shaping filters W1 and W2 placed at the output of the controller and the plant, respectively.

Finally, using the parameter dependent augmented plant, the gain scheduled controller is computed. The structure of this controller is given by

[Ak(O) Bk(O) ] _ L2 . [Ak(oJ BCI (oJ] Ck(O) Dk(O) - t=1 at CCI (OJ Dcl(oJ

The LPV controller is carried out by using the LMI toolbox function hinfgs

VII. Hoo LOOP-SHAPING DESIGN The loop-shaping design procedure described in this section is based on Hoo robust stabilization combined with classical loop-shaping as proposed in [7]. It is essentially a two stage design process. First, the open loop plant is augmented by pre and post­compensators to give a desired shape to the singular values of the open loop frequency response. Then the resulting shaped plant is robustly stabil ized using H 00

optimization. An important advantage is that no problem-dependent parameter varying or weight selection is required in the second step. The loop -shaping criterion is:

11::�sll < 1 Where the shaping filter is:

200 W1 = S + 0.0002

200 W - ---=-=--=-=-=-2 -

S + 0.0002

The LPV controller is computed such as the closed loop is stabilized. y witch guaranty the L2 gain of the system is bounded and equal to 1 (y=I.0002).

The resulting LPV controller exploits all available

information on ffi to adjust to the current plant (see Fig. 1 ). This provides smooth and automatic gain

scheduling with respect to the varying parameters ffi.

w z p (0))

w

Fig.I. LPV control of LPV Systems

Fig. 2 is a general block diagram of the suggested induction machine control scheme. As shown in this figure, we can see that only one PI speed controller is used.

Fig.2. Gain-scheduling controller structure

VIII. SIMULATIONS In order to validate our approach, simulation tests are

carried out using the proposed control scheme. The

testing conditions are as follows. The figure 3

represents the speed reference. The speed changed

from \00 rdls to - \00 rdls at 4 s, the machine is

loaded at 4Nm. For introducing the effects of

parameter variations, the rotor resistance is increased

of 100% compared to their normal values at 2s.

100 r

,

, , ----------,----------,----------,---------

:!:r:!::r!!:

Iii \111

50

i il � 0-w

-50

-100

6 Time (5) Fig.3. Speed various time.

Fig.3 show speed response versus time; we observe that a good tracking speed was achieved without any effect of variation of parameters. Also we can see in figs.4, 5 and 6 the good tracking current. Fig.7 represents the stator voltage input control.

40 r---._--_.--�----r---._--_.--�--� 30 --------r--------r-------r-------r--------r---------r---------r------­

: ·········!······I · l······]�··· I··.····1······. ·r······· � 0 .�

-10 ' , , r , , ,

::111:T11f Time (5)

FigA Stator current various time.

Time (5) 2.02 2.04 2.06 2.08

Fig.5. Zoom Stator current ant its reference various time

6 ----------!---------------r---------------!---------------!---------------r-------------5 ----------j---------------j----------------i---------------j---------------t--------------

4 ----------]---------------]----------------i---------------j---------------t--------------

3 ----:--:-:i-:-:--:-:--:J:-:--:-:--:-:--:r:--:-:--:----__ ( ____ :_: __ :_:L_: __ :_: __ :_:_

:!: 1 .�

O �Hl+fH+�H'+Hl+Hl'+Hl�Hl�Hl-HHfIHl-HH-HH

-1 -2 - - - - . - - - - - - - - - - - r , , , ,

: :::::::::I:::::::::::::L:::::::::::::t:::::::::::::::c::::::::::::r::::::::::::: 1.6 1.8 2.2 2.4 2.6

Time (5) Fig.6. Zoom Stator current according rotor resistance

variation.

600 400 200

o -200 -400 -600 -800

o

, , , , , , , ............... � ............ ............ ................................... � ......... . , . , , . , . · . , , . , . , . , , . , . , . , , . , . · . . , . , . , , , , , , , · . . . , . , ---'----------_ .. _---------_ ._---------_ ._----------,-----------'------------'----------· , , , , , . · . . , , , , , · . . . · . . . , . , . · . . . · , , , . . , ---:-----------�-----------:-----------:-----------:------------:------------:----------. . .

2 3 4 5 6 7 8 Fig.7 Stator voltage various time.

IX. CONCLUSION LMI Gain -scheduled tL, controller design for induction motor presented here has good properties in many ways. Scheduling of controller can be realized continuously and any variation in the plant dynamics can be covered by scheduling of the controller. Simulation results obtained are reasonable. One drawback of this approach is especially for DSP implementation due to the increasing degree of the controller. The result obtained in this study will be appreciated in the planned experimental works.

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APPENDIX The machine parameters are as follows:

Resistance e of the rotor; Rr = 4 n Resistance of the stator; Rs = 8 n Inductance of the rotor; Lr = 0.47 H Inductance of the rotor; Ls = 0.47 H Mutual inductance; M = 0.44 H Number of poles; p =2 Inertia; J = 0.04 kg.m2