control of electric motors and drives via convex optimization - nicholas moehle · 2018-08-27 ·...
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Control of Electric Motors and Drives
via Convex Optimization
Nicholas Moehle
Advisor: Stephen Boyd
February 5, 2018
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Outline
1. waveform design for electric motors
– permanent magnet
– induction
2. control of switched-mode converters
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Waveform design for electric motors
I traditionally:
– AC motors driven by sinusoidal inputs (and designed for this)1
– based on reference frame theory, c. 1930
I now:
– more computational power
– power electronics can generate near-arbitrary drive waveforms2
I our questions:
– given a motor, how to design waveforms to drive it?
– which waveform design problems are tractable? convex?
1Hendershot, Miller. Design of Brushless Permanent-Magnet Machines. 1994.2Wildi. Electrical Machines, Drives and Power Systems. 2006.
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Motor model
!; �
�
1
�
n
+
�
v
1
+
�
v
n
i
1
i
n
...
I
n windings, each with an RL circuit.
I electrical variables:
– voltage v(t) 2 Rn
– current i(t) 2 Rn
– flux �(t) 2 Rn
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Motor model
!; �
�
1
�
n
+
�
v
1
+
�
v
n
i
1
i
n
...
I the rotor has
– torque � (t)
– speed ! = onst: (high inertia mech. load)
– position �(t) = !t
I goal is to manipulate v to control �
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Stored energy
I stored magnetic energy is E(�; �)
– magnetic coupling depends on mechanical position
I
E is 2�-periodic in �
I inductance equation relates current and flux:
i = r
�
E(�; �)
I torque given by
� = �
�
��
E(�; �)
I in general, both are nonlinear in �
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Torque
I the average torque is:
�� = lim
T!1
1
T
Z
T
0
� (t) dt
I torque ripple is
r = lim
T!1
1
T
Z
T
0
�
� (t)� ��
�
2
dt
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Power loss
R
11
R
nn
...
I
R 2 Sn++
is the (diagonal) resistance matrix
I resistive power loss is iTRi
I average power loss is
p
loss
= lim
T!1
1
T
Z
T
0
i
T
Ri dt
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Circuit dynamics
�
1
�
n
+
�
v
1
+
�
v
n
i
1
i
n
...
I dynamics from Kirchoff’s voltage law, Faraday’s law:
v(t) = Ri(t) +
_
�(t)
I dynamics coupled across windings by inductance equation
i = r
�
E(�; �).
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Winding connection
+
�
u
1
+
�
u
m
......
v
1
v
n
I often, winding voltages v not controlled directly
I (e.g., wye/delta windings, windings contained in rotor)
I indirect control through terminal voltages u(t) 2 Rm
Ci(t) = 0; v(t) = C
T
e(t) +Bu(t);
I
C 2 Rp�n is the connection topology matrix
I
B 2 Rn�m is the voltage input matrix
I
e(t) 2 Rp are floating node voltages
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Winding connection examples
v
1
v
1
v
2
v
3
u
1
u
2
u
3
v
1
v
2
v
3
u
1
u
2
u
3
v
1
v
2
v
3
u
1
u
2
u
3
0
Ci(t) = 0; v(t) = C
T
e(t) +Bu(t);
I simple delta, wye, and independent winding connections
I some windings may be controlled only through induction
– e.g., windings on the rotor
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Optimal waveform design
I waveform design problem:
minimize p
loss
+ r
subject to �� = �
des
;
torque equation
inductance equation
circuit dynamics
winding pattern
I variables are i, v, u, e, �, � (all functions on R+
)I problem data:
– tradeoff parameter � 0
– resistance matrix R 2 Sn++
– energy function E : Rn � R+
! R+
– shaft speed ! 2 R
– desired torque �des
2 R
– winding connection parameters B 2 Rn�m and C 2 Rp�n
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I nonconvex in general, due to nonlinear torque and inductance
equations
I problem data 2�-periodic, but periodicity of solution not known
– in practice, solutions often not 2�-periodic in �
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Permanent magnet motor
NNSS
NN SS
u
1
u
2
u
3
u
4
u
5
I magnets in rotor change magnetic flux through windings as they
pass, producing voltage across the windings
I by simultaneously pushing current through the windings, electrical
energy is extracted (or injected)
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Permanent magnet motor
I energy function is quadratic:
E(�; �) = �
T
A� + b(�)
T
�
(quadratic part independent of rotor angle)
I inductance equation is linear:
� = Li+ �
mag
(�)
L is the inductance matrix, �mag
is the flux due to rotor magnets
I torque equation is affine:
� = k(�)
T
i+ �
og
(�)
k(�) is the motor constant, � og
is the cogging torque
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Permanent magnet motor
I dynamics, with �, are
v(t) = Ri(t) +
_
�(t)
I eliminating �:
v(t) = Ri(t) + L
di
dt
(t) + !k(�)
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Permanent magnet motor, waveform design
I optimal waveform design problem is convex
I
2�-periodicity of problem data with convexity implies 2�-periodicity
of a solution, if one exists3
3Boyd, Vandenberghe. Convex Optimization, page 189. 2004
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Permanent magnet motor, waveform design
I waveform design problem:
minimize
power lossz }| {
1
2�
Z
2�
0
i(�)
T
Ri(�) d�+
torque ripplez }| {
1
2�
Z
2�
0
(� (�)� �
des
)
2
�
d�
subject to1
2�
Z
T
0
� (�) d� = �
des
(av. torque)
� = k(�)
T
i+ �
og
(�) (torque)
v(�) = Ri(�) + !Li
0
(�) + !k(�) (dynamics)
Ci(�) = 0
(winding conn.)v(�) = C
T
e(�) +Bu(�)
I variables are i, v, u, e, � (all functions on [0; 2�℄)
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Permanent magnet motor, waveform design
I a periodic linear-quadratic control problem
– can discretize, solve by least squares
I in fact, many extensions retain convexity:
– voltage limits ju(�)j � u
max
– current limits ji(�)j � i
max
– nonquadratic definitions of torque ripple
I extensions typically involve solving a quadratic program
I more discussion in paper4:
– extensions/variations
– custom fast solver ! online waveform generation
4Moehle, Boyd. Optimal Current Waveforms for Brushless Permanent Magnet
Motors. 2015.
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Example
I
= 2 W=Nm
2
I left: ! = 300 rad/s, right: ! = 400 rad/s
0 100 200 300�5
0
5
0 100 200 300�50
0
50
0 100 200 3000.25
0.3
0.35
0 100 200 300�5
0
5
0 100 200 300�50
0
50
0 100 200 3000.25
0.3
0.35
� (
Æ
)
� (
Æ
)
i
1
(
A
)
i
1
(
A
)
u
1
(
V
)
u
1
(
V
)
�
(
N
m
)
�
(
N
m
)
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Induction motor
u
1
u
2
u
3
u
4
u
5
I rotor magnets replaced by more windings, which act as
electromagnets (with current)
I rotor current produced my magnetic induction (using stator currents)
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Induction motor
I Energy function is again quadratic:
E(�; �) = �
T
A(�)�
quadratic part dependent on � (affine part omitted for simplicity)
I inductance equation is linear:
� = L(�)i
I torque is (indefinite) quadratic:
� = �i
T
L
0
(�)i
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Induction motor, maximum torque problem
I general waveform design problem intractable
I we focus on the maximum torque problem ( = 0):
– torque ripple penalty disappears
– maximize average torque (a nonconvex quadratic function)
– power loss constraint (a convex quadratic function)
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Induction motor, maximum torque problem
I waveform design problem:
maximize
average torquez }| {
lim
T!1
1
T
Z
T
0
�i(t)
T
L
0
(!t)i(t) dt
subject to lim
T!1
1
T
Z
T
0
i(t)
T
Ri(t) dt � p
loss
(power loss)
v(t) = Ri(t) +
_
�(t) (dynamics)
Ci(t) = 0
(winding conn.)v(t) = C
T
e(t) +Bu(t)
�(t) = L(!t)i(t) (induction)
I variables are i, v, u, e, � (all functions on R+
)
I equivalent to minimizing ploss
with average torque constraint
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Induction motor, maximum torque problem
I can be converted to a nonconvex linear-quadratic control problemwith a quadratic constraint
– strong duality holds
– original proof due to Yakubovich5
I further details in our paper6
– equivalent semidefinite program (SDP)
– method for constructing optimal waveforms from SDP solution
– proof of tightness
5Yakubovich. Nonconvex optimization problem: The infinite-horizon linear-
quadratic control problem with quadratic constraints. 1992.6Moehle, Boyd. Maximum Torque-per-Current Control of Induction Motors via
Semidefinite Programming. 2016.
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Example
traditional, sinusoidally wound, 5-phase motor with wye winding:
u
1
u
2
u
3
u
4
u
5
desired torque �des
= 5 Nm, speed ! = 50 rad=s
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Example
0 2 4 6 8 10 12 14 16 18−100
−50
0
50
100
0 2 4 6 8 10 12 14 16 18−4
−2
0
2
4
0 2 4 6 8 10 12 14 16 180
2
4
6
8
u
(V
)
i
(A
)
�
(N�
m
)
� (rad)
power loss is 11 W per Nm torque produced
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Stator fault
Same motor, with open-phase fault:
u
1
u
2
u
4
u
5
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Stator fault
0 2 4 6 8 10 12 14 16 18−100
−50
0
50
100
0 2 4 6 8 10 12 14 16 18−5
0
5
0 2 4 6 8 10 12 14 16 180
2
4
6
8
u
(V
)
i
(A
)
�
(N�
m
)
� (rad)
power loss is 14 W per Nm torque produced
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Outline
1. waveform design for electric motors
– permanent magnet
– induction
2. control of switched-mode converters
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Controlling switched-mode converters
+
�source
load
I input are switch configurationsI traditionally:7
1. make discrete input continuous, by considering averaged switch
on-time (‘duty cycle’)
2. choose a duty cycle corresponding to desired equilibrium
3. linearize the resulting system around equilibrium, use linear control
I now:
– direct (switch-level) control7Kassakian. Principles of power electronics. 1991.
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Switched-linear circuit
+
�source
load
I state xt
2 Rn contains inductor currents, capacitor voltages– can be augmented to contain, e.g., reference signal
I for each switch configuration, we have a linear circuit
I switched-affine dynamics:
x
t+1
= A
u
t
x
t
+ b
u
t
; t = 0; 1; : : : ;
I dynamics specified by Ai, bi in mode i
I control input is the mode ut
2 f1; : : : ; Kg
I may include mode restrictions (e.g., for a diode)
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Switched-affine control
I switched-affine control problem is
minimizeP
T
t=1
g(x
t
)
subject to x
t+1
= A
u
t
x
t
+ b
u
t
x
0
= x
init
u
t
2 f1; : : : ; Kg
I constraints hold for all t
I variables are ut
and x
t
2 Rn
I problem data are dynamics Ai, bi, function g, and initial condition
x
init
I can be solved by trying out KT trajectories
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‘Solution’ via dynamic programming
I Bellman recursion: find functions Vt
such that
V
t
(x) = min
u2f1;:::;Kg
g(x) + V
t+1
(A
u
x+ b
u
)
for all x, for t = T � 1; : : : ; 0
I final value function V
T
= g
I optimal problem value is V0
(x
init
) at initial state xinit
I in general, intractable to compute (or store) Vt
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Model predictive control
I idea: solve switched-affine control problem, implement first control
action u
0
, measure new system state, and repeat
I called model predictive control (MPC) or receding horizon control
I given V = V
1
, MPC policy satisfies
�
mp
(x) 2 argmin
u2f1;:::;Kg
V (A
u
x+ b
u
)
(ties broken arbitrarily)
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Approximate dynamic programming policy
I in practice, MPC policy only works for T small
I (system response time measured in �s)
I instead, approximate V as a quadratic function ^
V
I given ^
V , ADP policy satisfies
�
adp
(x) 2 argmin
u2f1;:::;Kg
^
V (A
u
x+ b
u
)
I evaluating �adp
requires evaluating a few quadratic functions
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How to obtain ^
V ?
I quadratic lower bounds on V can be found via semidefinite
programming8
I compute V (x
(i)
) for many states x(i), fit best quadratic function ^
V
– we used this method
– subproblems solved using methods described in paper9
I use exact value function for approximate linear control problem(e.g., linear-quadratic control)
– provides a link to traditional methods
8Wang, O’Donoghue, Boyd. Approximate Dynamic Programming via Iterated
Bellman Inequalities. 2014.9Moehle, Boyd. A Perspective-Based Convex Relaxation for Switched-Affine
Optimal Control. 2015.
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Inverter example
+
�
V
d
L
1
L
2
L
1
L
2
L
1
L
2
C C C
I state xt
are inductor currents and capacitor voltages, and desired
output current phasors
I cost function is deviation of output currents from desired
(sinusoidally-varying) values
I model parameters Vd
= 700 V, L1
= 6:5 �H, L2
= 1:5 �H, C = 15
�F, Vload
= 300 V, and desired output current amplitude Ides
= 10
A.
I sampling time 30 �s
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Result
Policy State cost
ADP policy, 0.70
MPC policy, T = 1 1
MPC policy, T = 2 1
MPC policy, T = 3 1
MPC policy, T = 4 1
MPC policy, T = 5 0.45
I for T < 5 MPC policy is unstable
I running MPC with T = 5 takes several seconds on PC
I ADP takes few hundred flops (can be carried out in �s)
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Result
In steady state:
0 0.005 0.01 0.015 0.02 0.025 0.03−20
−10
0
10
20
0 0.005 0.01 0.015 0.02 0.025 0.03−400
−200
0
200
400
0 0.005 0.01 0.015 0.02 0.025 0.03−20
−10
0
10
20
Inputcurr.(A
)Cap
.voltag
e(V
)Outputcurr.( A
)
time (s)39
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Conclusions
I unconventional motors (asymmetrical, nonsinusoidally-wound,
non-rotary) can be controlled using optimization, by designing the
waveform to the motor
I modern techniques can be used to generate optimal controllers forpower electronic converters, which
– have fast response
– can easily incorporate constraints
– are intuitive to understand and tune
– make good use of modern microprocessor capabilities
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Sources
I motors– N. Moehle, S. Boyd. Optimal Current Waveforms for Brushless Permanent
Magnet Motors. International Journal of Control, 2015.– N. Moehle, S. Boyd. Maximum Torque-per-Current Control of Induction
Motors via Semidefinite Programming. Conf. on Decision and Control,2016.
– N. Moehle, S. Boyd. Optimal Current Waveforms for Switched-Reluctance
Motors. Multi-Conf. on Systems and Control, 2016.
I converters– N. Moehle, S. Boyd. A Perspective-Based Convex Relaxation for
Switched-Affine Optimal Control. Systems and Control Letters, 2015.– N. Moehle, S. Boyd. Value Function Approximation for Direct Control of
Switched Power Converters. Conf. on Industrial Electronics andApplications, 2017.
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Control of Electric Motors and Drives
via Convex Optimization
42