2016-IACC-0814
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Differential Flatness Based Speed/Torque Control with State-Observers of Permanent Magnet Synchronous Motor Drives '
4 12 13 l2 13 13 P. Thounthong , , S. Sikkabut , , N. Poonnoy , , P. Mungporn , , B. Yodwong , , P. Kumam 6 5 N. Bizon , B. Nahid-Mobarakeh , S. Pierfederid l Renewable Energy Research Centre (RERC) 2 Department of Teacher Training in Electrical Engineering (TE), Faculty of Technical Education 3 Thai-French Innovation Institute (TFII)
King Mongkut's University ofTechnology North Bangkok, 1518, Pracharat I Rd., Bangkok 10800, Thailand *e-mail:
[email protected]@gmai1.com
4 Department of Mathematics, King Mongkut's University of Technology Thonburi, Bangkok, Thailand 5
Faculty of Electronics, Communications and Computers, University ofPitesti, Arges 110040, Pitesti, Romania 6
GREEN Lab., Universite de Lorraine, 2 Vandceuvre-Ies-Nancy, Lorraine 54516, France
70
Abstract-- This paper introduces a nonlinear control scheme
based on the differential flatness approach for controlling the speed/torque
of
a
permanent
magnet
synchronous
motor
(PMSM) drive. The differential flatness estimation is a model based approach.
Then, two state-observers are proposed to
estimate a load
torque disturbance and a stator resistance
(represent losses in an inverter and PMSM) by means of its voltage drop. It can help to improve the PMSM drive system and
€
65
:>
60
OJ) ..
.s '0
�
� u Q
solutions to dynamics and stabilization problems. The design
1250
controller parameters are autonomous of the operating point; moreover, high dynamics in disturbance rejection is achieved. To laboratory,
and
digital
controller
estimation
DSll04
is
accomplished
platform.
with
Simulation
a
and
experimental results with a small-scale PMSM of 1000 W, 3000
rpm in a laboratory corroborate the excellent control scheme
during a motor-drive cycles.
Index
Terms--
Differential
algebraic,
permanent-magnet
synchronous machine (PMSM), digital control, motor control, motor drives, propulsion, torque control, variable speed drives
I.
INTRODUCTION
P
MSM drives are often used in many applications, particularly servo applications or electric vehicle applications such Toyota Camry hybrid car. Control, robustness, stability, efficiency, and optimization of PMSM drives remain an essential area of research. Classic work on controlling a PMSM drive is based on a field oriented control, where a linear control using PI compensator was proposed for speed/torque stabilization [1], [2]. Design controller parameters based on linear methods require a linear approximation, where this is dependent on the operating point. Because the PMSM/inverter drive model is nonlinear, it is
This work was supported by King Mongkut's University of Technology North Bangkok under Grant KMUTNB-GOV-58-06.
�
30')
PI
0
20
40
60
80
20
40
60
80
(PM
�
60")
100
120
140
160
180
200
100 (ms)
120
140
160
180
200
Time
(ms
1000
validate the proposed method, a hardware system is realized in a dSPACE
(PM
55 50
the efficiency. Using the flatness property, we propose simple
PI
�750 � 500 .... ..
0..
250
00
Time
Fig. I. Comparison of the flatness-based control law with a linear PI control law during a large load step.
natural to apply model-based nonlinear control strategies that directly compensate for system nonlinearity without requiring a linear approximation [3], [4]. The idea of differential flatness theory was first introduced by Fliess et al. [5]. This allowed an alternate representation of the system where trajectory planning and nonlinear controller design is clear cut. The advantages of the differential flatness approach are that the trajectories of the system are straightforwardly estimated by the trajectories of a flat output and its derivatives without integrating any differential equation. Currently, these ideas have lately been used in a variety of nonlinear systems across various engineering disciplines, including the following: 1) control of cathode pressure and oxygen excess ratio of a PEMFC system [6];
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PROPOSED DIFFERENTIAL FLATNESS BASED CONTROL OF
II.
PMSM DRIVES A.
Fig. 2. A three-phase inverter driving the PMSM, where VBu, is the DC bus voltage; is,,, the input inverter current; iA, iB, ic the motor phase currents.
Mathematic Model of the PMSMlinverter
Fig. 2 shows a system configuration of a three-phase inverter connected to the PMSM. The sinusoidal pulse-width modulation technique (SPWM) are applied to inverter in order to achieve a sinusoidal output voltage with minimal undesired harmonics. The power-invariant transformations from the stationary (abc) to the rotating reference frame (dq) are p-I applied, where [xa, xB]t =T32t·[Xa, Xb, xc]\ [Xd, Xqr = [xu> xB]t, and x are voltages, currents, flux linkages, etc; () is the rotor position (rad). The matrix transformations [10], [11] (named "Park's Transformation") may defme:
o
1 T32=
p =
Fig. 3. An equivalent circuit of PMSM drive.
2) reactive power and dc voltage tracking control of a three-phase voltage source converter [7]; 3) current control for three-phase three-wire boost converters [8]; 3) control of fuel cell/supercapacitor hybrid power plant.
[
J%
-
�
J3
2
2
1
COs( () J
(1)
2
J3 -
sin( () J
]
cos( () J ,
sin( () J
p_1
[
cos( () J -
sin( () J
]
sin( () J cos( () J .
(2)
Ignoring magnetic saturation, in dq-synchronous rotating frames, the equivalent circuit of PMSM inverter drive is shown in Fig. 3 and the differential equations of PMSM/inverter can be written as [12], [13]
(3) (4)
[9] .
At least, Fig. 1 shows a comparison of the flatness-based and traditional linear proportional-integral (PI) control methods, which uses a dc bus voltage regulation of 60 V in a fuel cell/supercapacitor (SC) hybrid power plant to step a dc load, as demonstrated by Thounthong et al. [9] . The parameters of the linear controller were tuned to obtain the best possible performance at phase margin (PM) at 30° and 60° . Fig. 1 shows the real experimental results obtained for both controllers during a load step. The flatness-based control shows good stability and an optimum response during dc bus voltage regulation. Using this data, one can conclude that the flatness-based control method provides better performance as compared to the classic PI controller. The paper is organized as follows. In section II the equations describing the PMSM model are reviewed. Differential flatness based speed/torque feedback control applied to PMSM is obtained. The two-nonlinear extended observers used to estimate PMSM's states and unknown load torque is briefly. System performance is evaluated in section III. Finally, conclusions are drawn in section IV.
=
(5) with,
Te = p . iq
. {tp
m
-
{Lq - Ld ). id )
(6) (7)
where, id and iq the direct and quadrature motor currents (A); 'Fm the permanent magnet flux linkage (Wb); Ld the d-axis inductance (H); Lq the q-axis inductance (H); (Oe is the electrical angular frequency (rad/s); COm the mechanical angular frequency (rad/s); p the number of pole pairs; Te the electromagnetic torque (Nm); TL the load torque (Nm); B is the friction coefficient (Nm·s/rad); and J is the moment of inertia of the rotor. It should be noted here that a PMSM is always drived by a three-phase inverter; for this reason, R is simplified resistance as losses in an inverter (static and dynamics losses; switching deadtime; voltage drops in IGBTs and Diodes) and in a PMSM (the stator winding resistance, hysteresis losses, and eddy current losses).
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B. Current (Torque) Control Loop For a non-salient machine and referred to (3) and (4), L = Ld = Lq. To prove that the system is flat [14, [15], one defines T the flat output y = [yJ, Y2]T , control variable u = [ UI ' U2] , and T state variable x = [x), X2] are defmed as follows: (8)
Then, the state variables
x
the transient current, a second order filter [18], [19] is used, such that the current command iCOM is always limited by, iREF (s )
iCOM (s )
-
_
---=-----
( )
--
s
2
--
Wn2
+
2S 2
--
Wn2
(18)
s +1
where (2 and Wn2 are the desired dommant damping ratio and natural frequency. C. Speed Control Loop
can be written as (9)
From (3) and (4), the control variables u can be calculated from the flat outputs y and its time derivatives (inverse dynamics [15]): (10)
The outer loop concerns the speed regulation where the flat output is chosen as Y3 = �n; a control variable U3 = iq, and a state variable X3 = � = /P3(Y3). So, the flatness based-speed controller output generates the command of the q-axis current iqCOM' According to mechanical equation (5) - (7) and on the assumption that iq (= Y2) = iqCOM because the inner current loop bandwidth is estimated to be faster than the bandwidth of the external speed loop, control variable U3 (=iqCOM) can be expressed in an inverse dynamics term as:
U3 =(J'@m +TL +B.wm )/p·'Pm =lf3(Y3,h )
= vdREF
(19)
= iqCOM
= VqREF Desired references for the dq-currents are represented by YlREF (= id REF) and Y2REF (= iqREF). Feedback control laws achieving an exponential tracking of the set-points are given by the following expression [16], [17]: t
(YI- YIREF ) +KII(YI- YIREF ) +K nf (YI- YIREF )dr = 0 o
(1 2 )
o
where
2 p(s ) = s + 2SI(Onis + (0; 1 ;
(14)
where Kit and Kt2 are the controller parameters. One may set as a desired characteristic polynomial:
Kll = 2sIwni ; KI2 = (0; 1
(15)
where ( I and Wnl are the desired dommant damping ratio and natural frequency. New variables are defined AI = YI and ,1,2 = 5'1, so that (1 2 ) and (13) become:
t
Al = YlREF +Kll(YlREF - Yl) +K12f (YlREF - Yl)dr
(16)
t
� = 5'1REF +KII(Y2REF - Yl) +KI2f (Y2REF - Y2)dr
(20)
( 21) (13)
o
o
t
,1,3 = Y3REF +K2I(Y3REF - Y3) +K22f (Y3REF - Y3)dr
t
(5'1 - Y2REF ) +KII(Y2 - Y2REF ) +K12f (Y2 - Y2REF )dr = 0
o
It is similar to the inner current control loops. A desired reference for the mechanical speed is represented by Y3REF (= �REF)' A feedback control law is given by the following expression:
(17)
The trajectory planning is an important step in the implementation of a flatness based-control. The flatness property allowed all the state and input variables to be written in function of a chosen flat output y. Thus, the flat output trajectory defines all the state or input variables trajectories. It is thus interesting to give a well-known waveform so that all the transient state behaviors can be predicted. Then, to limit
So that (3 and Wn3 are the desired dominant damping ratio and natural frequency. Finally, in view of the nature of the derived feedback control law (20), we need to generate the current command for the mverter. Since here our focus is on a smooth accelerator (known as a soft-start system), we restrict the reference profiles to smooth changes between stationary regimes. Then, the motion trajectory planning is defmed as:
wREF (s ) = wCOM (s )
( ) s
(22)
2
--
Wn4
+
2s 4
--
Wn4
s +1
where (4 and Wn4 are the desired dominant damping ratio and natural frequency. D. State-Observers The differential flatness approach is a model based estimation. Refer to the inverse dynamics equations (10), (11), and (19), the stator resistance R and load torque TL need to be known at least. For this reason, state observers for parameter estimation have to be considered. However, Vtq (= R-iq) is defined in place of R. To simplify the implementation and PMSM working in only constant torque region (id = 0), (4) and (5) may be written again:
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0Jm
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Observer I
Fig. 4. Proposed a differential flatness based speed/torque control with state-observers of a PMSM drive.
(23) (24) In this section, two new state observers are proposed. These state observer are dedicated to a specific subclass of nonlinear systems. The proposed state observer inspires from the subclass of state observers designated through the literature as "disturbance observers" [20], [21]. Disturbance observers are indeed very well adapted to the considered problematic consisting in losses estimation. Indeed, especially on the case of the PMSM drives [(23) and (24)], if considering disturb = (vtq rS, the disturbance observers are well adapted. The proposed state observers are dedicated to the subclass of nonlinear systems which can be described as follows:
(25)
where: 1) X E 91" x III is the vector of variable which is going to be estimated, and Y E 91" is the vector of measured variables; 2) x E 91" is the vector of the system state variables. Every state variable is supposed to be measured (i.e., Y = x) ; 3) P E 91111 is the vector of the unknown parameters to estimate. Parameters p are supposed to vary slowly compared to state variables x; 4) f and g are nonlinear functions of x and u (the command signal vector), respectively, of size 91" and 91" x m. Refer to (23) and (24), one may write
(27)
(28)
1). Proposed Observer 1: Asymptotically Stable For the subclass of nonlinear systems verifying (25), the proposed state observer I is defined through (29), considering the estimation errors ex (i - x ) and ep (p - p) =
[�l [ Ii
=
=
f(X,u)+ g(X,U)'P -S1 .ex - g l (x,u)'ex
]
(29)
with n n S) is the positive-definite matrix of size 91 x . Proof the derivative estimation errors ex and ep are written as follows:
ex ep
=
=
g(x,u).ep -Sl·ex
(30)
_g l (x,u).ex
(31)
Asymptotic stability of the estimation can be demonstrated with the classical Lyapunov approach. For this the Lyapunov candidate function, V is considered as follows: (32)
(26) The derivative of function 978-1-4799-8397-1/15/$31.00 © 2016 IEEE
V can
be expressed as
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2). Proposed Observer 11: Exponentially Stable The proposed non-linear state observer II is defined as
(36) with n n S2 is the positive-definite matrix of size m x . m P is the positive-definite matrix of size m x m. and, K =-p . g-1 ( x,) u
(37)
p
(38)
Ki=Kp .S2
Proof it is similar to the proposed observer I. The derivative of Lyapunov function V can be expressed as
Fig. 5. Test bench of the PMSM drive. Table 1. PMSMllnverter specification and parameters.
Rated
Power
Rated Speed
Prated
nrated
Rated Torque Trated Number of poles pair p Resi s tanc e (Motor+lnverter) Stator inductance L
Magnetic
flux 'F,n
=
Ld
=
R
VBus
Nm
'
n mH
Wb kg· m2 0.99x 10-3 Nm· s/rad V
540
10
Frequency Is
kHz
Table IT. Speed/torque regulation parameters.
SI
lUnl
1500
OJn2
150
1
S2 S3
15
OJn4
15
S4
iqMax iqMin
rad· pu.
ra d · pu .
1
lUn3
p u.
S- I -I
S
pu. rad· A
S-I
A
V =et x e· x + et p e· p ·
(33)
·
By combining (30), (31), and (33), V can be expressed as
_
t t V =etx ' g( x,) x,) u e) u . e p -e x ,Sl'ex + e p (g( x Finally,
.
V = -et x .SI .ex < 0
·
(34)
(35)
From (32) and (35), the estimation asymptotic stability [22], [23] can be ensured long as S, is positive-defmite matrix.
·
·
x, u ·ep -et x ,S2·ex () K p g( x,) u . e p et p ·Kp ,S2'ex t Ki ·ex _e p ' g t ( x,) u . ex
_
·
·
Then, by introducing Kp g ( x, u) results ·
ep
]
.
[
-S2 o
0 -P
][ ] .
ex ep
=
-P and Ki
(39)
=
Kp S2, it
