2016 International Conference and Exposition on Electrical and Power Engineering (EPE 2016), 20-22 October, Iasi, Romania
H-Infinity Control of Automatic Vehicle Steering
Razvan C. Rafaila
Gheorghe Livint
Faculty of Electrical Engineering and Applied Informatics
"Gheorghe Asachi" Technical University
Iasi, Romania
razvan.rafaila@gmail.com
Faculty of Electrical Engineering and Applied Informatics
"Gheorghe Asachi" Technical University
Iasi, Romania
glivint@tuiasi.ro
Abstract— The purpose of this paper is the development of
II.
H ∞ control strategy for the autonomous steering of ground
Typically, vehicle dynamics refer to vehicle motion in
longitudinal and lateral directions. Longitudinal dynamics
concern the vehicle behavior while accelerating, braking and
traction properties of the wheels on different road surfaces
under different conditions. Lateral dynamics is governed by the
vehicle stability analysis, cornering and road keeping. The
single-track model or bicycle model is widely used in the study
of vehicle motion and control [10], [11], and is also adopted in
this paper. In Fig. 1, the vehicle model is represented. The
dynamic model can be written using the following differential
equations:
vehicles. The focus is only on steering maneuvers, while the
vehicle is driving at constant speed. The obtained simulation
results show that the proposed H ∞ control approach could
provide very good performances in practical use.
Keywords—robust control; autonomous
control; vehicle model; loop-shaping
I.
vehicles;
VEHICLE DYNAMICS MODELLING
lateral
INTRODUCTION
The automotive community is focusing more and more to
bring complete autonomous vehicles to market. Intensive
research is carried out in both academic and industry fields for
new control concepts in order to tackle this task. The
challenges of implementing a best suited control system for
autonomous vehicles is the fact that the controller needs to
provide best performances in driving both longitudinal and
lateral vehicle dynamics, in the same time without decreasing
the passengers comfort and especially safety. In this situation,
the control system stability and robustness becomes crucial in
the development of autonomous vehicles.
mx = myψ + 2 Fxf + 2 Fxr ,
my = − mxψ + 2 Fyf + 2 Fyr ,
(1)
Iψ = 2aFyf − 2bFrf ,
where Fxf is the longitudinal force of the front wheel tire, Fxr
is the longitudinal force of the rear wheel tire, Fyf is the lateral
force of the front wheel tire, and Fyr is the lateral force of the
Research in automatic driving is ongoing for some years
and functionalities of partial autonomy are already available in
series production which are contributing to driver safety and
comfort, and interested readers may refer to [1], [2].
rear wheel tire. The states of the vehicle are as follows: the
longitudinal position x , the lateral position y and the yaw
angle ψ . Finally m , I , a and b are the vehicle mass,
vehicle inertia, distance from the front axle to the center of
gravity and distance from the rear axle to the center of gravity.
Multiple control system approaches are already studied in
academy and industry for the control of automatic driving
vehicles, and many of them rely on advanced control
techniques such as nonlinear control [3], robust control [4], [5]
and model predictive control [6], [7], [8], and [9].
It should be noted that the degrees of freedom of the
nonlinear model described by (1), are considered in the vehicle
coordinate system, which is centered in the vehicle's center of
gravity. An inertial representation of the vehicle motion can be
obtained by changing these coordinates to the inertial
coordinate system.
In this paper the loop shaping design method of an H ∞
controller is adopted, and the obtained closed loop performance
is studied in vehicle trajectory following. The desired trajectory
of the vehicle is considered already known., also it is assumed
that all needed sensors for measuring the vehicle motion and
actuators are available in the vehicle. Inclusion of all these
aspects is subject to future contributions.
The tire forces can be obtained by using the nonlinear
Pacejka Tire Model [12], [13], using the longitudinal slip
values - for calculating Fxf , Fxr , and the slip angles - for
calculating Fyf , Fyr . The Pacejka Tire Model tire model also
The rest of this paper is structured as follows: in Section II.
a short description of vehicle nonlinear dynamics and
modelling is presented; the H ∞ control design by loop-shaping
method is presented Section III; simulation results are
presented in section IV, and de conclusions are given in
Section V.
includes the influences that the normal forces acting on the
front and rear axles, and describes the tire forces as:
978-1-5090-6128-0/16/$31.00 ©2016 European Union
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Fxf = f xf (α f , κ f , Fzf ), Fxr = f xr (α r , κ r , Fzr ),
Fyf = f yf (α f , κ f , Fzf ), Fyr = f yr (α r , κ r , Fzr ),
Using the linear model described by (5) with (3) and (4),
the state-space formulation of the model gives:
(2)
z ( t ) = Az ( t ) + Bu ( t )
where α f denotes the front tire slip angle, α r is the rear tire
y ( t ) = Cz ( t )
slip angle, κ f is the longitudinal slip of the front tire, κ r is the
longitudinal slip of the rear tire, and Fzf , Fzr are the normal
,
(6)
with
forces acting on the front and rear axels. The nonlinearities that
have the highest impact in the vehicle dynamics, described by
equations (2), arise from the tire models, as the tire forces are
highly nonlinear for high slip quantities, which can occur when
the vehicle drives over slippery roads. The presented nonlinear
model will be further used to validate the controller
performance. For brevity, the nonlinear model will not be
presented here, and interested readers may refer to [7], [8], [12]
and [13] for more information regarding the vehicle nonlinear
modelling.
1
0

0 − 2C yf + 2C yr

mVx
A=
0
0

0 − 2aC yf − 2bC yr

IVx

 0 
 2C 
yf 

 m 
B=
 ,C
 0 
 2aC yf 


 I 

2aC yf − 2bC yr 
0 −Vx −

mVx
,
0
1


2
2
2a C yf + 2b C yr 
0 −

IVx

0
1
0
=
0

0
0
0 0 0
1 0 0 
0 1 0

0 0 1
and z = [ y , y ,ψ ,ψ ] is the state vector, u = δ and y are the
input and the output vector of the model, where δ is the front
wheel steering angle. For more details interested readers may
refer to [10] and [11].
T
III.
Fig. 1. Vehicle single-track model
DESIGN
In control theory, robust control concerns the design of
controllers, developed explicitly to account for the process
uncertainties.
For the proposed control design, a liniarization of the
nonlinear vehicle model is done, considering constant vehicle
speed. This linear model can be obtained by making linear
approximations of the tire forces, of the form:
Fyf = C yf α f , Fyr = C yr α r ,
where C yf
and C yr
The “standard” H ∞ optimal control problem is concerned
with the feedback system shown in Fig. 2, where w represents
the system inputs (setpoint and disturbance), y is the available
measurement, u is the control signal, and e is an error signal
that needs to be minimized. Considering the general case of the
multivariable systems, the plant is described by the transfer
matrix G , while H is the controller. In H ∞ context, G
represents not only the conventional plant to be controlled but
also any weighting functions included to specify the desired
performance, which will be discussed later. Suppose that G is
partitioned consistent with the inputs w , u and outputs e and
y as:
(3)
represent the cornering stiffness
coefficients characterizing the front and rear tires. Using small
angle approximations, the tire slip angles can be calculated as:
αf =δ −
y + aψ
y − bψ
, αr =
,
x
x
(4)
Finally, considering constant vehicle speed, the nonlinear
model described by (1) can be linearized to the form:
my = −mVxψ + 2Fyf + 2 Fyr ,
Iψ = 2aFyf − 2bFrf ,
LOOP SHAPING METHOD FOR H-INFINITY CONTROL
 G ( s) G12 (s) 
G(s) =  11

G21 ( s) G22 ( s) 
(5)
(7)
Then, the resulting transfer function of the closed loop is
given by:
where Vx = x represents the longitudinal vehicle speed.
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Twe = G11 + G12 H ( I − G22 H )−1 G12
(8)
 S ( jω ) ≤ ε , ∀ω ≤ ω0

 S ( jω ) ≤ M , ∀ω > ω0
(13)
where S is given by (10). Although this is generally
applicable, it is convenient to formulate the same performance
criteria using a weight function w given as a transfer function,
as follows:
w( jω ) S ( jω ) ≤ 1, ∀ω
Fig. 2.
H ∞ feedback control system
with
It should be reminded that the singular values of a
multivariable system, give a measure of the amplitude of the
respective system, and enable frequency domain analysis. For
SISO systems, the singular values are equivalent with the Bode
plot of the system. The H ∞ norm of a given system G is
defined as
G
∞
 sup σ ( G ( jω ) )
 1/ ε , ∀ω ≤ ω0
w( jω ) = 
1/ M , ∀ω > ω0
Applying such weighting functions on all the transfer
functions (10) to (12), the closed loop performance of the
system can be specified and tuned.
(9)
ω∈
Considering such weighting functions, the original plant
that needs to be controlled can be augmented for the H ∞
control design. The obtained closed loop structure, for the
augmented plant can be seen in Fig. 3.
where σ ( A ) represents the largest singular value of matrix A
. Essentially, this means that the H ∞ norm gives a measure of
the frequency gain of the system G . Thus the H ∞ control
problem is to design the stabilizing controller H , that
interacting with the plant, minimizes the H ∞ norm of the
closed loop transfer function, i.e. Twe
(14)
∞
. This can be achieved
by setting a certain threshold γ ∈ [1,1.5] such that Twe
∞
≤γ .
One H ∞ control design approach is the "loop shaping"
method, and interested readers may refer to [14], [15] and [16].
For simplicity, consider a SISO closed loop system. It is well
known that in the closed loop control system can be
characterized by the following transfer functions:
S (s) =
1
1 + H ( s )G ( s )
H (s)
R( s) =
1 + H ( s)G ( s )
G0 ( s ) =
H ( s )G ( s )
1 + H ( s )G ( s )
(10)
Fig. 3.
(11)
Having the performance specifications of the closed loop
system in time domain, the equivalent specifications from
frequency domain can be derived. Based on these frequency
specifications, the weighting functions can be designed as
follows:
(12)
Equation (10) describes the "sensitivity function" of the
control system, and (12) describes the overall closed loop
transfer function or "complementary sensitivity function".
The H ∞ design approach makes use of the functions
described by (10) to (12), by imposing the desired frequency
behavior of each of these functions. For example, the
performance of a given closed loop system can be specified as
requiring:
H ∞ closed loop with augmented plant
k
 s / k M s + ωb
w1 ( s ) = 
 s + ωb k ε1





 s + ωbc / k M u
w2 ( s ) = 
 k ε 2 s + ωbc





(14)
k
(15)
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 s + ωb 0 / k M 0
w3 ( s ) = 
 k ε 3 s + ωb 0





The calculation of the controller can then be made by using
Riccati approach or the LMI approach, see [14],[15]. Under
certain assumptions of controller existence and simplicity,
which are skipped here for brevity, the following theorems can
be formulated.
k
(16)
where, ωb is the system bandwidth, M s is the peak
sensitivity, ωbc is the controller bandwidth, M u is the peak
gain of the controller, ωb 0 is the closed loop bandwidth, and
M 0 is the peak gain of the closed loop system. ε1 , ε 2 and ε 3
Theorem 1: There exists a controller H such that Twe
if and only if:
 A
γ −2 B1 B1T − B2 B2T 
 T

− AT
 −C1 C1

eigenvalues on the imaginary axis;
∗
(1) the matrix
need to be set to small values. Increasing the parameter k ∈ 
results in a steeper rolloff of the frequency response of each
weighting function.
(2)
The function w1 is used to weight S , w2 is used to weight
R and w3 is used to weight G0 , as follows:
S ( jω ) ≤
1
w1 ( jω )
(17)
R( jω ) ≤
1
w2 ( jω )
(18)
1
w3 ( jω )
(19)
G0 ( jω ) ≤
there
X ∞ A + A X ∞ + X ∞ (γ
(4)
the
T
w1S , w2 R, w3G0 
∞
+ C1T C1
such
C1T C1
that
=0;
Y∞ > 0
T
− C2 C2 )Y∞ + B1 B1T
exists
−2
≤γ ,
has no
has no
such
that
=0;
(5) the spectral radius ρ ( X ∞Y∞ ) < γ 2 .
Theorem 2: If the necessary and sufficient conditions of
Theorem 1 the obtained controller has the following form:
 A∞
H =
F
 ∞
− Z ∞ L∞ 
0 

(23)
where
(20)
F∞ = − B2T X ∞
L∞ = −Y∞ C2T
(24)
Z ∞ = ( I − γ −2Y∞ X ∞ )−1
A∞ = A + B1 B1T X ∞γ −2 + B2 F∞ + Z ∞ L∞ C2
 w1S 
=  w2 R 
 w3G0 
control
H∞
there
AY∞ + Y∞ A + Y∞ (γ
By applying (8) to (20) it is obtained
thus,
B1 B1T ) X ∞
 AT
γ −2C1T C1 − C2T C2 


T
−A
 − B1 B1

eigenvalues on the imaginary axis;
Taking into account the weighting functions (14), (15) and
(16), the plant described by (7) can be written in the following
form:
Twe
−2
(3) the matrix
T
 w1 − w1G 
0
w2 
G ( s) = 
 0 w3G 


I
G 


X∞ > 0
exists
T
∞
(21)
problem
IV.
For implementing the proposed H ∞ controller of
autonomous vehicle steering, the linear vehicle model was
used, while the closed loop validation is done by using the
actual nonlinear model of the vehicle. The vehicle model is
liniarized at constant vehicle speed of Vx = 28 [ km / h] . The
control signal of the autonomous steering system is the front
wheels steering angle δ , and the output is the lateral position
y as presented in the second section of the paper. The
sensitivity function (10) and complementary sensitivity
function (12) were considered, where ωb = 15 [ rad / sec] ,
becomes
≤γ .
The solution of the H ∞ control problem is based on a state
space representation of the generalized plant G , which
includes also the weighting functions:
x = Ax + B1w + B2 u
e = C1 x + D1w + D2 u
IMPLEMENTATION AND SIMULATION RESULTS
ωb0 = 55 [ rad / sec] ,
(22)
ε1 = 0.001 ,
ε 3 = 0.0001
and
M s = M 0 = 1.9 . The Bode plot of inverse of the considered
y = C2 x + D3 w + D4 u
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weighting functions, which are penalizing the sensitivity and
complementary sensitivity functions as seen in (17) and (19),
can be seen in Fig. 4.Considering γ = 1.18 , the obtained closed
loop transfer function gives the Bode plot shown in Fig. 5. As
it can be seen, the closed loop transfer function Twe is
complying with the specified requirement described
Twe ∞ ≤ 1.18 . Also the sensitivity and complementary
10
S
-10
Gain - db
-20
sensitivity functions are shown in Fig. 6 and Fig. 7, where it
can be seen that (17) and (19) are satisfied.
-30
-40
10
-50
0
-60
-10
1/w 1 & 1/w 3 [db]
1/w1
0
-70
10-10
10 -5
-20
10 0
105
10 10
105
10 10
Frequency - Rad/Sec
Fig. 6. Comparison of 1 / w1 and S
-30
-40
20
-50
0
-60
-80
10-10
-20
1/w 1
1/w 3
10
-40
-5
10
0
10
5
10
10
Gain [db]
-70
Frequency [Rad/Sec]
Fig. 4. Bode plot of the inverse of the considered sensitivity functions
Finally, a six order H ∞ is found, that stabilizes the vehicle
lateral dynamics and fulfils the closed loop specifications. A
simulation of a double lane change maneuver is presented in
Fig. 8, in order to analyze the trajectory tracking of the closed
loop control system for the automatic vehicle.
-60
-80
-100
-120
1/w 3
-140
G0
-160
-180
10-10
As it can be seen from Fig. 8, the proposed H ∞ control
system is performing very well in vehicle trajectory tracking
applications, obtaining a very small tracking error.
10 -5
10 0
Frequency [Rad/Sec]
Fig. 7. Comparison of 1 / w3 and G0
0.2
0
actual lateral position
desired lateral position
0.1
y [m]
-0.5
-1
0
-0.1
-1.5
-2.5
-3
-3.5
-4
-4.5
10-10
0
5
10
15
20
25
20
25
Time [Sec]
Tracking Error [m]
T we [db]
-0.2
-2
10 -5
10 0
105
10 10
10-3
5
0
-5
0
5
10
15
Time [Sec]
Frequency [Rad/Sec]
Fig. 8. Double lane change maneuver at Vx = 28 km / h
Fig. 5. Bode plot of the closed loop transfer function
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Simulations were carried out also at different vehicle
speeds Vx = 36.65 [ km / h ] and Vx = 50 [ km / h ] such that the
behavior of the control system can be studied in situations
when the controlled system presents different dynamics than
the liniarized model. In Fig. 9 and Fig. 11 can be seen that the
automatic steering maneuvers are still performed with
sufficiently high accuracy, showing that the H ∞ controller is
robust.
to follow the desired trajectory with very low tracking error.
The closed loop behavior at different vehicle speeds, show that
the proposed controller is sufficiently robust, when the vehicle
is operating in a neighborhood of the liniarization point. The
overall analysis shows that the proposed H ∞ control method
promises very good performances in practical use.
ACKNOWLEDGMENT
The work of R. C. Rafaila was supported by a grant of the
Romanian National Authority for Scientific Research and
Innovation, CNCS – UEFISCDI, project number PN-II-RUTE-2014-4-0970.
0.2
actual lateral position
desired lateral position
y [m]
0.1
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0
-0.1
-0.2
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10-3
5
0
-5
0
5
10
15
Time [Sec]
Fig. 9. Double lane change maneuver at Vx = 36.65 km / h
V. CONCLUSIONS
In this paper a H ∞ control method was proposed for
controlling the steering maneuvers of an autonomous driving
vehicle. The H ∞ controller was designed using the loopshaping technique.
0.2
actual lateral position
desired lateral position
y [m]
0.1
0
-0.1
-0.2
0
5
10
15
20
25
Tracking Error [m]
Time [Sec]
0.01
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-0.01
0
5
10
15
20
25
Time [Sec]
Fig. 10. Double lane change maneuver at Vx = 50 km / h
Simulation results show that the proposed controller
performs very well in trajectory tracking, the vehicle being able
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