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Robust Vehicle Dynamics Control under Cornering
Stiffness Uncertainties with Insensitive H2 Theory
François GAY*, Philippe de LARMINAT**
*PSA Peugeot Citroën, 18, rue des Fauvelles 92250 La Garenne Colombes, FRANCE
**IRCCyN, 1, rue de la Noë, 44321 Nantes Cedex 03, FRANCE
Abstract: - This paper describes a multivariable robust controller aiming to control vehicle lateral displacement. The
proposed controller is based upon a new theory called Standard State Control (2SC) providing the control law designer
with few and intuitive tuning parameters. Though this kind of controller has great robust properties, specific
uncertainties can not be taken into account inside the methodology of 2SC. Then, we developed an insensitive H2
controller in order to be robust to specific uncertainties which are in our problem cornering stiffness due to unknown
adherence.
Key-Words: - Vehicle Control, H2 controller, uncertain parameters, insensitiveness
moment C z which can be obtained by differential

1 Introduction
Control vehicle dynamics is one of the great challenge of
these years. In order to achieve good handling vehicles
for security and comfort, car manufacturers trend to a
global control of the ground links systems such as
brakes, suspensions, rear steering and others. It is known
that instead of independently added controlled systems,
integrated control can greatly improve performances [1].
This means that we should consider multivariable
control law for better co-operation between actuators. An
other step of improvement is to provide controllers
which are robust to specific uncertainties. Some recent
theories propose solutions to make the controller robust
like H∞ theory and µ-synthesis. Nevertheless, these
methods are pessimistic and conservative. We propose in
this paper a robust multivariable control law based on H2
with specific insensitivity.
Vehicle behaviour while cornering depends on physics
parameters such as weight, mass distribution and above
all tyre characteristics. The dynamics may change a lot
and it yields to overreaction of the driver which can lead
to loss of stability of the loop driver/vehicle. Among
these situations, there are braking under µ-split
conditions, side wind gusts, braking while cornering,
excessive steering … Various systems can greatly
improve stability under critical situations. For example,
rear steering controlled by feedback on yaw velocity
enables good disturbance rejection [2]. Adding front
steering by robust decoupling of car steering results in
good yaw disturbance attenuation [3].
While cornering, yaw stabilisation depends on tyre
effectiveness and more precisely on the road friction
parameter µ. As a consequence, if µ changes the yaw
stabilisation may surprise the driver. Two chassis
systems are able to stabilise the vehicle : the yaw

braking and rear steering δ r . In this paper, these two
actuators are controlled to stabilise the vehicle under
uncertain road friction µ.

2 Problem Formulation
2.1 Objectives
The will of the driver when he steers is to follow a track
which is a function of the yaw velocity r and the lateral
velocity v . These measures are the two states of the
vehicle while lateral manoeuvres. We suppose that the
longitudinal speed u is a parameter of the system.
Disturbances acting on the vehicle can be summarised
into lateral force F and a yaw moment M . The Fig. 1
sums up this issue.

δf

u

VG
r

G
F

v
M

δf : steering angle

u : longitudinal speed
v : lateral velocity
r : yaw velocity

Weight : m
Inertia : C
Wheel base : L
Front wheel base : l 1
Rear wheel base : l 2

Fig. 1 : States Variables
The disturbing efforts F and M cause the real track to
be different from the expected one. The aim of C z and

δ r is to bring back the vehicle to the wanted track as
described on Fig. 2. In our specific problem, F and M

are generated by a variation of an unexpected road
friction variation µ. The control law structure used is
shown Fig. 3. For a better co-operation between C z and

δr ,

feedforward
separately.

and

feedback

are

synthesised

becomes δ r and C zFB becomes C z .

F

δf

M
Cz

The measurements of vehicle dynamics are the yaw
velocity r and also the lateral velocity v . This velocity
is not in reality measurable. But in this academic
context, we suppose it is. Moreover, it exists observers
that may estimate this velocity [5]. The feedback is
shown Fig. 4. As we only deals with feedback part, δ rFB

δr

δf

Expected
Track

F M

Cz
δr

Real Track
F
M

µ
r
v

Vehicle
K(s)

Fig. 4 : Feedback problem
Fig. 2 : Correction effects of C z and δ r
δf
δ rF F
δ rF B

Reference
Model

Vehicle

Feedback

+
Feedforward

CzFB

r*
v*

_+

r
_+

In the next section, we design a first controller K 0 ( s )
synthesised thanks to Standard State Control and
evaluate his behaviour with variations of µ. This
controller will be the first step of the insensitivity
algorithm exposed in section 4.

2.2 Equation of motion
The vehicle motions are described by the bicycle model
shown Fig. 5.
αf

v

Fig. 3 : Structure of the control law

F yf

From the wheel angle δ f , a reference is calculated
through a reference model with specific wanted
dynamics : it can be for example an attitude β
maintained to zero [4] without any restriction. Thanks to
a predictive model (which can be bicycle or other), a
feedforward part δ rFF of rear steering can be calculated
so that the vehicle measurements r and v are perfectly
equal to the reference model outputs r * and v* in case
of the predictive model perfectly described the real
vehicle. In such a situation, the feedback part is fed by
zero, so δ rFB and C zFB are equal to zero. With this
structure, feedback part only deals with unforeseen
disturbances such as F and M and is pure regulation
around zero without reference tracking. As the predictive
model used to calculate the feedforward part can never
fit the real car, the feedback part aims to compensate for
these errors.
An other advantage of this control law is that the
feedback law can be designed independently of the
feedforward part. In this paper, we focus on the feedback
part which aims to reject the disturbances F and M
supposed to model unforeseen disturbances such as
variation of µ.

βf

γt

Vf

G
Cz

δf

βr
δr αr

Vr

F yr

Fig. 5 : Bicycle Model
The equations are given by :

mγ t = m(v& + ru) = F yf + F yr

(1)

Cr& = l 1 F yf − l 2 F yr + C z

Transversal efforts F y are linked to the slip angle α by
the front and rear nominal cornering stiffness C 1f and

C r1 calculated on dry road with µ = 1 (2) :
F yf = C 1f α f
F yr = C α r
1
r

α
(2)

V
Fy

The front and rear slip angles α f and α r are calculated
from the front steer δ

f

and the rear steer δ r by (3) :

α f = δ f − β f and α r = δ r − β r

(3)

where β f and β r are respectively the front and rear
axle slip calculated from the chassis slip angle β
according to (4) :
(4)
β = (v + l r ) / u
f

In our problem measures y are directly the errors e that
the controller should bring back to zero this is to say r
and v . In order to specify the attenuation level of
disturbances F and M , we need to specify a predictive
model D(s ) for adding disturbances on control input
dαr and d Cz :
D (s)

1

β r = (v − l 2 r ) / u

δr
Cz

u

Then, linear equation of motion are obtained :

x& = Ax + Br δ r + Bz C z

(5)

y = Cx = ( v r ) T
with :

1
1


1
1

(
C
+
C
)

u
+
( l 2 C r1 − l 1C 1f )
f
r
 mu
u
A=

1
1 2 1
1
1
( l 1 C f + l 22 C r1 ) 
 ( l 2 C r − l 1C f ) −
Cu
 Cu
 (6)
 C r1 
0


Br =  m 1  , Bz =  1 
 C 
 − l 2C r 
 C 

+

Nominal
Vehicle
µ =1

r
v
r
v

e
y

Fig. 7 : Deterministic Model

D(s ) is chosen as D( s ) = s + 1 Td . As a result, the higher
is Td , the higher is the attenuation. Thus, d δr and d Cz
satisfies the relation :

D( s )dδr = 0 ⇒ d&δr + dδr Td = 0
D( s )d Cz = 0 ⇒ d&Cz + dCz Td = 0

(8)

extended model is called deterministic model :

C µf = C f ( µ ) = µC f ( µ = 1) = µC 1f
C rµ = C r ( µ ) = µC r ( µ = 1) = µC r1

(7)

Then, the matrices A and Br strongly depend on µ .

3 First Regulator

x& d = A d x d + Bd u
e = y = Cd x d

wy
wx

D (s)

δr
Cz

3.1 Design
As mentioned above, the first regulator is designed with
Standard State Control [6], [7]. This theory is strongly
based on methodology which enables the designer to
build standard problem specified on Fig. 6 from the Fig.
4 that can be H2 minimised.

z

G(s)
y
K(s)

Fig. 6 : Standard Problem

(9)

Fig. 8 defines the output errors v as v = u + d . The
next step consists in adding noise w composed of state
noise wx and measure noise w y .

u

u

+

Therefore, two states have been added to the model (5) :
these are the two exogenous states d δr and d Cz . This

We assume the following writing depending on µ of
cornering stiffness :

w

dδr
dCz

dδr
dCz
+
+

Nominal
Vehicle
µ =1

r
v
r
v

v
e
+

y

Fig. 8 : Full Standard Model
In order to get the final standard problem, weighting of
v by R c1 2 , e by S c1 2 , wx by Qo1 2 and w y by R o1 2 are
missing. In LQG context, R c and S c are tuning
parameters defining the criteria to minimise :

J =





0

( e T S c e + vT Rc v)dt

(10)

and Qo , R o have statistic meaning in terms of
variances. In H2 context, these matrices are all tuning
parameters. The disadvantage is a quite high number of
coefficients to choose and the designer has to know how

to fix them. 2SC recommends only two scalars
coefficients to calculate these four matrices and also
tuning rules. These rules are dictated by dual LTR (Loop
Transfer Recovery) effect which aims to recover
robustness of LQ law despite the observer. These
matrices are calculated by partial gramians.

S c = I , Rc =



Tc

Tc Gc GcT dt
0
−1

(11)

Ro = I , Qo =  To ∫ GoGoT dt 
 o

where Gc is the impulse response of ( A , B ) and Go is
To

( A dT

, C dT

the impulse response of
) . The higher is To ,
the better is the robustness of the loop. This is actually
the main tuning parameter. Tc enables to tune the
influence of noise measurement on the control input.
These parameters enable the designer to synthesise a
controller very easily and to tackle the delay margin
which is one of the most important robustness indicator.

3.2 Simulation results
The parameters are To = 1 s and Tc = 0.1 s which yield
a delay margin of Tr = 0.64 s which is rather
comfortable. This delay margin is defined by :

Tr = sT y ( s )

−1

(12)



where T y (s ) is the complementary sensitivity function.
The Fig. 9 presents simulation results with the bicycle
model with the nominal µ and µ = 0.6 . F is a step at
t = 1 s of 1800 N and M is a step of 2300 Nm at
t = 3s .

the controller with µ = 0.6 is quite different from the
nominal answer. The aim of the following section is to
attenuate this controller sensitivity.

4 Insensitive regulator
4.1 Parametric dependence formulation
The matrices ( A , B r ) (6) depend on µ. They can be
rewritten with (7) as :

µ
 µ
1
1
1
1 
− mu(C f + C r ) − u + u ( l 2 C r − l 1C f )
A=

µ
µ 2 1
 ( l 2 C r1 − l 1C 1f ) −
( l 1 C f + l 22 C r1 )
Cu
 (13)
 Cu
 µC r1 
0


Br =  m 1  , Bz =  1 
 C 
 − l 2 µC r 
 C 
µ can be expressed by µ = µ n + δ∆µ with µ n the
nominal value of µ , ∆µ the maximum variation of µ
and δ a proportion of ∆µ . Typically, µ n = 1.0 and
∆µ = 0.4 . As a result, A and B can be rewritten
approximately by :

A = A n + δA µ
B r = B n + δB µ

with

A µ = ∆µ( ∂A ∂µ ) µ = µn

(14)

Brµ = ∆µ( ∂Br ∂µ ) µ = µ n

Thus, the deterministic model (9) can be rewritten as :
(15)
x& = A x + B u + δ ( A x + B u )
dn



dn



Then we define the output ζ = A dµ x + Bdµ u and the
input ν = δζ so that the deterministic model is written
as :
(16)
x& = A x + B u + ν
dn

dn

Thus, the full standard model becomes :

δ

ν

w
Fig. 9 : K0 with µ = 1 [] and µ = 0.6 [--]
The above diagrams show the outputs v and r . The
bottom diagram show the inputs δ r and C z . We can see
that the loop is still stable. Nevertheless, the response of

Qo1/2 wx
wy
I

G

u

ζ

v
e

R1/2
c
I
y

Kr
Fig. 10 : Internal loop of uncertainties

z

where Kr is the robust controller. H2 control law
minimise the transfer H zw (K ) . As there is a new input

ν and a new output ζ , the new closed loop transfer is :
 H ζν
H( s ) = 
 H zν

This criteria J ( K r , K 0 ) corresponds to the minimisation
of :
(21)
J ( K ) = f ( K ) + g( K )h( K )
that can be minimised by the algorithm :

H ζw 
H zw 

(17)

Kri+1 = Arg min( J ( K , Kri )) with
J ( K , Kri ) = f ( K ) + g( K )h( Kri ) + g( Kri )h( K )

which is expressed as :
−1

H = H zw + δH zν ( I − δH ζν ) H ζw

(18)

≈ H zw + δH zν H ζw
Thus, the new criteria to minimise is :

J ( K ) = H zw

2

+ σ 2 H zν H ζw

(19)

2

where σ is a tuning parameter related to the uncertainty
δ.

4.2

(22)

K

Algorithm

We assume that we have a first regulator K 0 described
above. This regulator can be connected to the augmented
model with ζ and ν :

in case of f , g and h are scalars. We assume that this
procedure is still valid with matrices. The interest is that
we are able to H2 minimise the criteria (19) and the
algorithm can be computed.
Nevertheless, there is a problem of order because the
size of H ζw and H zν increase at each iteration. The
order of K0 is 4 . Then, the order of H ζw ( K 0 ) and

H zν ( K 0 ) is 8 . As a consequence the order of K r 1 is
8 + 8 + 4 = 20 and generally the order of K ri is :
(23)

dim( K ri ) = 2 i + 2 + 6i( i + 1)

We propose an order reduction of H ζw and H zν to a
simple gain at each iteration so that the order of K ri is

ν

w

ζ

u

permanently equal to the order K 0 , this is to say 4 .

z

G

4.3

y

K0

Order reduction

The aim of this reduction is to reduce H zν to a gain

Fig. 11 : Augmented Model
from which we can extract the two transfers H ζw ( K 0 )

Kzν and to reduce H ζw to a gain Kζw . The choice of a
reduction schema of H zν is shown Fig. 13 :

and H zν ( K 0 ) that we weight by the tune parameter σ .

K zν

The signals ν and ζ are respectively weighted by
σH ζw ( K 0 ) and σH zν ( K 0 ) yielding the noise wµ and
the error z µ :
W


w

ζ


σ Hζw(K 0) ν
u

G
Kr

ζ σ H (K )
zν 0


z

~


Kzν aims to minimise the H2 norm of ~
z µ wµ . We have
T
to write the quantity E (~
z ~
z ) . From Fig. 13, we can

y

µ

µ

write :

Neglecting the σ 4 term, the H2 norm of the transfer
Z W is :
2

Hζw



Fig. 13 : Reduction order of H zν

Z

Fig. 12 : New standard problem

J ( K r , K 0 ) = H zw

Hzν

^z
µ

+ σ 2 H zν ( K 0 )H ζw ( K r )

2

+ σ 2 H z ν ( K r )H ζ w ( K 0 )

2

(20)

~
z µ~
z µT = [ I − Kzν

 I 
z µ 
] [z µT ζ T ]
T 
ζ 
− K zν 

Hence z µ and ζ are the outputs of the system :

(24)



Hζw

The generalised margin gain is Tr = 0.08 s which is



Hzν

y

ζ

Fig. 14 : Outputs z µ and ζ
Therefore,

E ( yyT ) = CGc C T

where

C

is

the

observation matrix of this system and Gc the
controllability gramian of this system which exists as it
is stable. Hence, we have :

 P11
E (~
z µ z~µT ) = [ I − K zν ]
P21

P12   I 

T 
P22   − Kzν 

(25)

where CGc C T is partitioned into P11 , P12 , P21 and P22 .
The solution of this minimisation is given by :
(26)
−1

Kzν = P12 P22

In order to reduce H ζw we proceed by duality. The
equivalent schema of Fig. 13 is :

νµ

Hζw

ν

K ζw

Hzν

quite different from the margin of K 0 : Tr = 0.64 s .
Comparing Fig. 16 to Fig. 9, we can observe :
− the disturbances are attenuated quicker
with less overshoot
− above all, responses between µ = 1 and
µ = 0.6 are quite identical showing an
insensitivity of K r 1 with regard to the
unknown parameter µ .
The fact that rejection is better with K r 1 is maybe due
to the delay margin which has been drastically
decreased but can not justify the insensitivity of K r 1 :
this property is really due to the process of insensitivity
of the controller.
Fig. 17 show the response of controllers K ri obtained
after i iterations. We can observe :
− the algorithm converges
− after the second iteration, there are no
significant differences

z

Fig. 15 : Reduction order of H ζw
The method is the same : H ζw becomes H zTν , H zν
becomes H ζTw and the result Kζw is given by KzTν .

4.4 Simulation results
The same simulation than Fig. 9 is performed. We first
use the controller K r 1 obtained after the first iteration.
As we reduced its order, the order of K r 1 is 4 as well as

K 0 . The default value of σ is 0.5 .

Fig. 17 : Response of K ri
Fig. 18 show the two principal robustness indicators :
generalised delay margin and generalised margin gain.
This margin is defined by :

M % = 100 T y ( s )
expressed

in

%.

−1

(27)



Any

controller

should

give

M % > 70% . As we can see, there is no further

Fig. 16 : Response of Kr 1 with µ = 1 and µ = 0.6

evolution beyond the second iteration. Nevertheless,
insensitivity process decreases significantly delay
margin. At the second iteration, this margin is
0.1 s instead of 0.08 s after the first iteration without any
degradation of the temporal response as shown Fig. 17.
Hence, K r 2 would be the best controller.

outlined the tuning and methodology aspect that offers
Standard State Control. Moreover, the order of the
insensitive controller is reduced to the initial controller
to prevent too high order controller. In order to really
appreciate the robust property of the controller with
regard to µ, we compared the insensitive controller with
a controller having the same delay margin. Simulations
show that the insensitive algorithm brings parametric
robustness.

Fig. 18 : Robustness indicators
After these good results, we can wonder how a controller
K 0 ' with a delay margin of 0.1 s (like K r 2 ) would
behave with µ = 0.6 . If the response is significantly
deteriorated, then we could conclude that the
insensitivity algorithm really brings something more.
Such a controller K 0 ' can be obtained by 2SC with

To = 0.28 s . Fig. 19 show simulation results :

Fig. 19 : Response of K 0 '
As we can see, comparing to Fig. 16 the response of
K 0 ' is less insensitive than Kr 1 with oscillatory motions
of the input and r whereas the response of K r 1 is
completely damped and smoother for both inputs.

5 Conclusion
In this paper, we described the design of a robust
controller aiming to reject disturbances due to variation
of the tire/road friction parameter µ in the field of lateral
dynamics of a vehicle. This controller is the result of an
insensitiveness algorithm based on H2 minimisation. We

References:
[1] M. Yamamoto, Active Control Strategy for Improved
Handling and Stability, SAE 911902, 1991
[2] J. Ackermann, Active Steering for Better Safety
Handling and Comfort, AVCS’98
[3] J. Ackermann, Yaw Disturbance Attenuation by
Robust Decoupling of Car Steering, IFAC’96
[4] J.C. Whitehead, Four Wheel Steering :
Manoeuvrability and High Speed Stabilization, SAE
880642, 1988
[5] A. Hac, Estimation of Vehicle Side Slip Angle and
Yaw Rate, SAE 2000-01-0696, 2000
[6] Ph. de Larminat, “Automatique, Commande des
Systèmes Linéaires”, 2nd Ed. Hermes, 1995
[7] T. Cambois, Ph. de Larminat, The Standard State
Control (2SC) Application to Active Suspension
Design, CESA’98
[8] O. Begovich, Développement et Analyse d’Outils
pour la Conception des Systèmes de Commande
Robuste”, Thèse de Doctorat de l’Université de
Rennes, 1992

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