PWM LESS CURRENT CONTROL AT VSMM DRIVE
Žarko Čučej, Peter Cafuta, and Rajko Svečko University of Maribor, Slovenia
Keywords: electric motors, iM, Induction Motors, FOC, Field Oriented Control, current control, PWM inverters. Pulse Width Modulated inverters, VSI, Voltage Source Inverters, VSC, Variable Structure Control, BLSC, Boundary Layer Switching Controllers, SCC, Switching Current Control, time-discrete switching variable structure control, optimized mappings
Abstract: This paper discusses time-discrete field oriented variable structure current control of induction motor - voltage source inverter system without use of a pulse-width modulator. The controller is supplemented by feedfonA/ard selection of optimized mapping of controllers into voltage source inverter states. This proposed approach of direct inverter control depends on the boundary layer control, and the sign, nominal value and maximal values of the backe.m.f. estimate, it lessens back a.m.f. influence on chattering and makes it possible to extend the field angular velocity range of constant rotor field. Boundary layer control separates variable structure control modes. Supporting preassigned switching order and control objectives is assigned to each attraction domain.
Tokovna regulacija skupka napetostno izvorni pretvornik -izmenični motor brez uporabe modulatorja
Ključne besede: motorji električni, IM motorji indukcijski, FOC krmiljenje v orientaciji polja, krmiljenje tokovno, PWM inverterji modulirani impulzno širinsko, VSI inverterji napetostno izvorni, VSC krmiljenje s strukturo spremenljivo, BLSC, krmilniki komutaoijski plasti mejnih, SCC krmiljenje toka komuti-rajočega, krmiljenje s strukturo spremenljivo časovno-diskretno, preslikave optimirane
Povzetek: Članek obravnava časovno diskretno tokovno regulacijo s spremenljivo strukturo v poljskih koordinatah skupka asinhronski motor - napetostno izvorni pretvornik brez uporabe modulatorja. Regulatorje izpopolnjen s predkrmiljenjem izbiranja optimalne preslikave stanj v stanja pretvornika. Ta predlog direktnega krmiljenja pretvornikov je odvisna od režima regulacije in predznaka, minimalne ter maksimalne inducirane napetosti v motorju. Z njo se v veliki meri kompenzira vpliv inducirane napetosti motorja na drhtenje. V primerjavi z običajnimi rešitvami omogoči tudi razširitev vrtne hitrosti pri konstantnem rotorskem polju. Pasovni regulator razmeji režime regulacije s spremenljivo strukturo. Vsakemu režimu predpiše atrakcijsko domeno, ki omogoča predpisano zaporedje preklapljanja in kriterije kvalitete regulacije.
1 Introduction
For an Induction Motor (IM) the Pulse-Width Modulation (PWM) of the Voltage Source Inverter (VSI) plays an important role in the control system, since not only the level but also the phase of input signal must be controlled. These occur as certain disadvantages of PWM inverters because they are determined by the characteristics of PWM and are not addressable by controllers /1 / . These drawbacks can be eliminated by switching the variable structure (which has a direct control VSI - IM system) constructed by signals as the position, velocity, and currents, which contain information about disturbances and parameter variation, consequently the entire control system is completely accessible to control and can be optimized in a Variable Structure Control (VSC) design.
Unfortunately, time-discrete implementation of switching type VSC causes chattering, degrading all the benefits of direct switching control. In /2/,/3/ we describe chattering reduction using hysteresis controllers and in /4/ using finite automaton. The recent article /5/ proposes quasi VSC implemented with a first order deadbeat controller and a space-vector PWM. Regardless of the fact that the ob-
tained average current error is smaller than at direct switching control, this solution has some serious limitations. Disturbances should be smooth and limited, control robustness depends on the disturbance estimator used, and PWM properties are uncontrollable.
In this paper we present the Boundary Layer Switching Controller (BLSC) used in Field Oriented Control (FOC) adapted with a feedforward steering selection for mapping of the controllers states into VSI states. Using BLSC the fixed order switching is determined, and to each of its phases mapping is optimized in regard to the following control objectives:
sliding mode reaching time, average current error offset, chattering,
VSI switching losses.
Mapping optimization compared to /6/ involves zero voltage vectors. This introduced feedforward selection is similar to predictive current control /7/, except that: (i) the control is still VSC, and (ii) instead of back e.m.f., only information about the field velocity sign, and minimal and maximal values of back e.m.f. estimates are used.
This work was supported in part by the Slovene Research Ministry under Grant J2-1644-0796-99. 74
Tliis article is organized as follows. Problem presentation with description of VSI, IM model and a summary of VSC is contained in section 2. In section 3 the proposed control with compensation of back a.m.f., feedforward steering and design of ST is given. Results of simulations are collected in section 4. This article ends with conclusions and appendices containing derivation of zero dynamics, IM model and analysis of stability for use of zero voltages.
2. Problem presentation
IM is feed by a 3-leg bridge VSI supplied by DC voltage E and direct controlled by current switching controllers in the inner loop of FOG. VSI has eight states, which determine
six active voltage vectors u^,k = l,...6 with constant amplitude u^ =2E/3 and two zero voltage vectorse {«<(,>"7} (Fig. 1).
: E/2 —

«9 = i
m( 1,1,0); -£/2

^^ =/«(0,0,0) =i/(l,l,l)
> d
«(0,1,1)
Ma =

Fig. 1: Voltage vector representation from the d-q frame perspective, inverter, and load circuit.
A symmetrical, 3-phase IM with Y stator windings with a galvanic isolated central tap is considered. Its voltage model in d-q frame is described by the following MIMO system:
«0 = Ris + ^scrPho	{d,q},
(I;
where u = [d q^ is stator voltage vector, R^ and L^^ are rotor resistance and rotor leakage inductance respectively, e^ is sum of back e.m.f. and crosscoupled voltages
(= + e,/,. = V + ). and pi = di/dt. Latter, in the design we use a simple modified model:
^saPho ~ ^kO ' which is derived in Appendix C.
2.1	FOC
We assumed that FOC has a multi-loop structure and is sampled every T^ seconds, i.e. algorithms are implementa-ble in the digital signal processor. Outer loops contain state controllers and estimators for the load torque T, , rotor
magnetization current , and angle ß. They are responsible for the tracking of drive kinematics variables and the regulation of . Field weakening is not considered. The
selection of design coefficients c,, c,, and c, (elaborated in Appendix A and Appendix B) of outer control loops should guarantee stable zero dynamics with a stationary point:
0 = ^ = = O and ^ = (p = \j/^={). Outputs from outer
loops i'^"' and serve as command values for inner current control loops.
2.2	Current variable structure control
The control problem being considered is a determination of sequence and the duration of VSI states, so that by tracking and the drive FOC is satisfied:
s =
-iT

= Ic
•REF
-i, =0 ,
(3)
i.e. VSC is established in stator currents error space [] ^ where j'o == 0, Oe d,q are switching subspaces. This formulation embraces the control of stator currents and modulation of VSI outputs as one problem, which is solved by a MIMO VSC.
It is well known /9/, /10/, that the design of VSC can be done by satisfying a reaching condition. In FOC the torque and magnetization control are decoupled, meaning that the MIMO current control is designed by two independent controllers, which fulfill reaching conditions written in compact form as:
5,-
< 0

S^: n Sq by discontinuous control:
VÖ if s^>0
(4a) (4b) (4c)
Va =
Vo if ^o<0
(5)
with discontinuity on the switching subspaces S.. The set of control vectors v = v,, v,^ ^ j -h-■■4- is present-
ed in Fig. 2a. Control (5) ensuring (in ideal circumstances, when switching frequency is infinite) or sliding on with
reaching to S^, or sliding on Sq with reaching to S,^ either
sliding on Sg. Only in the last case is the request (1) completely satisfied.
The PWM's less mapping between control vector v and Stator voltage vector u gives IM stator voltage equation (2). Combining (4a), (4b) and (1) gives:
K| = 2ii/3=l 0,96

^ /
■^■■..60°
0,25


0......................... >
-0,96
Fig 2: Presentation of control vectors (a) and mapping range (b).
L

'Sa
S^<0, M,o > ^o
< O n	(6)
and can be simply implemented using digital signal processors. ST performs mapping where selection of columns addressed by p quantizer performs an inverse transformation from d-q to abc space.
Time discrete implementation of the decoupled SCC causes inverter outputs to have a finite pulse such as pulse width modulation. Their duration is equal to one sample interval and occasionally to an integer multiple of Ts. Therefore the decoupled SCC is never in SM. Consequently difference
in control vector amplitudes, i.e. | y^ |-|y+| =
cause
offset in I e I and subharmonic oscillation (in regard to fre-
quency 1/T^) of local average of | (determined by stator time constant) causing torque chattering.
If VSC, for control robustness is designed at max e^ , for mapping v ^ u from (6) it follows:
Vo=iho(P) =
when < - max
when u^^ > max
(7)
Due to the rotation of the active voltage vectors and their angular displacement at 60°, the mapping v links control vectors {v^.} with four angular sectors in d-q frame containing voltage vectors {m^} involved in mapping.
The sectors placements depend on terms e^ (or d;^, see (31) in Appendix C), but in the case of (7) only on their maximal amplitude. Therefore angle sectors are placed
symmetrically as are the symmetrical control vectors Vj (Fig. 2b).
2.3 Implementation of decoupled switching current control
This described current control is named decoupled Switching Current Control (SCC). It is of simple structure (Fig. 3)
Fig. 3: Realization of decoupled SCC.
The other important drawbacks of mapping (7) are: (i) zero vector voltages are uninvolved, and (ii) for components of the selected voltage vector the following holds
max u^i = max u^^ and min u^ = min u^^ , (8)
which limits the velocity range of nominal magnetization to:
max go ^0,25(25/3)
(9)
Absence of zero voltages cause high total switching frequency of VSI (at least twice that of l/T^,), consequently VSI switching losses are very high.
3 Proposed switching current control
Decoupled SCC performances are well enhanced with Fixed-order sliding mode switching scheme /10/, where the sliding mode takes its place in a preassigned order while state is traversing the state space:
moving along S^ only in initialization pliase during the establishment of nominal rotor magnetization; in regular control it mode moves in order
5
Q

(10)
and then maintaining limit cycle around Sg. Mappings supporting (10), should adopt attraction to VSC modes. For Sq the minimal reaching time is requested, after reaching Sq it is desirable to maintain the limit cycle around s-origin so, that the current ripple is redirected from torque into magnetization control (where it is well dumped by large rotor lag) and in VSI switching sequences the minimal number of switches is involved. This is achieved by deliberate use of zero voltage vectors in mapping.
3.1 Boundary Layer Switching Control
The fixed-order sliding mode switching scheme is made possible by the Boundary Layer Switching Control (BLSC), which separates the associated VSC modes. According to above description it has to differentiate between three VSC modes:
mode of moving along Sq
mode of limit cycle at , which has two phases: phase of obeying (4b). phase of obeying (4c).
These modes besides the mapping range presented in Fig. 2b, which accommodates attraction domain A^g, need two further domains:
domain A^, belonging , where	are mapped
into u with minimal amplitudes of Mj,,, fulfilling (6), and
vj,vj into t/with maximal amplitudes of
domain A^ belonging Sq where mapping requests
are opposite to requests for domain A^.
BLSC, which fulfills the above requirements, has three layers of coaxial arrangement with a center in s-plane origin (Fig. 4).

i
domain A
Q


7 domain A^q
-A
domain

-Mi
The borders between attraction domains are determined by the current error caused by maximal voltage amplitude, which arises in a stationary condition in one sampling interval. Using the estimation of i,, with difference Ai-^^T, for border between Aq and A^g we can state:
L
T.
(11)
'Sa
For inner border (between and A^) we consider in (11) maximal allowed backe.m.f.. Because
u
max e
1
■>2
(12)
Fig. 4: Borders and discrete VSC modes.
it follows, that in q direction the inner border \s b/2 . Boundaries in d direction we settled to ±b . This choice is based on a consideration that in the limit cycle around the error plane origin (s = 0) a sequence of voltage vectors could be formed, where zero voltage vectors follow each active vector. In this case the border in d direction should be at
least so far from the origin of s-plane that s^i, i.e. current
error in d direction can be zeroed by max | u^, \. Thus borders in d direction should be calculated similarly to borders in q direction, i.e. by (11) with consideration for adequate voltages in d direction. Because
max I u^i I = max | u^^ \ and max | e^, \< max | ej we justify
aforementioned choice of ±b for d direction.
Note, the boundaries are tied to the sampling interval T^, and their size actually determines the duration of sampling interval. In many IM control designs, the value of the sampling interval T^ is influenced more by desired rotor field angular accuracy, the rms value of stator current chattering, and invertervolume power density than by IM dynamic requests. One of this project's goals was to make the sampling interval as short as possible in regard to the limitations of the used digital signal processor. The target value was 25 /us.
3.2 Compensation of ßq
Use of zero voltage vectors Uq U-j in mapping during limit
cycle reduces offset and chatter of local average (in interval determined by the motor's time constants respectively), but for dump these phenomena, should be compensated.
Due to switching controllers they cannot be compensated by subtraction of e^ as is done for predictive controllers /
/7/. Rather, the slowly-occurring magnetization offset (caused by ) is minimized by rotor magnetization control
(see Appendix B), and oscillation of local average of s^^
causing torque chattering, is effectively minimized by mapping with property
V = V <7	'I
(13)
From (7) at e^^ > 0 follows v^^ = u^^ + e^^ and v* = u^^ - e^^
where u~,u* denotes voltages to which v~ and v* are mapped, consequently the above equation can only be
fulfilled if at least the sign, min and max of is known. By
rough linearization of u^ =|| Ji || sin p , considering e^^ >0
and limit min (i/ ) = max {e^^), after a short calculation we obtain:
mine	mine
sm/3ig>—, sm/?35>——i , (16)
u
u
Sector is placed in < < +f and similarly for
sector hold p^g < p < Pj^ -l- f . When hold p < 0 , then we obtain:
sin Aß ^
mm
>-1
, sin/?3ß<
mm e,.
u
(17)
The position of even sectors and are symmetrically over q-axe to position of and respectively.
Zero voltage vectors take place in A^ domain only and their selections is independent of field angle p . They are applied according to:
			
min		= - min	
			<7
(14)
This suboptimal solution enables a simple construction of the angular sectors distributions. Moreover, exact calculation in the case of min(e,J is by (7) limited to (14).
Mapping, considering (13) and (14) and the preassigned switching order is then:



when s^i > 0 when s^i < 0
(15a)
v; when M,,,
"kg
(15b)
V, when w,., < min e^
q	kq	q
v'l^ when > - max
(15c)
where (15b) and (15c) are valid for positive and negative e,, respectively.
The position of angular sectors , containing vectors u^
is determined by the geometric relationship between | |
and I u{p) I at pß and Pj,:, where angles p^ and Pj^
denote the beginning and end of angular sector (Fig.5).
At /9>0 for odd sectors /9,^ and /9,g can be generally written as:
if« [(n - 1)T3 ] = «7 V H, V u^ V Mg
then H (/7^3) = «7	(18)
else M(nTg) =
where nT^ is the sampling instant. This selection gives minimal VSI over-switching at the voltage vectors change [2]. As this is excellent for VSI and enables good compensation of e,,, it should be noted that at zero vector voltage the IM is effectively allowed to coast. Control seems to be lost during this time, but stability is still preserved because besides conditions (33) and (10) the following condition
Sign = Sign is also fulfilled (see stability analysis in Appendix D).
(19)
q'h
"se^^tor :

\
sector "öv / f \
«2
1/ " ■^n' --- J 1 f ♦	«1
i i^o ill
..................^

Q

-J
H ßf
sector -Öt :
mm| | = 0
sector ■Ö4:/
Fig. 5: Determination of Pj^ and p j,, in nominal fieid anguiar velocity range.
3.3 Implementation of proposed current control
The structure of the proposed current switching control is shown in Fig. 6. BLSC is implemented by a 2-dimensional
4-levelquantizerseparating the attractors domains /4,,, A^g
and Aq . Switching table ST is result of logic superposition of angular sectors determined in subsections 3.1 and 3.2 respectively. Feedforwarded p is quantized by a four-lev-el quantizer. It determines max and min value of back e.m.f. (because e,, = e.^^ + , whereas constant field is proportional to p and e,,, varies according to p and fast /5,,, it is simple to evaluate	and mine,^ for positive and neg-
ative angular velocity range of p). In the case, where distinguishing between and Ag domains is less important than maximal p with constant rotor field, the voltage range for A^ can be exploited in A^g domain. This in comparison to decoupled SCC enables to doubling of the p range of the constant rotor field, Table 1.

14=
BLŠC
is;

abc
m
			
switching	......N		.....N
table (ST)			V
VSI
feedfoPA'ard steering
i.......pquantizer""""
i (on 15° sectors)'
u{a,b,c)
p..
i,{cLq)
\abc dq \


Fig. 6: Proposed control with mapping and feedforward steering.
Table 1: Pairs maxe^^ - mme^^ in nominal and extended velocity ranges, p^^^. is maximal p range of SCC,
velocity range	max e <1	min e
	0,25(2E/3)	0
0>p>	0	- 0,25(25/3)
2pscc ^P^ Pscc	0,5(203)	0,25(2E/3)
-Pscc ^P^ -^Pscc	-0,25(203)	- 0,5(2E/3)
ST consists of 9 subtables, see Table 2. They are addressed by attractors domains A. and angular velocity range Pj. The rows in the subtable are determined by the signum functions of BLSC. Columns are addressed by p quantizer determining 24 angular quants . The inverse transformation from d-q to abc frame is the domain of columns. Determination of zero voltage vectors according to (18) follows ST.
Table 2: Switching table (ST). The VSI switches states are denoted by the corresponding voltage vectors.
Yjl. v„







9,: 0>p>"f
ß,, 9, 9, 9, 9, 9, 9, 9, 9,\9
subtable#1: A^^,; p^^c > P > ~P.
u.
u,
lU
«5 «6
U^ U^ «4	U^ Ih
Un
U,
M,
U,
Ih
U,
H,
U,
Ih
Ur,
Uj
subtable #2: A^ ; p^^c > P > O

lu
U,
u,
lu
Ih
u,
lU
u.
«3 «4 «4 i/4 i/4 «5 i/j Mj
i/.
i/.
lU
U,
H,
M,,
M,
U,
Ur,
subtable #3: A^ ; O > p > -Pscc
u.
u.

u.
u.

u,

U^
u,

u, u, u, u, «4 «5 U, U, u,
Ur,
u,
lu
u,
u.
lu
IL
u^

u,
u,,
u.
subtable #4: Ap ; p^^c > > O
u.
u.
Ur,
u
Ur Ur Ur U. U. U. U. Uj M, U
Ur.
Ur
U,
M,
u
zero voltage vectors, determined by (18)
subtable #5: Ap ; 0 > /) > -Pscc
zero voltage vectors, determined by (18)
u,
u^
u.
u^
"2
M,
«3 «4 U^ «4 «4 Mj M, «5 «/5 Mg



Ma
i/o
subtable #6: A^q , Aq ; /7 >
Mo
M.
M,
M,
M.
«3 M3 «3 M4 M4 M4 «4 Mj Mj M
U,
U,
U,
U^
subtable #7: A^g , A^ \ -p^cc ^ P
M,
M^
M.
Ur
i/o
M3 M4 M4 M4 M4 i/j Mj Mj Mj Mg
M,
M,
M,
Mo
Mo
Mo
M,
M,,
subtable #8: \ \ p> p
U,
u.
u.
Mg M) »I M, M, M^ Mp I M^ I M^ IM3
ii.
M.
Ur
zero voltage vectors, determined by (18)
subtable #9: Ap ; -p^cc > P
zero voltage vectors, determined by (18)
M,
Mo
Mo
M.
M,
U,
U,
Mo
M,
M,
Ur,
4. Simulation
A property of the proposed control with feedforward steering of optimized mappings has been evaluated by simulation. Data for IM, VSI, controllers, and perturbed tracking task considered in simulations are collected in Table 3.
Table 3: Motor, inverter, controllers, and task data.
Motor:	four pole, 1480 rpm, Y connected stator windings with isolated central tap
Parameters:	Lj=l,65mH T^, =4Nm T„=0,12S 7 =0,000656 kgm'
Inverter:	3-phase bridge, supply voltage £ = 310
Control:	SCO proposed
BLSC border: b - 3 A magnetization reference 4,75 A 4,75 A	
Kinematics:	c, =70000, Co =1000, c, =100
Perturbed tracking task:
4,71 [rad] ' position (p
angular verocity ej'
400 -la 300 -
"cl 200-
7). J
100-
o
10 20 30 40 50 [nisi
The chosen task enables testing of the proposed mapping in all VSC modes. Simulation starts by initialization, where
the nominal rotor magnetic field is established, then A^
and follow, which are disturbed in 15"' milliseconds with a change of the load torque. It pushes VSC into RM. The perturbation of the inertia J is used for evaluation of the robustness against IM parameter variation.
To show the properties of the proposed mapping, a comparison to see (characterized by Fig. 2b and presented in Fig. 3) has been made. The significant simulation results are collected in Table 4, and shown in Fig. 7.
From these results it follows that the proposed mapping with optimized voltage patterns has several advantages in comparison to SCC:
the number of switches involved in voltage vector switch-over and the number of voltage vector changes are reduced by 60 % and 40 % respectively,
chattering of is reduced by 40 %,
reduction of (i^ )„„, 1,73 % slightly reduces motor losses,
the angular velocity range of the constant rotor field is doubled,
tracking of kinematics variable is slightly improved.
Table 4: Simulation results.
parameter		SCC	proposed
nax min (f, J	[A]	20,26	10,83
	[A]	12,51	12,62
	[A]	13,39	12,97
max O'sJ-min	/h,	1,62	0,87
max min J	[A]	4,92	5,96
	[A]	0,93	0,57
	[A]	1,98	2,17
(L) [A] ^ ^ ^ rnu		13,39	13,15
u changes frequency total switch-over freq.	[Hz] [Hz]	19800 48200	11400 18800
end position error A^	[rad]	-0,087	-0,066
The complexity of the algorithms and the target sampling interval was tested by implementation of algorithms on the
floating point digital signal processor AT\&T C32 using a 40 MHz clock. The following results were achieved:
the boundary layer controller with feedforward steering needs about 4 ßs, and
the measurement of and , and the calculation of p need about 14 //s .
From the above it can be concluded that the sampling interval of T^ = 25//s at IM servo-drive system with proposed current control can be easily achieved.
5. Conclusion
The presented PWM less switched field oriented control with feedforward selection of subtable with optimized mapping has been investigated analytically and evaluated by simulations. The obtained results show that the proposed switching current control significantly reduces the drawbacks of decoupled SCC, i.e. chattering, offsets and inverter switching losses. The achieved improvements are based on the deliberate use of the reaching condition resulting in of preassigned switching order, time scale sepa-
reference solution: de-coupled SCC from Fig. 3
proposed solution
0" -0.01
-0.02 i
Aq)
7B0"	—

		
0	.(30	Q.ß^.....0T&4-
		
-0.01		
	/	
-0.02		
8 d=0 5
Fig. 7, Tracking the disturbed task simulation results for SCC (top), and BLSC (bottom). T^ = 25 //s, [A] A(p = (p"'^'' - (p [rad], s__d and s___q are Sj and respectively. Trajectories of are at (p = 4,75 [radj at the indicated 5 ms Intervals.
ration of FOC and the use of feedforwarded p Using these measures, more current ripples are directed into magnetization (where it is well dampened by rotor lag) than to torque generation.
Preassigned switching order also enables the employment of zero-voltage vectors. Their use reduces deviation of the
s local average at S^, thus reducing the chattering, as well as the total inverter switching frequency. With feedforward-ing p becomes possible the compensation for deviation
of the local average at active voltage vectors and also it
enables extension of jj with nominal rotor field.
Finally, it is worth mentioning that despite to all refinements, current control still has a relatively simple structure, which can be efficiently implemented in digital signal processors. The proposed VSC with voltage mapping is also imple-mentable in all symmetric 3-phase motor types.
Biblography
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Appendices
A. Kinematic tracking
The drive dynamics is described with
= (20) where (0 = (p is rotor angular velocity and J is drive inertia constant or variable as load torque T^^. By tracking, kinematics' variables (speed) and (p'"'-''' (rotor position) are prescribed functions. IM delivers electrical torque decoupled from magnetization by FOC:

''iurKSCI ^mhq ,
(21)
3 2 L, L,,
where P is number of the machine poles, , L,^ and L^
are mutual, rotor and stator inductances respectively, /„,„ present rotor magnetization current /8/. The rotor field angle:
P =
pdt =

Pi
\dt
iilR
(22)
is determined according to the rotor lag r„. If an error of position, angular speed and acceleration forms a linear combination, then the control task can be determined as:
c,e + c^e + e = 0
(23)
where e = (p- cp'"''', e = (p- (p""' and e = (p- are the position, angular speed and acceleration error respectively, and c) and c, are positive constants. From (23) follows
iREf
e = (p ~(p = c^e- c^e
(24)
and considering (20) and (4) the current reference is given as:
■ REF he,
k... k...
(25)
where f^ is estimated value of T^. From (25) follows, that
e , e, and e'are projected on a current error plane as current error in qr-axis direction of FOC. Maximal ratio between them and ||s Ijj is bounded by the selection of constants
Ci and Cj regarding stability and robustness of the controlled system.
B. Rotor magnetization control
If the rotor magnetization is represented by the rotor magnetization current /'„^ /8/, then the magnetization model is determined by:
PLR + hnR = i
Sd ■
(26)
Magnetization due to FOG is a slow process, in relation to torque production; thus, the offset caused by e^ can be minimized by a suitably designed magnetization current control. Using the same reasoning as kinematics tracking, the magnetization current error is expressed as

'mR
where „ is an estimate. The current reference is:
(27)
= ce
''Sd
f-mR
L
R.
(28)
C. Modified induction motor model
After calculating from (4), the equation (3) rewritten in the error form /11/ becomes:

(33)
Considering simplified IM model (2) at constant rotor field, where = 0 and consequently e,.,, = 0, condition (33) can be written as:
V


dq
+ -
■y,, r


<0
(34)
In the case of selected zero voltage vectors (34) becomes:


-e
dq


<0
(35)
and it is also evident that it is negative as long as /) 0
and the condition (19) for selecting of zero voltage vectors is fulfilled.
= R^^ - R^s, + L,, p/ff - +
(29)
and when rearranged is

The term

(30)
(31)
collects all the disturbances (exogenous and endogenous) acting on the system. It is known that the current tracking problem (3) is equivalent to the stabilization of (30). Because L^^ and R^ are semipositive values, the term -R^s^ is always stabilizing, i.e. any feedback that stabilized (31) with R^ =0 also stabilizes it with R^ >0. Moreover, the stater windings resistance in IM is usually low, and the term can be neglected. Furthermore, considering that
%
during the sampling interval is constant, its derivation
is zero. From (31) it follows that d^ = . Hence, the simplified model
is assumed in presented design.
(32)
D. Stability analysis
For attraction domain /i^ the reaching condition is derived by second Lyapunov method /9/ from quadratic form y >0 as:
In the case of p = 0, (34) is equal to zero, meaning that the system is at a standstill. But there is an assumption for
neglecting the influence of R^ for simplified IM model is
not valid anymore, consequently full IM model had to be considered in (33). Assuming selection of zero voltage and
that during sample interval is constant, (35) becomes:
y = —


h
'Sa
(36)
which is negative until the condition (10) is satisfied, since according to (19) signs of and e„ are reciprocal. Those facts can serve as proof of the following theorem:
Theorem: Zero voltage maintains sliding along switching lines, or limit cycle around the error plane origin, if and only if their use satisfies the conditions of (10) and (19).
Žarko Čučej, Peter Cafuta, and Rajko Svečko University of Maribor, Slovenia e-mail: zarko.cucej@uni-mb.s