model predictive control for automotive applications

Paolo Falcone (UNIMORE/CHALMERS)

Department of Signals and Systems

Automatic Control Course – Guest Lecture December 11th, 2019 1

Model Predictive Control for Automotive Applications

Dipartimento di Ingegneria “Enzo Ferrari”

Universita’ di Modena e Reggio Emilia

Paolo FalconeDepartment of Electrical EngineeringChalmers University of TechnologyGöteborg, Sweden




Predictive Vehicle Dynamics Control




Problem complexityHow does the logic change if further actuators

are added?Engine torque

Active front/rear steering Active

suspensionsActive

differentials




Global Chassis Control (GCC) problem

GCC

ylateral

z yaw

pitch

vertical

longitudinal roll

y

qx f Fx FyFz

ü Front steeringü Four brakesü Engine torqueüActive suspensionsüActive differential

ü Longitudinal, lateral and vertical veloctyü Yaw, roll and pitch angles/rates

Dri

ver

Com

man

dsCoordinating vehicle actuators in order to control

multiple dynamics




Testing scenario. Autonomous path followingProblem setup:

• Double lane change • Driving on snow/ice, with

different entry speeds

Control objective:

Minimize angle and lateral distancedeviations from reference trajectory by changing the front wheel steering angleand the braking at the four wheels

Controlling longitudinal, lateral and yaw dynamics by varying front steering angle and braking the four wheels




Challenges

• Highly nonlinear MIMO system with uncertainties– Tires characteristics

• 6 DOF model– Longitudinal, lateral, vertical, roll, yaw and

pitch dynamics• Hard constraints

– Rate limitations in the actuators, vehicle physical limits

• Fast sampling time– Typically 20-50 ms




Model Predictive Control

Main ingredients• Vehicle model• Optimization problem




Outline

• Introduction and motivations• Vehicle modeling• Problem formulation• Experimental results




Modeling: bicycle and four wheels modelsModeling the vehicle motion in an inertial frame subject to lateral,

longitudinal and yaw dynamics

10 states, 5 inputs




Tire modelingStatic tire forces characteristics

),,,( zcc FsfF µa=




Tire modeling

zF

zcl FcFF µ£+ 22




Outline





NMPC Control design

Optimization problem

Vehicle dynamics

max,min uuu tk ££Input constraints

1,, max,min

-+=

D£D£D

p

tk

Httkuuu

!

Constraints on input changes

€

J ξ t( ),U( ) = ηt+ i,t −ηreft+i ,t Q

2+ ut+ i,t R

2

i=1

H p

∑Cost function tHttt puuU ,, += !

( ) tosubj.

,min UJ tUx • Non Linear Programming

(NLP) problem

• Complex NLP solvers required

Real time implementation byü limiting the number of iterationsü using short horizons

Real time testing at low speed

( )( )tktk

tktktk

huf

,,

,,,1 ,xh

xx

=

=+

Experimentally tested




LTV-MPC controllerApproximating the non-linear vehicle model with a Linear Time Varying

(LTV) model At, Bt, Ct, Dt.*

tttt DCBA ,,,

•Convex optimization problem. QP solvers

•Easier real-time implementation

•Longer horizons

Performance and stability issues* due to linear approximation

* Kothare and Morari, 1995, Wan and Kothare, 2003

* Falcone et al 2008




Constraints on tire slip angle

mina maxa

p

tk

Httktktk

+=

££

! ,, max,min aaa

Controller performs well up to 21 m/s

The system is still nonlinear

Stability achieved through ad hoc state and input constraints




Stability of the LTV-MPC approach

( ))(),()1( (1) tutft xx =+

Consider the discrete time nonlinear system:

nRÎx mRuÎ

We consider the following linear approximation over the horizon N:

1,)()()1( ,,,

-+=

++@+

NttkdkuBkAk tktktk

!

xx

€

ξ0(k +1) = f ξ0(k),u(t −1)( ), ξ0(k) = ξ(t)dk,t = ξ0(k +1) − Ak,tξ0(k) − Bk,tu(t −1)€

Ak,t =∂f∂x ξ 0 (k )

u( t−1)

, Bk,t =∂f∂u ξ 0 (k )

u( t−1)




Stability of the LTV-MPC approach

€

(2) VN* ξ(t)( ) = min

ut ,t ,…ut+N−1,t

Qξt+ i,t 2

2+

i=1

N −1

∑ Rut+ i,t 2

2+

i=0

N −1

∑ Pξt+N ,t 2

2

subject to : ξk+1,t = Atξk,t + Btuk,t + dk,t , k = t,…,t + N −1 ξk,t ∈ X, k = t,…,t + N −1 uk,t ∈U, k = t,…,t + N −1 ξk+N ,t ∈ X f

ξt ,t = ξ(t)

€

(3) u t( ) = ut,t* ξ(t)( )




Stability of the LTV-MPC approachTheorem. The system (1) with the control law (2)-(3), where Xf=0, is uniformly asymptotically stable if

where:€

Q ˆ ξ t+N −1,t 2

2+ Rut+N −1,t 2

2≤ Qξt,t−1

*2

2+ Rut−1,t−1

*2

2− Q ˆ ξ t+ i,t − ξt+ i,t−1

*( )2

2− γ

i=1

N −2

∑

€

ˆ ξ k+1,t = Atˆ ξ k,t + Btuk,t−1

* + dk,t

k = t,…,t + N − 2ˆ ξ t ,t = ξ(t)

State and input convex constraints

Liu, 1968. Chen and Shaw. 1982. Mayne et al. 2000

€

γ = 2 Q ˆ ξ t+ i,t − ξ t+i ,t−1

*( )2Qξ

t+i ,t−1

*

2i=1

N −2

∑




What does that mean?

t

úû

ùêë

é

--

--

11

11

tt

tt

DCBA

Y

1-t t

úû

ùêë

é

tt

tt

DCBA

2-+ Nt

€

l ˆ ξ t+N −1,t ,ut+N −1,t( ) ≤ l ξt−1,t−1* ,ut−1,t−1

*( ) −Γ Q ˆ ξ t+ i,t − ξt+ i,t−1*( )

2

2& ' (

) * +

*1, -tkx

tk ,x̂




Simulation results at 21 m/s

The controller is able to stabilize the vehicle without any ‘ad hoc’ constraint.




Outline





Testing: Sault St. Marie in Upper Peninsula, MI, USA




Summary• Excellent performance • Limited tuning effort (less than 10 run ~ 1 hr)• Vehicle stabilized up to 70 Km/h on snowy tracks

• Coordination of steering and braking– Braking is delivered on the “same side” of the steering

• Front/rear braking distribution– Shifting the braking to the non saturated axle

• Countersteering– Steering in opposite direction of path following to prevent spinning




Test @ 40 Kph. Steering and braking coordination




Test @ 40 Kph. Steering and braking coordination

Left side braking Right side braking




Test @ 40 Kph. Countersteering manoeuvre.

Yaw instability induced by

large acceleration




Test @ 40 Kph. Countersteering manoeuvre




Experimental results




Acknowledgments

• Francesco Borrelli (UC Berkeley)• Eric Tseng (Ford)• Davor Hrovat (Ford)




Long Heavy Vehicles Combinations




Control Objectives

Reducing the yaw rate rearward amplifications

and , by means of the steering angles

while bounding the steering angles and rates of steering

Achieving the control objectives by solving a yaw rate tracking problem where

Reference models designed in order to remove resonance peaks in the yaw rates responses

refr ,2

11d

G21ref

refr ,3G31ref

€

r2 /r1

32 ,dd

€

r3 /r1




Reference Models




MPC Problem FormulationVehicle (linear) model

€

xk+1 = A(vx )xk + B(vx )uk + E(vx )dkyk = Cxk

€

x =

vyr1θ1

θ2˙ θ 1˙ θ 2

#

$

% % % % % % %

&

'

( ( ( ( ( ( (

úû

ùêë

é=

3

2

dd

u

€

vx =

vx1

vx2

vx3

"

#

$ $ $

%

&

' ' '

11d=d

€

y =

r2r3θ1θ2

#

$

% % % %

&

'

( ( ( (

€

J = Q(yt+ i − yref ,t+ i) 2

2+ Sut+ i−1 2

2+ RΔut+ i−1 2

2$ % & '

( )

i=1

N

∑

Cost functionYaw rate tracking problem translated into a cost function minimization problem




MPC Problem Formulation

€

minΔut ,…,Δut+N−1

Q(yt+ i − yref ,t+ i) 2

2+ Sut+ i−1 2

2+ RΔut+ i−1 2

2$ % & '

( )

i=1

N

∑

subject to

€

u*(t) = u(t −1) + Δu*(t,x(t))

Vehicle dynamics

Actuator limitations

The resulting state feedback steering control law is€

xk+1 = A(vx )xk + B(vx )uk + E(vx )dk uk = uk−1 + Δukyk = Cxk

$

% &

' &

€

umin ≤ uk ≤ umax

Δumin ≤ Δuk ≤ Δumax

$ % &




Experimental Results

Testing at Mira Test Center:Single lane change




Summary

1. RWA close to one– Good reference tracking

2. Smooth steering commands3. Small articulation angles at the end of the maneuver (Both

dolly and trailer aligned with the truck)– Please note the sensor offset




Yaw Rates

124 125 126 127 128 129 130 131 132-8

-6

-4

-2

0

2

4

6

8

Time (sec)

Yaw

rate

(Deg

/sec

)Yaw rates, truck tests. SLC Maneuver

Dolly yaw rateTrailer yaw rateTruck yaw rate




Dolly Yaw Rate

124 125 126 127 128 129 130 131 132-8

-6

-4

-2

0

2

4

6

8

Time (sec)

Yaw

rate

(Deg

/sec

)Yaw rate dolly, truck tests. SLC Maneuver

Dolly yaw rateDolly yaw rate reference




Trailer Yaw Rate

124 125 126 127 128 129 130 131 132-8

-6

-4

-2

0

2

4

6

Time (sec)

Yaw

rate

(Deg

/sec

)Yaw rate trailer, truck tests. SLC Maneuver

Trailer yaw rateTrailer yaw rate reference




Summary







Steering Angles

124 125 126 127 128 129 130 131 132-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Time (sec)

Stee

ring

(Deg

)Steerings, truck tests. SLC Maneuver

Dolly steering angleTrailer steering angleTruck steering angle




Summary







124 125 126 127 128 129 130 131 132-2

-1

0

1

2

3

Time (sec)

Artic

ulat

ion

angl

e (D

eg)

Dolly articulation angle, truck tests. SLC Maneuver

Dolly articulation angleReference signal

Dolly Articulation Angle

Sensor offset




Trailer Articulation Angle

124 125 126 127 128 129 130 131 132-3

-2

-1

0

1

2

3

4

Time (sec)

Artic

ulat

ion

angl

e (D

eg)

Trailer articulation angle, truck tests. SLC ManeuveR

Trailer articulation angleReference signal




Remarks

• Good tracking with minimal design and tuning efforts• Actuators physical limits included in control design• Possibility of easily

– Include constraints to guarantee RWA<1– Include constraints to limit lateral accelerations– Include constraints to limit off-tracking – Combine steering and braking commands– Guarantee perfect alignment of the combination despite of sensors

offsets




Acknowledgments

• Mathias Lidberg• Jonas Fredriksson• Kristoffer Tagesson (Volvo Trucks)• Leo Laine (Volvo Trucks)• Stefan Edlund (Volvo Trucks)• Richard Roebuck (Cambridge)• Andrew Odhams (Cambridge)

model predictive control for automotive applications

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