HAL Id: hal-00638497
https://hal.archives-ouvertes.fr/hal-00638497
Submitted on 7 Nov 2011
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Adaptive Nonlinear Control of Induction Motors through AC/DC/AC Converters
Abderrahim Elfadili, Fouad Giri, Abdelmounime El Magri, Luc Dugard, Fatima Zara Chaoui
To cite this version:
Abderrahim Elfadili, Fouad Giri, Abdelmounime El Magri, Luc Dugard, Fatima Zara Chaoui. Adap- tive Nonlinear Control of Induction Motors through AC/DC/AC Converters. Asian Journal of Con- trol, Asian Control Association (ACA) and Chinese Automatic Control Society (CACS) 2012, 14 (6), pp.1470-1483. �10.1002/asjc.478�. �hal-00638497�
Adaptive Nonlinear Control of Induction Motors through AC/DC/AC Converters
A. El Fadili*1, F. Giri1, A. El Magri1, L. Dugard2, F.Z. Chaoui1.
1. University of Caen Basse-Normandie, GREYC Lab UMR CNRS, 14032 Caen, France 2. GIPSA Lab UMR CNRS, Grenoble-INP, France
* Corresponding author: elfadili_abderrahim@yahoo.fr
Abstract. The problem of controlling induction motors, together with associated AC/DC rectifiers and DC/AC inverters, is addressed. The control objectives are threefold: (i) the motor speed should track any reference signal despite mechanical parameter uncertainties and variations; (ii) the DC Link voltage must be tightly regulated; (iii) the power factor correction (PFC) w.r.t. the power supply net must be performed in a satisfactory way. First, a nonlinear model of the whole controlled system is developed within the Park coordinates. Then, a multi-loop nonlinear adaptive controller is synthesized using the backstepping design technique. A formal analysis based on Lyapunov stability and average theory is made to exhibit the control system performances. In addition to closed-loop global asymptotic stability, it is proved that all control objectives (motor speed tracking, rotor flux regulation, DC link voltage regulation and unitary power factor) are asymptotically achieved, up to small but unavoidable harmonic errors (ripples).
Index Terms. Induction machines, speed and flux regulation, AC/DC/AC converters, PFC, Lyapunov methods, backstepping design technique.
1. INTRODUCTION
It is widely recognized that the induction motor has become a main actuator for industrial purposes.
Indeed, as compared to the DC machine, it provides a better power/mass ratio, simpler maintenance (as it includes no mechanical commutators) and a relatively lower cost. However, the problem of controlling induction motors is more complex because these are multivariable and highly nonlinear systems and some of their parameters are time-varying. From the technological viewpoint, a considerable progress has been made in power electronics over the last two decades. Reliable power converters have become available which makes technically possible flexible speed variation of electrical drives including induction machines. Accordingly, the speed variation of these machines is carried out by acting on the supply network frequency. Similarly, before the recent progress in modern power electronics, there was no effective and flexible way to vary the frequency of a supply network. In this respect, recall that the power networks may be either DC or AC but mono-phase (this is for instance the case in the electric traction domain). Therefore, three-phase DC/AC inverters turn out to be the only possible interface (between railway networks and 3-phase AC motors) due to their high capability to ensure flexible voltage and frequency variation.
The control problem at hand is to design a speed controller for the complex system including an AC/DC rectifier, a DC/AC inverter and an induction motor. The controller is expected to ensure a wide range speed regulation, despite mechanical parameters’ variations. Indeed, the rotor inertia moment, the viscous friction coefficient and the load torque are presently allowed to be unknown and step-like varying.
The rectifier is (of?) a pulse width modulation (PWM ) type that features power regeneration, controlled DC-link voltage, small filter, 4-quadrant operation (bidirectional power transmission).
Nowadays, the use of PWM rectifiers has considerably spread in industry and concerns a wide range spectrum of applications, [12], [4]. The point is that the association ‘rectifier-inverter-motor’ acts as a highly nonlinear load vis-à-vis to the main power AC network (which is supposed to provide almost perfect sinusoidal voltage to all other connected loads). Therefore, undesirable higher harmonics are generated (due to the load nonlinearity) and pollute the supply network. This harmonic pollution reduces the rectifier efficiency, induces voltage distortion in the AC supply network and induces electromagnetic compatibility problems [12].
In the light of the above considerations, it turns out that the control objective must not only be motor speed regulation but also the current harmonics rejection through power factor correction (PFC), [7].
In most previous works on induction machine speed control, the control problem was simplified by ignoring the dynamics of the AC/DC/AC converters. Accordingly, the machine is supposed to be directly controlled by stator voltages. The simplified control problem has been dealt with using several control strategies ranging from simple techniques, e.g. field-oriented control [5], to more sophisticated nonlinear approaches, e.g. passivity control [1], direct torque control [11], [3], or sliding mode control [10]. A control strategy that ignores the presence of the AC/DC rectifier suffers at least from two main drawbacks. First, the controller design relies on the assumption that the DC voltage (provided by the AC/DC rectifier) is perfectly regulated; the point is that perfect regulation of the rectifier output voltage cannot be ensured when ignoring the rectifier load which is nothing but the
‘inverter-motor’ set. The second drawback is concerned by the entire negligence of the PFC requirement. That is, from a control viewpoint, it is not judicious to consider separately the inverter- motor association, on one hand, and the power rectifier, on the other hand.
In the present work, we will develop a new control strategy that simultaneously accounts for all system components i.e. the AC/DC rectifier and the association ‘DC/AC inverter-motor’. Our control strategy is featured by its multi-loops nature. First, a current loop is designed so that the coupling between the power supply network and the AC/DC rectifier operates with a unitary power factor.
Then, a second loop is designed to regulate the output voltage of the AC/DC rectifier so that the DC- link between the rectifier and the inverter keeps on a constant voltage despite changes in the motor operation conditions. Finally, a bi-variable loop is designed to make the motor velocity track its
varying reference value and regulate the rotor flux norm to its nominal value. All control loops are designed using the Lyapunov and backstepping techniques [9]. Interestingly, the load torque, rotor inertia and friction coefficient are presently allowed to be unknown parameters subject to step-like changes. This parametric uncertainty is coped with, by providing the controller with a parameter adaptation capability. It will be formally proved that the proposed multi-loop nonlinear adaptive controller actually stabilizes (globally and asymptotically) the controlled system and meets its tracking objectives with a good accuracy. Specifically, the motor speed and rotor flux norm will be shown to perfectly track their references, whatever the initial conditions. The rectifier input current and output voltage are shown to match well their reference values. More precisely, the steady-state tracking errors for these two variables are both harmonic signals with amplitudes depending, among others, on the supply network frequency. These theoretical results are obtained by making a judicious use of adequate control theory tools, e.g. averaging theory and Lyapunov stability [8].
The paper is organized as follows: the system under study (i.e. the AC/DC/AC converter and induction motor association) is modeled and given a state space representation in Section 2; the controller design and the closed-loop system analysis are presented in Section 3; the controller performances are illustrated through numerical simulations in Section 4. For convenience, the main notations used throughout the paper are described in Table II given at the end of the paper.
2. MODELING ‘AC/DC/AC CONVERTER-INDUCTION MOTOR’ ASSOCIATION
The controlled system scheme is represented by Fig 1. It includes an AC/DC boost rectifier, on one hand, and an ‘inverter-induction motor’ combination, on the other hand. The inverter is a DC/AC converter operating, like the AC/DC rectifier, according to the known pulse width modulation (PWM) principle.
Fig.1. AC/DC/AC drive circuit with three-level inverter
~
L1
ic
ia ib ic
ve(t) vdc 2C
S
S S’
S’
S1 S2 S3
S1’ S2’ S3’
Inverter Rectifier
rter
Filter is s.ie
IM ie
2.1 AC/DC Rectifier Modeling
The power supply network is connected to an H-bridge converter consisting of four IGBT’s with anti- parallel diodes for bidirectional power flow mode. The converter should be controlled so that two main tasks are accomplished: (i) providing a constant DC link voltage; (ii) ensuring an almost unitary power factor connection with the power network. Applying Kirchhoff’s laws, this subsystem is described by the following set of differential equations:
dc e
e sv
L L v dt di
1 1
1
(1a)
s e
dc i
C i s C dt
dv
2 1 2
1
(1b)
where ie is the current in inductor L1, vdc denotes the voltage in capacitor 2C , is designates the input current inverter, ve 2.E.cos(et) is the sinusoidal network voltage (with known constants E,e) and s is the switch position function taking values in the discrete set {1,1}. Specifically:
ON is S and OFF is S if
OFF is S and ON is S if s
1
1 (1c)
The above (instantaneous) model describes accurately the physical inverter. Then, it is based upon to build up converter simulators. However, it is not suitable for control design due to the switched nature of the control inputs. As a matter of fact, most existing nonlinear control approaches apply to systems with continuous control inputs. Therefore, the control design for the above converter will be performed using the following average version of (1a-b) [6]:
2 1 1 1
1 1
x u L L v dt
dx e
(2a)
is
C x u C dt dx
2 1 2
1
1 1
2 (2b)
where:
ie
x1 , x2 vdc, u1=s (2c)
are the average values over the cutting periods of ie, vdc and s, respectively.
2.2 Inverter-Motor Modeling
The induction model is based on the motor equations in the rotating -and- axes and reads as:
J i T
i J m J
f dt
d L
r s r
s
)
( (3a)
r r s s
s ba bp i m v
dt di
1
(3b)
r r s s
s ba bp i mv
dt di
1
(3c)
r s
sr r
r a aM i p
dt
d (3d)
r s
sr r
r a aM i p
dt
d (3e)
where is,is,r,r,, and, TL, are the stator currents, rotor fluxes, angular speed, and load torque, respectively. Wherever they come in, the subscripts s and r refer to the stator and rotor, respectively.
That is, Rsand Rrare the stator and rotor resistances; Ls andLr are the self-inductances. Msr denotes the mutual inductance between the stator and rotor windings. p designates the number of pole-pairs,
J the inertia of the motor-load set, and f is the friction coefficient. The remaining parameters are defined as follows:
r r
L a R ,
r s
sr
L L b M
, (Lr2Rs Msr2Rr)/LsLr2, 1(Msr2 /LsLr),
r sr
L p M
m ,
Ls
m
1
1 .
In (3a-e), vs,vs are the stator voltage in the -coordinates (Park’s transformation of the three phase stator voltages). The inverter is featured by the fact that the stator α- and β-voltages can be controlled independently. To this end, these voltages are expressed in function of the corresponding control action (see e.g. [2]):
u3
v
vs dc , vs vdc u2 (4a)
where (u2,u3) represent the average α- and β-axes (Park’s transformation) of the three phase duty ratio system (s1,s2,s3). The latter are defined by (1c) replacing there (S ,S') by (Si,Si') (i1,2,3).
Now, let us introduce the state variables:
3
x , x4 is , x5 is,x6 r , x7 r , (4b) where the bar refers to signal averaging over cutting periods (just as in (2c)). Using the power conservation principle, the power absorbed by the DC/AC inverter is given by the usual expression
s
ai x i
P 2 . On the other hand, the power released by the inverter is given by Prm x2(u2x4u3x5). As
rm
ai P
P , it follows that:
u2x4 u3x5
is (4c)
Then, substituting (4a-c) in (3a-e) yields the following state-space representation of the association
‘inverter-motor’:
J x T x x x J x m J
f dt
dx L
3 ( 5 6 7 4)
3 (5a)
2 2 1 4 7 3 6
4 bax bpx x x mu x
dt
dx (5b)
2 3 1 5 6 3 7
5 bax bpx x x m u x
dt
dx (5c)
7 3 4 6
6 ax aM x px x
dt dx
sr
(5d)
6 3 5 7
7 ax aM x px x
dt dx
sr
(5e)
The state space equations thus obtained are put together to get a state-space model of the whole system including the AC/DC/AC converters and the induction motor. For convenience, the whole system’s model is rewritten here for future reference:
2 1 1 1
1 1
x u L L v dt
dx e
(6a)
) (
2 1 2
1
5 3 4 2 1
1
2 u x u x
C x u C dt
dx (6b)
J x T x x x J x m J
f dt
dx L
3 ( 5 6 7 4)
3 (6c)
2 2 1 4 7 3 6
4 bax bpx x x mu x
dt
dx (6d)
2 3 1 5 6 3 7
5 bax bpx x x mu x
dt
dx (6e)
7 3 4 6
6 ax aM x px x
dt dx
sr
(6f)
6 3 5 7
7 ax aM x px x
dt dx
sr
(6g)
3. CONTROLLER DESIGN 3.1 Control Objectives
There are two operational control objectives:
(i) Speed regulation: the machine speed must track, as closely as possible, a given reference signal
ref , despite the parametric uncertainties (concerning the load torque TL, rotor inertia J and friction coefficient f ).
(ii) PFC requirement: the rectifier input current ie must be sinusoidal and in phase or opposed phase with the AC supply voltageve.
As there are three control inputs at hand, namely u1, u2 and u3, two more control objectives are added:
(iii) Controlling the continuous voltage vdc making it track a given reference signalvdcref .
(iv) Regulating the rotor flux norm r x62x72 to a reference valueref , preferably equal to its nominal value.
3.2 AC/DC rectifier control design
3.2.1 Controlling rectifier input current to meet PFC
The PFC objective means that the input current rectifier should be sinusoidal and in phase (or opposite phase) with the AC supply voltage. Therefore, one seeks a regulator that enforces the current x1 to tack a reference signal x1* of the form:
ve
k
x1* (7)
At this point, k is any real parameter that is allowed to be time-varying. This parameter is positive when the induction machine operates in motor mode and negative in the generator mode. Introduce the current tracking error:
* 1 1
1 x x
z (8)
In view of (6a), the above error undergoes the following equation:
* 1 1 2 1 1
1 v /L u x /L x
z e (9)
To get a stabilizing control law for this first-order system, consider the quadratic Lyapunov function
2 1 1 0.5z
V . It can be easily checked that the time-derivative V1 is a negative definite function of z1 if the control input is chosen as follows:
2* 1 1 1
1 1
1 L c z (v /L ) x /x
u e (10)
with c1 0 is a design parameter. The properties of such a control law are summarized in the following proposition, the proof of which is straightforward.
Proposition 1. Consider the system, next called current (or inner) loop, composed of the current equation (6a) and the control law (10) where c1 0 is arbitrarily chosen by the user. If the reference
ve
k
x1* and its first time derivative are available, then one has the following properties:
1) The current loop undergoes the equation z1 c1 z1 with z1 x1x1*. As c1 is positive, this equation is globally exponentially stable, i.e. z1 vanishes exponentially, whatever the initial conditions.
2) If in addition k converges (to a finite value), then the PFC requirement is asymptotically fulfilled in average i.e. the (average) input current x1 tends (exponentially fast) to its reference kve as t 3.2.2 DC link voltage regulation
The aim is now to design a tuning law for the ratio kin (7) so that the rectifier output voltage x2 vdc
is steered to a given reference value vdcref . As mentioned above, vdcref is generally (but not mandatory) set to the nominal value of the stator voltage amplitude. The first step in designing such a tuning law is to establish the relation between the ratio k (control input) and the output voltagex2 . This is the subject of the following proposition.
Proposition 2. Consider the power rectifier described by (6a-b) together with the control law (10).
Under the same assumptions as in Proposition 1, one has the following properties:
1) The output voltage x2 varies, in response to the tuning ratiok, according to the equation:
) (
2 ) 1 (
2 1
5 3 4 2 1
2 2
2 u x u x
C v
z v k x C dt dx
e
e
(11)
2) The squared voltage (y x22) varies, in response to the tuning ratiok, according to the equation:
) , ( /
/ 1
2 C zv C x t
v k dt dy
e
e
(12)
with
C x u x u x t
x, ) ( )/
( 2 2 4 3 5
(13)
Proof. The power absorbed by the AC/DC rectifier is given by the well known expression
e
absorbed x v
P 1 . On the other hand, the power released by the rectifier (toward the load including the capacity and the inverter) is given by Preleased u1x1x2. Using the power conservation principle, one
has Pabsorbed Preleased or, equivalently:
2 1 1
1v u x x
x e
(14) Also, from (7)-(8), one immediately gets that x1 kve z1 which together with (14) yields
2 1 2 1
1x (kv z v ) x
u e e . This establishes (11) due to (6b). Deriving y x22 with respect to time and using (11) yields relation (12) and completes the proof of Proposition 2
The ratio k stands up as a control signal in the first-order system defined by (12). As previously mentioned, the reference signal dcref2
def
ref v
y (of the squared DC-link voltagex2 vdc) is chosen to be constant (i.e. yref 0), it is given the nominal value of stator voltage amplitude. Then, it follows from (12) that the tracking error z2 y yref undergoes the following equation:
ref e
et C E z t C x t y
k E C k E
z2 2 / 2 cos(2 )/ 2 1cos( )/ ( , ) (15) where one used the fact that ve 2.E.cos(et) and ve2 E2(1cos(2et)). To get a stabilizing control law for the system (15), consider the following quadratic Lyapunov function:
2 2 2 0.5z
V (16)
It is easily checked that the time-derivative V2 can be made negative definite in the state z2 by letting:
( , )
) cos(
2 ) 2
cos( 1 2 2
2
2 k E t E z t C c z x t
E
k e e C yref (17a)
where c2 0 is a design parameter. The point is that such an equation involves a periodic singularity due to the mutual neutralization of the first two terms on the left side of (17a). To get off this singularity and, besides, to avoid an excessive chattering in the solution, the two terms in cos(.) on the left side of (17a) are ignored, leading to the following approximate simpler solution:
c2z2 (x,t)/E2 C
k C yref /E2 (17b)
Bearing in mind the fact that the first derivative of the control ratio k must be available (Proposition 1), the following filtered version of the above solution is proposed:
c2z2 (x,t)/E2 C
d k d
k dC y /E2
ref
(18)
At this point, the regulator parameters d,c2 are any positive real constants. The proof of the forthcoming proposition 3 (?) will make it clear how these should be chosen for the control objectives to be achieved. For now, let us summarize the main findings (?) in the following proposition.
Proposition 3. Consider the inner control loop consisting of the AC/DC rectifier described by (6a-b) together with the control laws (10) and (18). Using Proposition 1 (Part 1), it turns out that the inner loop undergoes, in the (z1,z2,k)-coordinates, the following equation, where z1 x1x1* and
yref
y
z2 :
0
) 2 cos(
) 2 cos(
0 )
, ( 1
0
0
0 0
0 0
1 2
2 2
1
2 2
2 1
2 1
t z
E C t k
C y E
t x E
C k d
z z
d c E
C d
C E c
k z z
e e
ref
(19)
3.3 Motor speed and rotor flux norm regulation
The problem of controlling the rotor speed and flux norm is now addressed for the induction machine described by (5a-e). The speed reference ref is any bounded and derivable function of time and its two first derivatives are available and bounded. These properties can always be achieved filtering the reference through second-order linear filters. The flux reference ref is fixed to its nominal value. The controller design will now be performed in two steps using the tuning-functions backstepping adaptive technique [9]. First, introduce the tracking errors:
3
3 x
z ref (20)
) ( 62 72
2 ref
4 x x
z
(21) Step 1. It follows from (5a) and (5d-e) that the errors z3 and z4 undergo the differential equations:
J x f J T J x x x x m
z3 ref ( 6 5 7 4)/ L / 3/ (22)
) (
2
2 ref ref 6 4 7 5
4 aM x x x x
z sr 2a(2ref z4) (23)
In (22) and (23), the quantities m(x6x5x7x4)and 2aMsr(x6x4x7x5) stand up as virtual control signals. If these were the actual control signals, the error system (22)-(23) could be globally asymptotically stabilized by letting m(x6x5x7x4) 1 and 2aMsr(x6x4x7x5)1 with:
) (
)
( 3 3 3
1 J c z ref TL f ref z
def
(24a)
On the other hand, the factJ , TLand f are unknown suggests the certainty equivalence from of equations (24a).
)) ˆ(
) ˆ ˆ(
3 3
3
1 J c z ref TL f ref z
def
(24b)
) (
2
2 ref ref 2ref 4
4 4
1 c z a z
def
(25)
where c3 and c4 are any positive design parameters and Jˆ, TˆLand fˆ are the estimates of J , TLand
f respectively. Indeed, considering the Lyapunov function:
) (
5 .
0 32 42
3 z z
V (26)
one would get from (22)-(23), letting m(x6x5x7x4)1 and 2aMsr(x6x4x7x5) 1:
2 4 4 2 3 3
3 c z c z
V (27)
This would prove the global asymptotic stability of the system (22)-(25). As the quantities
) (x6x5 x7x4
m and 2aMsr(x6x4x7x5) are not the actual control signals, they cannot be let equal to 1
