sliding mode control for a permanent magnet synchronous ... - DergiPark

ISTANBUL UNIVERSITY – JOURNAL OF ELECTRICAL & ELECTRONICS ENGINEERING

YEAR VOLUME NUMBER

: 2005 :5 :2

(1427-1433)

SLIDING MODE CONTROL FOR A PERMANENT MAGNET SYNCHRONOUS MACHINE FED BY THREE LEVELS INVERTER USING A SINGULAR PERTURBATION DECOUPLING Ahmed MASSOUM1 Mohamed Karim FELLAH1 Abdelkader MEROUFEL1 Patrice WIRA2 Baghdad BELLABES1 1

ICEPS Laboratory, Sidi Bel Abbes University. ALGERIA Mips-trop Laboratory, Haute Alsace University. Mulhouse Cedex FRANCE

2

E-Mail: [email protected]

ABSTRACT In this paper, we present the singular perturbation for decoupling a permanent magnet synchronous machine that deals with a separation of the variables in disjoined subset or two separate models: one having a slow dynamics, and the other a fast dynamics. The control speed and the Id current are carried out by sliding mode regulators. A qualitative analysis of the evolution of the principal variables describing the behavior of the global system (PMSM-Inverter (PWM) -Control) and its robustness is development by several tests of digital simulation Keywords: PMSM, singular perturbation, three levels inverter (PWM), sliding mode controllers.

1. INTRODUCTION The technique of the vectorial control allows comparing the PMSM to the D.C machine with separate excitation from the point of view couples. The flux vector must be concentrated on the D axis with the Id current null [1].However the exact knowledge of the rotoric flux position gives up a precision problem [2].Thus, it is possible to control independently the speed and the forward current Id .The traditional algorithms of control (PI or PID) prove to be insufficient where the requirements in performances are very severe. Several methods of control are proposed in the technical literature, among them, the sliding mode control. Its nonlinear algorithm give the robustness properties with respect to the Received Date : 07.01.2005 Accepted Date: 13.05.2005

parametric variations and well adapts to the modeled systems [4,5].To this end, we are interested in the application of sliding mode for the control of the PMSM decoupled by the singular perturbation technique. The work is composed by a PMSM modeling in the Park frame and an overview of the singular perturbation technique in order to decouple the machine model. Then, a brief outline on the sliding control and its application to the speed and the Id current control of the PMSM supplied with the three levels inverter. In the last step, a comment on the results obtained in simulation and a conclusion where we emphasize the interest and the contribution of this method of order.

1428

Sliding Mode Control For A Permanent Magnet Synchronous Machine Fed By Three Levels Inverter Using A Singular Perturbation Decoupling

2. THE NL MODEL OF THE PMSM With the simplifying assumptions relating to the PMSM, the model of the machine expressed in the reference frame of Park, in the form of state is written [3, 5, 6]. m (1) x =F(x)+ ∑ gi(x)Ui i=1 With

 x  I   1  d  U  U  x = x = I  ; U = 1 = d  2 q i U  U2   q   x  Ω  

 3 (2)  1   L    0  d g = 0 ; g = 1  1  0  2  Lq     0     pL   −Rx + q x x   1 2 3 Ld Ld    f1(x)   pLd p Φf  R   − F(x)= f2(x) = x − xx − x  (3) f (x) Lq 2 Lq 1 3 Lq 3  3   −f x +p(Ld −Lq) x x +p Φf x −Cr  J 3  1 2 J J 2 J 

( ) z(t 0 )=η(ε )

x t 0 =ξ(ε )

The model is known as singularly disturbed because: The introduction of a small parameter ε is considered as perturbation. The particular value ε=0 introduces a singularity. Generally, this depends on the decomposition of the original system in two reduced dimensions subsystems. By rearranging the state vector to decompose it into two sub-vectors, the following model is obtained: x = f ( x, u , t ) (6) y = h(x) With

[

T

T

T

1

2

][

x(0)=x

T

T

T

1f

2s

2f

]

(7)

0

The first state sub-vector has a negligible fast part with respect to the slow part, therefore: T

T

T

x =x =x s 1s s T x =0

(8)

f

The second sub-vector has the negligible slow component with respect to the fast component, therefore:

The variables to be controlled are current Id and mechanical speed Ω

x =x =x

 y (x )   h (x )   x   I  Y(x )= 1 = 1 = 1 = d   y2(x )  h 2(x )  x 2   Ω 

This leads to:

T

T

T

2 T

2f

f

x =0

(9)

2s

(4)

3. SINGULAR PERTURBATIONS The interest of this method relates especially the determination of composite form control. The global model is decoupled in a slow sub-model and a fast sub-model. The singularly perturbed systems analyzed by this technique must have special form called standard. This form is written as follows [8,9,10] x = f (x , z, u , t , ε ) (5) εz = g(x , z, u , t , ε ) y = h (x, z ) With

T

1s

x = x ,x = x + x ,x +x

T

[

T

T

s

f

x = x ,x

]

(10)

Let’s take ε defined as being the ratio between the time-constants of fast variable and the slow variable: τ (11) ε= f τs The model becomes: x =f x s,x f ,u, t,ε

( ) εz =g(x s,x f ,u, t,ε ) y=h(x s, x f )

(12)

With x s(0)= x s0

x f (0)= x f0

[ ]

ε = 0,1

Ahmed MASSOUM, Mohamed Karim FELLAH, Abdelkader MEROUFEL, Patrice WIRA, Baghdad BELLABES

(13)


The slow model is found by supposing that ε =0; f 2(x s,x f ,u, t,0)=0 (14)

For the fast model, one takes ε ≠ 0: εx f =f2 x s,x f ,u, t,ε

)

(16)

4. SINGULAR PERTURBATIONS APPLIED TO PMSM By considering that the variables Ids and Iqs are fast and that ωr is slow and choosing: Ids = Idsr ; Iqs = Iqsr ; ωr = ωrl One will have [5,8,9,10]: dωrl p 2 p (18) = φf Iqsr − Cr − f ωrl dt J J J Lq dI R ε dsr =−ε s Idsr +ε ωrlIqsr + ε vds Ld Ld dt Ld dIqsr R L (19) ε =−ε s Iqsr −ε d ωrlIdsr + ε vqs dt Lq Lq Lq

Slow model: 2

 Rs dω rs p Φf  v qss = 2  dt 2  Rs   Ld Lq J ( ) + ω rs  (20)  Ld  R Φ p 1 f ω rl v qss ) − C r − ω rs − s f ω rs − Ld Lq Ld J J

y s = ω rs With ωrs (0) =0

•

Rs L ε I qsf − ε d ω rs I dsf + v qsf Lq Lq Lq

φ − ε f ω rs Lq

(22)

With Idsr(0)=0, et Iqsr(0)=0

v  U = ds vqs 

T

v + v  = dss dsf vqss + vqsf 

T

(23)

A judicious choice of the controls such as: vds = vdsf avec vdss = 0 vqs = vqss avec vdsf = 0 (24) deals to the two decoupled models according to: • dωrs dt

−

The slow model: 2

=

p φf

2(

Rs

 R 2  Ld Lq J ( s ) +ωrs   Ld 

R sφf Ld Lq

vqss

(25)

p ωrs)− Cr − f ωrs J J

y s = ω rs

φf ω Lq rl

By rearranging the equations, one obtains: •

= −ε

y f = I dsf

The fast reduced system is obtained by transforming the time scale T of the original system into a fast time scale τ , such as: t −t (17) τ= 0 ε

−ε

dI qsf

(15)

y = H(x s , u , t )

(

Lq R dI dsf ε ωrf I qsf + v dsf (21) = −ε s I dsf + ε Ld Ld Ld dτ dτ

One will have: x s = F(x s , u , t )

1429

With ωrs(0) = 0 • Fast model: d Idsf R =−ε s Idsf + ε vdsf dτ Ld Ld d Iqsf dτ

= −ε

Rs Lq

Iqsf

yf =Idsf

With Idsf (0) = 0

Fast model:


(26)

1430


5. SLIDING MODE CONTROL The sliding mode control algorithm design is to determine three different stages as follow[ 2,3,4]:

5.1 Commutation Surface J. Slotine proposes a form of general equation to determine the sliding surface [ 3,4 ].

( )

r −1

(27) S(x)= d + λ e dt e = x d − x : variation; λ :positive coefficient ;

6. THE SLIDING MODE APPLICATION 6.1. Basic structure of the sliding mode regulator By using the principle of the flux orientation and by neglecting the time-constant of the converter, we can represent the model of the PMSM in the reference frame d-q in two independent subsets [2,3,4 ,6]: - Control according to the d axis and q axis as shown in figures 1 and 2:

r: relative degree ; x d : desired value.

5.2 Convergence Condition The convergence condition is defined by the equation of Lyapunov [ 3,4 ]. S(x).S (x) < 0 (28) Figure 1. d axis control

5.3 Control Calculation The control algorithm includes two terms, the first for the exact linearization, the second discontinuous one for the system stability [ 3,4 ]. u c = u eq + u n (29) •

u eq is

calculated

expression S(x)=0

starting

from

-

Control according to q axis

the (30)

• u n is given to guarantee the attractivity of the variable to be controlled towards the commutation surface. The simplest equation is the form of relay: u n = ksgnS(x) ; k > 0 (31) k : high can cause the ‘chattering ‘ phenomenon.

6.2 Regulator synthesis

5.4 The ‘CHATTERING’ Phenomenon elimination

S1(I d ) = I dref − I d S 2 (I q ) = I qref − I q

The high frequency oscillation phenomenon can be reduced by replacing the function ` sgn' by a saturation function [3,4 ].  k S(x) si S(x) ε ε >0 

Figure 2. q axis control 6.2.1 Surface choice We choose the sliding surface according to the relation of Slotine and the output relative degree [ 2,3,4 ].

(33)

S 3 (Ω r ) = Ω ref − Ω r

For the cascade control, we distribute the surface control S3(Ωr ) between speed and the component Iq .



6.2.2 Control calculation a. SMC relating to the d axis S1(Id)=0 ⇒ udeq= 1 (Idref − f) g1  kd S (I ) si S1(Id) ≤εd (34) S (I ).S (I )ε 1 d 1 d d  d finally the control law relating to the d axis is written as: u dc = u deq + u dn = 1 (Idref − f1)+ k dsgn S1(I d ) (35) g1 b. SMC relating to the q axis SMC relating to speed  kΩ S (Ω ) si S3(Ωr) ≤εΩ S (Ω ).S (Ω )ε 3 r Ω 3 3  Ω −f ) S 3(Ω r )=0 ⇒ Iqref = 1 (Ω g 3 ref 3 (36) (Ld − Lq )Iq φf +p g3 =p J J 6.2.3 Stability factor determination The functions coefficients ‘sgn(s) ‘ must be quite selected to ensure the stability of the system and to satisfy the sliding mode conditions[ 3,4]. k d