JSAE Review 22 (2001) 435–444
Control of electrical power assist systems: H N design, torque
estimation and structural stability
Rakan C. Chabaana, Le Yi Wangb,*
a
Advanced Steering Controls and Electronic Design, Ford Motor Company, 20000 Rotunda Dr., Mail Drop 5021,
Dearborn, MI 48121-2053, USA
b
ECE Department, Wayne State University, Detroit MI 48202, USA
Received 26 March 2001
Abstract
In this paper, the H N method is employed to design controllers for improved performance and robustness of electrical power
assist steering systems (EPAS). EPAS systems are nonlinear MIMO systems with multiple objectives, including fast response to the
driver torque command, good driver feel, and attenuation of load disturbance and sensor noises. Since the EPAS system has
nonlinear frictions, component deviations and load disturbances, its linearized model is subject to significant modeling errors and
external disturbances. Consequently, EPAS controllers must provide substantial robustness. In this paper, a control design method
is introduced which employs the boost curve to form a feedforward control and an MISO H N optimal feedback to ensure refined
performance, robustness, and disturbance attenuation. Since the driver’s torque command cannot be directly measured, due to
packaging and cost issues, a torque estimator is introduced to generate the driver’s torque command by using other measured signals
which are normally available in the vehicle. Introduction of the torque estimator leads to an additional feedback loop, affecting
system robust stability and performance in a fundamental manner. Robust stability under this circumstance is established. The
controllers are evaluated via simulation on both linearized systems and original nonlinear systems, and verified on vehicle testing
data. r 2001 Society of Automotive Engineers of Japan, Inc. and Elsevier Science B.V. All rights reserved.
1. Introduction
In this paper, the H N method is employed to design
controllers for improved performance and robustness of
electrical power assist steering systems (EPAS). In the
recent years, the automotive industry has pursued
rigorously to improve automotive fuel economy and
provide variable assist capabilities. EPAS systems are
introduced as one potential solution to meet these
challenges. EPAS systems have several major advantages over conventional hydraulic systems: ð1Þ increased
fuel economy up to 4%; ð2Þ weight savings; ð3Þ
complexity reduction, due to elimination of fluid filling
operation and reduction of mechanical components; ð4Þ
package simplification due to elimination of hoses or
fluid cooler; ð5Þ flexibility in providing adaptive and
responsive on-demand steering assist; ð6Þ software
programmable steering characteristics; ð7Þ increased
*Corresponding author.
E-mail addresses: rchabaan@ford.com (R.C. Chabaan),
lywang@ece.eng.wayne.edu (L.Y. Wang).
reliability through self-diagnostic protection features.
To development engineers, the most important feature
of an EPAS system is that desired steering feel can be
changed without major physical modifications to valve
shape, torsion bar stiffness and boost pressure.
An EPAS system is a control system that electrically
amplifies the driver steering torque inputs to the vehicle
for improved steering comfort and performance. An
EPAS system consists of a steering wheel, a column, a
rack, an electric motor, a gearbox assembly, as well as
some torque, position and speed sensors. The essential
operation of an EPAS system can be depicted in the
system diagram shown in Fig. 1. Currently, all EPAS
systems employ a pinion torque sensor, between the
steering column and pinion, to determine the amount of
the torque assist to the driver which is calculated via a
tunable nonlinear boost curve. Then, this signal is used
as control command to the electric motor to achieve the
appropriate level of assist.
EPAS systems are nonlinear MIMO systems with
multiple objectives, including fast response to the driver’s torque command, good driver feel, and attenuation
0389-4304/01/$20.00 r 2001 Society of Automotive Engineers of Japan, Inc. and Elsevier Science B.V. All rights reserved.
PII: S 0 3 8 9 - 4 3 0 4 ( 0 1 ) 0 0 1 2 6 - 6
JSAE20014496
436
R.C. Chabaan, L.Y. Wang / JSAE Review 22 (2001) 435–444
on robust stability is investigated in Section 6. Computer simulation results are presented in Section 7 to
demonstrate the effectiveness of the torque estimator
and robustness of the closed-loop EPAS system.
Verification is further conducted by employing actual
vehicle testing data in Section 8. Finally, some conclusions are drawn in Section 9.
The key results of this paper were presented at the
AVEC 2000 Conference [9,10].
1.1. Literature
Fig. 1. EPAS models.
of load disturbance and sensor noises. Since the EPAS
system has nonlinear frictions, component deviations
and load disturbances, its linearized model is subject to
significant modeling errors and external disturbances.
Consequently, EPAS controllers must provide substantial robustness. In this paper, a control design method is
introduced which employs the boost curve to form a
feedforward control and an MISO H N optimal feedback to ensure refined performance, robustness, and
disturbance attenuation.
To facilitate the application of the H N method, the
nonlinear EPAS model is first linearized around its
nominal values. The multiple objectives, including
steering feel, disturbance attenuation and assist torque
performance, are formally introduced into a frequencydomain performance index. On the basis of the
performance index, the linearized model of the plant is
augmented into a state space model of higher dimensions. This system is then controlled by a feedforward
compensator via the boost curve for fast response and
an H N optimal feedback for enhanced robustness. Due
to the lack of direct measurement of the driver’s torque,
an estimator is developed to estimate the driver’s
command torque from the measured pinion torque.
For a production EPAS system, the overall performance
of the designed control system is evaluated by simulation and verified on vehicle testing data.
The paper is organized as follows. Section 2 presents
a generic EPAS model. Its components and their
roles are explained. The control problem is formulated
in Section 3. After a discussion on the basic performance
requirements of an EPAS system, a control system
structure is introduced which includes a feedforward
compensator and a feedback controller. The H N
optimal controller is designed in Section 4. It is shown
that the original plant model must be further expanded
to satisfy the basic requirements of the H N design.
Additional channels and state variables are hence
introduced. The original feedforward and feedback
control uses the driver torque Td as an input signal.
Since this signal is not measured, an estimator is
designed in Section 5. The implication of the estimation
Analysis of stability of EPAS systems has been
reported on several EPAS products. In [1], analytical
expressions of transfer functions are derived to model
assist dynamics, steering system compliance, and driver
feel of the road for a dual-pinion system.1 The analytical
estimation of stability boundary in terms of assist
stiffness-inertia envelopes was presented, which gives
an approximation for the maximum torque KaoKm G
that can be generated by the assist system. In [2,3,11],
vibration control techniques are used for dynamic
analysis of a single-pinion system.2 Two lead-lag
compensators were employed for control design. The
controller synthesis using constrained optimization
procedure is further explored. One of the advantages
of this method is that they can reduce the torque ripple
and optimize steering feel for given assist steering gains.
On the other hand, the design processes rely on several
additional sensors such as steering wheel position,
steering wheel velocity, motor position, motor velocity,
and torque sensor at pinion.
Along a similar direction, analysis of stiffness and feel
was reported in [4]. Mathematical equations, which
relate rack force to the driver torque and steering angles,
were developed to help EPAS gear engineers design an
appropriate boost curve. These mathematical equations
can assist in determining the regions of boost curves for
which the steering feel varies as a function of vehicle
speed. The paper [5] studied the problem of power
steering ‘road feel’. The relationship between the pump
pressure vs. steering torque was derived. It was argued
that a gear can be designed on the basis of a set of boost
curves which can be adapted as a function of vehicle
speed. These boost curves can accommodate the
desirable reduction of effort at parking and increased
effort at straight driving as the vehicle speed increases.
Numerous feedback control techniques have been
applied to EPAS systems, ranging from traditional leadlag compensation to more advanced control techniques.
For example, the paper [6] investigated control design
by employing an H N -type criterion to provide an
1
In the dual-pinion arrangement, the driver input acts through one
pinion and the power assist acts through another.
2
In the single-pinion arrangement, the driver input and power assist
act through one pinion only.
R.C. Chabaan, L.Y. Wang / JSAE Review 22 (2001) 435–444
appropriate steering feel based on road information. It
was revealed that the key element for the EPAS system
was frequency rectification between the rack force and
the reaction torque. In [7], several key points in
implementing EPAS system in vehicle were described.
A three-degree-of-freedom model of the EPAS was
developed. It was shown that by eliminating certain
higher frequency components a reduced order model
could be obtained. An analysis on the closed-loop
system was performed to understand compromises in
system design. In [8], it was explained that increasing
assist gain reduces steering torque for drivers. However,
it may cause undesirable steering vibration at specified
frequencies around 30 Hz: Motor angular velocity was
needed for this control algorithm. An estimator design
was investigated to obtain this signal without installing
such a sensor. Two methods were investigated: Observer
437
and back e.m.f. The authors pointed out that back e.m.f.
was superior to the observer method and no additional
cost was needed.
2. EPAS systems
Following Newton’s laws of motions, we can establish
a nonlinear dynamic model of the EPAS system as
follows (see Fig. 2):
Kc
Jc y. c þ Bc y’ c þ Kc yc ¼ xr þ Td þ fc ðyc ; y’ c Þ;
rp
Kc
1
Km G
G
M x. r þ Br x’ r ¼
yc xr þ
ym x r
rp
rp
rp
rp
Ft þ fr ðxr ; x’ r Þ;
Jm y. m þ Bm y’ m ¼ Tm Ta þ fm ðym ; y’ m Þ;
Fig. 2. A nonlinear EPAS model.
ð1Þ
438
R.C. Chabaan, L.Y. Wang / JSAE Review 22 (2001) 435–444
where Td and Ta are the driver torque ðN mÞ and
assist torque ðN mÞ: yc ; xr ; ym are steering wheel rotary
position (rad), rack position (m), and assist motor
rotary position (rad), respectively. Plant parameters
include Jc (steering column rotational moment of inertia
ðkg m2 ÞÞ; Kc (steering column stiffness (N/rad)), Bc
(steering column viscous damping (N s=rad)), M (mass
of the rack (kg)), Br (viscous damping on the rack
(N s=rad)), rp (steering column pinion radius (m)), G
(motor gear ratio), Km (motor shaft and gearbox
rotational stiffness (N/rad)), Jm (motor rotational
moment of inertia ðkg m2 ÞÞ; Bm (motor shaft and
gearbox viscous damping ðN s=radÞÞ; Kt (tire or rack
centering spring rate (N/m)). fc ; fr ; and fm are nonlinear
friction terms in the model.
Here,
G
Ta ¼ Km ym xr
rp
is the assist torque. The motor electromagnetic torque
input Tm ðN mÞ is related to motor current input by an
inner loop Tm ¼ Gin ðsÞu where u is the motor input and
Gin is the inner-loop transfer function. In the frequency
band of interest, Gin 61 (per normalization of mechanical units). As a result, one may simply view Tm as the
control input. Also, the resistance term Ft ¼ Kt xr þ Fr :
We note that the resistance force on the rack is mainly
Kt xr : The remaining disturbances from road condition
changes are denoted by Fr :
Generically, an EPAS system has two main external
inputs: The driver torque Td which is not measured and
the assist motor input current u which is a control signal.
Due to variations in road and tire conditions, sensor
inaccuracy and measurement noise, the EPAS system is
subject to external disturbances Fr (resistance force
on the rack) and d (sensor noises). The dimension of
d is identical to the number of sensors utilized in the
system.
The key outputs which are to be controlled include
assist torque Ta and the pinion torque Tc : Ta is not
measured directly. On the other hand, in all current
EPAS systems a torque sensor is used to measure Tc :
Other measured signals vary with hardware configurations. The methodology we introduce in this paper is
generic and applicable to all types of sensor configurations. Based on a production EPAS system, we shall
assume that the column angle yc and assist motor
angular velocity y’ m are measured.
Upon an approximation of the nonlinear components fc ; fr ; fm by their linear nominal expressions, a
linearized state space can be obtained in the state space
form
x’ ¼ Ax þ B1 Td þ B2 Fr þ B3 u;
y ¼ Cx þ D1 Td þ D2 Fr þ D3 u
ð2Þ
or transfer function form
2 3 2
3
Ta
G11 G12 G13 2 3
Td
6 7 6
6 Tc 7 6 G21 G22 G23 7
76 7
6 7¼6
74 u 5:
6 yc 7 4 G
G33 5
31 G32
4 5
Fr
G41 G42 G43
y’ m
ð3Þ
3. Control problems
3.1. Design objectives
An EPAS system responds to a driver’s torque
command Td and delivers an appropriate level of assist
torque Ta via a tunable boost curve by controlling the
motor input u: The system is subject to external
disturbances Fr and sensor noises dT on Tc ; dc on yc
and dm on y’ m : It follows that a typical EPAS system can
be generically described as in Fig. 3.
The goals of EPAS design can summarized as follows.
Assist torque generation: The desired level of assist is
defined by a boost curve which relates the driver torque
to the desired assist torque. The boost curve can be
expressed as
Ta* ¼ BðTd ; vÞ;
where v is the vehicle speed (mph). A typical boost curve
is shown in Fig. 4.
Due to dynamic delays of the EPAS system in
responding to changes in Td ; the actual Ta generated
by the EPAS may be significantly different from the
desired Ta* ; depending on control designs. The first
objective of a well-designed EPAS system is to deliver Ta
that is reasonably close to Ta* in a frequency band
½0; w0 ; where w0 is about 40 Hz to deliver a desirable
steering feel.
Driver’s feel: When road condition changes, the
resistance force Fr on the rack will vary. To provide a
Fig. 3. Inputs and outputs of EPAS systems.
439
R.C. Chabaan, L.Y. Wang / JSAE Review 22 (2001) 435–444
closed-loop pole positions tend to move towards
unstable regions when Ka becomes large. A common
remedy for this situation is to design a very conservative
EPAS control. Consequently, system performance is
compromised.
3.2. Control structures
We will start with the control structure when Td is
available. Afterwards, Td will be replaced by its estimate
T#d to derive an implementable control system.
Under nominal operating conditions with Fr ¼ 0;
dT ¼ dc ¼ dm ¼ 0; the EPAS system can be linearized
and represented by
Ta ¼ P11 Td þ P12 u;
ð4Þ
y ¼ P21 Td þ P22 u;
ð5Þ
where y ¼ ½Tc ; yc ; y’ m T is the measurement vector,
and P11 ¼ G11 ; P12 ¼ G12 ; P21 ¼ ½G21 ; G31 ; G41 T ; P22 ¼
½G22 ; G32 ; G42 T :
Suppose that the nominal value of the input torque is
Td0 : From the boost curve, we can derive desired assist
torque Ta0 ¼ BðTd0 Þ (for a given vehicle speed). From
Eqs. (4) and (5), the steady-state relationship
Ta0 ¼ P11 ð0ÞTd0 þ P12 ð0Þu0
Fig. 4. Boost curve for a given vehicle speed.
desirable level of driver’s feel of driving conditions, the
driver must perceive a suitable amount of changes in the
twist torque Tc : In other words, one can define a desired
mapping from Fr to Tc as
Tc ¼ Gf Fr :
For instance, one may require Gf ¼ Kf ¼ 0:15 in a
specified low frequency range.
Robustness: The EPAS system contains nonlinear
components such as frictions and damping. Also,
unit-to-unit deviations and component wearing will
introduce parameter deviations from the nominal
models. Furthermore, road condition changes and
sensor noises introduce uncertainties during its operation. An EPAS system must be designed to provide
satisfactory performance in the presence of such
uncertainties.
A main challenge in EPAS control system design is
that the controller must perform well under all possible
operating conditions of the boost curve. Using Tc in
place of Td results in a feedback loop whose loop gain
depends on the slope Ka of the boost curve. In a
traditional hydraulic EPAS system, Ka can range from 0
to 50: In a simple feedback system where Tc is sent
through the boost curve to generate control input, the
ð6Þ
can be used to determine the steady-state control value
u0 :
While u0 can be directly computed from Eq. (6) for
the given Td0 and Ta0 ; u0 is in fact approximately BðTd0 Þ:
Indeed,
u0 ¼
Ta0 P11 ð0ÞTd0 BðTd0 Þ P11 ð0ÞTd0
¼
:
P12 ð0Þ
P12 ð0Þ
Without modeling errors, P12 ð0Þ ¼ 1 and usually BðTd0 Þ
is much larger than P11 ð0ÞTd0 : Hence, we may view
u0 ¼ BðTd0 Þ
as the nominal feedforward control. Then, y0 can be
calculated from
y0 ¼ P21 ð0ÞTd0 þ P22 ð0Þu0 :
Correspondingly, the equilibrium value x0 of the state
variable from Eq. (2) can be determined. Now, by
defining
*
x ¼ x0 þ x;
Td ¼ Td0 þ T*d ;
Ta ¼ Ta0 þ T*a ;
u ¼ u0 þ u;
*
y ¼ y0 þ y;
*
we can derive a perturbation system
x’* ¼ Ax* þ B1 T*d þ B2 Fr þ B3 u;
*
y* ¼ C1 x* þ D1 T*d þ D2 Fr þ D3 u:
*
440
R.C. Chabaan, L.Y. Wang / JSAE Review 22 (2001) 435–444
Fig. 6. H N control structure.
Fig. 5. Control structure.
The goal of feedback control is to employ a feedback
u* ¼ CðsÞy*
such that the effects of disturbances T*d ; Fr ; dT ; dc ; dm on
T*a is minimized. CðsÞ will eventually be designed by
using H N control. It follows that the control structure
consists of a feedforward and a feedback controllers, as
illustrated in Fig. 5.
4. H N design
4.1. Design procedure
The purpose of the feedback C is to provide
robust stability of the closed-loop system and minimize
the effect of Fr ; dT ; dc ; dm on T*a : To facilitate the
design of C by using the H N design tools, we
must modify the system state space models. The original
plant is
x’ ¼ Ax þ B1 Td þ B2 Fr þ B3 u;
y ¼ Cx þ D1 Td þ D2 Fr þ D3 u:
The control objectives can be qualitatively stated as:
ð1Þ Reduction of jjWa T*a jj2 for desired torque production, where T*a ¼ Ta Ta* and jj jj2 is the H 2 norm. ð2Þ
Reduction of jjWf ðTc Kf Fr Þjj2 ; for appropriate road
feel. ð3Þ Reduction of control jjKu ujj2 for fuel economy.
Here, Wa and Wf are weighting functions that define
frequency ranges in which the reductions are to be
effective. Typical examples of the weighting functions
are
Wa ðsÞ ¼ ca
b1 s þ 1
;
a1 s þ 1
Wf ðsÞ ¼ cf
b2 s þ 1
:
a2 s þ 1
Since the weighting functions are dynamic systems,
they must be incorporated into the state space model. A
state space model for Wa can be derived as follows: Let
ea ¼ Wa T*a : Then
1
b1
b1
w’ a ¼ wa þ ca 1 1 Td ;
C1 x þ Ca Ka
a1
a1
a1
1
b1
Ca b1 Ka
Td
ea ¼ wa þ ca C1 x a1
a1
a1
is a state space model for the weighting function.
Similarly, a state space model for Wf can be derived as
1
b2
b2
w’ f ¼ wf þ cf 1 1 Kf Fr ;
C2 x þ cf
a2
a2
a2
1
b2
b2
ef ¼ wf þ cf C2 x cf Kf Fr :
a2
a2
a2
By expanding the state variable to
2 3
wa
6 7
x1 ¼ 4 wf 5;
x
the inputs to u1 ¼ ½d; uT with d ¼ ½dT ; dc ; dm ; Td ; Fr ;
outputs to y1 ¼ ½w; yT with controlled output w ¼
½eu ; ea ; ef and measured output y ¼ ½Tc ; yc ; y’ m ; as
shown in Fig. 6, we can obtain an augmented state
space model
* 1 þ B* 1 u þ B* 2 d;
x’ 1 ¼ Ax
" #
d
*
*
y1 ¼ Cx1 þ D
;
u
where
2
a11
6
A* ¼ 6
4 0
0
a12
0
3
0
6 7
B* 1 ¼ 4 0 5;
B3
0
ca ð1 ba11 ÞC1
3
7
cf ð1 ba22 ÞC2 7
5;
A
2
B* 2 ¼ ½BT ; Bc ; Bm ; Bd ; BF :
R.C. Chabaan, L.Y. Wang / JSAE Review 22 (2001) 435–444
441
Here
2
0
0
2
6
6
6
6
6
6
6
Bm ¼ 6
6
6
6
6
6
6
4
1
a2
0
0
01
6
6
60 0
6
6
60 0
D* ¼ 6
6
6e 0
6
60 e
4
0 0
7
7
7
7
7
0 7
7
7;
0 7
7
GKm 7
7
rp 7
7
0 5
Km
0
0
3
7
ca ba11 C1 7
7
cf ba22 C2 7
7
7;
C3 7
7
7
C4 5
0
0 0
3
2
0
0
0
3
3 2
0
0
7
6 7 6 b2
7
BF ¼ 4 0 5 þ 6
4 cf ða2 1ÞKf 5;
B2
06
1
3
0
6 7
Bd ¼ 4 0 5;
B1
61
6 a1
6
60
6
C* ¼ 6
60
6
6
40
6 0 7
7
6
7
6
6 0 7
7
6
6 K 7
6
c7
7
Bc ¼ 6
6 0 7;
7
6
6 Kc 7
6 rp 7
7
6
7
6
4 0 5
0
0
2
0
2
2
6 0 7
7
6
7
6
6 0 7
7
6
7
6
6 1 7
7;
6
BT ¼ 6
7
6 0 7
6 1 7
6 rp 7
7
6
7
6
4 0 5
0
2
3
3
C5
0
0
0
0
Ca ba11 Ka
0
0
0
cf ba22 Kf
0
0
0
0
0
0
e
0
0
Ku
3
7
0 7
7
7
0 7
7;
7
0 7
7
0 7
5
0
where e is a small positive number which makes the
lower left 3 5 submatrix of D* full rank. Also the
nonzero Ku makes the top right 3 1 submatrix of D*
full rank. As a result, the H N design problem is
nonsingular.3
It follows that the design problem can be formulated
as an H N minimization problem: Find the feedback
matrix u ¼ CðsÞy which minimizes w in the presence of
disturbance d: The controller can be designed by using
the m-Tools Toolbox [12].
5. Estimation of Td
The desired level of assist torque is defined by a boost
curve which relates the driver torque to the desired assist
torque. However, since the driver torque is not
3
The reader is referred to the user’s manual of the m-Tools Toolbox
for more detailed explanations of conditions on state space H N
solutions.
Fig. 7. Td and Tc in open-loop responses.
measured, the boost curve cannot be directly implemented to generate the desired assist torque. Currently, a
pinion torque sensor is installed to measure Tc : In most
EPAS systems, Tc signal is used as an approximation of
Td for boost curve input (feedforward control) and
feedback control implementations.
However, although at steady state, Tc and Td are
essentially identical, they differ dramatically during
dynamic transitions. The difference is mostly pronounced when the assist torque is present. Without
compensation of this dynamic difference, the dynamic
behavior and performance of the EPAS will be severely
compromised. Figs. 7 and 8 demonstrate their differences in open and closed-loop configurations. From the
preliminary analysis and test data, the column, steering
shaft torsion stiffness, and moment of inertia are shown
to be difficult to determine exactly due to the nonlinearity of the system. Also, the center of gravity of the
steering wheel varies when the driver rotates the wheel
while driving. Hence, the driver torque and pinion
torque are different during dynamic transitions if Jc and
Kc are not modeled correctly.
In this paper we introduce an estimator for Td by
using the signal Tc and plant models. Since
Tc ¼ G21 Td þ G22 u;
442
R.C. Chabaan, L.Y. Wang / JSAE Review 22 (2001) 435–444
feedback loop has variable loop gain, depending on Ka ;
which changes with the vehicle speed and input torque.
To guarantee stability of the entire system, it is
necessary to establish robust stability conditions under
which the closed-loop system is stable for all possible
values of Ka : These conditions are given in the following
theorem.
Note that by the H N design, both 1=ð1 þ CP22 Þ
and C=ð1 þ CP22 Þ are stable. From the plant model and
estimator design, we have the following relationships:
y ¼ P12 Td þ P22 u; u ¼ Ka F 0 T#d Cy;
T#d ¼ Qy þ Q2 u;
where the boost-curve gain 0pKa pkmax and F 0 is a
stable filter for frequency-domain loop shaping.
Theorem 1. Suppose that the uncertainty on the system is
given by 0pKa pkmax and jjWDjjN pe: If
1 0 W F 1
1 þ CP ok e;
22 N
max
then the closed-loop system is robustly stable.
Proof. Since the output Ta is related to the inputs by
Ta ¼ P11 Td þ P12 u and P11 and P12 are stable systems,
Ta will be bounded whenever u is bounded. As a result,
for stability analysis, we only need to establish the
boundedness of u:
Fig. 8. Td and Tc in closed-loop responses.
u ¼ Ka F 0 ðQy þ Q2 uÞ Cy
the exact relationship between Td and Tc is
Td ¼ G1
21 ðTc G22 uÞ:
¼ ðKa F 0 Q CÞy þ Ka F 0 Q2 u
It is noted, however, that G1
21 is improper and unstable.
Since the frequency band in which Td must be accurately
estimated is relatively narrow, it is possible that a proper
and stable estimator can be found which approximate
G1
21 over the frequency band of interest. For a production EPAS system, such an estimator was found as
T#d ¼ Q1 Tc þ Q2 u ¼ Qy þ Q2 u;
where Q1 and Q2 are proper and stable, and Q ¼
½Q1 ; 0; 0:
From the design of the estimator, it is easy to verify
that D ¼ QP22 Q2 is small over the low frequency
band of interest. More accurately, there exists a bi-stable
weighting function W (both W and W 1 are stable)
such that
jjWDjjN pe:
ð7Þ
6. Stability analysis
When Tc is measured and used to predict Td and used
as an input to the boost curve to generate control
signals, an additional feedback loop is formed. This
¼ ðKa F 0 Q CÞðP12 Td þ P22 uÞ þ Ka F 0 Q2 u
¼ ðKa F 0 Q CÞP12 Td þ ððKa F 0 Q CÞP22 þ Ka F 0 Q2 Þu
¼ ðKa F 0 Q CÞP12 Td þ ðKa F 0 D CP22 Þu:
It follows that
ð1 þ CP22 Ka F 0 DÞu ¼ ðKa F 0 Q CÞP12 Td
and
W 1 F 0
ðKa F 0 Q CÞP12
1 Ka DW
Td :
u¼
1 þ CP22
1 þ CP22
By the H N design, the right-hand side ðKa F 0 Q CÞ
P12 =ð1 þ CP22 Þ is stable with
ðKa F 0 Q CÞP12 pZoN:
1 þ CP22
N
Furthermore, by hypothesis
1 0 W F W 1 F 0 o1:
m ¼ Ka DW
pk
jjDWjj
max
N 1 þ CP22 N
1 þ CP22 N
It follows from the Banach Space Contraction Principle
that 1 Ka DW½ðW 1 F 0 Þ=ð1 þ CP22 Þ is invertible in
R.C. Chabaan, L.Y. Wang / JSAE Review 22 (2001) 435–444
443
H N and
1 W 1 F 0
1
:
1 Ka DW
p
1m
1 þ CP22
N
Therefore, the closed-loop system is stable and
1
W 1 F 0
ðKa F 0 Q CÞP12 Z
:
p
1 Ka DW
1m
1 þ CP22
1 þ CP22
N
&
7. Computer simulation
Simulations were performed on the linearized system
as well as the original nonlinear EPAS system (see
Fig. 1) to ensure that the linear controller can be
implemented.
Fig. 9 depicts the responses of the linearized EPAS
system to the driver’s torque command for a step with a
sine wave to simulate a worst-case condition to the
EPAS system. As the figure shows, the H N controller
provides very good response of desired torque (Tadesired) vs. produced torque by an EPAS to achieve the
desired steering feel.
Fig. 10 is the responses of the original nonlinear
system. The figure shows that the RMS error has
increased from 5:49% of linear system to 11:9% of
nonlinear. However, the increase in RMS error will not
hinder the steering feel since the difference is 0:14 N m
which cannot be detected by the driver due to some
unfiltered vibration in steering column and road
disturbances.
Fig. 11 shows the relationship between the driver
torque and the assist torque, produced by the closedloop system. The shape approximates closely the desired
boost curve.
Fig. 9. Controlled responses on linearized systems.
Fig. 10. Controlled responses on original systems.
Fig. 11. Relationship between the driver torque and the assist torque.
8. Vehicle data verification
To test H N control algorithms and the driver torque
estimator, a small vehicle was instrumented with a
single-pinion EPAS system as shown in Fig. 1. Several
measurement channels were installed to measure vehicle
speed (mph), driver torque (nm), steering angle (deg),
rack displacement (m), motor position (deg), motor
velocity (deg/s), and boost curves. These signals were
measured by using a Dspace box at 500 Hz with vehicle
speeds ranging from 10 to 30 mph for 60 s: The
measured driver torque, steering angle, and rack
displacement are shown in Fig. 12.
The measured data are then inputed to the control
system and torque estimator to generate torque estimation.
Fig. 13 compares the measured driver torque to the estimated torque. It is apparent that the torque estimator provides sufficiently accurate estimates for the driver torque.
The performance of the H N controller is verified by
employing the measured driver’s torque as the command
input. Fig. 14 compares the desired assist torque from
444
R.C. Chabaan, L.Y. Wang / JSAE Review 22 (2001) 435–444
Fig. 12. Measured signals.
Fig. 14. Vehicle data verification: desired assist torque vs. generated
assist torque.
production of satisfactory robust controllers, leading to
reduction of development time.
The controllers introduced in this paper are currently
undergoing technology transfer to production EPAS
systems and patent applications.
References
Fig. 13. Vehicle data verification: measured driver torque vs. estimated
torque.
the boost curves and the actual assist torque generated
by the closed-loop system, as well as their differences.
9. Concluding remarks
The EPAS system is of unique importance in
automotive system development. It is an important
technology advancement from the classical hydraulic
systems. Understanding its control requirements and
developing generic design methodologies will facilitate
rapid development of control strategies following hardware alternations. Unlike traditional PID or lead-lag
compensations which require substantial tuning and
manual calibration on individual systems, the method
introduced in this paper can potentially lead to fast
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