Robust Multivariable Linear Parameter Varying ... - Google Sites

Vehicle modeling & Validation

Semi-active suspensions Control

Global Chassis Control

(Some) conclusions & perspectives

Appendix

Robust Multivariable Linear Parameter Varying Automotive Global Chassis Control C. Poussot-Vassal PhD. defense, September 26th 2008 GIPSA-lab, Control Systems Department, Grenoble, France

Jury de thèse: Rapporteurs: Examinateurs: Co-directeurs:

Brigitte d’Andréa-Novel (Professeur, Ecole des Mines de Paris) Sergio M. Savaresi (Professeur, Politecnico di Milano) Michel Basset (Professeur, Université de Haute Alsace) Peter Gáspár (Directeur de Recherche, Académie des Sciences de Budapest) Luc Dugard (Directeur de Recherche CNRS, GIPSA-lab) Olivier Sename (Professeur, Grenoble INP, GIPSA-lab)

C. Poussot-Vassal - PhD. defense [GIPSA-lab / SLR team]

1/57





Appendix

PhD. context & objectives Continuation of: I

D. Sammier, 2001 (semi-active suspension modeling and control)

I

A. Zin, 2005 (active suspension control toward global chassis control)

Investigations on: I

Vehicle dynamics modeling & analysis

I

(Semi-)active suspensions modeling & control

I

Global Chassis Control (GCC) involving suspensions, steering & braking systems

I

LPV robust control design (H∞ , H2 , multi-criteria)


2/57





Appendix

PhD. context & objectives Continuation of: I

D. Sammier, 2001 (semi-active suspension modeling and control)

I

A. Zin, 2005 (active suspension control toward global chassis control)

Investigations on: I

Vehicle dynamics modeling & analysis

I

(Semi-)active suspensions modeling & control

I

Global Chassis Control (GCC) involving suspensions, steering & braking systems

I

LPV robust control design (H∞ , H2 , multi-criteria)


2/57





Appendix

PhD. collaborations & results Jozsef Bokor, Peter Gáspár, Zoltan Szabó - MTA SZTAKI, Hungary: I

Semi-active suspensions control [IFAC SSC 2007], [Control Engineering Practice 2008]

I

Half vehicle suspension control through qLPV/H∞ /H2 control [VSDIA 2006]

I

Gain scheduled Braking & Active suspensions control [IFAC AAC 2007], [IFAC WC 2008], [IEEE trans. CST (under review)]

Ricardo Ramirez-Mendoza - Tecnologico de Monterrey, Mexico: I

Semi-active suspensions [IFAC Mechatronics 2006]

I

Adaptive active suspension control & multi-body modeling [IFAC WC 2008]

"Grenoble team" I

Analysis of semi-active suspension [IFAC WC 2008]

I

Skyhook & Anti-roll bar distribution [International Journal of Vehicle Autonomous Systems 2009]

I

ABS discussion [European Journal of Control 2008]

I

Braking & steering control [IEEE CDC 2008]


3/57





Appendix

About this presentation Today’s presentation focus: I

Vehicle modeling & "validation" I

I

LPV semi-active suspension control design I

I

[coll. MIAM, Mulhouse]

vehicle dynamic toolbox (still under development)

[coll. SZTAKI, Budapest]

in Control Engineering Practice 2008

GCC involving braking & steering systems I

in IEEE Conference on Decision and Control 2008


4/57





Appendix





Appendix


5/57





Appendix

Vehicle modeling & Validation Steering system

Actuators specification (suspensions, braking, steering) Suspension system

Braking system

Vehicle modeling (dynamical equations)

Vehicle path 150

Validation (experimental tests)

y [m]

100

50

NL model Measure 0 0

10

20

30

40

50

60

70

80

x [m]


6/57





Appendix

Suspension system Objective I

Link between unsprung and sprung masses

I

Influences comfort / road-holding performances

I

Involves vertical (zs , zus ) dynamics Nonlinear Renault Mégane front spring force

Nonlinear Renault Mégane front damper force

4000

1500

3000 1000 2000

Fc [N]

Fk [N]

1000 0

500

0

−1000 −2000

−500 −3000 −4000 −0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

zdef [m]


0.04

0.06

−1000 −1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

z’def [m/s]

7/57





Appendix


Link between unsprung (mus ) and sprung (ms ) masses

I


I

Involves vertical (zs , zus ) dynamics

Passive quarter vehicle model

{ms , zs } {mus , zus }


7/57





Appendix



I


I


Semi-active quarter vehicle model



7/57





Appendix



I


I


Active quarter vehicle model



7/57





Appendix

Suspension system Half and Full vertical vehicle models I

Extends the quarter vehicle model

I

Involves vertical (zs , zus ), pitch (φ) dynamics

{ms , zs , φ}


8/57





Appendix

Suspension system Half and Full vertical vehicle models I

Extends the quarter vehicle model

I

Involves vertical (zs , zus ), pitch (φ) and roll (θ) dynamics

{ms , zs , φ}


{ms , zs , θ}

8/57





Appendix

Vehicle model - dynamical equations (1/3) Full vertical model

z¨s züsij θ¨ ¨ φ

= = = =

− Fszf l + Fszf r + Fszrl + Fszrr /ms Fszij − Ftzij /musij (Fszrl − Fszrr )tr + (Fszf l − Fszf r )tf /Ix (Fszrr + Fszrl )lr − (Fszf r + Fszf l )lf /Iy


9/57





Appendix

Wheel & Braking system Objective I

Link between wheel and road

I

Influences safety performances

I

Involves longitudinal (xs ) rotational (ω)and slipping (λ =


v−Rω ) max(v,Rω)

dynamics

10/57





Appendix

Wheel & Braking system Objective I

Link between wheel and road (zr , µ)

I


I

Involves longitudinal (xs ) rotational (ω)and slipping (λ =

v−Rω ) max(v,Rω)

dynamics

Normalized longitudinal tire force

Extended quarter vehicle model Dry 1

Cobblestone

Ftx/Fn

0.8

0.6

Wet 0.4

0.2

Icy 0 0


0.1

0.2

0.3

0.4

0.5

λ

0.6

0.7

0.8

0.9

1

10/57





Appendix

Vehicle model - dynamical equations (2/3) Full vertical model


= = = =

− Fszf l + Fszf r + Fszrl + Fszrr /ms Fszij − Ftzij /musij (Fszrl − Fszrr )tr + (Fszf l − Fszf r )tf /Ix (Fszrr + Fszrl )lr − (Fszf r + Fszf l )lf /Iy


11/57





Appendix

Vehicle model - dynamical equations (2/3) Full vertical and longitudinal model

x ¨s z¨s

(Ftxf r + Ftxf l ) + (Ftxrr + Ftxrl ) /m − Fszf l + Fszf r + Fszrl + Fszrr /ms Fszij − Ftzij /musij (Fszrl − Fszrr )tr + (Fszf l − Fszf r )tf /Ix (Fszrr + Fszrl )lr − (Fszf r + Fszf l )lf +mh¨ xs /Iy

züsij θ¨ ¨ φ

= = = = =

λij

=

vij −Rij ωij max(vij ,Rij ωij )

ω ˙ ij

=

(−RFtxij (µ, λ, Fn ) + Tb )/Iw


11/57





Appendix

Wheel & Steering system Objective I

Wheel / road contact

I


I

Involves lateral (ys ), side slip angle (β) and yaw (ψ) dynamics


12/57





Appendix

Wheel & Steering system Objective I

Wheel / road contact

I


I

Involves lateral (ys ), side slip angle (β) and yaw (ψ) dynamics

Bicycle model ys

6 Ftyf ]

− → v

1 Ftxf * K β

- xs

ψ


12/57





Appendix

Vehicle model - dynamical equations (3/3) Full vertical and longitudinal model

(Ftxf r + Ftxf l ) + (Ftxrr + Ftxrl ) /m

x ¨s

=


= = = =

− Fszf l + Fszf r + Fszrl + Fszrr /ms Fszij − Ftzij /musij (Fszrl − Fszrr )tr + (Fszf l − Fszf r )tf /Ix (Fszrr + Fszrl )lr − (Fszf r + Fszf l )lf + mh¨ xs /Iy

λij

=

vij −Rij ωij max(vij ,Rij ωij )

ω ˙ ij

=

(−RFtxij (µ, λ, Fn ) + Tb )/Iw


13/57





Appendix

Vehicle model - dynamical equations (3/3) Full model

x ¨s y¨s z¨s züsij θ¨ ¨ φ ¨ ψ

= = = = = = =

λij

=

ω ˙ ij

=

βij

=

(Ftxf r + Ftxf l )cos(δ) + (Ftxrr + Ftxrl )−(Ftyf r + Ftyf l ) sin(δ) + mψ˙ y˙ s /m (Ftyf r + Ftyf l ) cos(δ) + (Ftyrr + Ftyrl ) + (Ftxf r + Ftxf l ) sin(δ) − mψ˙ x˙ s /m − Fszf l + Fszf r + Fszrl + Fszrr /ms Fszij − Ftzij /musij (Fszrl − Fszrr )tr + (Fszf l − Fszf r )tf −mh¨ ys + (Iy − Iz )ψ˙ φ˙ /Ix (Fszrr + Fszrl )lr − (Fszf r + Fszf l )lf + mh¨ xs +(Iz − Ix )ψ˙ θ˙ /Iy (Ftyf r + Ftyf l )lf cos(δ) − (Ftyrr + Ftyrl )lr + (Ftxf r + Ftxf l )lf sin(δ) +(Ftxrr − Ftxrl )tr + (Ftxf r − Ftxf l )tf cos(δ) − (Ftxf r − Ftxf l )tf sin(δ) +(Ix − Iy )θ˙ φ˙ /Iz vij −Rij ωij cos βij max(vij ,Rij ωij cos βij )

(−RFtxij (µ, λ, Fn ) + Tb )/Iw x ˙ arctan y˙ ij

ij C. Poussot-Vassal - PhD. defense [GIPSA-lab / SLR team]

13/57





Appendix

Vehicle model - dynamical equations (3/3) Full model

x ¨s y¨s z¨s züsij θ¨ ¨ φ ¨ ψ

= = = = = = =

λij

=

ω ˙ ij

=

βij

=

(Ftxf r + Ftxf l )cos(δ) + (Ftxrr + Ftxrl )−(Ftyf r + Ftyf l ) sin(δ) + mψ˙ y˙ s −Fdx /m (Ftyf r + Ftyf l ) cos(δ) + (Ftyrr + Ftyrl ) + (Ftxf r + Ftxf l ) sin(δ) − mψ˙ x˙ s −Fdy /m − Fszf l + Fszf r + Fszrl + Fszrr +Fdz /ms Fszij − Ftzij /musij ˙ (Fszrl − Fszrr )tr + (Fszf l − Fszf r )tf −mh¨ ys + (Iy − Iz )ψ˙ φ+M dx /Ix ˙ (Fszrr + Fszrl )lr − (Fszf r + Fszf l )lf + mh¨ xs +(Iz − Ix )ψ˙ θ+M dy /Iy (Ftyf r + Ftyf l )lf cos(δ) − (Ftyrr + Ftyrl )lr + (Ftxf r + Ftxf l )lf sin(δ) +(Ftxrr − Ftxrl )tr + (Ftxf r − Ftxf l )tf cos(δ) − (Ftxf r − Ftxf l )tf sin(δ) ˙ +(Ix − Iy )θ˙ φ+M dz /Iz vij −Rij ωij cos βij max(vij ,Rij ωij cos βij )

(−RFtxij (µ, λ, Fn ) + Tb )/Iw x ˙ arctan y˙ ij

ij C. Poussot-Vassal - PhD. defense [GIPSA-lab / SLR team]

13/57





Appendix

Vehicle model - synopsis

q

?

?

z˙s , zs z˙us , zus

-

Suspensions



Fszij

xs



-  ys 

-

zs

Chassis  

 x ¨s , y¨s ˙ ψ, v,  Fsz , zus

6

? -

Wheels

6


Ftx,y,z   λij - β  ij ωij (tire, wheel dynamics)

(vehicle dynamics) 

-

 θ  φ  ψ

14/57





Appendix

Vehicle model - synopsis Fdx,y,z & Mdx,y,z

(external disturbances)

q

?

?


-

Suspensions



Fszij

xs



-  ys 

-

zs

Chassis  

 x ¨s , y¨s ˙ ψ, v,  Fsz , zus

6

? -

Wheels

6

(road characteristics) [µij , zrij ]




-

 θ  φ  ψ

14/57





Appendix

Vehicle model - synopsis Fdx,y,z & Mdx,y,z (suspensions control)

(external disturbances)

uij

q

?

?


-

Suspensions δ



 x ¨s , y¨s ˙ ψ, v,  Fsz , zus

6

xs



-  ys 

-

zs

(braking & steering control) [Tbij , δ] 



Fszij

? -

Wheels

6

(road characteristics) [µij , zrij ]


Chassis



-

 θ  φ  ψ

14/57





Appendix

Experimental validation [coll. MIAM, Mulhouse] Double line change manoeuver at 60km/h Yaw rate (ψ’)

Lateral speed (vy)

50

6

40 4

30 20

ψ’ [deg/s]

vy [km/h]

2

0

−2

10 0 −10 −20 −30

−4

−6 44

NL model Measure 45

46

−40 47

48

49

50

51

52

−50 44

53

NL model Measure 45

46

47

48

Time [s]

49

50

51

52

53

Time [s] Vehicle path

Roll speed (θ’) 150

1.5

1

100

y [m]

θ’ [deg/s]

0.5

0

50

−0.5

−1

−1.5 44

NL model Measure 45

46

NL model Measure 47

48

49

50

51

Time [s]


52

53

0 0

10

20

30

40

50

60

70

80

x [m]

15/57





Appendix

Semi-active suspensions Control Model and actuator

Control design

SER representation 0.2 0.15 0.1

u [N]

0.05

Active H∞ Clipped H∞ LPV H∞

Validation (simulations tests)

SH−ADD ADD

0 −0.05 −0.1 −0.15 −0.2 −0.05

−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.05

zdef’ [m/s]


16/57





Appendix

Framework & Objectives Semi-active suspension control research I

LQ clipped: complex, involve state measurement Tseng et al. [VSD, 1994]

I

H∞ & skyhook clipped: Sammier et al. [VSD, 2003]

I

MPC based: involve optimization, state measurement, robustness? Canale et al. [Trans. CST, 2006], Giorgetti et al. [IJRNLC, 2006], Guia et al. [VSD, 2004]

I

ADD, Mixed SH-ADD: simple structure, comfort oriented Savaresi et al.[ASME, 2005, 2007]

Objectives I

Enhance passenger comfort & road-holding

I

Ensure semi-active constraint

I

Simplify controller structure


17/57





Appendix

Framework & Objectives Semi-active suspension control research I

LQ clipped: complex, involve state measurement Tseng et al. [VSD, 1994]

I

H∞ & skyhook clipped: Sammier et al. [VSD, 2003]

I

MPC based: involve optimization, state measurement, robustness? Canale et al. [Trans. CST, 2006], Giorgetti et al. [IJRNLC, 2006], Guia et al. [VSD, 2004]

I

ADD, Mixed SH-ADD: simple structure, comfort oriented Savaresi et al.[ASME, 2005, 2007]

Objectives I

Enhance passenger comfort & road-holding

I

Ensure semi-active constraint

I

Simplify controller structure


17/57





Appendix

Quarter vehicle model Passive & Controlled damper case zs

ms Fk

zus

kt

>

mus

zus

kt zr


u

Fk

Fc

mus

zs

ms

zr

18/57





Appendix

Quarter vehicle model Passive & Controlled damper case zs

ms u

Fk

>

mus

zus

kt zr


18/57





Appendix

Controller structure philosophy Principle The idea is to design a controller I

where the control input is limited when the required force is achievable by the semi-active actuator

I

synthesized on the quarter vehicle model

Methodology We use the H∞ synthesis, extended to LPV systems Shamma et al. [Automatica, 1991], Scherer et al. [TAC, 1997] and Scherer [IJRNLC, 1996]. ||z||2 ||w||2

I

H∞ synthesis: frequency based performance criteria, disturbance rejection)

I

LPV: Linear Parameter Varying, to handle nonlinearity or derive adaptive controller


(as pole placement,

19/57





Appendix

Controller structure philosophy Principle The idea is to design a controller I

where the control input is limited when the required force is achievable by the semi-active actuator

I

synthesized on the quarter vehicle model

Methodology We use the H∞ synthesis, extended to LPV systems Shamma et al. [Automatica, 1991], Scherer et al. [TAC, 1997] and Scherer [IJRNLC, 1996]. ||z||2 ||w||2

I

H∞ synthesis: frequency based performance criteria, disturbance rejection)

I

LPV: Linear Parameter Varying, to handle nonlinearity or derive adaptive controller


(as pole placement,

19/57





Appendix

Implementation scheme & principle

u

- SA actuator

−

+ -

v

ε

Ω

3

z˙def

? ρ(ε) ρ u

?

c0 z˙def + uH∞ (ρ)


[zdef , z˙def ]

20/57





Appendix

Scheduling strategy u

-SA actuator + − ?v ρ(ε)

Scheduling strategy

u

Ω

ρ

? C(ρ)

1 z˙ def

[zdef , z˙ def ]

Scheduling parameter ρ(ε) 10 9 8 7

ρ(ε)

6

I

5

ρ(ε) ∈

0.1

10

4 3

8

µ = 10

2

7

µ = 10

1 0 −1

6

µ = 10 −0.8

−0.6

−0.4

−0.2

0

ε=u−v

0.2


0.4

0.6

0.8

1 −3

x 10

21/57





Appendix

LPV control design (1/5)

w1

- Wzr

zs

- zr

zdef z3 -

- Wu (ρ) -

u

C(ρ)

6

?

ρ(ε)

+

6

- z1

- Wzdef - z2

Σ y

- u

ρ

- Wzs

z˙def

v

−

ε

?n +


Wn

w2

22/57





Appendix


w1

- Wzr

zs

- zr

zdef z3 -

- Wu (ρ) -

u

C(ρ)

6

?

ρ(ε)

+

6

- z1

- Wzdef - z2

Σ y

- u

ρ

- Wzs

z˙def

v

−

ε

?n +


Wn

w2

22/57





Appendix


w1

- Wzr

zs

- zr

zdef z3 -

- Wu (ρ) -

u

C(ρ)

6

?

ρ(ε)

+

6

- z1

- Wzdef - z2

Σ y

- u

ρ

- Wzs

z˙def

v

−

ε

?n +


Wn

w2

22/57





Appendix


w1

- Wzr

zs

- zr

zdef z3 -

- Wu (ρ) -

u

C(ρ)

6

?

ρ(ε)

+

6

- z1

- Wzdef - z2

Σ y

- u

ρ

- Wzs

z˙def

v

−

ε

?n +


Wn

w2

22/57





Appendix


w1

- Wzr

zs

- zr

zdef z3 -

- Wu (ρ) -

u

C(ρ)

6

?

ρ(ε)

+

6

- z1

- Wzdef - z2

Σ y

- u

ρ

- Wzs

z˙def

v

−

ε

?n +


Wn

w2

22/57





Appendix

LPV control design (2/5) u

-SA actuator

⇒ Wu (ρ) is a ρ-parameter dependent weight + − ?v ρ(ε) u

System & Weights

Ω

ρ

? C(ρ)

1 z˙ def

[zdef , z˙ def ]

The system is LTI, and the parameter dependency comes in the weight functions. . . I

Wzs =

s ω11 s ω12

+1 +1

, chassis performance objective

1

, suspension performance objective

I

Wzdef =

I

Wzr = 7.10−2 , road model

I

Wn = 10−4 , noise model

I I

s ω21

Wu (ρ) = ρ ρ ∈ 0.01

+1

1

s +1 1000

10

, control attenuation


23/57





Appendix

LPV control design (2/5) u

-SA actuator

⇒ Wu (ρ) is a ρ-parameter dependent weight + − ?v ρ(ε)

Ω

1

ρ

u

System & Weights

z˙ def [zdef , z˙ def ]

? C(ρ)

Wu (ρ) Bode Diagram

Bode Diagram

80

40

1/Wz

Gain (dB)

20

s

1/Wz

60 def

increasing ρ

50

Gain (dB)

30

70

10

0

40 30 20 10

−10

0 −20 −10 −30 −1 10

0

10

1

10

Pulsation (rad/sec)


2

10

3

10

−20 −1 10

0

10

1

10

2

10

3

10

4

10

Pulsation (rad/sec)

23/57





Appendix

LPV control design (3/5) LTI system/controller/closed-loop    

x(t) ˙ Σ: y(t)    z(t) x˙ c (t) C: u(t)  x(t) ˙    x˙ c (t) CL :   z(t) 


= =

w(t) u(t) w(t) Cx(t) + D u(t) Ax(t) + B

= =

Ac xc (t) + Bc y(t) Cc xc (t) + Dc y(t) x(t) = A + Bw(t) xc (t) x(t) = C + Dw(t) xc (t)

24/57





Appendix

LPV control design (3/5) LPV system/controller/closed-loop    

x(t) ˙ Σ(ρ) : y(t)    z(t) x˙ c (t) C(ρ) : u(t)  x(t) ˙    x˙ c (t) CL(ρ) :   z(t) 


= =

w(t) u(t) w(t) C(ρ)x(t) + D(ρ) u(t) A(ρ)x(t) + B(ρ)

= =

Ac (ρ)xc (t) + Bc (ρ)y(t) Cc (ρ)xc (t) + Dc (ρ)y(t) x(t) = A(ρ) + B(ρ)w(t) xc (t) x(t) = C(ρ) + D(ρ)w(t) xc (t)

24/57





Appendix

LPV control design (4/5) H∞ criteria Apkarian et al. [TAC, 1995] Stabilize system CL(ρ) (find K > 0) while minimizing γ∞ .   A(ρ)T K + KA(ρ) KB∞ (ρ) C∞ (ρ)T 2 I  B∞ (ρ)T K −γ∞ D∞ (ρ)T  < 0 C∞ (ρ) D∞ (ρ) −I

Infinite set of LMIs to solve (ρ ∈ Ω) (Ω is convex) LPV control designs Arzelier [HDR, 2005], Bruzelius [Thesis, 2004] LFT, Gridding, Polytopic


25/57





Appendix

LPV control design (4/5) H∞ criteria Apkarian et al. [TAC, 1995] Stabilize system CL(ρ) (find K > 0) while minimizing γ∞ .   A(ρ)T K + KA(ρ) KB∞ (ρ) C∞ (ρ)T 2 I  B∞ (ρ)T K −γ∞ D∞ (ρ)T  < 0 C∞ (ρ) D∞ (ρ) −I

Infinite set of LMIs to solve (ρ ∈ Ω) (Ω is convex) LPV control designs Arzelier [HDR, 2005], Bruzelius [Thesis, 2004] LFT, Gridding, Polytopic


25/57





Appendix

LPV control design (5/5) Polytopic approach Solve the LMIs at each vertex of the polytope formed by the extremum values of each varying parameter, with a common K Lyapunov function. i

C(ρ) =

2 X

αk (ρ)

k=1

Ack Cck

Bck Dck

where, Qi αk (ρ) =

j=1

Qi

|ρ(j) − C c (Ωk )j |

j=1 (ρ(j)

− ρ(j))

,

i

2 X

αk (ρ) = 1 , αk (ρ) > 0

k=1


26/57





Appendix

LPV control design (5/5) Polytopic approach Solve the LMIs at each vertex of the polytope formed by the extremum values of each varying parameter, with a common K Lyapunov function. i

C(ρ) =

2 X

αk (ρ)

k=1

Ack Cck

Bck Dck

ρ2 ρ2

ρ2

C(ω2 )

6

C(ω1 ) ρ1


C(ω4 ) C(ρ)

C(ω3 )

- ρ1

ρ1

26/57





Appendix

Bode diagram for frozen ρ z¨s /zr (C1)

zs /zr (C2)

60

10 Bode Diagram

Bode Diagram

55 0

COMFORT

50

increasing ρ −10 Magnitude [dB] (dB)

Magnitude [dB] (dB)

45 40

−20

increasing ρ

35

−30 30 −40 25 20

1 10 Pulsation [rd] (rad/sec)

−50

2

10

0

10

1 Pulsation [rd] 10(rad/sec)

zus /zr (RH1)

2

10

zdef /zr (RH2)

10

10 Bode Diagram Bode Diagram

0

5

increasing ρ

increasing ρ

−10 Magnitude [dB]

Magnitude [dB] (dB)

increasing ρ 0

increasing ρ

−20

−5 −30

ROAD HOLDING

−10

−15

−40

0

10

1

Pulsation [rd] 10(rad/sec)


2

10

−50

0

10

1 Pulsation [rd]10(rad/sec)

2

10

27/57





Appendix

Performance evaluation on the nonlinear model Criteria used for evaluation (frequency based) sZ P SD{f1 ,a1 }→{f2 ,a2 } (x) =

f2

f1

Z

a2

x2 (f, a)da · df

a1

Performances & PSD metric zs

ms u

Fk

>

mus

zus

kt

I

(C1) Comfort at high frequencies: z¨s /zr , [4-30]Hz

I

(C2) Comfort at low frequencies: zs /zr , [0-5]Hz

I

(RH1) Road-holding: zus /zr , [0-20]Hz

I

(RH2) Suspension constraints: zdef /zr , [0-20]Hz

zr C. Poussot-Vassal - PhD. defense [GIPSA-lab / SLR team]

28/57





Appendix

Performance evaluation on the nonlinear model Improvement rate Improvement rate =

P SDpassive − P SDcontrolled P SDpassive

Results for nonlinear simulation Signal (C1) z¨s /zr [4-30]Hz (C2) zs /zr [0-5]Hz (RH1) zus /zr [0-20]Hz (RH2) zdef /zr [0-20]Hz

Active H∞ 4.8% 52.8% 3.2% 5.3%


Clipped H∞ 3.8% 23.5% 4.2% 5.7%

LPV H∞ -4.4% 18.9% 9.9% 10.4%

ADD 10% 16.9% −4.9% −7.8%

SH-ADD 10.8% 36.2% −5.8% −4.5%

28/57





Appendix

Nonlinear simulation - pseudo-Bode z¨s /zr (C1)

zs /zr (C2)

700

1.8

600

1.6

Passive Active H∞

Clipped H

1.4

Clipped H

LPV H∞

1.2

LPV H∞

∞

500

Magnitude

Magnitude

COMFORT

Passive Active H∞

SH−ADD ADD

400

300

∞

SH−ADD ADD

1 0.8 0.6 0.4

200 0.2 100

5

10

15

20

25

0 0

30

0.5

1

1.5

Frequency [Hz]

Magnitude

1.8

ROAD HOLDING

1.6 1.4

3.5

4

4.5

5

2

Clipped H∞ LPV H∞ SH−ADD ADD

1.2

1.5

Passive Active H∞

1

Clipped H

∞

1 0.8

LPV H∞

0.5

SH−ADD ADD

0.6 0.4 0

3

2.5

Passive Active H∞

Magnitude

2

2.5

zdef /zr (RH2)

2.4 2.2

2

Frequency [Hz]

zus /zr (RH1)

2

4

6

8

10

12

14

Frequency [Hz]


16

18

20

0 0

2

4

6

8

10

12

14

16

18

20

Frequency [Hz]

29/57





Appendix

Nonlinear simulation - time SER representation 0.2

Active H 0.15 0.1

u [N]

0.05

∞

Clipped H∞ LPV H

∞

SH−ADD ADD

0 −0.05 −0.1 −0.15 −0.2 −0.05

−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.05

zdef’ [m/s] C. Poussot-Vassal - PhD. defense [GIPSA-lab / SLR team]

30/57





Appendix

Global Chassis Control GCC design

Validation (simulation tests)


31/57





Appendix

Framework & Objectives GCC research Recent research area Shibahata [ARC, 2005]: I

Inverse model and optimization Andreasson et al. [VSD, 2006]

I

Nonlinear approach involving Suspension & Braking Chou et al. [VSD, 2005]

I

MPC based involving Braking & Steering Falcone [IEEE CDC, 2007]

I

LPV approach involving Anti-roll bar, Braking and Suspensions Gáspár et al. [IEEE CDC, 2005]

I

LPV approach involving Suspension & Braking Poussot-Vassal et al. [IFAC WC, 2008]

Objectives I

Improve passengers comfort & safety in critical situations

I

Multi actuators (steering & braking), with fault tolerant properties

I

Supervise available resources


32/57





Appendix

Framework & Objectives GCC research Recent research area Shibahata [ARC, 2005]: I

Inverse model and optimization Andreasson et al. [VSD, 2006]

I

Nonlinear approach involving Suspension & Braking Chou et al. [VSD, 2005]

I

MPC based involving Braking & Steering Falcone [IEEE CDC, 2007]

I

LPV approach involving Anti-roll bar, Braking and Suspensions Gáspár et al. [IEEE CDC, 2005]

I

LPV approach involving Suspension & Braking Poussot-Vassal et al. [IFAC WC, 2008]

Objectives I

Improve passengers comfort & safety in critical situations

I

Multi actuators (steering & braking), with fault tolerant properties

I

Supervise available resources


32/57





Appendix

GCC - structure & principle

δd

-

AS

ABS

δ+

-+

Tb0

(controlled output)

Vehicle ψ˙

Tbrj

rj

- EMB

6

˙ y¨s - ψ,

-

-

Tb∗

rj

GCC(ξ) δ0 ξ EMB: Electro Mechanical Braking AS: Active Steering

6

Monitor

ψ˙ ref (v) Tb∗ − Tbrj rj

Local ABS strategy Tanelli et al. [EJC 2008]


33/57





Appendix

GCC - LTI synthesis model (Vehicle) δd

ys

6 Ftyf ]

− → v

1 Ftxf * M

-

AS ABS

Tb0 rj

-

β

-xs

-+ -

δ+

6

-

-

EMB Tb∗ rj

˙ ψ

rj -

GCC(ξ) δ0 ξ

ψ

Vehicle

Tb

6

Monitor

˙

ψref (v) ∗

Tb − Tb rj rj

LTI bicycle model    y¨s y˙ s  ψ¨  = A  ψ˙  + B1 δ 0 + B2 Tb∗ rj β β˙ 


34/57





Appendix

GCC - LPV controller (GCC) (1/3) δd

Principle

-

To stabilize the system, the GCC provides:

ABS

∗ Mdz

I

a stabilizing moment

I

. . . converted in braking torque Tb∗

AS

Tb0 rj

-

6

-

Vehicle

Tb EMB Tb∗ rj

˙ ψ

rj -

GCC(ξ) δ0

rj

I

-+ -

δ+

ξ

and an additive steering angle δ 0 (if necessary)

6

Monitor

˙

ψref (v) ∗

Tb − Tb rj rj

H∞ parameter dependent performances - Weψ˙

z1 -

-

WM ∗ dz Wδ0 (ξ)

ψ˙ ref (v)

-+ -−

- GCC(ξ)

- Bicycle

-

z3 z4

- Wv˙ y - z2

∗ , δ0 } {Mdz

ψ˙ C. Poussot-Vassal - PhD. defense [GIPSA-lab / SLR team]

35/57





Appendix

GCC - LPV controller (GCC) (2/3)

- Weψ˙

z1 -

-

WM ∗ dz Wδ0 (ξ)

ψ˙ ref (v)

-+ -−

- GCC(ξ)

- Bicycle

-

z3 z4

- Wv˙ y - z2

∗ , δ0 } {Mdz

ψ˙ I I I

Weψ˙ = 10 Wv˙ y =

I

10−3

WM ∗ = dz

s/500+1 s/50+1

, error performance objective

, lateral acceleration performance objective

s/10$+1 10−5 s/100$+1

Wδ0 (ξ) =

s/κ+1 ξ s/10κ+1

, $, braking actuator bandwidth

, κ, steering actuator bandwidth


36/57





Appendix

GCC - LPV controller (GCC) (3/3)

Bode Diagram

105

100

Bode Diagram

−18

LPV ξ=0.1 LPV ξ=10

−20 −22

95 Magnitude (dB)

Magnitude (dB)

−24 90

−26 −28

85 −30 80 −32 75 −1 10

0

10

1

10

2

10

−34 −1 10

LPV ξ=0.1 LPV ξ=10 0

10

Pulsation (rad/sec)

(a) Brake control

(b) Steer control

I

ξ = ξ: steering action is not penalized

I

ξ = ξ: steering action is highly penalized


1

10

Pulsation (rad/sec)

2

10

37/57





Appendix

Supervisor (Monitor) δd

-

e = max(|TbABS

rj

− Tb∗ |), where j = {l, r}

-+ -

δ+ Tb0 rj

ABS

"Braking efficiency measure" I

AS

-

6

-

Vehicle

Tb EMB Tb∗ rj

˙ ψ

rj -

GCC(ξ) δ0

rj

ξ

6

Monitor

I

If the error is "low" ⇒ ξ 7→ ξ ⇒ only braking system is used

I

If the error is "high" ⇒ ξ 7→ ξ ⇒ braking system and steering system are used ξ ξ

Brake only

˙

ψref (v) ∗

Tb − Tb rj rj

Brake & Steer

6

ξ χ C. Poussot-Vassal - PhD. defense [GIPSA-lab / SLR team]

- e χ 38/57





Appendix

Nonlinear simulations - time (1/4) Moose test (again. . . ) I

Initial speed: 100km/h

I

Wet road

I

Safe (left) and faulty (right) braking actuator

Remark: the faulty actuator (left brake) can only provide a maximal torque of 50Nm (instead of 1200Nm).


39/57





Appendix

Nonlinear simulations (faulty) - time (2/4)1

1

Thanks to P. Bellemain!


40/57





Appendix

Nonlinear simulations - time (3/4) Actuator control signals 300 250 200 150 100 50 0

0

0.5

1

1.5

2

2.5

400

400

350

350

300 250 200 150 100 50 0

3

0

0.5

1

2

2.5

300 250 200 150 100 50 0

3

0.5

1

1.5

2

2.5

350 300 250 200 150 100 50 0

3

0

0.5

1

Time [s]

3

3

2.5

2.5

2 1.5 1 0.5 0 −0.5 −1

0

Time [s]

δ+ [deg]

δ+ [deg]

Time [s]

1.5

400

Tb right [Nm]

350

Tb left [Nm]

Tb right [Nm]

Tb left [Nm]

400

1.5

2

2.5

3

Time [s]

2 1.5 1 0.5 0 −0.5

0

0.5

1

1.5

2

Time [s]

2.5

3

−1

0

0.5

1

1.5

2

2.5

3

Time [s]

Figure: Safe actuator (left) & Faulty left braking actuator (right)


41/57





Appendix

Nonlinear simulations - time (4/4) Yaw rate



42/57





Appendix



43/57





Appendix

Conclusions About today’s presentation. . . I

A new semi-active suspension strategy Poussot-Vassal et al. [CEP, 2008] Flexible design: possibility to apply H∞ , H2 , Pole placement, Mixed etc. criterion Measurement: only the suspension deflection sensor is required Computation: synthesis leads to two LTI controllers & simple scheduling strategy (no on-line optimization process involved) Robustness: internal stability & robustness Problems: implementation issues, numerical solution (engineering)

I

An approach to the GCC problem Poussot-Vassal et al. [CDC, 2008] Flexible design: integration of different sub-controllers Computation: synthesis leads to two LTI controllers & simple scheduling strategy (no on-line optimization process involved) Robustness: internal stability & fault tolerant Problems: implementation issues, numerical solution (engineering)


44/57





Appendix

Conclusions . . . and the PhD. thesis Modeling: vehicle dynamics analysis Suspension: analysis & control (both active & semi-active) GCC: different structures design & analysis Theory: robust control theory Tools: LMI, LPV problems (conservatism, multi-objectives investigations)


44/57





Appendix

Perspectives About today’s presentation. . . LPV: implementation issues of LPV controllers Toth et al. [IFAC WC, 2008] Engineering: design simplification (in progress. . . ) Performances: add other performance parameters Robustness: analyze robustness (µssv analysis) Zin et al. [VSD, 2008] Extension: extend the GCC, by using steer, brake and suspension systems (in progress. . . ) Comparisons: compare the GCC to other designs Chou et al. [VSD, 2005] and Falcone et al. [CST, 2007]


45/57





Appendix

Perspectives . . . and the PhD. thesis Modeling: enhance the vehicle modeling step and identify the key parameters [with the MIAM team] Suspension: extend the proposed semi-active suspension structure to real suspension (e.g. the SOBEN one), by including in the design, the structural dynamics of the considered system LPV design: methodology to design scheduling strategies for LPV controller Constraint: include in the controller synthesis, the saturation constraints or anti-windup strategies (Henrion, Biannic, Grimm)


45/57





Appendix

Merci pour votre attention Grazie per la sua attenzione Köszönöm figyelemüket Gracias por su attenzion


46/57





Appendix

Robust Multivariable Linear Parameter Varying Automotive Global Chassis Control C. Poussot-Vassal PhD. defense, September 26th 2008 GIPSA-lab, Control Systems Department, Grenoble, France

Jury de thèse: Rapporteurs: Examinateurs: Co-directeurs:

Brigitte d’Andréa-Novel (Professeur, Ecole des Mines de Paris) Sergio M. Savaresi (Professeur, Politecnico di Milano) Michel Basset (Professeur, Université de Haute Alsace) Peter Gáspár (Directeur de Recherche, Académie des Sciences de Budapest) Luc Dugard (Directeur de Recherche CNRS, GIPSA-lab) Olivier Sename (Professeur, Grenoble INP, GIPSA-lab)


47/57





Appendix

Appendix


48/57





Appendix

Actuators & Vehicle dynamics Suspensions (Comfort, Road-holding) I

Involved dynamics: zs , θ, φ

I

Mainly influence the attitude behavior

Braking (Safety) I

Involved dynamics: xs , ys , ψ, λ, ω

I

Must avoid wheel slipping

I

Mainly influence the longitudinal and lateral behavior

Steering (Safety) I

Involved dynamics: ys , ψ, β

I

Mainly influence the lateral behavior


49/57





Appendix

Vehicle model - assumptions I

The direction column is not considered

I

The auto-aligning moments are neglected

I

The kinematic effects due to suspension geometry are neglected

I

The gyroscopic effects of the sprung masses are neglected

I

The tire cambering is neglected

I

The anti-roll bars are not considered

I

The vehicle chassis plane is considered parallel to the road

I

The aerodynamical and wheel resistive effects are neglected


50/57





Appendix

Experimental validation & limitation (1/5) Double line change manoeuver at 60km/h [coll. MIAM, Mulhouse] System input: longitudinal speed (vx) [km/h]

System input: steering angle (δ) [deg] 8

59

6 58.5 4 58

vx [km/h]

δ [deg]

2 0 −2

57.5

57 −4 56.5 −6 −8 44

45

46

47

48

49

50

51

Time [s]

52

53

56 44

45

46

47

48

49

50

51

52

53

Time [s]

Called "Moose test" (in French: "test à l’élan"): avoid an object on the road


51/57





Appendix

Experimental validation & limitation (2/5) Double line change manoeuver at 60km/h [coll. MIAM, Mulhouse] Longitudinal acceleration (vx’)

Longitudinal speed (vx)

3

59

NL model Measure

2

58.5

1

vx [km/h]

vx’ [m/s2]

58 0

−1

57.5

57 −2 56.5

−3

−4 44

45

46

47

48

49

50

51

Time [s]


52

53

56 44

NL model Measure 45

46

47

48

49

50

51

52

53

Time [s]

52/57





Appendix

Experimental validation & limitation (3/5) Double line change manoeuver at 60km/h [coll. MIAM, Mulhouse] Lateral acceleration (vy’)

Lateral speed (vy)

15

6

4

10

vy [km/h]

vy’ [m/s2]

2 5

0

0

−2 −5

−4

NL model Measure −10 44

45

46

47

48

49

50

51

52

53

−6 44

Time [s]

NL model Measure 45

46

47

48

49

50

51

52

53

Time [s]

⇒ differences mainly due to lateral tire modeling


53/57





Appendix

Experimental validation & limitation (4/5) Double line change manoeuver at 60km/h [coll. MIAM, Mulhouse] Yaw rate (ψ’)

Yaw (ψ)

50

40

40 30 30 20

10

ψ [deg]

ψ’ [deg/s]

20

0 −10 −20

10

0

−10

−30 −40 −50 44

−20

NL model Measure 45

46

47

48

49

50

51

52

53

−30 44

Time [s]

NL model Measure 45

46

47

48

49

50

51

52

53

54

Time [s]

⇒ differences mainly due to lateral tire modeling


54/57





Appendix

Experimental validation & limitation (5/5) Double line change manoeuver at 60km/h [coll. MIAM, Mulhouse] Vehicle path

Roll speed (θ’) 150

1.5

1

100

y [m]

θ’ [deg/s]

0.5

0

50

−0.5

−1

−1.5 44

NL model Measure 45

46

NL model Measure 47

48

49

50

51

52

53

0 0

10

20

Time [s]

30

40

50

60

70

80

x [m]

⇒ roll rate differences due to center of gravity & anti-roll bar


55/57





Appendix

GCC - Nonlinear simulations Brake efficiency measure & Scheduling ξ 1000

br

max(eT ,eT )

800

600

bl

bl

br

max(eT ,eT )

1000

400

200

0

0

0.5

1

1.5

2

2.5

800

600

400

200

0

3

0

0.5

1

Time [s]

1.5

2

2.5

3

2

2.5

3

Time [s]

8

8

6

6

ξ

10

ξ

10

4

4

2

0

2

0

0.5

1

1.5

2

Time [s]

2.5

3

0

0

0.5

1

1.5

Time [s]



56/57





Appendix

GCC - Nonlinear simulations Vehicles path



57/57

Robust Multivariable Linear Parameter Varying ... - Google Sites

Robust Multivariable Linear Parameter Varying ... - Google Sites

Suggest Documents

Robust and Linear Parameter-Varying Control of ... - tub.dok

Linear Parameter Varying Control of Induction Motors

Linear Parameter-Varying Descriptions of Nonlinear Systems

Linear Parameter Varying Identification of Dynamic

Linear Parameter Varying Control of Induction Motors

Masters Thesis: Quasi-Linear Parameter Varying ... - CiteSeerX

analysis and control of linear parameter-varying

Persistency of excitation criteria for linear, multivariable, time-varying

Robust Synthesis for Linear Parameter Varying Systems ...www.researchgate.net › post › attachment › download

Linear Parameter Varying Control of Permanent Magnet Synchronous ...

Solution of Parameter-Varying Linear Matrix Inequalities in Toeplitz Form

Constrained Time-Optimal Control of Linear Parameter-Varying Systems

Linear Parameter Varying Approach to Wind ... - Semantic Scholar

Linear parameter-varying control and H-infinity synthesis dedicated to ...

On the fractional closed-loop linear parameter varying system

Uniform observer design for linear parameter varying ... - CiteSeerX

Linear Parameter-Varying Modeling for Gain ... - Science Direct

Linear, Parameter-Varying Wheel Slip Control for Two ... - IEEE Xplore

Linear Parameter Varying Induction Motor Control with ... - IEEE Xplore

Linear Parameter-Varying Feedforward Control: A Missile Autopilot ...

Quasi-Linear Parameter Varying Modeling of Variable-Speed Pitch ...

On Pointwise Feedback Invariants of Linear Parameter-varying Systems

Parameter Arrangement in Multivariable FLC Tuning - Google Sites

state observer to multivariable time varying systems