Preprints of the 18th IFAC World Congress
Milano (Italy) August 28 - September 2, 2011
Sensorless Speed Control of Non–salient
Permanent Magnet Synchronous Motors
Dhruv Shah ∗ Gerardo Espinosa–Pérez ∗∗ Romeo Ortega ∗
Mickael Hilairet ∗∗∗
∗
Laboratoire des Signaux et Systèmes, CNRS–SUPELEC, 91192
Gif–sur–Yvette, France (e–mail: shah{ortega}@lss.supelec.fr)
∗∗
Facultad de Ingenierı́a – UNAM, A.P. 70-256, 04510 México D.F.,
MEXICO (gerardoe@servidor.unam.mx)
∗∗∗
LGEP - CNRS / SUPELEC , 91192 Gif sur Yvette, France
(e–mail: mickael.hilairet@supelec.fr)
Abstract: In this paper it is presented an asymptotically stable sensorless controller for a
non–salient permanent magnet synchronous motor that is perturbed by an unknown constant
load torque. The proposed scheme is a fourth order nonlinear observer–based scheme that
does not rely on—intrinsically nonrobust—operations like open–loop integration of the systems
dynamical model nor signal differentiation, and can be easily implemented in real time. The
controller is easy to commission, with the tuning gains directly determining the convergence
rates of the position, speed and load torque observers. Experimental results that illustrate the
usefulness of the scheme are presented.
Keywords: Sensorless control; PMSM Control; Nonlinear control.
1. INTRODUCTION
assignment passivity–based controller (IDA–PBC) of [8],
yields a (locally) asymptotically stable closed–loop system.
Control of electrical machines is a topic that has attracted
the attention of the control community for many years
now. Rigorous mathematical analysis of the usual strategies applied in industrial applications and the proposition
of novel control schemes have been reported in the literature [1, 2, 13]. In this paper we are interested in the
challenging problem of eliminating the use of sensors for
the mechanical variables (position and speed), the so–
called sensorless control [3], for non–salient permanent
magnet synchronous motors (PMSM).
To the best of our knowledge, this is the first time
a complete answer is given to a sensorless problem—
under reasonable practical assumptions. Schemes that
rely on, heuristically conceived schemes abound in the
literature [11]. Many results are also available for the
(practically unrealistic) case of known initial position [9].
In [10] a probably stable sensorless scheme for wound rotor
synchronous motors is proposed. A key difference of the
latter machine with the PMSM is the availability of flux
measurements that considerably simplifies the observation
problem. The reader is referred to [4] for a more detailed
review of the sensorless control literature for PMSMs, both
in the control, and the drives communities.
For the successful speed control of PMSMs three variables must be estimated out of the measurement of the
electrical coordinates: rotor position and speed, and load
torque—the latter assumed constant. A position observer
for PMSM that exhibits strong (exponential) stability
properties was recently reported in [4]. In [5] the position
observer was combined with an ad–hoc speed estimator
and a standard field–oriented controller, which are often
used in applications, yielding very encouraging experimental results. However, no theoretical justification was provided for the overall observer–controller scheme. In [6], see
also [4], the problem of simultaneous observation of speed
and load torque, assuming position is known, was solved
using immersion and invariance (I&I) techniques [7]. The
purpose of this paper is to propose a variation of the latter
estimator, which combined with the position observer of [4]
and the (full state–feedback) interconnection and damping
⋆ The work of Gerardo Espinosa was supported by DGAPA–UNAM,
CONACYT (51050), and the SUPELEC Foundation. The work of
Dhruv Shah was supported by the Indo–French project No. 3602-1,
under the aegis of IFCPAR.
Copyright by the
International Federation of Automatic Control (IFAC)
Caveat: This is an abridged version of the full paper
available upon request from the authors.
2. PMSM MODELS AND PROBLEM FORMULATION
The classical fixed–frame (αβ) model of the unsaturated
non–salient PMSM is given by [13]
L
diαβ
= −Riαβ − np ωΦJ ραβ + vαβ
dt
J ω̇ = np ΦiTαβ J ραβ − τL
(1)
(2)
ρ̇αβ = np ωJ ραβ
(3)
0 −1
, iαβ = col(iα , iβ ) and vαβ =
where J :=
1 0
col(vα , vβ ) are the stator currents and motor terminal
voltages, respectively, ω is the angular velocity, L is
the stator inductance, R is the stator resistance, nP is
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Preprints of the 18th IFAC World Congress
Milano (Italy) August 28 - September 2, 2011
the number of pole pairs, J is the moment of inertia
(normalized with nP ) and Φ is the magnetic flux, τL is
the load torque, which is assumed constant but unknown
and
ρα
cos(θ)
ραβ :=
.
(4)
=
ρβ
sin(θ)
The model (1–3) can be written in rotating (dq) coordinates by means of the transformation
h
i
cos(θ) − sin(θ)
Jθ
(5)
= ρα I2 ... ρβ J ,
e =
sin(θ) cos(θ)
with I2 the 2 × 2 identity matrix, to obtain
di
= −(RI2 + np ωLJ )i − np ωΦJ e1 + v
dt
J ω̇ = np Φi2 − τL
L
θ̇ = np ω,
where
while
1
np θ
(6)
is the position corresponding to the velocity ω
i
1
; i = e−J θ iαβ = 1
0
i2
v
v = e−J θ vαβ = 1 .
v2
e1 = e−J θ ραβ =
Our main result is the following proposition.
Proposition 1. Consider the PMSM model (1)–(3) with
some desired constant speed ω ⋆ 6= 0. Assume
A.1 The only variables available for measurement are iαβ .
A.2 The load torque τL is constant but unknown.
A.3 The parameters R, L, Φ and J are known.
There exists a fourth–order observer–based speed regulator of the form
(8)
The main contribution of the paper is the solution of the
following problem.
An asymptotically stable sensorless controller.
Consider the PMSM model (1)–(3) with some desired
constant speed ω ⋆ 6= 0. Assume that only the electrical
signals, i.e., iαβ , are available for measurement, that
the load torque τL is an unknown constant, and the
parameters R, L, Φ, J are known. Design an output–
feedback controller that ensures asymptotic regulation
of ω for a well–defined set of initial conditions.
3. CONTROLLER STRUCTURE AND MAIN RESULT
The proposed controller is a fourth–order certainty–
equivalent version of the full–information globally asymptotically stabilizing IDA–PBC controller [14]
vαβ = q(ραβ , ω, τL , iαβ )
and is obtained replacing ραβ , ω, τL by their estimates.
The dynamics of the controller are due to the different
observers that generates the estimates that we denote
ρ̂αβ , ω̂, τ̂L , respectively.
In order to carry–out the stability analysis of the proposed
scheme, the error coordinates are lumped into a seventh–
dimensional vector defined as 1
The constants L and J are introduced because—consistent with
the Hamiltonian formulation—the IDA–PBC is derived with the
motor dynamics represented using the energy variables, flux and
momenta.
(9)
which is a mixture of regulation errors, (·) − (·)∗ , and
c − (·). In addition, notice that the
estimation errors, (·)
errors in both, the currents and the vector ραβ , are defined
in the dq coordinates.
(7)
Remark 1. The main advantage of the dq–model is that it
transforms the periodic orbits associated to the constant
speed operation of the αβ model of the PMSM into
equilibrium points. However, it must be recalled that the
input is vαβ while the measurable output is given by
iαβ = eJ θ i.
1

L(i − i∗ )
∗
J(ω − ω )




χ =  e−J θ (ρ̂αβ − ραβ )  .


ω̂ − ω
τ̂L − τL

ρ̂αβ
η̇ = h(η, iαβ )
T
ω̂˙ τ̂˙L = g(η, iαβ )
vαβ = q(ρ̂αβ , ω̂, τ̂L , iαβ )
such that the closed–loop error dynamics is described by
a differential equation of the form
χ̇ = f (χ),
(10)
with zero a (locally) asymptotically stable equilibrium.
More precisely, there exists ǫ > 0 such that for all initial
conditions in the ball |χ(0)| ≤ ǫ, the trajectories satisfy
limt→∞ χ(t) = 0.
4. FULL INFORMATION CONTROL
In this section we present a simplified version of the full–
information IDA–PBC derived in [8], which will serve as a
basis for our certainty–equivalent design. For this purpose,
we write the system dynamics in port–Hamiltonian form
[14], considering
x12
Li
x=
(11)
=
x3
Jω
and the energy function H(x) = 12 xT Qx, with


1
 I2 0 
Q=L
(12)
1 .
0
J
Then, the dq system (6) can be written in the form
v
ẋ = F (x)∇H(x) +
(13)
−τL
∂ ⊤
) and the interconnection and damping
where ∇ = ( ∂x
matrices are lumped into
−RI2
−J np (x12 + Φe1 )
.
F (x) =
np (x12 + Φe1 )T J T
0
The assignable equilibrium set for (13) is given by {x∗ ∈
L
R3 | x∗2 = Φn
τL }, with x∗1 and x∗3 arbitrary. Consistent
p
with engineering practice, we will fix x∗1 = 0 in the sequel.
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Preprints of the 18th IFAC World Congress
Milano (Italy) August 28 - September 2, 2011
The objective of IDA–PBC is to design a control law
v = v(x) to obtain a closed–loop system of the form
ẋ = Fd (x)∇Hd (x),
(14)
where the desired energy function, say Hd (x) satisfies
x∗ = arg min Hd (x) while Fd (x) + FdT (x) ≤ 0. The
controller that achieves this objective is presented in the
next proposition.
Proposition 2. Consider the PMSM dq model (13) with a
desired equilibrium point
T
L
∗
∗
τL , x3
x = 0,
np Φ
The full–information control


L
−
τL x3


JΦ
(15)
vF I = 

R
∗
τL
Φω +
np Φ
renders x∗ globally asymptotically stable.
Proof. Define the desired closed–loop energy function as
the quadratic in the errors form
1 T
χ12
x12 − x∗12
, (16)
Hd (x) = χ13 Qχ13 ; χ13 =
=
x3 − x∗3
χ3
2
where Q is as in (12). Equating the right hand sides of (13)
and (14), i.e. obtaining the so–called matching equation,
it can be noticed that the equations that depend on the
control inputs v1 and v2 can be easily satisfied with a
suitable selection of these variables. In order to look for a
solution of the underactuated equation notice that, given
the definition of x∗2 , it can be written in an equivalent way
as
np Φ
1
1
1
χ2 = F31 χ1 + F32 χ2 + F33 χ3 ,
L
L
L
J
then it is immediate to recognize that a solution is given
by
F31 = F33 = 0; F32 = np Φ.
where Fij are the blocks of Fd (x), which has been partitioned in a conformal way. The skew–symmetry condition
on Fd (x) suggests to define F23 = −np Φ. Replacing in the
second row of the matching equation yields
np
np
R
x2 −
x1 x3 −
Φx3 + v2 =
L
J
J
1
1
np Φ
F21 χ1 + F22 χ2 −
χ3 .
L
L
J
∗
Now x1 = 0, hence χ1 = x1 , which suggests the solution
Ln
F21 = − J p x3 , F22 = −R and
R
np Φ ∗
v2 = x∗2 +
x .
L
J 3
With the definitions given up to this point, the first row of
the matching equation can be satisfied taking F11 = −R,
Ln
n
F12 = J p x3 , F13 = 0 and v1 = − Jp x∗2 x3 . Finally, the
closed–loop system takes the desired port–Hamiltonian
form (14) with energy function (16) and


Lnp
−R
x3 0


J

Ln
(17)
Fd (x) = 
 − p x3 −R −np Φ  ,
J
0
np Φ
0
hence the equilibrium x∗ is globally stable. Asymptotic
stability follows verifying that, there exists α > 0 such
that Ḣd ≤ −α|χ12 |2 and that |χ12 |2 is a detectable output
for the closed–loop system (14).
∇∇∇
Considering the result above, the sensorless certainty–
FI
equivalent version of the controller vαβ
takes the form
i
.
ρ̂α I2 .. ρ̂β J v̂


L
− τ̂L ω̂


Φ
v̂ := 
(18)

R
np Φω ∗ +
τ̂L
np Φ
where the first equation appears due to the necessity to
transform the dq controller into the αβ frame. In the
next two sections the observers required to generate the
different estimates are introduced.
vαβ =
h
5. POSITION OBSERVER OF [4]
In this section the observer presented in [4], which estimates the position via the observation of the flux, is briefly
revisited.
In PMSMs the stator flux, λ, is given by [12]
λ = Liαβ + Φραβ ,
leading, considering (1), to the expression
(19)
λ̇ = −Riαβ + vαβ ,
(20)
whose right hand side has the important property to be
measurable. On the other hand, the vector function
η(λ) := λ − Liαβ ,
(21)
satisfies |η(λ)| = Φ.
In [4] it is shown that
˙
λ̂ = −Riαβ + vαβ + γη(λ̂)[Φ2 − |η(λ̂)|2 ],
(22)
where γ > 0 is an observer gain, is a gradient descent observer for the flux that guarantees the following remarkable
stability properties
P1 (Global stability) For arbitrary speeds, the disk {λ̃ ∈
R2 | |λ̃| ≤ 2Φ}, with λ̃ := λ̂ − λ, is globally attractive.
P2 (Local exponential stability) The zero equilibrium λ̃ =
0 is exponentially stable if there exists constants
T, ∆ > 0 such that
Z
1 t+T 2
ω (s)ds ≥ ∆,
T t
for all t ≥ 0.
P3 (Stability in regulation) If the speed is constant and
satisfies |ω| > 14 γΦ2 , then the origin λ̃ = 0 is the
unique equilibrium of the observer error dynamics
and it is globally asymptotically stable.
−
Instrumental for the development of the proposed controller is the following (not implementable) representation
in terms of ραβ .
Proposition 3. From (19) and the observer (22) define the
estimate
1 λ̂ − Liαβ
ρ̂αβ =
Φ
and the error ρ̃αβ := ρ̂αβ − ραβ . The observer (22) may be
written as
(23)
ρ̂˙ αβ = −γΦ2 |ρ̂αβ |2 − 1 ρ̂αβ + np ωJ ραβ ,
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Preprints of the 18th IFAC World Congress
Milano (Italy) August 28 - September 2, 2011
while the estimation error ρ̃αβ satisfies
ρ̃˙ αβ = −γΦ2 |ρ̃αβ |2 + 2ρ̃Tαβ ραβ (ρ̃αβ + ραβ ) .
(24)
Proof. First, notice that observer error dynamics can be
written as
˙
λ̃ = −γ[|λ̃|2 + 2Φλ̃⊤ ραβ (t)][λ̃ + Φραβ (t)],
(25)
which, considering that λ̃ = Φρ̃αβ , yields
˙
λ̃ = −γΦ3 |ρ̃αβ |2 + 2ρ̃Tαβ ραβ (ρ̃αβ + ραβ )
leading directly to (24). Now, notice that
|ρ̃αβ |2 + 2ρ̃Tαβ ραβ = |ρ̂αβ |2 − 1,
while ρ̃˙ αβ = ρ̂˙ αβ − np ωJ ραβ which replaced in (24) yields
(23).
∇∇∇
Proof (sketch). Following the I&I procedure [7], consider
the coordinate
ω
χ67 := ξ −
+ ζ(ρ̂αβ ),
(30)
τL
whose dynamics is given by
χ̇67 = ξ˙ − ∇ζ[γΦ2 |ρ̂αβ |2 − 1 ρ̂αβ + np ωJ ρ̂αβ ] +
#
"
np Φ T
τL
−
i
J
ρ̂
αβ
+ J
J αβ
0
Then, if
ξ˙ = A33 (ξ + ζ) + γΦ2 |ρ̂αβ |2 − 1 ∇ζ ρ̂αβ +
"
#
np Φ T
i
J
ρ̂
αβ
αβ
+
J
0
6. AN I&I SPEED AND LOAD TORQUE OBSERVER
In this section an I&I observer [7] for the unmeasurable
variables ω and τL is designed. The construction proceeds
along the following steps.
S1 The parametrization of the mechanical dynamics—in
terms of ραβ —given in (2), as well as the representation of the flux observer (22) given in (23), are used.
S2 The term ραβ in both equations is decomposed as
the sum of its estimate ρ̂αβ and the error ρ̃αβ , and we
treat the latter as a perturbation.
S3 A globally exponentially convergent I&I observer of
ω and τL is designed neglecting the perturbation in
the system. 2
The mechanical equation (2) and the position observer
(23) can be written in the “perturbed” form
J ω̇ = np ΦiTαβ J ρ̂αβ − τL − (np ΦiTαβ J ρ̃αβ )
ρ̂˙ αβ = −γΦ2 |ρ̂αβ |2 − 1 ρ̂αβ + np ωJ ρ̂αβ − (np ωJ ρ̃αβ )
τ̇L = 0,
(26)
where, for completeness, the last (trivial) equation has
been added.
Proposition 4. Consider the unperturbed system given by
(26) with ρ̃αβ = 0 and the speed and load torque observer
"a
#
"
#
np Φ T
2
2
−
n
a
i
J
ρ̂
p
1
αβ
ξ˙ = A33 ξ + J
A (ρ̂αβ ) +
J αβ
np a1 a2
0
a1
ω̂
A(ρ̂αβ )
(27)
=ξ +
−a2
τ̂L
where A(·) is an operator defined in Appendix A and A33
is the Hurwitz matrix
"
#
1
−n
a
−
p
1
A33 :=
(28)
J , a1 , a2 > 0.
np a2 0
For some α > 0 and for all initial conditions (ω(0), ξ(0)) ∈
R × R2 ,
αt ω̂(t) − ω(t) lim e = 0.
(29)
τ̂
(t)
−
τ
t→∞
L
L That is, (27) is a globally exponentially convergent speed
and load torque observer for the unperturbed system
obtained from (26) with ρ̃αβ = 0.
2 The perturbation term that is neglected in this section is lumped
into the overall error dynamics, whose stability is analyzed in Section
7.
and
(31)
ρ̂β
a1
arctan
.
(32)
−a2
ρ̂α
it is obtained that χ̇67 = A33 χ67 yielding to the fact
that χ67 (t) → 0, generating therefore an estimate of
ω
given by ξ + ζ(ρ̂αβ ). The proof is completed by
τL
replacing the function arctan by the operator A in (32),
and noting that the derivations above remain valid after
this substitution.
∇∇∇
Remark 2. If instead of the operator A(·) the arctan(·)
function is used in the implementation of the observer,
some Dirac delta functions might appear in the speed
estimation and the error dynamics due to the instantaneous jumps exhibited by this function. The operator
A(·), which is widely used in the drives community, is
“essentially” equal to arctan(·), and is introduced to avoid
these undesirable behavior.
ζ(ρ̂αβ ) =
7. PROOF OF THE MAIN RESULT
In order to state the stability properties of the closed–
loop system, their dynamics are described using the error
coordinates (9). For ease of reference, these equations are
sequentially derived for χ13 , χ45 and χ67 to express, later
on, the error dynamics in the form
χ̇ = Aχ + Γ(χ),
(33)
where A is the system matrix of the linearized system and
the elements of the vector Γ(χ) contain (second or higher
order) products of the components of χ. The proof of
the claim of asymptotic stability of Proposition 1, follows
(invoking Lyapunov’s indirect method) showing that A is
a Hurwitz matrix.
For the currents and speed tracking errors, χ13 , it is
possible to write their dynamics as
χ̇13 = A11 χ13 + A12 χ45 + A13 χ67 + Γ13 (χ)
where
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(34)
Preprints of the 18th IFAC World Congress
Milano (Italy) August 28 - September 2, 2011
A11 = Fd (x∗ )Q


np Φ ∗
R
L
τL x∗3
−
x3 −
τL
−

JΦ
J
np Φ 


R
L
A12 =  np Φ ∗

∗
x
+
τ
−
τ
x


L
L 3
J 3 np Φ
JΦ
0
0


L
L ∗
 − Φ τL − JΦ x3 


R
A13 =  0
(35)


np Φ 
0
0
with Fd (x) and Q defined in (17) and (12), respectively,
and Γ13 (χ) is such that ∇Γ13 (0) = 0. Moreover, the matrix
A11 given by


R np ∗
x3
0
−


J
 npL
R
np Φ 
.
Fd (x∗ )Q = 
−
x
−
−
 J 3
L
J 


np Φ
0
0
L
is Hurwitz. This can be proved by showing that its characteristic polynomial is of the form s3 + c1 s2 + c2 s + c3 ,
with the coefficients ci > 0 and verifying c1 c2 > c3 .
Concerning the estimation error for ραβ , χ45 of the error
vector χ, it can be obtained that their dynamics can be
written as
χ̇45 = A22 χ45 + Γ45 (χ)
(36)
with


np ∗
−2γΦ2
x3

A22 =  np ∗ J
(37)
− x3 0
J
and Γ45 (χ) is such that ∇Γ45 (0) = 0. It is important to
notice that the matrix A22 is Hurwitz for all x∗3 6= 0.
Finally, for the speed and load torque estimation errors,
their dynamics take the form
χ̇67 = A23 χ45 + A33 χ67 + Γ67 (χ),
(38)
where


Φx∗2
np
∗
a
x
−
0
−
1 3

L
A23 =  J n
,
p
a2 x∗3
0
J
A33 the Hurwitz matrix defined in (28), and Γ67 (χ) is such
that ∇Γ67 (0) = 0.
Combining the results presented above, we obtain that the
error vector χ satisfies a differential equation of the form
(33) with
#
#
"
"
Γ13 (χ)
A11 A12 A13
0 A22 0 , Γ(χ) = Γ45 (χ) .
A=
Γ67 (χ)
0 A32 A33
Recalling that ∇Γ(0) = 0, it only remains to prove that
A is Hurwitz. For, we notice that A is similar to a block
triangular Hurwitz matrix. More precisely, with
"
#
I3 0 0
T = 0 0 I2 ,
0 I2 0
we get
#
"
A11 A12 A13
0 A33 A32 ,
T AT −1 =
0 0 A22
which is Hurwitz due to the fact that A11 , A22 and A33
are Hurwitz matrices.
8. EXPERIMENTAL RESULTS
In this section the current results obtained towards the
complete experimental validation of the proposed controller are presented. They include the response of the
closed–loop system composed by the motor and the state–
feedback version of the controller, and the response of
both the position and the speed/load torque observers. To
carry out this evaluation an electromechanical assembly
consisting in a PMSM, its associated power converter and
a power brake (used as a load) available in the Laboratoire
de Genie Electrique de Paris was used.
The nonlinear observers as well as the control algorithm were implemented in a DSPACE DS1103 card. The
pulsewidth modulation switching frequency was set at
20KHz and the dead time 4.3µs. The sampling time for
electrical parameters was fixed at 50µs while the sampling
time for speed control was set at 1ms. The measurement
of phase currents was done using two Hall effect sensors
equipped with a 2nd order Raunch filter for conditioning
purposes. The real position θ was measured by a 3600
pulses per revolution encoder.
The considered motor parameters were L = 0.0075H,
R = 0.83Ω, Φ = 0.17W b, np = 3 and J = 0.012kg.m2.
Moreover, in order to evaluate the scheme under stringent
conditions, the motor was at standstill at the beginning of
the experiments while the initial values for the estimated
speed and load torque were set to zero. On the other hand,
ρ̂α (0) = Φ and ρ̂β (0) = 0. Regarding the load torque, it
was maintained constant during the experiment but its
value was not known.
In Figure 1(a) the reference speed ω ∗ , the actual speed ω
and the estimated speed ω̂ are shown when the reference
speed changes from -500 rpm to +500 rpm. This reference
was chosen in order to evaluate the observer behavior
when the speed crosses the zero value. As can be notice,
the performance achieved by the state–feedback controller
is remarkable. In addition, at this point is interesting
to mention that this response was achieved with stator
currents values less than 3A and stator voltage values less
than 50V .
Concerning the different observers response, in Figure
1(b) and Figure 2 the estimated load torque τ̂L and
the estimated functions ρ̂αβ (using a time–window that
facilitates their evaluation when the speed crosses the zero
value) are shown, respectively, while as already mentioned,
the estimated speed is presented in Figure 1(a). In these
figures it can be noticed how the position and speed
observers achieve the convergence desired behavior while
the estimated load remains bounded.
9. FUTURE RESEARCH
The effect of the operator A(·) must be further elaborated.
In addition characterization of an estimate of the region
of attraction of the equilibrium point is necessary. Concerning robustness, it could be enhanced exploiting some
degrees of freedom of the design that were skipped for readability. Namely, we used the simplest version of IDA–PBC,
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Preprints of the 18th IFAC World Congress
Milano (Italy) August 28 - September 2, 2011
ω, ω̂ and ω ∗
500
rpm
[2]
[3]
0
[4]
−500
0
1
2
3
4
5
6
7
8
9
t
τˆL
10
[5]
Nm
5
0
[6]
−5
−10
0
1
2
3
4
5
6
7
8
9
t
[7]
Fig. 1. Speed and Load torque measured and observed
[8]
ˆ
cos(θ), cos(θ)
1.5
cos(θ)
1
ˆ
cos(θ)
0.5
0
[9]
−0.5
−1
−1.5
0.6
0.8
1
1.2
1.4
1.6
1.8
t
[10]
ˆ
sin(θ), sin(θ)
1.5
sin(θ)
1
ˆ
sin(θ)
[11]
0.5
0
−0.5
−1
−1.5
0.6
0.8
1
1.2
1.4
1.6
[12]
1.8
t
[13]
[14]
Fig. 2. Position at zero crossing and low speed
e.g., without damping injection—see [8] for further details.
Also, (32) is the simplest solution for the observer design,
to which we can add some arbitrary functions without
altering the result. The impact of these modifications on
the systems performance is currently under investigation.
Regarding the experimental evaluation, the operation of
the controller together with the different observers is under
evaluation and its finalization is imminent.
REFERENCES
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Appendix A. OPERATOR A
The output value e of the arctan(u) function is wrapped
in the set (−π, π] leading to discontinuities. The purpose
of the operator A is is to avoid them, task that is done by
including at its output an additional block which generates
a modified output y = e + 2nπ where n is a counter,
initialized at zero, that is increased by 1 each time the
e > π or decreased by 1 if e < −π, i.e. unwrapping the
original output.
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