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Online Model Predictive Torque Control forPermanent Magnet Synchronous Motors
Gionata Cimini∗, Daniele Bernardini∗, Alberto Bemporad∗ and Stephen Levijoki†
∗ODYS Srl
†General Motors Company
2015 IEEE International Conference on Industrial Technology
Gionata Cimini, [email protected] 1 / 20
Outline
1 Introduction and Motivations
2 Model Predictive Control Design
3 Experimental Results
4 Conclusions
Gionata Cimini, [email protected] 2 / 20
Company Overview (www.odys.it)
ODYS is a private SME founded in 2011, originally as a university spin-off
Core business: consultancy and software for the development of advancedcontrol systems, with special focus on model predictive control (MPC)
Main expertise: advanced multivariable control design, efficient real-timeoptimization algorithms, and tools for their deployment in embedded systems
Application domainsautomotive
aerospace
energy
process control
Gionata Cimini, [email protected] 3 / 20
Background
The standard control method for PMSMs is the Field Oriented Control (FOC)(cascade structure, with three linear regulators)
→The work deals with the application of Model Predictive Control (MPC), toPMSM torque control.
MPC in electrical drives control is, mainly, motivated by two facts:
mathematical models of these systems are relatively well known
several constraints are needed for safety reasons
The main idea of MPC
Obtain the control actions by solving, at each sampling time, a finite-horizonoptimal control problem
Gionata Cimini, [email protected] 4 / 20
Background
The standard control method for PMSMs is the Field Oriented Control (FOC)(cascade structure, with three linear regulators)
→The work deals with the application of Model Predictive Control (MPC), toPMSM torque control.
MPC in electrical drives control is, mainly, motivated by two facts:
mathematical models of these systems are relatively well known
several constraints are needed for safety reasons
The main idea of MPC
Obtain the control actions by solving, at each sampling time, a finite-horizonoptimal control problem
Gionata Cimini, [email protected] 4 / 20
Model Predictive Control
Output Setpoint
MeasuredOutput
ControlInput
k +Nu k +Npk
Past
Linear prediction model
Control and Prediction Horizon
Hard input and soft output constraints
Gionata Cimini, [email protected] 5 / 20
Model Predictive Control
Output Setpoint
PredictedOutput
MeasuredOutput
PredictedInput
ControlInput
Control HorizionPrediction Horizion
k +Nu k +Npk
Past Future
Linear prediction model
Control and Prediction Horizon
Hard input and soft output constraints
Gionata Cimini, [email protected] 5 / 20
Model Predictive Control
Output Setpoint
MeasuredOutput
ControlInput
k + 1
Past Future
k +Nu k +Np
Linear prediction model
Control and Prediction Horizon
Hard input and soft output constraints
Gionata Cimini, [email protected] 5 / 20
Existing solutions
Explicit MPCa: the optimization problem is pre-solved offlinevia multiparametric Quadratic Programming (mpQP)
! it is an easy-to-implement PieceWise Affine (PWA) function ofthe parameters
! fast binary search tree algorithm can be used
% it is applicable only to small problems
% high memory requirements
Finite Control Setb: exploits the discrete nature of thesystem by manipulating the inverter switch positions directly
! MPC is solved on line, by enumeration
! very fast with short prediction horizon
% computational load scales exponentially with the horizon
% high sampling frequency, not decoupled from control frequency
% variable frequency
————————————————————-a Bolognani, S.; Bolognani, S.; Peretti, L.; Zigliotto, M., ”Design and Implementation of Model Predictive Control for ElectricalMotor Drives”b Rodriguez, J. and Kazmierkowski, M.P. and Espinoza, J.R. and Zanchetta, P. and Abu-Rub, H. and Young, H.A and Rojas,”C.A, State of the Art of Finite Control Set Model Predictive Control in Power Electronics”
Gionata Cimini, [email protected] 6 / 20
Existing solutions
Explicit MPCa: the optimization problem is pre-solved offlinevia multiparametric Quadratic Programming (mpQP)
! it is an easy-to-implement PieceWise Affine (PWA) function ofthe parameters
! fast binary search tree algorithm can be used
% it is applicable only to small problems
% high memory requirements
Finite Control Setb: exploits the discrete nature of thesystem by manipulating the inverter switch positions directly
! MPC is solved on line, by enumeration
! very fast with short prediction horizon
% computational load scales exponentially with the horizon
% high sampling frequency, not decoupled from control frequency
% variable frequency
————————————————————-a Bolognani, S.; Bolognani, S.; Peretti, L.; Zigliotto, M., ”Design and Implementation of Model Predictive Control for ElectricalMotor Drives”b Rodriguez, J. and Kazmierkowski, M.P. and Espinoza, J.R. and Zanchetta, P. and Abu-Rub, H. and Young, H.A and Rojas,”C.A, State of the Art of Finite Control Set Model Predictive Control in Power Electronics”
Gionata Cimini, [email protected] 6 / 20
MPC with online optimization
Embedded QP solver:Model Predictive Control → Quadratic Programming with affine constraints
! MPC is solved online
! Low memory Requirements
! Switching frequency decoupled from sampling frequency
! Fixed switching frequency
% High computational burden
Main Objective
Verify the feasibility of embedded, online MPC
Application in low-power DSP, commonly used for motion control
Gionata Cimini, [email protected] 7 / 20
Standard PI Field Oriented Control
Inverter
PMSMPMSMPWM
abc
dq
iq
Kt
τ
∫θ
ω
ωrefPI
τref
id
id,ref
PI
PI
abc
dq
ud
uq
The aim is replacing the standard PI-based torque controller, with . . .
Gionata Cimini, [email protected] 8 / 20
Proposed MPC Torque Control
Inverter
PMSMPMSMPWM
abc
dq
iq ∫θ
ω
ωrefPI
τref
id
abc
dq
ud
uq MPCid,ref
. . . an MPC torque control, solved with online optimization.
Gionata Cimini, [email protected] 9 / 20
Outline
1 Introduction and Motivations
2 Model Predictive Control Design
3 Experimental Results
4 Conclusions
Gionata Cimini, [email protected] 10 / 20
Prediction Model
Consider an isotropic PMSM , with one pole pair. The nonlinear mathematical model is:
id(t) = −R
Lid(t) + ω(t)iq(t) +
1
Lud(t)
iq(t) = −R
Liq(t)− ω(t)id(t) +
1
Luq(t)−
λ
Lω(t)
ω(t) =B
Jω(t) +
Kt
Jiq(t)−
1
Jτl(t),
——————————————————————————–The linearized model of the electrical subsystem is obtained imposing nominal speed ω(t) = ω0.
x(k + 1) = Ax(k) +Bu(k) +Gv(k)
y(k) = Cx(k)
A = eAcTs , B =
∫ Ts
0eAcτdτBc, G =
∫ Ts
0eAcτdτGc, C = Cc
Ac =
−RL ω0
−ω0 −R
L
Bc =
1
L0
01
L
Gc =
[0
−λ
L
]Cc =
[1 00 Kt
]
Gionata Cimini, [email protected] 11 / 20
Prediction Model
Consider an isotropic PMSM , with one pole pair. The nonlinear mathematical model is:
id(t) = −R
Lid(t) + ω(t)iq(t) +
1
Lud(t)
iq(t) = −R
Liq(t)− ω(t)id(t) +
1
Luq(t)−
λ
Lω(t)
ω(t) =B
Jω(t) +
Kt
Jiq(t)−
1
Jτl(t),
——————————————————————————–The linearized model of the electrical subsystem is obtained imposing nominal speed ω(t) = ω0.
x(k + 1) = Ax(k) +Bu(k) +Gv(k)
y(k) = Cx(k)
A = eAcTs , B =
∫ Ts
0eAcτdτBc, G =
∫ Ts
0eAcτdτGc, C = Cc
Ac =
−RL ω0
−ω0 −R
L
Bc =
1
L0
01
L
Gc =
[0
−λ
L
]Cc =
[1 00 Kt
]
Gionata Cimini, [email protected] 11 / 20
System Constraints
The most critical constraints concern the electrical components
u ∈ U = {u ∈ R2 : ‖u‖2 ≤ Vmax} (1)
x ∈ X = {x ∈ R2 : ‖x‖2 ≤ Imax} (2)
ud
uq
Vmax
(a) Voltage constraints
id
iq
Imax
(b) Current constraints
Vmax → Depending from the maximum DC-bus and from the modulation scheme(Vmax = VDC/
√3 for PWM and SVM)
Imax → Chosen to prevent overheating. Peaks larger than this limit are allowedfor short time intervals .
Gionata Cimini, [email protected] 12 / 20
System Constraints
The most critical constraints concern the electrical components
u ∈ U = {u ∈ R2 : ‖u‖2 ≤ Vmax} (1)
x ∈ X = {x ∈ R2 : ‖x‖2 ≤ Imax} (2)
Uud
uq
Vmax
(c) Voltage constraints
X
id
iq
Imax
(d) Current constraints
Vmax → Depending from the maximum DC-bus and from the modulation scheme(Vmax = VDC/
√3 for PWM and SVM)
Imax → Chosen to prevent overheating. Peaks larger than this limit are allowedfor short time intervals .
Gionata Cimini, [email protected] 12 / 20
How about embedded online optimization?
MPC problem formulation
min∆u
N−1∑i=0
‖Q(yk+i|k − r(k))‖22+
Nu−1∑j=0
‖R∆uk+j|k‖22+
+ ‖P (yk+N|k − r(k))‖22s.t. xk|k = x(k)
xk+i+1|k = Axk+i|k +Buk+i|k +Gv(k)
yk+i+1|k = Cxk+i+1|k
uk+i|k ∈ U, xk+i+1|k ∈ X
i = 0, 1, . . . , N − 1
j = 0, 1, . . . , N −Nu − 1
Quadratic problem (QP)
minz
q(z) ,1
2z′Hz + p(t)′F ′z
s.t. Gz ≤W + Sp(t)
convex quadratic objectivefunction
polyhedral constraints
QPs are easier to solve thanstandard formulation
We developed an embedded QP solver
Gionata Cimini, [email protected] 13 / 20
How about embedded online optimization?
MPC problem formulation
min∆u
N−1∑i=0
‖Q(yk+i|k − r(k))‖22+
Nu−1∑j=0
‖R∆uk+j|k‖22+
+ ‖P (yk+N|k − r(k))‖22s.t. xk|k = x(k)
xk+i+1|k = Axk+i|k +Buk+i|k +Gv(k)
yk+i+1|k = Cxk+i+1|k
uk+i|k ∈ U, xk+i+1|k ∈ X
i = 0, 1, . . . , N − 1
j = 0, 1, . . . , N −Nu − 1
Quadratic problem (QP)
minz
q(z) ,1
2z′Hz + p(t)′F ′z
s.t. Gz ≤W + Sp(t)
convex quadratic objectivefunction
polyhedral constraints
QPs are easier to solve thanstandard formulation
We developed an embedded QP solver
Gionata Cimini, [email protected] 13 / 20
How about embedded online optimization?
MPC problem formulation
min∆u
N−1∑i=0
‖Q(yk+i|k − r(k))‖22+
Nu−1∑j=0
‖R∆uk+j|k‖22+
+ ‖P (yk+N|k − r(k))‖22s.t. xk|k = x(k)
xk+i+1|k = Axk+i|k +Buk+i|k +Gv(k)
yk+i+1|k = Cxk+i+1|k
uk+i|k ∈ U, xk+i+1|k ∈ X
i = 0, 1, . . . , N − 1
j = 0, 1, . . . , N −Nu − 1
Quadratic problem (QP)
minz
q(z) ,1
2z′Hz + p(t)′F ′z
s.t. Gz ≤W + Sp(t)
convex quadratic objectivefunction
polyhedral constraints
QPs are easier to solve thanstandard formulation
We developed an embedded QP solver
Gionata Cimini, [email protected] 13 / 20
Outline
1 Introduction and Motivations
2 Model Predictive Control Design
3 Experimental Results
4 Conclusions
Gionata Cimini, [email protected] 14 / 20
Processor In the Loop Experiments
Inverter
PMSMPMSMPWM
abc
dq
iq ∫θ
ω
ωrefPI
τref
id
abc
dq
ud
uq MPCid,ref
Simulation model: MBE.300.E500 PMSM, commercially available from Technosoft c©
Platform: Low cost F28335 Delfino DSP by Texas Instruments c©. (32-bit, 150 MHz CPUand IEEE-754 single-precision Floating- Point Unit)
This board is commonly used for motion control!
Gionata Cimini, [email protected] 15 / 20
Processor In the Loop Experiments
Inverter
PMSMPMSMPWM
abc
dq
iq ∫θ
ω
ωrefPI
τref
id
abc
dq
ud
uq MPCid,ref
Simulation model: MBE.300.E500 PMSM, commercially available from Technosoft c©
Platform: Low cost F28335 Delfino DSP by Texas Instruments c©. (32-bit, 150 MHz CPUand IEEE-754 single-precision Floating- Point Unit)
This board is commonly used for motion control!
Gionata Cimini, [email protected] 15 / 20
Processor In the Loop Experiments
Inverter
PMSMPMSMPWM
abc
dq
iq ∫θ
ω
ωrefPI
τref
id
abc
dq
ud
uq MPCid,ref
Simulation model: MBE.300.E500 PMSM, commercially available from Technosoft c©
Platform: Low cost F28335 Delfino DSP by Texas Instruments c©. (32-bit, 150 MHz CPUand IEEE-754 single-precision Floating- Point Unit)
This board is commonly used for motion control!
Gionata Cimini, [email protected] 15 / 20
Processor In the Loop Experiments
Inverter
PMSMPMSMPWM
abc
dq
iq ∫θ
ω
ωrefPI
τref
id
abc
dq
ud
uq MPCid,ref
Simulation model: MBE.300.E500 PMSM, commercially available from Technosoft c©
Platform: Low cost F28335 Delfino DSP by Texas Instruments c©. (32-bit, 150 MHz CPUand IEEE-754 single-precision Floating- Point Unit)
This board is commonly used for motion control!
Gionata Cimini, [email protected] 15 / 20
Control Performance
torque control with MPTC
0 0.1 0.2 0.3 0.4 0.5-0.03
-0.02
-0.01
0
0.01
0.02
0.03
time (s)
τ(N
·m)
τl
τref
MPC
τMPC
MPTC vs PI-FOC torque transient
0.3 0.31 0.32 0.33 0.34-0.03
-0.025
-0.02
-0.015
time (s)
τ(N
·m)
τl
τPI
τMPC
MPTC vs PI-FOC speed tracking
0 0.1 0.2 0.3 0.4 0.5-500
0
500
1000
1500
2000
2500
3000
3500
4000
time (s)
ω(rpm)
ωref
ωPI
ωMPC
MPC provides 2.3% and 4.2% improvementin speed and torque tracking, respectively
(ISE index)
Gionata Cimini, [email protected] 16 / 20
Control Performance
torque control with MPTC
0 0.1 0.2 0.3 0.4 0.5-0.03
-0.02
-0.01
0
0.01
0.02
0.03
time (s)
τ(N
·m)
τl
τref
MPC
τMPC
MPTC vs PI-FOC torque transient
0.3 0.31 0.32 0.33 0.34-0.03
-0.025
-0.02
-0.015
time (s)
τ(N
·m)
τl
τPI
τMPC
MPTC vs PI-FOC speed tracking
0 0.1 0.2 0.3 0.4 0.5-500
0
500
1000
1500
2000
2500
3000
3500
4000
time (s)
ω(rpm)
ωref
ωPI
ωMPC
MPC provides 2.3% and 4.2% improvementin speed and torque tracking, respectively
(ISE index)
Gionata Cimini, [email protected] 16 / 20
Control Performance
torque control with MPTC
0 0.1 0.2 0.3 0.4 0.5-0.03
-0.02
-0.01
0
0.01
0.02
0.03
time (s)
τ(N
·m)
τl
τref
MPC
τMPC
MPTC vs PI-FOC torque transient
0.3 0.31 0.32 0.33 0.34-0.03
-0.025
-0.02
-0.015
time (s)
τ(N
·m)
τl
τPI
τMPC
MPTC vs PI-FOC speed tracking
0 0.1 0.2 0.3 0.4 0.5-500
0
500
1000
1500
2000
2500
3000
3500
4000
time (s)
ω(rpm)
ωref
ωPI
ωMPC
MPC provides 2.3% and 4.2% improvementin speed and torque tracking, respectively
(ISE index)
Gionata Cimini, [email protected] 16 / 20
Control Performance
torque control with MPTC
0 0.1 0.2 0.3 0.4 0.5-0.03
-0.02
-0.01
0
0.01
0.02
0.03
time (s)
τ(N
·m)
τl
τref
MPC
τMPC
MPTC vs PI-FOC torque transient
0.3 0.31 0.32 0.33 0.34-0.03
-0.025
-0.02
-0.015
time (s)
τ(N
·m)
τl
τPI
τMPC
MPTC vs PI-FOC speed tracking
0 0.1 0.2 0.3 0.4 0.5-500
0
500
1000
1500
2000
2500
3000
3500
4000
time (s)
ω(rpm)
ωref
ωPI
ωMPC
MPC provides 2.3% and 4.2% improvementin speed and torque tracking, respectively
(ISE index)
Gionata Cimini, [email protected] 16 / 20
Online Optimization
The sampling interval is 0.3ms
Dimension of the QP problem: 5 decision variables and 41 constraints
DSP with 150MHz processor
One hardware multiplier 32× 32 bit
Single precision
0 0.1 0.2 0.3 0.4 0.50
50
100
150
200
time (s)
t sol(µs)
Memory occupancy: 2.5kB out of the 34kB of memory provided by the DSP
Gionata Cimini, [email protected] 17 / 20
Online Optimization
The sampling interval is 0.3ms
Dimension of the QP problem: 5 decision variables and 41 constraints
DSP with 150MHz processor
One hardware multiplier 32× 32 bit
Single precision
0 0.1 0.2 0.3 0.4 0.50
50
100
150
200
time (s)
t sol(µs)
Memory occupancy: 2.5kB out of the 34kB of memory provided by the DSP
Gionata Cimini, [email protected] 17 / 20
Outline
1 Introduction and Motivations
2 Model Predictive Control Design
3 Experimental Results
4 Conclusions
Gionata Cimini, [email protected] 18 / 20
Conclusions
Summary
The feasibility of MPC with online optimization in terms of
required computational power
memory occupancy
has been demonstrated through processor-in-the-loop tests on a low power DSP
Future Activities
Extension of the proposed strategy to consider the speed loop
Flux weakening handling
Evaluation of the effects of winding resistance increase due to temperature rise
Application to automotive problems, such as Electronic Throttle Controland Hybrid Electric Motor Control
Gionata Cimini, [email protected] 19 / 20