Model Predictive Control for Automotive Applications

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Paolo Falcone (UNIMORE/CHALMERS) Automatic Control Course – Guest Lecture December 11^th, 2019 1

Model Predictive Control for Automotive Applications

Dipartimento di Ingegneria “Enzo Ferrari”

Universita’ di Modena e Reggio Emilia

Paolo Falcone

Department of Electrical Engineering Chalmers University of Technology Göteborg, Sweden

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Paolo Falcone (UNIMORE/CHALMERS)

Department of Signals and Systems

Automatic Control Course – Guest Lecture December 11^th, 2019 2

Predictive Vehicle Dynamics Control

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Problem complexity

How does the logic change if further actuators are added?

Engine torque

Active front/rear

steering Active

suspensions Active

differentials

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Global Chassis Control (GCC) problem

GCC

lateral

y z

_yaw

pitch vertical

longitudinal rol l

y

x

^f

^F

^x

_F _F

_y q

z

ü Front steering ü Four brakes ü Engine torque ü Active suspensions ü Active differential

ü Longitudinal, lateral and vertical velocty

ü Yaw, roll and pitch angles/rates

Driver Commands

Coordinating vehicle actuators in order to control

multiple dynamics

(5)

Testing scenario. Autonomous path following

Problem setup:

• Double lane change

• Driving on snow/ice, with different entry speeds

Control objective:

Minimize angle and lateral distance deviations from reference trajectory

by changing the front wheel steering angle and the braking at the four wheels

Controlling longitudinal, lateral and yaw dynamics by varying

front steering angle and braking the four wheels

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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

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Model Predictive Control

Main ingredients

• Vehicle model

• Optimization problem

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Outline

• Introduction and motivations

• Vehicle modeling

• Problem formulation

• Experimental results

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Modeling: bicycle and four wheels models

Modeling the vehicle motion in an inertial frame subject to lateral, longitudinal and yaw dynamics

10 states, 5 inputs

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Tire modeling

Static tire forces characteristics

) ,

, ,

(

_z

c

f s F

F = a µ

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Tire modeling

F

z

z c

l

cF F

F

²

+

²

£ µ

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Outline

• Introduction and motivations

• Vehicle modeling

• Problem formulation

• Experimental results

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NMPC Control design

Optimization problem

Vehicle dynamics

max ,

min u u

u £ _k _t £ Input constraints

1 ,

,

_min _, _max

- +

=

D

£ D

p t

k

H t

t k

u u

u

! Constraints on

input changes

€

J

(

ξ

( )

t ^,U

)

⁼ ^η^{t +i,t} ⁻^η^reft +i ,t Q 2

+ u_{t +i,t R}²

i=1 H_p

Cost function

∑

Û = û_t_,_t!û_t₊_H_p_,_t

( )

to subj.

, min J _t U

U x

• Non Linear Programming

(NLP) problem

• Complex NLP solvers required

Real time implementation by ü limiting the number of iterations

ü using short horizons

Real time testing at low speed

( )

_k _t

t k

t k t k t

k

h

u f

, ,

, , ,

1 ,

x h

x x

=

+ =

Experimentally tested

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LTV-MPC controller

Approximating the non-linear vehicle model with a Linear Time Varying (LTV) model A

_t

, B

_t

, C

_t

, D

_t

.*

t t t

t

B C D

A , , ,

•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

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Constraints on tire slip angle

a

min a_max

p t

k

H t

t k

t k t

k

+

=

£

!

,

, , max

min

a a

a

Controller performs well up to 21 m/s

The system is still nonlinear

Stability achieved through ad hoc state and input constraints

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Stability of the LTV-MPC approach

( ⁽ ^), ⁽ ⁾ )

) 1 (

(1) x t + = f x t u t

Consider the discrete time nonlinear system:

R

n

x Î _u _Î _R

^m

We consider the following linear approximation over the horizon N:

1 ,

) ( )

( )

1 (

_, _, _,

- +

=

+ +

@ +

N t

t k

d k

u B k

A

k

_k _t _k _t _k _t

! x x

€

ξ

₀

(k +1) = f ( ξ

₀

(k),u(t −1) ) ^, ^ξ

⁰

^{(k) =} ^ξ ^(t)

d

_k,t

= ξ

₀

(k +1) − A

_k,t

ξ

₀

(k) − B

_k,t

u(t −1)

€

A

_k,t

= ∂f

∂x

^ξ_{u(t −1)}⁰^{(k )}

, B

_k,t

= ∂f

∂u

^ξ_{u(t −1)}⁰^{(k )}

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Stability of the LTV-MPC approach

€

(2) V

_N^*

( ξ(t) ) ⁼ ^min

u_{t ,t},…u_{t +N −1,t}

Qξ

_{t +i,t 2}²

+

i=1 N −1

∑ ^Ru

t +i,t 2

2

+

i=0 N −1

∑ ^Pξ

t +N,t 2

2

subject to :

ξ

_{k +1,t}

= A

_t

ξ

_k,t

+ B

_t

u

_k,t

+ d

_k,t

, k = t, …,t + N −1 ξ

_k,t

∈ X, k = t, …,t + N −1

u

_k,t

∈U, k = t, …,t + N −1 ξ

_{k +N ,t}

∈ X

_f

ξ

_t,t

= ξ(t)

€

(3) u t ( ) ^{= u}

t,t

*

( ξ ^(t) )

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Stability of the LTV-MPC approach

Theorem. The system (1) with the control law (2)-(3), where X

_f

=0, is uniformly asymptotically stable if

where:

€

Q ˆ ξ

_{t +N −1,t}

2

+ Ru

t +N −1,t 2

2

≤ Q ξ

_{t,t −1}^* ₂²

+ Ru

_{t −1,t −1}^* ₂²

− Q ˆ ( ξ

_{t +i,t}

− ξ

_{t +i,t −1}^*

)

₂²

⁻ ^γ

i=1 N −2

∑

€

ξ ˆ

_{k +1,t}

= A

_t

ξ ˆ

_k,t

+ B

_t

u

_{k,t −1}^*

+ d

_k,t

k = t, …,t + N − 2

ξ ˆ

_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}

)

₂

^Q ^ξ

^*^{t +i,t −1} ²

i=1 N −2

∑

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What does that mean?

t

úû ê ù

ë é

- -

1 1

t t

D C

B A

Y

- 1

t t

úû ê ù

ë é

t t

D C

B A

- 2 + N t

€

l ˆ ( ξ

_{t +N −1,t}

^,u

_{t +N −1,t}

) ^{≤ l} ⁽ ^ξ

^{t −1,t −1}^*

^,u

^{t −1,t −1}^*

⁾ ^{− Γ Q ˆ} ^& _' ⁽ ( ^ξ

^{t +i,t}

⁻ ^ξ

^{t +i,t −1}^*

)

₂²

⁾ _* ⁺

* 1 ,t-

x

k

t

ˆ

k ,

x

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Simulation results at 21 m/s

The controller is able to stabilize the vehicle

without any ‘ad hoc’ constraint.

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Outline

• Introduction and motivations

• Vehicle modeling

• Problem formulation

• Experimental results

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

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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

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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

• Front/rear braking distribution

• Countersteering

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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

• Front/rear braking distribution

• Countersteering

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Test @ 40 Kph. Steering and braking coordination

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Test @ 40 Kph. Steering and braking coordination

Left side braking Right side braking

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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

• Front/rear braking distribution

• Countersteering

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Test @ 40 Kph. Countersteering manoeuvre.

Yaw

instability induced by

large

acceleration

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Test @ 40 Kph. Countersteering manoeuvre

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Test @ 70 Kph. Countersteering manoeuvre

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Test @ 70 Kph. Countersteering manoeuvre

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Test @ 70 Kph. Countersteering manoeuvre

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Experimental results

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Acknowledgments

• Francesco Borrelli (UC Berkeley)

• Eric Tseng (Ford)

• Davor Hrovat (Ford)

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Long Heavy Vehicles Combinations

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Long Heavy Vehicles Combinations

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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

r

₂_,ref

d

11

G

_21ref

r

₃_,ref

G

_31ref

€

r

₂

/r

₁

3 2

, d d

€

r

₃

/r

₁

(40)

Reference Models

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MPC Problem Formulation

Vehicle (linear) model

€

x

_{k +1}

= A(v

_x

)x

_k

+ B(v

_x

)u

_k

+ E(v

_x

)d

_k

y

_k

= Cx

_k

€

x = v

_y

r

₁

θ

₁

θ

₂

θ ˙

₁

θ ˙

₂

#

$

%

&

' ( ( ( ( ( ( (

ú û ê ù ë

= é

3 2

d u d

€

v

_x

= v

¹_x

v

_x²

v

_x³

"

#

$

%

&

' ' '

d

11

= d

€

y = r

₂

r

₃

θ

₁

θ

₂

#

$

%

&

' ( ( ( (

€

J = Q(y

_{t +i}

− y

_{ref ,t +i}

)

2

+ Su

_{t +i−1 2}²

+ RΔu

_{t +i−1 2}²

$

% & '

( )

i=1 N

∑

Cost function

Yaw rate tracking problem translated into a cost function

minimization problem

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MPC Problem Formulation

€

Δu_t,…,Δu

min

_{t +N −1}

Q(y

_{t +i}

− y

_{ref ,t +i}

)

2

+ Su

_{t +i−1 2}²

+ RΔu

_{t +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

€

x

_{k +1}

= A(v

_x

)x

_k

+ B(v

_x

)u

_k

+ E(v

_x

)d

_k

u

_k

= u

_{k −1}

+ Δu

_k

y

_k

= Cx

_k

$

% &

' &

€

u

_min

≤ u

_k

≤ u

_max

Δu

_min

≤ Δu

_k

≤ Δu

_max

$ %

&

(43)

Experimental Results

Testing at Mira Test Center:

Single lane change

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Summary

1. RWA close to one

– Good reference tracking

2. Smooth steering commands

3. Small articulation angles at the end of the maneuver (Both dolly and trailer aligned with the truck)

– Please note the sensor offset

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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 rate Trailer yaw rate Truck yaw rate

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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 rate

Dolly yaw rate reference

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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 rate

Trailer yaw rate reference

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Summary

1. RWA close to one

2. Smooth steering commands

3. Small articulation angles at the end of the maneuver (Both dolly and trailer aligned with the truck)

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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)

Steering (Deg)

Steerings, truck tests. SLC Maneuver

Dolly steering angle Trailer steering angle Truck steering angle

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Summary

1. RWA close to one

2. Smooth steering commands

3. Small articulation angles at the end of the maneuver (Both dolly and trailer aligned with the truck)

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124 125 126 127 128 129 130 131 132

-2 -1 0 1 2 3

Time (sec)

Articulation angle (Deg)

Dolly articulation angle, truck tests. SLC Maneuver

Dolly articulation angle Reference signal

Dolly Articulation Angle

Sensor offset

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Trailer Articulation Angle

124 125 126 127 128 129 130 131 132

-3 -2 -1 0 1 2 3 4

Time (sec)

Articulation angle (Deg)

Trailer articulation angle, truck tests. SLC ManeuveR

Trailer articulation angle Reference signal

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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