sliding mode control of pmsm drives subject to torsional oscillations in the mechanical load
DESCRIPTION
Stephen J Dodds School of Computing and Technology. Jan Vittek University of Zilina Slovakia. Sliding Mode Control of PMSM Drives Subject to Torsional Oscillations in the Mechanical Load. OVERVIEW OF PRESENTATION. Motivation Brief overview of sliding mode control The plant and its model - PowerPoint PPT PresentationTRANSCRIPT
Sliding Mode Control of PMSM Drives Subject to Sliding Mode Control of PMSM Drives Subject to Torsional Oscillations in the Mechanical LoadTorsional Oscillations in the Mechanical Load
Jan VittekUniversity of Zilina
Slovakia
Stephen J DoddsSchool of Computing
and Technology
OVERVIEW OF PRESENTATIONOVERVIEW OF PRESENTATION
Motivation
Brief overview of sliding mode control
The plant and its model
The case for separate single input, single output sliding mode controllers
Formulation of a practicable general sliding mode controller
Plant rank determination for correct SMC selection
Zero dynamics for rotor angle control
The set of three sliding mode controllers
Presentation of simulation results
Conclusions and recommendations
MOTIVATION OF THE RESEARCH
The tuning needed for conventional motion controllers at the commissioning stage and whenever changes in the driven mechanical load occur is, in general, very time consuming and requires knowledge and experience of dynamical systems and control. When significant mechanical vibration modes are present this problem is not only exacerbated but it may not even be possible to tune conventional controllers to attain satisfactory performance. Through its non-reliance on plant models, sliding mode control has been investigated with a view to finding a simple solution readily acceptable in industry.
Brief Overview of Sliding Mode Controlfor Single Input, Single Output Plants
Basic sliding mode controller:
r 1
r 1 r 1ed y
dt
D
erivative Estim
ator
maxu
maxuS0
r 1a
1a
S
demy 0e
2a2
2
2
d y
dte
1dy
dte
0
Switching boundary in r-1 dimensional error space:
0 r 1e , eS 0
‘p’ region:
maxu
0eu
S
‘n’ region: maxu
0eu
S
r 1dem 1 r 1
In the sliding mode,
ideally, S 0e
y 1sy s 1 a s a s
u yPLANTorder: n
rank:r n
The Plant to be Controlled
Lm
d mi
ai
r m cdemu bdemu ademu
d demu
q demu
L
Load Mass
PMSM Inverter
Clark-Park Trans.
Flexible drive shaft
Shaft Encoder & Processing
PWM
cmi
bmi
bi ci
Inverse Clark-Park Trans.
Shaft Encoder & Processing
r
Lm
r m r m
The Plant Model:The Two-Mass, one Motor System
Two control problems will be addressed:
a) The control of the rotor angle.
b) The control of the load mass angle.
both in the presence ofan external load torque applied to the load mass.
Flexible shaft
(torsional compliance)
Load inertia
Motor rotor
inertia
Plant Model
Lresr e rr
rs
L
L L er
L
K
K
1
1
J
J
r
1
J sc
Le
1
s
L
1
J s1
s
sK
r
1x
3x4x
2x
LLL LJ
rr rJ L
Ls
Lre
Load moment of inertia
Electro-magnetic
torque
e
Inertial datum
Spring constant
sK
External load mass
load torque
Le
Rotor moment of inertia
rJ
LJ
L
r
External rotor load torque,Lre
Mechanical Part:
di dt Ai B udi Fd d r q Electrical Part:
di dt Ci E Di Gq d r uq q
qc drJ H iKi Control Variables
Complete Block Diagram Model of Plant
r
1
J sc
Le
1
s
L
1
J s1
s
sK
r
qi
LL LJ
rr rJ L
Ls
Lre1
s
1
s
diA
D
L
rJ
r
F
G
du
qu H
K
B
C
E
As will be seen, despite the interaction in the plant rendering the control problem a multivariable one, separate single input, single output sliding mode control loops will suffice. The argument for this will be presented next.
Single Input, Single Output Sliding Mode Controllers
The signal, r q, may be regarded as a disturbance
input to the direct axis current control loop. So the plant simplifies to
the following for the direct axis current control:
r
1
J sc
Le
1
s
L
1
J s1
s
sK
r
qi
LL LJ
rr rJ L
Ls
Lre1
s
1
s
diA
D
L
rJ
r
F
G
du
qu H
K
B
C
E
1
s di
A
Fdu
didi
r q' Disturbance ' B i This leaves only r or L to be
controlled using uq.
This may also be achieved by single input,
single output sliding mode controllers.
It is now clear that a single input, single
output sliding mode controller may be
designed for controlling id using ud.
Formulation of Practicable Sliding Mode ControllerFirst, return to the basic sliding mode controller:
r 1y
Derivative E
stimator
maxu
maxuS0
r 1a
1a
S
demy
2a
y
y
u yPLANTorder: n
rank:r n
The closed-loop phase portrait and a typical trajectory are illustrated here for r = 2:
demy y
y
0
max
S 0
u u
max
S 0
u u
Switching Boundary
To overcome this problem, the control chatter may be eliminated by replacing the switching boundary with a boundary layer giving a continuous transition of u between –umax and +umax between the sides of the boundary :
y
0
max
S 0
u u
max
S 0
u u
Boundary Layer
demy y
Formulation of Practicable Sliding Mode Controller
The problem with this is that the control chatter during the sliding motion may interact adversely with the switching of the inverter, so the control accuracy and stator current waveforms could be poor.
demy y
y
0
max
S 0
u u
max
S 0
u u
Switching Boundary
Formulation Practicable Sliding Mode ControllersAn ‘Ideal’ derivative estimator would amplify high frequency components of measurement noise. This problem may be overcome, however, by combining a low pass filter with each differentiator, but there is a trade-off between the degree of filtering and robustness of the SMC.
r 1y
maxu
maxu
S
r 1a
1a
S
demy
2ay
y
u yPLANTorder: n
rank:r n
K
s
Ideal Differentiator
s
Ideal Differentiator
s
Ideal Differentiator
Formulation Practicable Sliding Mode ControllersAn ‘ideal’ derivative estimator would amplify high frequency components of measurement noise. This problem may be overcome, however, by combining a low pass filter with each differentiator, but there is a trade-off between the degree of filtering and robustness of the SMC.
r 1y
maxu
maxu
S
r 1a
1a
S
demy
2ay
y
u yPLANTorder: n
rank:r n
K
f
s
1 sT
f
s
1 sT
f
s
1 sT
r 1y
maxu
maxu
S
r 1a
1a
S
demy
2ay
y
u yPLANTorder: n
rank:r n
K
s
Ideal Differentiator
s
Ideal Differentiator
s
Ideal Differentiator
Plant Rank Determination for SMC Design
To determine the rank w.r.t. a selected output, the number of integrators in each forward path from every control input and that output may be counted.
Then the rank is equal to the
smallest integrator count.
1
Rank w.r.t. id:: ri = 1
23
r
1
J sc
Le
1
s
L
1
J s1
s
sK
r
qi
LL LJ
rr rJ L
Ls
Lre1
s
1
s
diA
D
L
rJ
r
F
G
du
qu H
K
B
C
E
Rank w.r.t. r:: r = 3
33
43
44
Plant Rank Determination for SMC DesignTo determine the rank w.r.t. a selected output, the number of integrators in each forward path from every control input and that output may be counted.
Then the rank is equal to the smallest integrator count.
r
1
J sc
Le
1
s
L
1
J s1
s
sK
r
qi
LL LJ
rr rJ L
Ls
Lre1
s
1
s
diA
D
L
rJ
r
F
G
du
qu H
K
B
C
E
r
1
J sc
Le
1
s
L
1
J s1
s
sK
r
qi
LL LJ
rr rJ L
Ls
Lre1
s
1
s
diA
D
L
rJ
r
F
G
du
qu H
K
B
C
E
Rank w.r.t. L::
Minimum integrator count = 3 to this point
rL = 5
Plant Rank Determination for SMC DesignTo determine the rank w.r.t. a selected output, the number of integrators in each forward path from every control input and that output may be counted.
Then the rank is equal to the smallest integrator count.Hence minimum integrator count to the output, L , is 5.
Le
L
1
J s1
s
sK
LL LJ
Ls
L
r
1
J sc
1
s
r
qi
rr rJ L
Lre1
s
1
s
diA
D
rJ
r
F
G
du
qu H
K
B
C
E
Zero Dynamics for Rotor Angle ControlSuppose r has been brought to zero by the sliding mode controller. Then an uncontrolled subsystem may be identified in the plant block diagram, as follows:
The only input to this subsystem is Le once r = 0. So the remainder of the plant can be ignored.
0
Zero Dynamics for Rotor Angle ControlSuppose r has been brought to zero by the sliding mode controller. Then an uncontrolled subsystem may be identified in the plant block diagram, as follows:
Le
L
1
J s1
s
sK
LL LJ
Ls
L
The only input to this subsystem is Le once r = 0 and the remainder of the plant is ignored.
0
The eigenvalues (poles) of this uncontrolled subsystem are therefore
1,2 s Ls j K J
Hence the subsystem is subject to uncontrolled oscillations at a frequency of
s LK J rad / s
In simple terms, the control system holds the rotor fixed but allows the load mass to oscillate, restrained by the torsion spring but without damping.
The characteristic equation of this subsystem, from the determinant of Mason’s formula, is:
2s s2
LL
K K1 0 s 0
JJ s
For direct axis current control, ri = 1, so the order of the highest derivative to feed back is ri – 1 = 0. In this case no output derivatives are needed and the ideal SMC has no closed loop dynamics. The practicable version of the SMC then reduces to a simple proportional controller with a high gain.
The Set of Sliding Mode Controllers
d mi
r m
Plant
qdemu
ddemu
Lm
ddemi
0
U dK
U
U dK
U
1ra
2ra
Der
ivat
ive
Est
imat
or
r dem
r m
r m
Ldem 4La
3La
2La
1La Lm
Lm
Lm
Lm
r m
Lm D
eriv
ativ
e E
stim
ator
rS
LS
For rotor angle control, rr = 3, so the order of the highest derivative to feed back is rr – 1 = 2. The first derivative is the rotor speed and assumed to be produced by the shaft encoder software, so a derivative estimator is only needed for the second derivative. The closed loop dynamics is of second order.
r2
21r 2r s r
s
S 0 characteristic equation :
21 a s a s 1 T s
9
using the Dodds 5% T formula
sr
1r
4Ta
9
2sr
2r
4Ta
81
The Set of Sliding Mode Controllers
For load mass angle control, rL = 5, so the order of the highest derivative to feed back is
rL – 1 = 4. The first derivative is the load mass angular velocity and assumed to be
produced by the shaft encoder software, so a derivative estimator is only needed for the second, third and fourth derivatives. The closed loop dynamics is of fourth order.
d mi
r m
Plant
qdemu
ddemu
Lm
d demi
0
U dK
U
U dK
U
1ra
2ra
Der
ivat
ive
Est
imat
or
r dem
r m
r m
Ldem 4La
3La
2La
1La Lm
Lm
Lm
Lm
r m
Lm D
eriv
ativ
e E
stim
ato
r
rS
LS
L
2 3 41 2 3 4
4sL
s
S 0 characteristic equation:
1 a s a s a s a s
2T using the Dodds1 s
5% T formula:15
sL1L
8Ta
15
3sL
3L
32Ta
3375
2sL
2L
24Ta
225
4sL
4L
16Ta
50625
Rotor moment of inertia Jr = 0,0003 kgm2
Direct axis inductance Ld = 53.8 mH
Quadrature axis inductance
Lq = 53.8 mH
Permanent mg. flux PM = 0.262 Wb
Stator resistance Rs = 33.3
No. of pole pairs p = 3
Motor
Load moment of inertia JL = 0,0003 kgm2
Torsion spring constant Ks = 9 Nmr/rad
External load torque L(t) = 20 Nm/s ramp to
constant value of 20 Nm, starting at t = 0,6 s
Load
PARAMETERS FOR SIMULATION
Settling Times (5% criterion) Ts = Ts = TsL = 0,2 s
Filtering time constant Tf = 100 s
Gain of control saturation element
K = 200
Control saturation limit = Inverter DC voltage
Umax = 360 V
Controller
SIMULATION SIMULATION OF ROTOR OF ROTOR
ANGLE ANGLE CONTROLCONTROL
0 0.2 0.4 0.6 0.8 1 -2
0
2
4
6
8
10
12
r , r id , e=id - r [rad]
t [s]
r
r id
e=2(r id - r )
0 0.2 0.4 0.6 0.8 1 9
9.2
9.4
9.6
9.8
10
10.2
10.4
10.6
10.8
11 r ,
L , [rad]
r id
r
t [s]
0 0.2 0.4 0.6 0.8 1-2
0
2
4
6
8
10
12 r ,
L [rad]
r ,
L
t [s]
0 0.2 0.4 0.6 0.8 1 -20
-15
-10
-5
0
5
10
15
20
Ls , Le [Nm]
t [s]
Le
Ls
0 0.2 0.4 0.6 0.8 1-10
-5
0
5
10
15
id , iq [A]
t [s]
iq
id
0 0.2 0.4 0.6 0.8 1 9
9.2
9.4
9.6
9.8
10
10.2
10.4
10.6
10.8
11 L , L id , [rad]
L
t [s]
L id
0 0.2 0.4 0.6 0.8 1 -2
0
2
4
6
8
10
12
L , L id , e=id - L [rad]
t [s]
L id L
e=2(L id - L )
SIMULATION SIMULATION OF LOAD MASS OF LOAD MASS
ANGLE ANGLE CONTROLCONTROL
0 0.2 0.4 0.6 0.8 18
8.5
9
9.5
10
10.5
11
11.5
12 L , r
[rad]
t [s]
r
L ,
0 0.2 0.4 0.6 0.8 1 -10
-5
0
5
10
15
20
25
Ls , Le [Nm]
t [s]
Le
Ls
0 0 . 2 0 . 4 0 . 6 0 . 8 1- 5
0
5
1 0 i d , i q [ A ]
i d , i q [ A ]
i d , i q [ A ]
t [ s ]
i q
i d
CONCLUSIONS AND RECOMMENDATIONSCONCLUSIONS AND RECOMMENDATIONS
The simulations predict robustness for sliding mode control of rotor angle and also load mass angle in that the ideal responses are followed with moderate accuracy.
The differences between the simulated and ideal responses are attributed to the finite gains of the control saturation elements within the boundary layers..
The vector control condition of keeping the direct axis stator current component to negligible proportions is very effectively maintained.
It is recommended that the potential accuracy of the method is ascertained by exploring the design limits regarding sampling frequency, saturation element gain, and the derivative estimation filtering time constant, in the presence of measurement noise.
Other derivative estimation methods should also be investigated, such as the high gain multiple integrator observer.
Extension to the control of mechanisms with more than one uncontrolled vibration mode would be of interest.
The results obtained here are sufficiently promising to warrant experimental trials, which will attract potential industrial users.