Model Predictive Control
in automotive systems
R. Scattolini, A. Miotti
Dipartimento di Elettronica e Informazione
2
Outline
• Gasoline
engines
• Diesel engines
• Hybrid fuel cell vehicles
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Dip. di Elettronica e Informazione
3
Gasoline engines
• System description
• Engine model and validation (VW Lupo)
• Design of the engine static maps
• Design of a predictive control scheme
• State and unknown parameters observers
• Feedback linearization + MPC
• Conclusions
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Dip. di Elettronica e Informazione
4
Air path scheme
Throttle
EGR
VVT
AIR FLOW
METER
Measurable
variable
ECU
SPEED
SENSOR
PRESSURE
SENSOR
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Actuators
5
A multivariable system
Brake
pedal
Driver
ECU
Accelerator
Advance
Throttle
EGR
VVT
Fuel
Torque
Air supply
system
Vehicle
dynamics
Engine
Manifold
pressure
Input air
flow
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Gear
Engine
speed
Vehicle
speed
Lambda sensor
Emissions
[NOX , CO, HC]
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6
System characteristics
• Multivariable system
• Strongly non linear
• Heavily coupled
• Few measurements available on-board
• No measurements of performance variables (torque,
emissions)
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Dynamic model
and assumptions
7
Lumped-parameter, time-based, nonlinear, Mean Value
Model (MVM)
Uniform thermodynamic properties;
Homogeneous mixtures of air and combustion (ideal)
gas;
Stoichiometric combustion of air and fuel;
Heat transfer effects neglected;
Flow through restrictions modeled as isentropic
compressible flow.
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Dip. di Elettronica e Informazione
8
Throttle
Algebric Model
wth is the flow through throttle Measured by means of an air flow meter
From experimental data the parameter kAth (α
α) has to be identified (also as
function of N).
Problem: The downstream pressure pth is unknown
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Dip. di Elettronica e Informazione
pth=pman
9
Intake manifold
wair : from throttle equation
wegr: from EGR model
weng: flow coming in the engine
3 state variables
wair
mair
wegr
megr
Tman
weng
xegr: egr percentage coming in engine
Assumption: Ideal gas law
m air = wair − weng
Mass equations
Tman =
Energy equation
megr
mair
, m egr = wair − weng
mair + meng
mair + meng
Rman
(c pair wairTair + c pegr wegrTegr
Vmancvman
− c pair wengTman (1 − xegr ) − c pegr wengTman xegr )
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Dip. di Elettronica e Informazione
10
Static maps
• volumetric efficiency;
• EGR valve;
• torque;
• emissions;
• engine temperature.
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Dip. di Elettronica e Informazione
11
Vehicle dynamics
Transmission ratio shaft/wheels : τt = τc τp
Re : Effective wheel radius
M : Mass of the vehicle
Jr : Inertia of e wheel
Jat is the total momentum of inertia
Power at shaft
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Engine losses
Road Load
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12
Air path validation (1)
Manifold pressure [Pa]
Intake manifold pressure
Air flow [kg/h]
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Dip. di Elettronica e Informazione
13
Validation of the vehicle model
Engine speed [rpm]
Vehicle speed [km/h]
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Dip. di Elettronica e Informazione
14
Mathematical model
The model, ignoring some additional dynamics and transport
delays, takes the form
x = f ( x, u )
y = h ( x, u )
where
 N 
Torque 
Tman 
 α 
 NO 


x

 , y=
x =  mair  , u = 
VVT
HC




megr 
 EGR 
 CO 




Main problems: some state and output variables are not measured.
Many static functions are derived from data and do not have an
explicit mathematical form.
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Design of an industrial
control scheme
driver:
pedal request
αo
Static
engine map
Torque
request Γ o
15
u
Static
control maps
N
Engine
Computed with the simulator
Built with the simulator
by solving a suitable
optimization problem
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Dip. di Elettronica e Informazione
Design of the
static control maps
16
For any point (Γ 0,N) on a prescribed grid, the engine
simulator has been used to solve the following (static)
optimization problem
0
2
min λ1 (Γ − Γ ) + λ2 NOx + λ3CO + λ4 HC
α , EGR ,VVT
N=N
0 ≤ EGR ≤ 250
0.85α 0 ≤ α ≤ 1.15α 0
0 ≤ VVT ≤ 40 0
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Dip. di Elettronica e Informazione
17
Control maps
Table and breakpoints data for block:
PROVA_VVT_ottimnt/Mappa
VV
o
T Ottimizzazione /farf − nt
Throttle
α
20
10
0
100
4000
50
15
←15←−5
←20 ←−10
←25
←30←10
←70
←65
←95
←35
←75
←80
←60
←90
←55
←50
←85
←45
←40
←5
10
5
←−15
←0
←−20
←−25
←−30
←−35
←−40
←−45
4000
3000
0
100
3000
50
2000
0
Γ
20
1000
o
−50
Column breakpoints
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0
Annotations denote column breakpoints
30
Table data
←85
←80
←75
←70
←65
←60
←55
←50
←45
←40
←35
←30
←25
←20
←15
←10
←−10
←−5
←5
←−15
←−20
←0 ←−25
←−30
←−35
←−40
←−45
40
Annotations denote column breakpoints
←95
←90
50
Table data
EGR
Table and breakpoints data for block:
PROVA_VVT_ottimnt/Mappa VV
T Ottimizzazione /EGR − nt
2000
0
1000
rpm
Row breakpoints
Column breakpoints
−50
0
Row breakpoints
Dip. di Elettronica e Informazione
Design of an
MPC regulator
18
Designed on a linear identified model (N = 2500 rpm, Γ = 30 Nm)
MPC
regulator
Torque
request Γ o
δu
u
Static
control maps
N
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Torque
Dip. di Elettronica e Informazione
Air path
Torque
Fuel cons.
NOx
CO
HC
19
MPC design
State estimator + MPC state feedback control law
Ny = 40 (prediction horizon) , Nu = 1 (control horizon) :
Performance index
J = J1 + J 2
Np
[
J1 = ∑ µ1 (Γ o ((h + l )Tc ) − Γ ((h + l )Tc ))
l =1
Np
2
[
2
]
2
2
J 2 = ∑ µ 2 NOx ((h + l )Tc ) + µ3δα (hTc ) + µ 4δEGR (hTc )
l =1
]
δα min ≤ δα ( hTc ) ≤ δα max
Constraints
δEGRmin ≤ δEGR ( hTc ) ≤ δEGRmax
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20
Experiments
30
With MPC
Without MPC
Set point
28
120
With MPC
Without MPC
Set point
110
100
24
Vehicle speed [Km/h]
Vehicle speed [Km/h]
26
22
20
18
90
80
70
16
60
14
50
12
5
10
15
20
25
30
Time [s]
35
40
50
45
Low speed
5
60
70
Time [s]
80
90
High speed
With MPC
Without MPC
100
With MPC
Without MPC
8
6
4
Torque error [Nm]
Torque error [Nm]
0
−5
2
0
−2
−4
−10
−6
−8
−15
5
10
15
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20
25
30
Time [s]
35
40
45
Dip. di Elettronica e Informazione
50
60
70
Time [s]
80
90
100
21
Results
Cost on
emissions
Cost on torque
error
Static maps
MPC weight
on torque
error
MPC
weight on
torque +
emissions
Static maps
MPC weight
on torque
error
MPC
weight on
torque +
emissions
Exp. 1
776.3
781.1
740.3
874.1
27.78
32.98
Exp.2
291.5
289.2
260.6
30
0.6287
5.085
Exp. 3
206.3
202.8
185.1
53.37
3.757
13.06
Exp. 4
1091
1096
1044
1042
56.21
282.6
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22
State feedback linearization
State feedback law
Simplified model
Constraints on fictitious inputs v
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23
Nonlinear observers
STATES ARE NOT MEASURABLE
SLIDING MODE NONLINEAR OBSERVER
The most uncertainty parameter is the
volumetric efficiency
NONLINEAR INPUT OBSERVER
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Dip. di Elettronica e Informazione
24
Linear MPC + F.L. + nonlinear observer
1/s
Torque req. poman
Linear
MPC
Feedback Wth, Wegr
linearization
pman,,NOx
engine
Nonlinear
state
observer
Main problem: constraints handling
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Dip. di Elettronica e Informazione
25
Pressure transient
Perturbed model
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Dip. di Elettronica e Informazione
26
Turbocharged Diesel engines
• Engine description
• Engine model
• Model analysis and validation
• Control problems
• “Classical” control structures
• Advanced control design
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Dip. di Elettronica e Informazione
27
Air path
Intake manifold
Exhaust manifold
Turbocharger
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Dip. di Elettronica e Informazione
28
Dynamic model
Dynamic equations
Mass balances in the manifolds
2 = w1e −wegr−w2t +wfuel
m
1 = w c1 − w1e + w egr
m
Energy balances
1
(w c1 (T c1 c pair − T1 c vair ) − w 1e T1 R + w egr (T egr c pegr − T1 c vegr ))
m 1 c vair
1
T2 =
(w1e(Tengcpair − T2cvair) − w2tT2R + wfuel(Tengcpegr − T2cvegr) - wegrT2R - Kconv(T2 - Tamb))
m2cvair
T 1 =
Turbocharger
P
− Pcompressor
N tc = turbine
N tc I tc
Pturbine
1− γ


  p2  γ 
= w 2t c pegr T2 ηt 1 −   
  p0  


Pcompressor
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γ −1


  p1  γ

= w c1c pair Tamb    − 1 ηc
  p0 



Dynamic model - 2
29
Variables to be determined:
Recirculating gas flow rate (wegr)
EGR valve opening (KAegr)
Gas flow rate to the engine (w1e)
Volumetric efficiency (ηvol)
Engine temperature (Teng)
Compressor air flow (wc1)
Compressor efficiency (ηc)
Turbine efficiency (ηt)
Static map
Turbine flow rate (w2t)
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Turbine flow rate
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31
Model characteristics
Inputs
Engine speed (N)
Fuel flow rate (wfuel)
EGR
VGT
Outputs
Intake manifold pressure (p1)
Compressor air flow rate (wc1)
Control variables
EGR
VGT
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32
Step responses - 1
VGT step variations, EGR constant
N=2500 [rpm]
N=3000 [rpm]
Wfuel=1.6 [g/s]
Wfuel=1.6 [g/s]
Different magnitudes
Change of signo
N=2000 [rpm]
Wfuel=2.3 [g/s]
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33
Step responses - 2
EGR step variations, VGT constant
N=2500 [rpm]
N=3000 [rpm]
Wfuel=1.6 [g/s]
Wfuel=1.6 [g/s]
N=2000 [rpm]
Wfuel=1 [g/s]
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34
Static gains
VGT-MAF
VGT-MAP
EGR-MAF
EGR-MAP
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Control design - 1
Feedforward control
PI + feedforward (limited operating range - N=2500[rpm] wfuel=1.6[g/s])
H2 + feedforward (limited operating range - N=2500[rpm] wfuel=1.6[g/s])
Feedforward
Torque/fuel
EGR=0
VGT=constant
rpm
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36
Control design - 2
Feedforward + PI control
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37
Nonlinear control design
State observer (sliding mode observer + unknown input observer)
+
Feedback linearization + linear MPC
or
Nonlinear MPC + interpolation
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38
Nonlinear MPC
1. Solve off-line (MUSCOD) a dynamic optimization problem for a great
number of initial conditions in the operating space (torque, rpm)
2. Use the results to find a suitable interpolation (NN – RBF - …) of the
state-feedback control law
3. Use it in conjunction with a state observer
mapo,,mafo
Interpolated EGR, VGT
N-MPC
Nonlinear
state
observer
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Diesel
engine
map,,maf
39
Hybrid fuel cell vehicle
•
A vehicle with a fuel cell stack and a battery
•
A detailed simulation model has been developed
•
A low level scheme controls the fuel cell acting on the compressor
command and on the anode and cathode valves
•
The high level MPC controller manages the power fluxes, i.e. the
power absorbed by the motor and the one provided by the fuel cell
•
The main constraints are related to the battery charge (SOC)
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40
Control strategy
Power
request
High level
control
iBatt
iFC
Low level
control
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Pmot = PFC + PBatt
41
High level control
Constraints:
min
max
δPmot
≤ δPmot (k ) ≤ δPmot
min
max
δI FC
≤ δI FC (k ) ≤ δI FC
SOCmin ≤ SOC (k ) ≤ SOCmax
Pmot
Power
Request
FCCurrent
Pmot = PFC + PBatt
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42
High level control - 3
System to be controlled
Pmot
FC System
Power
+
IFC
SOC
Battery
Performance index:
2
2
(k ) + r2 I FC
(k )
J = ∑ q1 (Power o (k ) Power (k )) + q2 (SOC o (k ) SOC (k )) + r1 Pmot
N 1
[
2
i =0
min
max
δPmot
≤ δPmot (k ) ≤ δPmot
Constraints:
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min
max
δI FC
≤ δI FC (k ) ≤ δI FC
SOCmin ≤ SOC (k ) ≤ SOCmax
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2
]
43
High level control - 2
Sampling time: 0.05 s
Two tunings:
• Low use of the battery
• High use of the battery
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Simulations - 1
Battery – high use
Battery – low use
Pmot = PFC + PBatt
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45
Simulations - 2
Battery – high use
Battery – low use
Pmot = PFC + PBatt
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