Vehicle modeling & Validation
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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)
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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)
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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)
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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]
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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
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(Some) conclusions & perspectives
Appendix
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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]
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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]
C. Poussot-Vassal - PhD. defense [GIPSA-lab / SLR team]
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]
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Suspension system Objective I
Link between unsprung (mus ) and sprung (ms ) masses
I
Influences comfort / road-holding performances
I
Involves vertical (zs , zus ) dynamics
Passive quarter vehicle model
{ms , zs } {mus , zus }
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Suspension system Objective I
Link between unsprung (mus ) and sprung (ms ) masses
I
Influences comfort / road-holding performances
I
Involves vertical (zs , zus ) dynamics
Semi-active quarter vehicle model
{ms , zs } {mus , zus }
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Suspension system Objective I
Link between unsprung (mus ) and sprung (ms ) masses
I
Influences comfort / road-holding performances
I
Involves vertical (zs , zus ) dynamics
Active quarter vehicle model
{ms , zs } {mus , zus }
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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}
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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}
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{ms , zs , θ}
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Vehicle model - dynamical equations (1/3) Full vertical model
z¨s z¨usij 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
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Wheel & Braking system Objective I
Link between wheel and road
I
Influences safety performances
I
Involves longitudinal (xs ) rotational (ω)and slipping (λ =
C. Poussot-Vassal - PhD. defense [GIPSA-lab / SLR team]
v−Rω ) max(v,Rω)
dynamics
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Wheel & Braking system Objective I
Link between wheel and road (zr , µ)
I
Influences safety performances
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
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0.1
0.2
0.3
0.4
0.5
λ
0.6
0.7
0.8
0.9
1
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Vehicle model - dynamical equations (2/3) Full vertical model
z¨s z¨usij 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
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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
z¨usij θ¨ ¨ φ
= = = = =
λij
=
vij −Rij ωij max(vij ,Rij ωij )
ω ˙ ij
=
(−RFtxij (µ, λ, Fn ) + Tb )/Iw
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Wheel & Steering system Objective I
Wheel / road contact
I
Influences safety performances
I
Involves lateral (ys ), side slip angle (β) and yaw (ψ) dynamics
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Wheel & Steering system Objective I
Wheel / road contact
I
Influences safety performances
I
Involves lateral (ys ), side slip angle (β) and yaw (ψ) dynamics
Bicycle model ys
6 Ftyf ]
− → v
1 Ftxf * K β
- xs
ψ
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Vehicle model - dynamical equations (3/3) Full vertical and longitudinal model
(Ftxf r + Ftxf l ) + (Ftxrr + Ftxrl ) /m
x ¨s
=
z¨s z¨usij 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 + mh¨ xs /Iy
λij
=
vij −Rij ωij max(vij ,Rij ωij )
ω ˙ ij
=
(−RFtxij (µ, λ, Fn ) + Tb )/Iw
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Vehicle model - dynamical equations (3/3) Full model
x ¨s y¨s z¨s z¨usij 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]
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Vehicle model - dynamical equations (3/3) Full model
x ¨s y¨s z¨s z¨usij 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]
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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
C. Poussot-Vassal - PhD. defense [GIPSA-lab / SLR team]
Ftx,y,z λij - β ij ωij (tire, wheel dynamics)
(vehicle dynamics)
-
θ φ ψ
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Vehicle model - synopsis Fdx,y,z & Mdx,y,z
(external disturbances)
q
?
?
z˙s , zs z˙us , zus
-
Suspensions
Fszij
xs
- ys
-
zs
Chassis
x ¨s , y¨s ˙ ψ, v, Fsz , zus
6
? -
Wheels
6
(road characteristics) [µij , zrij ]
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Ftx,y,z λij - β ij ωij (tire, wheel dynamics)
(vehicle dynamics)
-
θ φ ψ
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Vehicle model - synopsis Fdx,y,z & Mdx,y,z (suspensions control)
(external disturbances)
uij
q
?
?
z˙s , zs z˙us , zus
-
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 ]
C. Poussot-Vassal - PhD. defense [GIPSA-lab / SLR team]
Chassis
Ftx,y,z λij - β ij ωij (tire, wheel dynamics)
(vehicle dynamics)
-
θ φ ψ
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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]
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52
53
0 0
10
20
30
40
50
60
70
80
x [m]
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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]
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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
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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
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Quarter vehicle model Passive & Controlled damper case zs
ms Fk
zus
kt
>
mus
zus
kt zr
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u
Fk
Fc
mus
zs
ms
zr
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Quarter vehicle model Passive & Controlled damper case zs
ms u
Fk
>
mus
zus
kt zr
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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
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(as pole placement,
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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
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(as pole placement,
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Implementation scheme & principle
u
- SA actuator
−
+ -
v
ε
Ω
3
z˙def
? ρ(ε) ρ u
?
c0 z˙def + uH∞ (ρ)
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[zdef , z˙def ]
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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
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0.4
0.6
0.8
1 −3
x 10
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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 +
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Wn
w2
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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 +
C. Poussot-Vassal - PhD. defense [GIPSA-lab / SLR team]
Wn
w2
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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 +
C. Poussot-Vassal - PhD. defense [GIPSA-lab / SLR team]
Wn
w2
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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 +
C. Poussot-Vassal - PhD. defense [GIPSA-lab / SLR team]
Wn
w2
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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 +
C. Poussot-Vassal - PhD. defense [GIPSA-lab / SLR team]
Wn
w2
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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
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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)
C. Poussot-Vassal - PhD. defense [GIPSA-lab / SLR team]
2
10
3
10
−20 −1 10
0
10
1
10
2
10
3
10
4
10
Pulsation (rad/sec)
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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)
C. Poussot-Vassal - PhD. defense [GIPSA-lab / SLR team]
= =
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)
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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)
C. Poussot-Vassal - PhD. defense [GIPSA-lab / SLR team]
= =
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)
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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
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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
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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
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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
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C(ω4 ) C(ρ)
C(ω3 )
- ρ1
ρ1
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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)
C. Poussot-Vassal - PhD. defense [GIPSA-lab / SLR team]
2
10
−50
0
10
1 Pulsation [rd]10(rad/sec)
2
10
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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]
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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%
C. Poussot-Vassal - PhD. defense [GIPSA-lab / SLR team]
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%
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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]
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16
18
20
0 0
2
4
6
8
10
12
14
16
18
20
Frequency [Hz]
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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]
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Global Chassis Control GCC design
Validation (simulation tests)
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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
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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
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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]
C. Poussot-Vassal - PhD. defense [GIPSA-lab / SLR team]
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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˙
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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]
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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
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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
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1
10
Pulsation (rad/sec)
2
10
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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
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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).
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Nonlinear simulations (faulty) - time (2/4)1
1
Thanks to P. Bellemain!
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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)
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Nonlinear simulations - time (4/4) Yaw rate
Figure: Safe actuator (left) & Faulty left braking actuator (right)
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(Some) conclusions & perspectives
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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)
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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)
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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]
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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)
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Appendix
Merci pour votre attention Grazie per la sua attenzione Köszönöm figyelemüket Gracias por su attenzion
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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)
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Appendix
Appendix
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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
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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
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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
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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]
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56 44
NL model Measure 45
46
47
48
49
50
51
52
53
Time [s]
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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
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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
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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
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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]
Figure: Safe actuator (left) & Faulty left braking actuator (right)
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GCC - Nonlinear simulations Vehicles path
Figure: Safe actuator (left) & Faulty left braking actuator (right)
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