ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
Contents lists available at ScienceDirect
ISA Transactions
journal homepage: www.elsevier.com/locate/isatrans
Research article
Sensorless H1 speed-tracking synthesis for surface-mount permanent
magnet synchronous motor
Ramón Ramírez-Villalobos a, Luis T. Aguilar b,n, Luis N. Coria a
a
b
Tecnológico Nacional de México – Instituto Tecnológico de Tijuana, Calz. del Tecnológico S/N, Tomas Aquino, 22414 Tijuana, BC, México
Instituto Politécnico Nacional – CITEDI, Ave. Instituto Politécnico Nacional 1310, Nueva Tijuana, Tijuana, BC 22435, Mexico
art ic l e i nf o
a b s t r a c t
Article history:
Received 28 October 2015
Received in revised form
8 September 2016
Accepted 5 January 2017
In this paper, a sensorless speed tracking control is proposed for a surface-mount permanent magnet
synchronous motor by using a nonlinear  ∞-controller via stator currents measurements for feedback.
An output feedback nonlinear  ∞ -controller was designed such that the undisturbed system is uniformly asymptotically stable around the desired speed reference, while also the effects of external
vanishing and non-vanishing disturbances, noise, and input backlash were attenuated locally. The rotor
position was calculated from the causal dynamic output feedback compensator and from the desired
speed reference. The existence of the proper solutions of the perturbed differential Riccati equations
ensures stabilizability and detectability of the control system. The efficiency of the proposed sensorless
controller was supported by numerical simulations.
& 2017 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords:
Nonlinear H1 control
Permanent magnet synchronous motor
Sensorless speed-tracking synthesis
1. Introduction
Due to the fast dynamic performance, high efficiency, large
torque-to-current ratio and advancement in magnetic materials
the permanent magnet synchronous motors (PMSMs) have an
important role in variable speed applications [1–5]. For a robust
and high precision control is necessary to provide an accurate
measurement of the mechanical variables (rotor position and
speed). Typically, tachometers or optical encoders are used to
measure these mechanical variables. However, these sensors increase cost and size of the drive systems, and the reliability is
reduced [6–8]. The so-called sensorless control problem for
PMSMs, in which only stator current and voltage measurements
are available for feedback, has been a challenging problem for
PMSMs in the last decades.
Generally, the sensorless methods can be classified in two
main categories according to the speed range: estimation
through high-frequency signal injection and by employing the
dynamical model. The first method detects the rotor position
through a high-frequency carrier signal superimposed on the
pulse-width modulated waveform of the power inverter. These
techniques are suitable for very low speed and zero speed [9].
However, requires an unit signal to generate an external
n
Corresponding author.
E-mail addresses: ramon.ramirez@tectijuana.edu.mx (R. Ramírez-Villalobos),
laguilarb@ipn.mx (L.T. Aguilar), luis.coria@gmail.com (L.N. Coria).
excitation, and may induce high-frequency noise [8,10]. In
contrast, the model-based methods are a proper solution for
high speed operations. Based on the dynamic model, the rotor
position is estimated from the back electromotive force (EMF),
by employing the arctangent method. This method is mainly
useful for feedback control. However, the performance of these
techniques are sensitive to the variation of motor parameters
and external load disturbance. In addition, the arctangent
method fails at very low speed range, mainly during the EMF
zero crossing [5,11,12]. For the aforementioned reasons, a wide
speed robust control scheme against parameters variation and
external disturbances is necessary to improve the performance
of PMSMs.
Existing literature about model-based estimation for PMSM
drives can be grouped into several techniques, such as backstepping control [13], full-order and reduced-order observer
[14], sliding mode sliding mode observer [1,5,12], extended
Kalman filter [6], intelligent control [4,8], predictive control
[15,16], and adaptive control [7,17]. Nonetheless, some of them
are sensitive to parameter variations, disturbances or nonlinear dynamics, and offer a low performance at standstill and
low speed ranges. On the other hand, a robust position sensorless method based on sliding mode observer has been presented, by employing the arctangent method, with a well
performance at high speed ranges. In order to overcome the
limitations at low speed ranges; in Refs. [1,18] a combination of
high-frequency techniques and model-based techniques at low
http://dx.doi.org/10.1016/j.isatra.2017.01.007
0019-0578/& 2017 ISA. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: Ramírez-Villalobos R, et al. Sensorless H1 speed-tracking synthesis for surface-mount permanent magnet
synchronous motor. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.01.007i
R. Ramírez-Villalobos et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
2
and high speed ranges, respectively, have been proposed. Additionally, a sensorless speed control strategy based on a sliding mode observer has been proposed in Ref. [5], where the
rotor position is obtained from a Lyapunov stability analysis,
instead of using the arctangent method. In contrast to the
aforementioned techniques,  ∞ control method is suitable to
deal with problems involving multivariable systems considering control specifications as disturbances attenuation, asymptotic tracking, and robust stability into a single control problem
[19,20].
Consequently, the motivation of this paper lies on propose
an output feedback nonlinear  ∞-controller to solve the sensorless speed-tracking control problem for PMSMs in spite of
the above mentioned external disturbances and uncertainties.
The controller is designed such that the undisturbed system is
uniformly asymptotically stable around the speed desired reference, while the influence of external disturbances and/or
parameter variations are minimized. In order to avoid the restriction at zero speed and low speed range the arctangent
method is avoided, the information of rotor position has been
obtained from the causal dynamic output feedback compensator and from the desired speed reference. Under appropriate
assumptions the existence of solutions of perturbed Riccati
differential equations, appearing in solving the  ∞ control
problem for the linearized system, a local solution of the  ∞
control problem was guaranteed. Thus, local stabilizability and
detectability properties were ensured.
The paper is organized as follows. The nonlinear mathematical model of PMSM is introduced in Section 2. A background about time-varying  ∞ control synthesis is presented
in Section 3. The speed-tracking control problem of a PMSM
and its state equations are introduced in Section 4, while desired trajectory synthesis procedure is also discussed. Furthermore, a nonlinear  ∞ output control for time-varying
systems is synthesized. The sensorless  ∞ control scheme is
presented in Section 5. Numerical simulations illustrate the
performance of the proposed controller in Section 6. Finally,
conclusions are presented in Section 7.
2. PMSM mathematical model
The two-axis stator voltage state equations of the PMSM in a
rotating (d,q) coordinates is expressed as follows
⎡ vd ⎤ ⎡ R + sL d − pLq ωe ⎤ ⎡ id ⎤ ⎡ 0 ⎤
⎥⎢ ⎥ + ⎢
⎥,
⎢⎣ v ⎥⎦ = ⎢
q
⎢⎣ pLd ωe R + sL q ⎥⎦ ⎣ iq ⎦ ⎣ ϕωe ⎦
(1)
where vd(t) and vq(t) are the d-q stator voltages (which constitute
the control inputs), id(t) and iq(t) are the scalars d-q stator currents,
ωe (t ) is the rotating speed of the magnet flux, s¼ d/dt is the differential operator. The constant parameter R > 0 is the stator
winding resistance, L d > 0 and L q > 0 are the stator winding inductances on d-q axis, ϕ > 0 is the permanent-magnet flux linkage, and p is the number of pole pairs.
The electromagnetic torque of the PMSM is given by
Te =
3p
(L d − L q ) id iq + KT iq,
2
dω r
+ Fω r = Te − TL,
dt
where J > 0 is the rotor inertia, F ≥ 0 is the viscous friction
coefficient, TL(t) is the load torque, and ωr (t ) is the rotor speed.
Under assumptions of symmetry between phases, linear magnetic circuits and negligible magnetic hysteresis in the PMSM,
the system represented by (1) and (3) can be expressed by the
following state-space representation
did
R
v
= piq ω r − id + d ,
dt
L
L
diq
vq
ϕ
R
= − pid ω r − iq − ω r + ,
dt
L
L
L
dω r
KT
F
TL
= iq − ω r − .
dt
J
J
J
(4)
It is assumed that the positive constant motor parameters R, L,
ϕ, KT, J, and the non-negative constant parameter F are known.
The load torque T L (t) is an unknown but uniformly bounded
function.
In most cases, the PMSM drive is based on the field oriented
control (FOC) scheme (see Fig. 1). In such scheme, the stator current id is set to zero in order to decouple the system (4) and control
the PMSM as DC motor. Thereby, the rotor position and speed can
be controlled by forcing the stator current iq to track a current
reference i*, which can be considered as virtual control input
[6,10,17].
The objective addressed in this paper is to design an output
feedback nonlinear  ∞-controller for system (4) in order to
asymptotically track the rotor speed ωr (t ) to a desired reference ω* (t ) and asymptotically stabilize the stator current id
to zero, that is,
lim ∥ ω r (t ) − ω* (t )∥ = 0,
t →∞
lim ∥ id (t )∥ = 0,
t →∞
(5)
starting the PMSM from any initial condition and despite of
external disturbance TL (t ) ∈  . The stator currents of the PMSM
are the only available measurements for feedback, which are
perturbed by the vector (wd, wq )T ∈ 2.
3. Preliminaries
(2)
where KT > 0 is the motor torque constant. The mechanical
equation of the motor is
J
Fig. 1. Fundamental field-oriented control scheme.
Consider a non-autonomous nonlinear system of the form
ẋ = f (x, t ) + g1 (x, t ) w + g2 (x, t ) u,
z = h1 (x, t ) + k12 (x, t ) u,
(3)
y = h2 (x, t ) + K21 (x, t ) w ,
(6)
Please cite this article as: Ramírez-Villalobos R, et al. Sensorless H1 speed-tracking synthesis for surface-mount permanent magnet
synchronous motor. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.01.007i
R. Ramírez-Villalobos et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
where x (t ) ∈ n is the state space vector, t ≥ 0 is the time
variable, u (t ) ∈ m is the control input, w (t ) ∈ r is the unknown disturbance, z (t ) ∈ l is the unknown output to be
controlled, and y (t ) ∈ p is the only available measurement on
the system.
For technical reasons, the following standard assumptions are
assumed for the generic system (6)
(A1) The functions f (x, t ), g1 (x, t ), g2 (x, t ), h1 (x, t ), h2 (x, t ), k12 (x, t ),
k21 (x, t ) are piecewise continuous in t for all x and locally
Lipschitz continuous in x for all t.
(A2) f (0, t ) = 0, h1 = (0, t ) = 0, and h2 = (0, t ) for almost t.
T
(A3) h1T (x, t ) k12 (x, t ) = 0 , k12
(x, t ) k12 (x, t ) = I , k21 (x, t ) g1T (x, t ) = 0,
T
and k21 (x, t ) k21
(x , t ) = I .
3
(1) The equation
−P ̇ (t ) = P (t ) A (t ) + AT (t ) P (t ) + C1T (t ) C1 (t )
⎡ 1
⎤
+ P (t ) ⎢ 2 B1B1T − B2 B2T ⎥ (t ) P (t ),
⎣γ
⎦
(11)
possesses a uniformly bounded positive semidefinite symmetric solution P(t) such that the system
ẋ = ⎡⎣ A − B2 B2T − γ −2B1B1T ⎤⎦ (t ) x (t )
(
)
(12)
is exponentially stable.
1
(2) Being specified with A1 (t ) = A (t ) + 2 B1 (t ) B1T (t ) P (t ), the equation
γ
Z ̇ (t ) = A1 (t ) Z (t ) + Z (t ) A1T (t ) + B1 (t ) B1T (t )
According to [21], p. 82, Assumption (A1) guarantees the wellposedness of the above dynamic system, while being enforced by
integrable exogenous inputs. Assumption (A2) ensures that the
origin is an equilibrium point of the non-driven (u = 0) disturbance free (w = 0) dynamic system (6). Assumption (A3) is a
simplifying assumption inherited from the standard  ∞ control
problem.
The  ∞-control problem is stated as follows. Given a system of
the form (6) and a real number γ > 0, it required to find (if any) a
causal dynamic output feedback compensator
u =  (ξ , t ),
ξ ̇ =  (y , ξ ),
(7)
with internal state ξ ∈ s , such that the undisturbed closed-loop
system is uniformly asymptotically stable around the origin and its
 2 gain less than γ if the response z, resulting from w for initial
state x (t0 ) = 0 and ξ (t0 ) = 0 satisfy
∫t
ti
z (t ) 2 dt < γ 2
0
∫t
t1
w (t ) 2 dt,
0
(13)
possesses a uniformly bounded positive semidefinite symmetric
solution Z(t), such that the system
ẋ = ⎡⎣ A1 − Z C2T C2 − γ −2B2 B2T P ⎤⎦ (t ) x (t ),
(
)
(14)
is exponentially stable.
According to the time-varying strict bounded real lemma [22,23],
Conditions (C1) and (C2) ensure that there exist a positive constant ε0
such that the system of the perturbed Riccati equations
−Pε̇ (t ) = Pε (t ) A (t ) + AT (t ) Pε (t ) + C1T (t ) C1 (t )
⎡ 1
⎤
+ Pε (t ) ⎢ 2 B1B1T − B2 B2T ⎥ (t ) Pε (t ) + εI ,
⎣γ
⎦
(15)
Zε̇ (t ) = Aε (t ) Zε (t ) + Zε (t ) AεT (t ) + B1 (t ) B1T (t )
⎡ 1
⎤
+ Zε (t ) ⎢ 2 Pε B2 B2T Pε − C2T C2 ⎥ (t ) Zε (t ) + εI ,
⎣γ
⎦
(16)
(8)
for all t1 > t0 and all piecewise continuous functions w(t). A
locally admissible controller (7) constitutes a local solution  ∞
control problem if there exist a neighborhood U of the equilibrium such that inequality (8) is satisfied for all t1 > t0 and all
piecewise continuous functions w(t) for which the state trajectory of the corresponding closed-loop system, starting from
the initial point x (t0 ) = 0 and ξ (t0 ) = 0, remains in U for all
t ∈ [t0, t1].
Under Assumptions (A1)–(A3), coupled together, the corresponding Hamilton-Jacobi-Isaacs inequalities are subsequently
linearized and a local solution of the  ∞ control problem is obtained. The subsequent local analysis involves the linear  ∞
control problem for the system
ẋ = A (t ) x + B1 (t ) w + B2 (t ) u,
z = C1 (t ) x + D12 (t ) u,
y = C2 (t ) x + D21 (t ) w ,
⎡ 1
⎤
+ Z (t ) ⎢ 2 PB2 B2T P − C2T C2 ⎥ (t ) Z (t ),
⎣γ
⎦
has a unique uniformly bounded, positive definite symmetric
solution ( Pε (t ) , Zε (t ) ) for each ε ∈ (0, ε0 ) where Aε (t ) = A (t ) +
γ −2B1 (t ) B1T (t ) Pε (t ) . The above equations are now utilized to derive a
local solution of the  ∞ control problem for system (6). The following Theorem is used to design an  ∞ speed-tracking controller
for the PMSM.
Theorem 1. Consider a system of the form (6) with Assumptions
(A1)–(A3). Let conditions (C1) and (C2) be satisfied with a certain
γ > 0 and let ( Pε (t ) , Zε (t ) ) be a uniformly bounded positive symmetric
solution of (15), (16) under some ε > 0. Then, the causal dynamic
output feedback compensator
(9)
where
∂f
(0, t ), B1 (t ) = g1 (0, t ), B2 (t ) = g2 (0, t ),
∂t
∂h
∂h
C1 (t ) = 1 (0, t ), C2 (t ) = 2 (0, t ),
∂t
∂t
A (t ) =
D12 (t ) = k12 (0, t )
D21 (t ) = k21 (0, t ).
(10)
Such a problem is now well understood if the linear system (9) is
stabilizable and detectable from u and y, respectively.
The following conditions are necessary and sufficient for a solution of the problem to exist
Fig. 2. Sensorless control scheme.
Please cite this article as: Ramírez-Villalobos R, et al. Sensorless H1 speed-tracking synthesis for surface-mount permanent magnet
synchronous motor. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.01.007i
R. Ramírez-Villalobos et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
4
⎡ 1
⎤
ξ ̇ = f (ξ , t ) + ⎢ 2 g1 (ξ , t ) g1T (ξ , t ) − g2 (ξ , t ) g2T (ξ , t ) ⎥ Pε (t ) ξ (t )
⎣γ
⎦
+ Zε (t ) C2T (t )(y (t ) − h2 (ξ , t )),
u = − g2T (ξ , t ) Pε (t ) ξ (t ),
(17)
is a local solution of the  ∞ control problem with the disturbance
attenuation level γ.
Proof. Proof of Theorem 1 can be found in [21], Thm. 24.
Table 1
Motor parameters.
Symbol
Parameter
Value
Unit
R
L
ϕ
p
F
Stator resistance
Stator inductance
Permanent-magnet flux linkage
Pole-pairs
Viscous friction coefficient
4.3
359
24.5
1
Ω
mH
mWb
0.157 × 10−3
Nms
J
Rotor inertia
1.1 × 10−6
kgm2
(18)
□
4. ∞ speed-tracking synthesis
4.1. Problem statement
It is assumed that a desired speed ω* (t ) for the PMSM is twice
continuously differentiable and the functions ω* (t ), ω̇* (t ), ω̈* (t ) are
uniformly bounded in t. If there were no external disturbance in
the PMSM the rotor speed can be forced to track ω* (t ), defining the
Fig. 3. Simulations results of control drive without  ∞ controller: (a) Time evolution of stator current id, (b) stator current vector q-component tracking error and (c) speedtracking error.
Please cite this article as: Ramírez-Villalobos R, et al. Sensorless H1 speed-tracking synthesis for surface-mount permanent magnet
synchronous motor. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.01.007i
R. Ramírez-Villalobos et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
⎛⎛
⎞
R
d
ϕ⎞
vq = L ⎜ ⎜ pid + ⎟ ω* (t ) + i* (t ) − k q ( i q − i* (t ) ) +
i* (t ) + u q ⎟ ,
⎝⎝
⎠
L⎠
L
dt
control inputs as follows
5
(23)
vd = − L (k d id + piq ω* (t )),
(19)
⎛⎛
⎞
R
d
ϕ⎞
vq = L ⎜ ⎜ pid + ⎟ ω* (t ) + i* (t ) − k q ( iq − i* (t ) ) +
i* (t ) ⎟,
⎝⎝
⎠
L⎠
L
dt
(20)
that imposes on the disturbance-free speed desired stability
properties around ωr (t ) while also locally attenuating the effect of
the disturbance. Thus, the controller to be constructed consists of
the speed feed-forward compensator (19)–(20) and attenuators
(ud (t ) , uq (t )), internally stabilizing the closed-loop system around
the desired speed.
This approach it is confined to the speed-tracking control
problem where
(21)
(i) The output to be controlled is given by
with the virtual control input i* (t ) defined as
i* (t ) =
⎞
J ⎛F
⎜ ω* (t ) + ω̇* (t ) ⎟.
KT ⎝ J
⎠
The objective is to design a controller of the form
(
)
vd = L −k d id − piq ω* (t ) + ud ,
(22)
⎡
id ⎤
⎥
⎢
⎢ iq − i* ⎥ ⎡ 03 × 2 ⎤ ⎡ ud ⎤
+⎢
z = ρ⎢
⎥ ⎢ ⎥,
ω − ω*⎥ ⎣ I2 ⎦ ⎣ uq ⎦
⎥
⎢ r
⎣ 02 × 1 ⎦
(24)
Fig. 4. Simulations results adding  ∞ control drive: (a) Time evolution of stator current id, (b) stator current vector q-component tracking error and (c) speed-tracking error.
Please cite this article as: Ramírez-Villalobos R, et al. Sensorless H1 speed-tracking synthesis for surface-mount permanent magnet
synchronous motor. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.01.007i
R. Ramírez-Villalobos et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
6
Fig. 5. Simulations results adding  ∞ control drive with a non-vanishing perturbation function: (a) Time evolution of stator current id, (b) stator current vector q-component
tracking error and (c) speed-tracking error.
with a positive weight coefficient ρ. Here, In is the n × n
identity matrix and 0n × m is the n × m matrix of zeros.
(ii) The stator currents measurements
⎡ id ⎤ ⎡ wd ⎤
⎥+⎢ ⎥
y=⎢
⎣ iq − i*⎦ ⎣ wq ⎦
(25)
are the only available measurements, and these measurements
are corrupted by the vector [wd, wq ]T .
The  ∞ speed-tracking control problem for PMSM can formally be stated as follows. Given the system representation (4),
(5), (19)–(25), a desired speed ω* (t ) to track, and a real number
γ > 0, it is required to find (if any) a causal dynamic feedback
controller (7) such that the undisturbed closed-loop system is
uniformly asymptotically stable around ω* (t ), and its  2-gain is
locally less than γ, that is, inequality (8) is satisfied for all t1 > t0
and all piecewise continuous functions w (t ) = (wd, wq, TL )T for
which the corresponding state trajectory of the closed-loop
system, initialized at the origin, remains in some neighborhood
of this point.
4.2.  ∞ synthesis
For the synthesis of the  ∞ speed-tracking controller, consider
the state-space vector x = (x1, x2, x3 )T = (id , iq − i*, ωr − ω*)T , the
unknown disturbance
w = (wd, wq, TL )T , and the control input
u = (ud , uq )T . Then system (4), (24) and (25), represented in terms
of the state-space vector x, can be specified as time-varying nonlinear system (6) with
Please cite this article as: Ramírez-Villalobos R, et al. Sensorless H1 speed-tracking synthesis for surface-mount permanent magnet
synchronous motor. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.01.007i
R. Ramírez-Villalobos et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
⎤
⎡ ⎛R
⎞
⎢ − ⎜ + k d ⎟ x1 + px2 x3 + pi* (t ) x3⎥
⎝
⎠
L
⎥
⎢
⎥
⎢
⎛R
⎞
ϕ
f (x, t ) = ⎢ − ⎜ + k q ⎟ x2 − x3 − px1x3 ⎥,
⎝
⎠
L
L
⎥
⎢
⎥
⎢
KT
F
x2 − x3
⎥
⎢
J
J
⎦
⎣
⎡0 0 0 ⎤
⎢
⎥
g1 (x, t ) = ⎢ 0 0 0 ⎥,
⎢⎣ 0 0 − J−1⎥⎦
system (9) where matrices are described by
⎤
⎡ ⎛R
⎞
0
pi* (t )⎥
⎢ − ⎜ + kd ⎟
⎝
⎠
L
⎥
⎢
⎢
⎛R
⎞
ϕ ⎥
⎜
⎟
⎥,
A (t ) = ⎢
−
+ kq
−
0
⎝L
⎠
L ⎥
⎢
⎢
KT
F ⎥
− ⎥
0
⎢
J
J ⎦
⎣
⎡ I2 ⎤
g2 (x, t ) = ⎢
⎥,
⎣ 01 × 2 ⎦
⎡ x ⎤
,
h1 (x, t ) = ρ ⎢
⎣ 02 × 1⎦⎥
⎡ x1 ⎤
h2 (x, t ) = ⎢ ⎥,
⎣ x2 ⎦
⎡ 03 × 2 ⎤
k12 (x, t ) = ⎢
⎥,
⎣ I2 ⎦
k21 (x, t ) = ⎡⎣ I2 02 × 1⎤⎦.
7
⎡0 0 0 ⎤
⎥
⎢
B1 (t ) = ⎢ 0 0 0 ⎥,
⎢⎣ 0 0 − J−1⎥⎦
(26)
A solution to the  ∞-speed tracking controller synthesis involves
the standard  ∞ control problem for the time-varying linearized
⎡ I2 ⎤
B2 (t ) = ⎢
⎥,
⎣ 01 × 2 ⎦
⎡ I3 ⎤
C1 (t ) = ρ ⎢
⎥,
⎣ 02 × 3 ⎦
C2 (t ) = ⎡⎣ I2 02 × 1⎤⎦,
⎡ 03 × 2 ⎤
D12 (t ) = ⎢
⎥,
⎣ I2 ⎦
D21 (t ) = ⎡⎣ I2 02 × 1⎤⎦.
(27)
Finally, by applying Theorem 1 to system (26) thus specified, it
Fig. 6. Simulations results with adding  ∞ control drive under the presence of noise and input backlash: (a) Time evolution of stator current id, (b) stator current vector qcomponent tracking error and (c) speed-tracking error.
Please cite this article as: Ramírez-Villalobos R, et al. Sensorless H1 speed-tracking synthesis for surface-mount permanent magnet
synchronous motor. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.01.007i
R. Ramírez-Villalobos et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
8
is derived a local solution of the  ∞-tracking control problem.
Theorem 2. Let the following conditions be satisfied
(1) The desired speed ω* (t ) is twice continuously differentiable and
its first two time derivatives are uniformly bounded in t.
(2) (C1) and (C2) hold for the matrix functions A , B1, B2, C1, C2 governed by (9), (27).
and let (Pε (t ) , Zε (t )) be the corresponding bounded positive definite
solution of (15), (16) under some ε > 0. Then the output feedback
⎡ 1
⎤
ξ ̇ = f (ξ , t ) + ⎢ 2 g1 (ξ , t ) g1T (ξ , t ) − g2 (ξ , t ) g2T (ξ , t ) ⎥ Pε (t ) ξ (t )
⎣γ
⎦
+ Zε (t ) C2T (t )(y (t ) − h2 (ξ , t )),
u = − g2T (ξ , t ) Pε (t ) ξ (t ),
(28)
(29)
subject to (26) is a local solution of the  ∞ - speed tracking control
problem for the permanent magnet synchronous motor (4), (5), (19)–(25).
Remark 1. The validation of Theorem 2 is confined to the specification of Theorem 24 from [21] to the present case.
5. Proposed sensorless control scheme
The drive system is based on the field-oriented control scheme.
The overall block diagram of the proposed sensorless control
system is illustrated in Fig. 2.
Notice in this scheme that the speed controller (21) receives the desired speed reference ω* (t ) and brings the required current i* (t ) for the  ∞-controller and the current
controllers. The stator current d–component and the stator q–
component error are injected into the block  ∞-controller,
while it provides the output feedback to the current controllers (22) and (23). It can be noted in Fig. 2, that is not
Fig. 7. Robustness of the  ∞-tracking controller against external disturbances and rotor resistance variation: (a) Time evolution of stator current id, (b) stator current vector
q-component tracking error and (c) speed-tracking error.
Please cite this article as: Ramírez-Villalobos R, et al. Sensorless H1 speed-tracking synthesis for surface-mount permanent magnet
synchronous motor. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.01.007i
R. Ramírez-Villalobos et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
9
Fig. 8. Robustness of the  ∞-tracking controller against external disturbances and stator inductance variation: (a) Time evolution of stator current id, (b) stator current
vector q-component tracking error and (c) speed-tracking error.
designed an observer to estimate the rotor position and the
velocity. Besides, the speed is not feedback to the speed controller. The information of rotor position θ is obtained by integrating the desired speed reference ω* (t ) plus the internal
state ξ3 (t ), from (28), that is
θ (t ) =
∫ (ξ3 + ω*) dt,
(30)
and then it is injected to the Park and inverse Park transformations (see Ref. [11]). Due to a continuous estimate of rotor
position is taken from (30), the proposed sensorless control
scheme is suitable for full speed ranges. In contrast with literature dealing with the same problem, to design a switching
scheme with separate control strategies for low speed ranges
and high speed ranges is not needed.
6. Numerical simulations
The performance of the controller (28), (29) was studied by
simulation using MATLAB/SIMULINK ® applied to an industrial
PMSM benchmark, given in Ref. [8], whose nominal parameters
are provided in Table 1. The parameters selected for the controller
are γ = 1 × 107 , ρ = 1, ε = 1 × 104 , kd = 25 × 103, and kq = 36 × 103.
The signal, specified as
⎧− 0
if t < 0.5,
⎪
if 0.5 ≤ t < 1.5,
⎪ − 750t − 375
⎪ − 750
if 1.5 ≤ t < 3.5,
⎪
if 3.5 ≤ t < 4.5,
⎪ − 750t − 1875
⎪
ω* (t ) = ⎨ − 1500
if 4.5 ≤ t < 6.5,
⎪
⎪ − 2250t + 16125 if 6.5 ≤ t < 7.5,
⎪ − 750
if 7.5 ≤ t < 9,
⎪
⎪ − 1500t − 14250 if 9 ≤ t < 9.5,
⎪
⎩− 0
if 9.5 ≤ t < 10,
(31)
is chosen as desired speed reference. The load torque was chosen
as
TL (t ) = 15 × 10−3·sin (10πt ).
(32)
Please cite this article as: Ramírez-Villalobos R, et al. Sensorless H1 speed-tracking synthesis for surface-mount permanent magnet
synchronous motor. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.01.007i
10
R. Ramírez-Villalobos et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
Fig. 9. Robustness of the  ∞-tracking controller against external disturbances and viscous friction coefficient variation: (a) Time evolution of stator current id, (b) stator
current vector q-component tracking error and (c) speed-tracking error.
The perturbations
wd (t ) = wq (t ) = 1 × 10−2 ·cos (0.1πt ) exp ( − 2t ),
(33)
are applied to each available outputs. The initial conditions selected for simulations are id (0) = iq (0) = ωr (0) = 0 and the initial
condition for compensator state ξ (0) ∈ 3 was set also to zero.
The effect of external disturbances (load torque and perturbed
outputs) on the speed-tracking responses for the nominal case of
PMSM without using  ∞-controller are shown in Fig. 3. It can be
seen in Fig. 3(a) that the stator current id stays around the origin
and reaches an approximate maximum value of 13 × 103 A. In
Fig. 3(b) it can be seen that the stator current vector q-component
error is varying around at zero where the speed-tracking error is
oscillating between 7 75 rpm, as shown in Fig. 3(c).
The response for the nominal case adding the  ∞ control drive
is depicted in Fig. 4. It can be seen in Fig. 4(a) that the stator
current id reaches an approximate maximum value of 2.5 × 10−3 A,
and remains around the origin, while the stator current q-component has an error with average values equal to zero as shown in
Fig. 4(b). Fig. 4(c) highlights that the  ∞-controller has a speedtracking error that remains within 715 rpm.
Fig. 5 illustrates the performance of the closed-loop system
under a non-vanishing external disturbance
wd (t ) = wq (t ) = 1 × 10−2 ·cos (πt ).
(34)
In this figure it can be observed that stator current id remains
bounded between ±5 × 10−3 A, while the stator current q-component error and the speed-tracking error oscillate within ±0.4 A
and 715 rpm, respectively, as shown in Fig. 5(b) and (c),
respectively.
Sensor noise and input backlash are phenomena that could
appear in real-world scenarios (see, e.g., [24]). Therefore, Fig. 6
shows the response of the closed-loop system under the presence
of 15 dB Gaussian noise level for the available measurements and
Please cite this article as: Ramírez-Villalobos R, et al. Sensorless H1 speed-tracking synthesis for surface-mount permanent magnet
synchronous motor. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.01.007i
R. Ramírez-Villalobos et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
input backlash with 0.25 V deadzone level. Notice that stator
current id, the stator current q-component error and the speederror tracking still remains bounded around zero. Therefore, it is
evident that the  ∞-controller can attenuate the effects of undesired measurement noise and input backlash.
As corroborated in Fig. 4–6, the influence of external disturbances and noise was attenuated at the output of the PMSM by
the  ∞-controller. Performing a comparison from Fig. 3–6 was
noted that the effects introduced by the load torque and the perturbed outputs on the stator currents id, iq, and the rotor speed ωr
are smaller when  ∞-controller is added.
For sake of comparison, robustness of the  ∞-controller
against parameter variations was presented. First, the response of
PMSM under an exponential variation of 40% for the rotor resistance R is presented in Fig. 7. As shown in Fig. 7(a) and (b), the
stator current id remains bounded between −6 × 10−3 A and
3 × 10−3 A, while stator current vector q-component tracking error
remains within ±0.5 A. Fig. 7(c) shows that the speed-tracking
error remains bounded.
Fig. 8 shows the response of PMSM under a time-varying
parametric variation of ±50% for stator inductance L. It can be seen
in Fig. 8(a) that the stator current id reaches a minimum
value equal to zero and a maximum value of 130 × 10−3 A.
The stator current vector q-component tracking error
oscillates within ±0.4 A, as is shown in Fig. 8(b) and (c) shows that
the speed-tracking error remains bounded approximately within
720 rpm.
Finally, the robustness with respect to a periodic variation of
±25% for the viscous friction coefficient F is presented in Fig. 9. As
is shown in Fig. 9(a)–(c), that the stator current id presents small
oscillations. On the contrary, the stator current vector q-component tracking and the speed-tracking errors stay bounded between
1 A and 0.5 A, and 100 and 50 rpm, respectively.
Figs. 7–9 show that the  ∞-controller is also suitable to attenuate the effect introduced by the nonlinear variations of the
parameters R, L and F. Consequently, as seen from the results in
Figs. 4–9, robustness of the  ∞-tracking controller against external
disturbances and parameter variations are corroborated.
7. Conclusions
An output feedback nonlinear  ∞-controller was considered in
this paper, in order to solve the speed sensorless problem for a
surface-mount permanent magnet synchronous motor operating
under uncertain conditions, assuming the stator currents as the
only available measurement for feedback. The proposed approach
drives the rotor speed of the PMSM to a desired speed reference in
presence of external vanishing and non-vanishing disturbances,
noise, and input backlash, while the rotor position is calculated
from the causal dynamic output feedback compensator and from
the desired speed reference. Numerical simulations support the
effectiveness of the sensorless  ∞ control scheme.
11
[3] Magri AE, Giri F, Besançon G, Fadili AE, Dugard L, Chaoui F. Sensorless adaptive
output feedback control of wind energy systems with PMS generators. Control
Eng Prac 2013;21(4):530–43. http://dx.doi.org/10.1016/j.
conengprac.2013.11.005.
[4] Öztürk N, Çelik E. Speed control of permanent magnet synchronous motor
using fuzzy controller based on genetic algorithms. Int J Electr Power Energy
Syst 2012;43(1):889–98. http://dx.doi.org/10.1016/j.ijepes.2012.06.013.
[5] Ren J, Liu Y, Wang N, Liu S. Sensorless control of ship propulsion interior
permanent magnet synchronous motor based on a new sliding mode observer. ISA Trans 2015;54:15–26. http://dx.doi.org/10.1016/j.isatra.2015.08.008.
[6] Kunga Y, Thanha N, Wang M. Design and simulation of a sensorless permanent
magnet synchronous motor drive with microprocessor-based PI controller and
dedicated hardware EKF estimator. Appl Math Model 2015:79–104. http://dx.
doi.org/10.1016/j.apm.2015.02.034.
[7] Bifaretti S, Iacovone V, Rocchi A, Tomei P, Verrelli C. Nonlinear speed tracking
control for sensorless PMSMs with unknown load torque from theory to
practice. Control Eng Pract 2012;20(7):714–24. http://dx.doi.org/10.1016/j.
conengprac.2012.03.010.
[8] Accetta A, Cirrincioneb M, Puccia M. TLS EXIN based neural sensorless control
of a high dynamic PMSM. Control Eng Pract 2012;20(7):725–32. http://dx.doi.
org/10.1016/j.conengprac.2012.03.012.
[9] Alsofyany I, Idris N. A review on sensorless techniques for sustainable reliability and efficient variable frequency drives of induction motors. Renew
Sustain Energy Rev 2013;24:111–21. http://dx.doi.org/10.1016/j.
rser.2013.03.051.
[10] Verrelli C. Synchronization of permanent magnet electric motors new nonlinear advanced results. Nonlinear Anal Real World Appl 2012;13(1):395–409.
http://dx.doi.org/10.1016/j.nonrwa.2011.07.051.
[11] Ahmad M. High performance AC drives: modelling analysis and control.London: Springer-Verlag; 2010.
[12] Chi W, Cheng M. Implementation of a sliding-mode-based position sensorless
drive for high-speed micro permanent-magnet synchronous motors. ISA Trans
2014;53(2):444–53. http://dx.doi.org/10.1016/j.isatra.2013.09.017.
[13] Hamida M, Glumineau A, de Leon J. Robust integral backstepping control for
sensorless IPM synchronous motor controller. J Frankl Inst 2012;349(5):1734–
57. http://dx.doi.org/10.1016/j.jfranklin.2012.02.005.
[14] Solsona J, Valla MI, Muravchick C. A nonlinear reduced order observer for
permanent magnet synchronous motors. IEEE Trans Ind Appl 1996;43:492–7.
http://dx.doi.org/10.1109/41.510641.
[15] Chai S, Wang L, Rogers E. Model predictive control of a permanent magnet
synchronous motor with experimental results. Control Eng Pract 2013;21
(11):1584–93. http://dx.doi.org/10.1016/j.conengprac.2013.07.008.
[16] Chen J, Liu T. Implementation of a predictive controller for a sensorless interior
permanent-magnet synchronous motor drive system. IET Electr Power Appl
2012;6(8):513–25. http://dx.doi.org/10.1049/iet-epa.2011.0242.
[17] Hamida M, Glumineau A, de Leon J, Loron L. Robust adaptive high order
sliding-mode optimum controller for sensorless interior permanent magnet
synchronous motors. Math Comput Simul 2014;105:79–104. http://dx.doi.org/
10.1016/j.matcom.2014.05.006.
[18] Omrane I, Etien E, Dib W, Bachelier O. Modeling and simulation of soft sensor
design for real-time speed and position estimation of pmsm. ISA Trans
2015;57:329–39. http://dx.doi.org/10.1016/j.ijepes.2012.06.013.
[19] Isidori A, Astolfi A. Disturbance Attenuation and  ∞-Control Via Measurement
Feedback in Nonlinear Systems. IEEE Trans Autom Control 1992;37(9):1283–
93. http://dx.doi.org/10.1109/9.159566.
[20] van de Schaft A.  2 -Gain Analysis of Nonlinear Systems and Nonlinear State
Feedback  ∞ Control. IEEE Trans Autom Control 1992;37(6):770–84. http:
//dx.doi.org/10.1109/9.256331.
[21] Orlov Y, Aguilar L. Advanced  ∞ control: towards nonsmooth theory and
applications.New York: Birkhäuser; 2014.
[22] Orlov Y, Acho L, Solis V. Nonlinear  ∞-control of time varying systems. In:
Proceedings of the 38th Conference on Decision and Control, Phoenix, USA;
1999, p. 3764–3769.
[23]  ∞ control of linear time-varying systems: a state-space approach, SIAM J
Control Optim 29; 1991, p. 1394–1413.http://dx.doi.org/10.1137/0329071.
[24] Ponce I, Bentsman J, Orlov Y, Aguilar L. Generic nonsmooth  ∞ output
synthesis application to a coal-fired boiler/turbine unit with attenuator
deadzone. IEEE Trans Control Syst Technol 2015;23(6):2117–28. http://dx.doi.
org/10.1109/TCST.2015.2399672.
References
[1] Chen S, Luo Y, Pi Y. PMSM sensorless control with separate control strategies
and smooth switch from low speed to high speed. ISA Trans 2015;21(11):650–
8. http://dx.doi.org/10.1016/j.isatra.2015.07.013.
[2] Grouz F, Sbita L, Boussak M, Khlaief A. FDI based on an adaptive observer for
current and speed sensors of PMSM drives. Simul Model Pract Theory
2013;35:34–9. http://dx.doi.org/10.1016/j.simpat.2013.02.006.
Please cite this article as: Ramírez-Villalobos R, et al. Sensorless H1 speed-tracking synthesis for surface-mount permanent magnet
synchronous motor. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.01.007i