Local weak observability conditions of sensorless AC drives

0.33 MB·

by  Mohamad Koteich

Local Weak Observability Conditions of Sensorless
AC Drives
Mohamad Koteich, Student Member, IEEE,
Abdelmalek Maloum, Gilles Duc and Guillaume Sandou
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J KEYWORDS

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Sensorless control, Synchronous motor, Induction Motor, AC machine.
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A
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. ABSTRACT
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a Alternating current (AC) electrical drive control without mechanical sensors is an active
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research topic. This paper studies the observability of both induction machine and synchronous
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1 machine sensorless drives. Observer-based sensorless techniques are known for their deteriorated
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9 performance in some operating conditions. An observability analysis of the machines helps
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1 understanding (and improving) the observer’s behavior in the aforementioned conditions.
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1 I. INTRODUCTION
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Eco-friendly technologies have attracted worldwide attention due to several environmental
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v issues. In this context, electric motors have become a serious competitor with combustion engines
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in many industrial applications, especially in automotive industry.
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The use of AC drives has been pioneered by the recent advances in power semiconductor
switching frequencies, and power converter topologies [1]. High performance control of AC
Mohamad Koteich is with Renault S.A.S. Technocentre, 78288 Guyancourt, France, and also with L2S - CentraleSupe´lec -
CNRS - Paris-Sud University, 91192 Gif-sur-Yvette, France (e-mail: [email protected]).
Abdelmalek Maloum is with Renault S.A.S. Technocentre, 78288 Guyancourt, France (e-mail: abdel-
[email protected]).
Gilles Duc and Guillaume Sandou are with L2S - CentraleSupe´lec - CNRS - Paris-Sud University, 91192 Gif-sur-Yvette,
France (e-mail: [email protected]; [email protected]).
drives can be achieved using vector control [2], which requires the measurement of the rotor
position. For many reasons, mainly for cost reduction and reliability increase, mechanical sensor-
less techniques have attracted the attention of researchers as well as many large manufacturers
[3]. These techniques consist of sensing the motor currents and voltages, and using them as
inputs to an estimation algorithm (such as the state-observer algorithm) that estimates the rotor
angular speed and/or position.
One limitation of the use of sensorless techniques is the deteriorated performance in some
operatingconditions:namelythelow-speedoperationinthecaseofsynchronousmachines(SMs),
and the low-frequency input voltage in the case of induction machines (IMs) [4]. Usually, this
problem is viewed as a stability problem, and sometimes is treated based on experimental results.
However, the real problem lies in the so-called “observability conditions” of the system.
Over the past few years, a promising approach, based on the local weak observability concept
[5], has been used in order to better understand the deteriorated performance of sensorless AC
drives. Several papers have been published on this topic: authors in [6] [7] [8] [9] [10] study
the local observability of IMs in some operating conditions, namely for zero acceleration and
low-frequency input voltages. Among SMs, only the permanent magnet synchronous machine
(PMSM) is studied in the literature [9] [10] [11] [12] [13].
A unified approach of AC drives observability analysis is proposed in this paper. Sufficient
conditions for both IM and SM’s observability are studied. Concerning the IM, a more general
study, that covers a wider region of operating conditions, is done; the speed is not considered
to be constant, which requires the estimation of the resistant torque. Concerning the SMs, the
wound-rotor synchronous machine (WRSM) is studied, and considered to be the general case
of SMs; its model and observability conditions are easily extended to the permanent magnet
synchronous machine (PMSM) and the synchronous reluctance machine (SyRM).
The main purpose of this paper is to contribute to a better understanding of the deteriorated
performance of sensorless techniques. The paper is divided into five sections; after this introduc-
tion, thelocal weak observability concept is presented insection 2. Sections3 and 4 arededicated
to the observability study of IMs and SMs respectively. Conclusions are made in section 5.
II. LOCAL WEAK OBSERVABILITY THEORY
The local weak observability concept [5], based on the rank criterion, is introduced in this
section.
A. Problem statement
Systems of the following form (denoted Σ) are considered:

x˙ = f (x(t),u(t))
Σ : (1)
y = h(x(t))
where x ∈ X ⊂ Rn is the state vector, u ∈ U ⊂ Rm is the control vector (input), y ∈ Rp is the
output vector, f and h are C∞ functions.
The observation problem can be then formulated as follows [14]: Given a system described
by a representation (1), find an accurate estimate xˆ(t) for x(t) from the knowledge of u(τ), y(τ)
for 0 ≤ τ ≤ t.
The system observability is required for observer design. For nonlinear systems, global ob-
servability is not practical, since the observer often requires the system to be instantaneously
observable in a certain neighborhood of the state trajectories; the system should be locally weakly
observable.
B. Observability rank condition
The system Σ is said to satisfy the observability rank condition at x , if the observability
0
matrix, denoted by O (x), is full rank at x . O (x) is given by:
y 0 y
∂ (cid:104) (cid:105)T
O (x) = L0h(x) L h(x) L2h(x) ... Ln−1h(x) (2)
y ∂x f f f f x=x0
where Lkh(x) is the kth-order Lie derivative of the function h with respect to the vector field
f
f.
C. Observability theorem
A system Σ (1) satisfying the observability rank condition at x is locally weakly observable
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at x . More generally, a system Σ (1) satisfying the observability rank condition for any x , is
0 0
locally weakly observable [5]. Rank criterion gives only a sufficient condition for local weak
observability.
III. OBSERVABILITY ANALYSIS OF INDUCTION MACHINES
This section deals with the local observability conditions of induction machines. First, the
machine model is presented, then its local weak observability is studied. In this study, the speed
is not considered to be constant, therefore the resistant torque, which is assumed to vary slowly,
should be added to the estimated state vector.
A. Machine model
The IM can be modeled as a three-phase stator and a three-phase rotor. The stator is supplied
by a three-phase source, whereas the rotor windings are in short-circuit (see Figure 1). The IM
state-space model in the two-phase stator reference frame (α β ) can be written as follows:
s s
(cid:20) (cid:18) (cid:19) (cid:21)
dI 1 M 1
s = V −r I + I −ω J Ψ (3a)
s s s 2 e 2 r
dt σL L τ
s r r
(cid:18) (cid:19)
dΨ 1 M
r = − I −ω J Ψ + I (3b)
2 e 2 r s
dt τ τ
r r
dω 3p2M p
e = ITJ Ψ − T (3c)
dt 2 J L s 2 r J r
r
dT
r
= 0 (3d)
dt
where
M2 M2 L
r
r = R +R ; σ = 1− ; τ = (4)
s s r L2 L L r R
r s r r
R, L and M stand respectively for the resistance, inductance and mutual inductance. The indices
s and r stand for stator and rotor parameters. p is the number of pole pairs, J is the inertia of
the rotor with the associated load, ω is the electrical speed of the rotor and T is the resistant
e r
torque. I , Ψ and V stand for the stator currents, rotor fluxes and stator voltages in the stator
s r s
reference frame (α β ) :
s s
(cid:104) (cid:105)T (cid:104) (cid:105)T (cid:104) (cid:105)T
I = i i ; Ψ = ψ ψ ; V = v v (5)
s sαs sβs r rαs rβs s sαs sβs
The model (3) can be fitted to the structure Σ (1) by taking:
(cid:104) (cid:105)T
x = IT ΨT ω T ; u = V ; y = I (6)
s r e r s s
I is the n×n identity matrix, and J is the π/2 rotation matrix :
n 2
   
1 0 0 −1
I2 =   ; J2 =   (7)
0 1 1 0
bs βr βs βr βs
q
s
v
bs
αr
θ
e
αs ds
v as
as
θr αr
vcs θ
s
θ
e
α
s
cs
(a) Schematic representation (b) Vector diagram
Fig. 1: Schematic representation (a) and vector diagram (b) of the induction machine. (α β ),
s s
(α β )and(d q )arerespectivelythetwo-phasestator,rotor,androtatingmagneticfieldreference
r r s s
frames.
B. Observability study
The system (3) is a 6-th order system. The local observability study requires the evaluation
of derivatives up to the 5-th order of the output. However, regarding the equations complexity,
only the first and second order derivatives are evaluated for the IM in this paper. Higher order
derivatives of the output are too lengthy, and very difficult to deal with. As the observability
rank criterion provides sufficient conditions, information contained in the first and second order
derivatives is rich enough to study the IM observability. It should be noticed that, in practice,
observer-based techniques do not usually need information on higher than second order deriva-
tives.
Symbolic math software is used te evaluate complex expressions. Moreover, the following
change of variables is made in order to make the study easier:
I(cid:101) = σL I (8)
s s s
M
Ψ(cid:101) = Ψ (9)
r r
L
r
The system (3) becomes:
dI(cid:101)
s
= V +aI(cid:101) +γ(t)Ψ(cid:101) (10a)
s s r
dt
dΨ(cid:101)
r
= −γ(t)Ψ(cid:101) −(a−b)I(cid:101) (10b)
r s
dt
dω c p
e = I(cid:101)TJ Ψ(cid:101) − T (10c)
dt J s 2 r J r
dT
r
= 0 (10d)
dt
with
r R 3p2 (cid:18) 1 (cid:19) dγ dω
a = − s ; b = − s ; c = ; γ(t) = I −ω J ; = − eJ (11)
2 e 2 2
σL σL 2σL τ dt dt
s s s r
The scaled output is:
 
y = I(cid:101)s = (cid:101)isαs (12)
(cid:101)isβs
its first order derivative is:
dI(cid:101)
s
y˙ = = V +aI(cid:101) +γ(t)Ψ(cid:101) (13)
s s r
dt
Adding (10a) and (10b) gives:
dI(cid:101) dΨ(cid:101)
s r
+ = V +bI(cid:101) (14)
s s
dt dt
then:
dΨ(cid:101) dI(cid:101)
r s
= V +bI(cid:101) − (15)
s s
dt dt
The second order derivative of the output can be then written as:
d2I(cid:101) dV(cid:101) dI(cid:101) dΨ(cid:101) dγ
s s s r
= +a +γ(t) + Ψ(cid:101) (16)
dt2 dt dt dt dt r
dV dI(cid:101) dI(cid:101) dγ
s s s
= +a +γ(t)V +γ(t)bI(cid:101) −γ(t) + Ψ(cid:101) (17)
s s r
dt dt dt dt
dV dI(cid:101) dγ
= s +γ(t)V +(aI −γ(t)) s +γ(t)bI(cid:101) + Ψ(cid:101) (18)
s 2 s r
dt dt dt
The observability study is done using the scaled output and its derivatives:
y = I(cid:101) (19)
s
y˙ = V +aI(cid:101) +γ(t)Ψ(cid:101) (20)
s s r
(cid:18) (cid:19)
y¨ = dVs +aV +(cid:0)a2I −(a−b)γ(t)(cid:1)I(cid:101) + dγ +aγ(t)−γ(t)2 Ψ(cid:101) (21)
s 2 s r
dt dt
The IM observability matrix, evaluated for these derivatives, can be written as:
 
1 0 0 0 0 0
 
 0 1 0 0 0 0 
 
 
 a 0 1 ω ψ(cid:101) 0 
OIM =  τr e rβs  (22)
y  
 0 a −ω 1 −ψ(cid:101) 0 
 e τr rαs 
 
d d e e f f 
11 12 11 12 11 12
 
d d e e f f
21 22 21 22 21 22
with:
a−b c c
d = a2 − − ψ(cid:101)2 ; d = −(a−b)ω + ψ(cid:101) ψ(cid:101) (23)
11 τ J rβs 12 e J rαs rβs
r
a−b c c
d = a2 − − ψ(cid:101)2 ; d = (a−b)ω + ψ(cid:101) ψ(cid:101) (24)
22 τ J rαs 21 e J rαs rβs
r
a 1 c ω dω c
e11 = τ − τ2 +ωe2 + J(cid:101)isβsψ(cid:101)rβs ; e12 = aωe −2τe + dte − J(cid:101)isαsψ(cid:101)rβs (25)
r r r
a 1 c ω dω c
e22 = τ − τ2 +ωe2 + J(cid:101)isαsψ(cid:101)rαs ; e21 = −aωe +2τe − dte − J(cid:101)isβsψ(cid:101)rαs (26)
r r r
(cid:18) (cid:19)
2 p
f11 = 2ωeψ(cid:101)rαs −(a−b)(cid:101)isβs + a− τ ψ(cid:101)rβs ; f12 = −Jψ(cid:101)rβs (27)
r
(cid:18) (cid:19)
2 p
f21 = 2ωeψ(cid:101)rβs +(a−b)(cid:101)isαs − a+ τ ψ(cid:101)rαs ; f22 = Jψ(cid:101)rαs (28)
r
The matrix OIM (22) is a 6×6 matrix, its determinant is the following:
y
∆ = p M2 (cid:20) 1 dωe (cid:0)ψ2 +ψ2 (cid:1)−(cid:18)ω2 + 1 (cid:19)(cid:18)dψrαsψ − dψrβsψ (cid:19)(cid:21) (29)
IM J L2 τ dt rαs rβs e τ2 dt rβs dt rαs
r r r
The expression (29) gives a sufficient local weak observability condition of the system (3); the
value of ∆ has to be non-zero to ensure the local observability of sensorless IM drives.
IM
1) Resultsinterpretation: Theexpression(29)evokesinthefollowingtwoparticularsituations
(to ensure the local weak observability of the system):
• ω˙e = 0: For constant rotor speed, the rotor fluxes should be neither constant (or slowly
varying) nor linearly dependent, i.e. ψ (cid:54)= k.ψ , for all constant k.
rαs rβs
˙ ˙
• ψrαs = ψrβs = 0:Iftherotorfluxesareslowlyvarying(constant),whichusuallycorresponds
to low-frequency input voltages, the speed has to vary (rapidly).
2) Geometrical interpretation: The IM is locally weakly observable if the determinant ∆
IM
(29) is different from zero, this means:
τ ω˙ dψrαsψ − dψrβsψ d d (cid:18)ψ (cid:19)
r e (cid:54)= dt rβs dt rαs ⇐⇒ arctan(τ ω ) (cid:54)= arctan rβs (30)
1+τ2ω2 ψ2 +ψ2 dt r e dt ψ
r e rαs rβs rαs
Knowing that (Figure 1b):
(cid:18) (cid:19)
d ψ d
arctan rβs = θ (31)
s
dt ψ dt
rαs
the condition (30) becomes:
d
ω (cid:54)= arctan(τ ω ) (32)
s r e
dt
where ω is the angular frequency of the rotating magnetic field. The relationship (32) tells that
s
if the rotor speed ω is slowly varying, the input voltage frequency should not be very low.
e
Nevertheless, in a wide range of applications, the rotor speed is (almost) constant during the
most of machine operation time, the main conclusion to ensure the IM local weak observability
is that the input voltage frequency should not be very low.
IV. OBSERVABILITY ANALYSIS OF SYNCHRONOUS MACHINES
An SM is a three-phase machine, having the same stator structure as the IM. Depending
on the rotor structure, there exist three main SM sub-families (Figures 2, 3, and 4): wound-
rotor (WRSM), permanent-magnet (PMSM) and reluctance type SyRM synchronous machines.
Both WRSM and PMSM can be either salient type (non cylindrical) rotor or non-salient type
(cylindrical) rotor. Interior PMSM (IPMSM) is a salient machine, surface-mounted PMSM
(SPMSM) is a non-salient machine. In this section, WRSM is studied, and considered to be
the general case of SMs.
β β
q
b
q
Ψ
O
v
b
d
L i
∆ q
θ
α d
a
v
a
v f
L i
θ δ d
O M i
v f f
c
θ
α
c
(a) Schematic representation (b) Vector diagram showing the (dashed) observability vector
Fig. 2: Schematic representation (a) and vector diagram (b) of the wound-rotor synchronous
machine. (αβ) and (dq) are respectively the stator and rotor reference frames.
A. Wound-rotor synchronous machine model
The state-space model of the WRSM is written in the two-phase (αβ) stationary reference
frame (see Figure 2a), in a way to be fitted to the structure (1):
dI
= −L−1R I +L−1V (33a)
eq
dt
dω p p
= T − T (33b)
m r
dt J J
dθ
= ω (33c)
dt
where the state, input and output vectors are respectively:
(cid:104) (cid:105)T (cid:104) (cid:105)T (cid:104) (cid:105)T
x = IT ω θ ; u = V = v v v ; y = I = i i i (34)
α β f α β f
I and V are the current and voltage vectors. Indices α and β stand for stator signals, index f
stands for rotor (field) ones. ω stands for the electrical rotor angular speed and θ for the electrical
angular position of the rotor. L is the (position-dependent) matrix of inductances, and R is the
matrix of resistances:
   
L +L cos2θ L sin2θ M cosθ R 0 0
0 2 2 f s
  ∂L  
L =  L sin2θ L −L cos2θ M sinθ ; R = R+ ω ; R =  0 R 0 
 2 0 2 f  eq ∂θ  s 
   
M cosθ M sinθ L 0 0 R
f f f f
J is the moment of inertia of the rotor with its associated load, p is the number of pole pairs,
T is the resistant torque and T is the motor torque:
r m
3 3
(cid:2) (cid:3)
T = pM i (i cosθ−i sinθ)− pL (i2 −i2)sin2θ−2i i cos2θ (35)
m 2 f f β α 2 2 α β α β
β β
d d
v v
β β
d d
θ θ
α α
v v
α α
(a) IPMSM (b) SPMSM
Fig. 3: Schematic representation of the IPMSM (a) and the SPMSM (b)
B. Other SMs models
The other SMs can be seen as special cases of the salient-type WRSM; the IPMSM (Figure
3a) model can be derived by considering the rotor magnetic flux to be constant:
di
f
= 0 (36)
dt
and by substituting M i by the permanent magnet flux ψ :
f f r
ψ
r
i = (37)
f
M
f