Title: Sliding Mode Control of PMSM Drives Subject to Torsional Oscillations in the Mechanical Load
1Sliding Mode Control of PMSM Drives Subject to
Torsional Oscillations in the Mechanical Load
2OVERVIEW 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
3MOTIVATION 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.
4Brief Overview of Sliding Mode Controlfor
Single Input, Single Output Plants
Basic sliding mode controller
5The Plant to be Controlled
6The Plant ModelThe Two-Mass, one Motor System
Load inertia
Two control problems will be addresseda) 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)
Motor rotor inertia
7Plant Model
8Complete Block Diagram Model of Plant
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.
9Single Input, Single Output Sliding Mode
Controllers
The signal, B wr Iq , 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
It is now clear that a single input, single
output sliding mode controller may be designed
for controlling id using ud.
This leaves only qr or qL to be controlled using
uq.
This may also be achieved by single input, single
output sliding mode controllers.
10Formulation of Practicable Sliding Mode Controller
First, return to the basic sliding mode
controller
11Formulation of Practicable Sliding Mode Controller
12Formulation Practicable Sliding Mode Controllers
An 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.
13Formulation Practicable Sliding Mode Controllers
An 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.
14Plant 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.
Rank w.r.t. id
Then the rank is equal to the smallest integrator
count.
15Plant 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.
Rank w.r.t. qr
rq 3
Then the rank is equal to the smallest integrator
count.
16Plant 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.
Rank w.r.t. qL
17Zero Dynamics for Rotor Angle Control
Suppose qr 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 GLe once qr
0.
So the remainder of the plant can be ignored.
0
18Zero Dynamics for Rotor Angle Control
Suppose qr has been brought to zero by the
sliding mode controller. Then an uncontrolled
subsystem may be identified in the plant block
diagram, as follows
19The Set of Sliding Mode Controllers
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.
20The 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.
21PARAMETERS FOR SIMULATION
Motor
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 W
No. of pole pairs p 3
Load
Load moment of inertia JL 0,0003 kgm2
Torsion spring constant Ks 9 Nmr/rad
External load torque GL(t) 20 Nm/s ramp to constant value of 20 Nm, starting at t 0,6 s
Controller
Settling Times (5 criterion) Tsi Tsr TsL 0,2 s
Filtering time constant Tf 100 ms
Gain of control saturation element K 200
Control saturation limit Inverter DC voltage Umax 360 V
22SIMULATION OF ROTOR ANGLE CONTROL
23SIMULATION OF LOAD MASS ANGLE CONTROL
24CONCLUSIONS 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.