CONTROL
usable (highly simplified) nominal model for control, it must be validated by data. Val-idation, however, gives no hints on how to model the uncertainty itself, it gives only a
"no" or "not impossible" answer to validity. The D-K-W iteration method presented in Chapter 4 tunes an unfalsified uncertainty model of a given structure. Sections 6.2.1 - 6.2.3 present the simplified nominal vehicle model that describes the effects of asym-metric front wheel braking on the steering system and the yaw motion of the vehicle.
The structure of the uncertainty model is also specified in Section 6.2.4.
6.2.1 Derivation of physical equations
The idea of this chapter on how to use the brake system for steering comes from the following observation. It can be experienced with a loaded heavy-truck that applying a strong (10bar) brake pressure on one of the rear wheels has practically no effect on the yaw motion of the vehicle. (This kind of intervention, however, can be efficient on passenger cars.) The reason is related to the load distribution. A more reliable and robust cornering can be executed through turning aside the steering system by applying small brake pressure to one of the front wheels. The steering facility of braking is much weaker than steering by the handwheel, therefore, the driver can easily overwrite the steering command. Thus the steer-by-brake problem is solvable if the driver releases the handwheel, which is assumed hereafter.
The aim if the section is to describe the steer-by-brake mechanism by physical equations. The derivation can be performed by applying simplification assumptions to the HFVS dynamic model and is detailed in Appendix B. The HFVS is linearized by using first order Taylor-series approximation around state-variables of zero. In a front-wheel-braking scene only small longitudinal wheel slips and steering angles are assumed. Nonlinear adhesion function is linearized in the region of small slip values.
The cross-effects of longitudinal, roll and heave motions are neglected. These, and some technical, modelling assumptions result in the model scheme plotted in Figure 6.3 and described by linear model
M0 : x˙ =A0x+B0u y=C0x
wherex= [β r δδ˙∆vR]T,u= ∆p,y= [r δ∆vR]T and the state-space matrices depend on constant parameters and forward velocity as given by (B.15).
Braking one of the front wheels with a nonzero pressure difference ∆p at the front wheels develops wheel velocity difference (∆vR) described by differential wheel model (B.14). The emerging slip difference (∆s) generates a torque (Ms) that acts as input for the steering system. The increasing steering angle cause the vehicle cornering as described by the single-track model. The yaw motion develops stabilizing (aligning) torquenRKFy through side force Fy.
6.2. MODELLING FOR STEER-BY-BRAKE CONTROL
f r r r f f f
2
2 2
f f
r r f f r r f f
z z z
c c c l c l c
mv 1 mv mv
c l c l c l c l c l r r
J J v J
+ −
− − +
β
β = + δ
− +
−
ɺ ɺ
f
S s RK y
M rc s n F
= 2∆ + f eff2 pT eff
R
w w
c r c r
v s p
2J J
∆ɺ = ∆ − ∆
v r l v s ∆vR + w
−
=
∆ FB Fy
v
nR nK
Fy FB
rs
δ−β−
= v
r c l
Fy f f
δ r
r
∆p
r Fy ∆s
MS
δ δ
δm
CS
MS kS
δm
S S S m S
M +J δ =ɺɺ C (δ − δ −) k δɺ
d m m m S m
m C ( ) k T
J δɺɺ =− δ −δ− δɺ + Td
f r r r f f f
2
2 2
f f
r r f f r r f f
z z z
c c c l c l c
mv 1 mv mv
c l c l c l c l c l r r
J J v J
+ −
− − +
β
β = + δ
− +
−
ɺ ɺ
f
S s RK y
M rc s n F
= 2∆ + f eff2 pT eff
R
w w
c r c r
v s p
2J J
∆ɺ = ∆ − ∆
v r l v s ∆vR + w
−
=
∆ FB Fy
v
nR nK
Fy FB
rs
δ−β−
= v
r c l
Fy f f
δ−β−
= v
r c l
Fy f f
δ r
r
∆p
r Fy ∆s
MS
δ δ
δm
CS
MS kS
δm
S S S m S
M +J δ =ɺɺ C (δ − δ −) k δɺ
d m m m S m
m C ( ) k T
J δɺɺ =− δ −δ− δɺ + Td
δ δm
CS
MS kS
δm
S S S m S
M +J δ =ɺɺ C (δ − δ −) k δɺ
d m m m S m
m C ( ) k T
J δɺɺ =− δ −δ− δɺ + Td
Figure 6.3: Modelling scheme for steer-by-brake control 6.2.2 Structure estimation and model reduction
Physically parameterized continuous-time LPV state-space family is chosen in predictor form
˙ˆ
x(t) = A(θ, v)ˆx(t) +B(θ, v)u(t) +L(θ)e(t), ˆ
y(t) = C(θ, v)ˆx(t), (6.1)
e(t) = y(t)−y(t),ˆ
whereθdenotes the vector of unknown parameters. Advantages of physical parametriza-tion:
• Known dependence on the scheduling variable avoids searching in an infinite func-tion space
• Less parameters then in case of black-box models
• Physical insight into the relevance of the model components
• Physical insight into the role of parameters might help to estimate parametric uncertainty to the nominal model (e.g. the change of load, load distribution and cornering stiffness)
CONTROL
M
0M
1M
4M
3M
2A1
A3 A2
A2 A3
M
0M
1M
4M
3M
2A1
A3 A2
A2 A3
Figure 6.4: Structure graph generated from M0 by assumptions A1-A3
A disadvantage is that the parameters in canonical forms tend to be ill-conditioned, criterion function for identification may be quasi-convex.
It was stated in [RB05a] that small cornering does not excite sufficiently the yaw dynamics for an identification of the cornering stiffness parameters when only the yaw-rate can be measured. The reason is that there is small difference in the dynamics of the systems δ 7−→ β and δ 7−→ ψ. This means that a simpler model than the single-˙ track model is sufficient to describe the yaw motion with small steering excitation.
Similarly, the steering system’s dynamics is excited through the fast wheel dynamics.
Fast dynamical components, which are not excited by the slower dynamics of a yaw-rate controller, can be neglected. FromM0 four different LPV model structures can be derived by applying the following simplification assumptions.
• A1. In simulations the sideslip angle and yaw rate appeared to be approximately proportional. That is why state β is omitted and β(t)r(t), approximated as
β(s) r(s)
s=0
= lr
v −mlf
crl v, is inserted in the expression forr.˙
• A2. The wheel model is assumed to be fast for the excitation bandwidth and
∆vR(t) is replaced by the DC component of its Laplace-transform:
∆vR(s)|s=0 =lwr−2 CpTv cfref f∆p
• A3. The steering system is assumed to be fast for the excitation bandwidth, δ(t)≈δ(s)|s=0 = (l
v − rslw
2nRKv−mlf
crl v)r+ rs
2nRKv∆vR
Applying the simplifications from A1 to A3, the model graph on Figure 6.4 is gen-erated. Assumption A1. is required for the state-space models to have an identifiable parametrization. Models from M1 to M4 are in observer canonical representation, which is an identifiable structure according to [81, Appendix 4A].
The filter gain L can be directly parameterized. Because of the ill-conditioned parametrization of the canonical structure, in case of full-parameterized 3x2 L matrix
6.2. MODELLING FOR STEER-BY-BRAKE CONTROL
the criterion function is almost quasi-convex and the parameter vector θdoes not con-verge. A sufficient parametrization isL=
0 0 0 0 0 p7
T
, soθ=
p1 · · · p7 , which means that the steering angle error is fed back.
The models are identified by using prediction error method with the criterion V(θ) =
nv
X
i=1
1 Ni
Ni
X
k=1
εi(kTs, θ)TΛ−1i εi(kTs, θ), Λ−1i = diagh w
1
a1,i, . . . , awny
ny ,i
i
, (6.2)
ap,i = 1 Ni
Ni
X
k=1
yp,i2 (kTs), p= 1,2, . . . , ny,
where εi(kTs, θ) = yi(kTs)−yˆi(kTs, θ) is the prediction error vector at time t = kTs, Ni is the number of data-points in the ith experiment. The velocity vi is constant over each experiments. The ny is the number of outputs, nv is the number of exper-iments composing the criterion function. The model outputs yˆi(kTs, θ) are computed by transforming the continuous-time linear models (6.1) with v = vi into the discrete time-domain assuming zero-order-holder as follows
ξ(k+ 1) = F(θ, vi)ξ(k) +G(θ, vi)u(k) +H(θ)ν2(k) z(k) = C(θ, vi)ξ(k) +D(θ, vi)u(k),
ξ(0) = ˆx(0) = 0, F(θ, vi) = eA(θ,vi)Ts, G(θ, vi) =
Z Ts
0
eA(θ,vi)(Ts−τ)B(θ, vi)dτ, H(θ) =
Z Ts
0
eA(θ,vi)(Ts−τ)L(θ)dτ, theny(kTˆ s) :=z(k),k= 1, . . . , Ni.
ForΛa constant weighting is applied wherewiscalars reflect the relative importance of the outputs in the matching.
The identification is performed with experiments with the HFVS. (More details of the identification procedure is presented in [RB05a] where real-life experimental data from a MAN truck was used.)
First, the resulted structures are validated. On figure 6.5 the pole-location of the four LPV models (M1,...,M4) are pictured with fixed scheduling variable fromv= 8m/s to 20m/s. The input bandwidth is also plotted as a circle with radius of 8rad/s. Those structures can be accepted of which poles are placed inside the circle. Poles outside the circle imply the presence of not excited, fast dynamics. According to figure 6.5 it can be concluded that the wheel dynamics is too fast. Since M2 is better in criterion than M4,M2 is selected as a candidate for controller design.
On figure 6.6M2 is validated on the time domain. On the left side the velocity was 8.05 m/s during the experiment, on the right it was 17.49m/s. The measured input
CONTROL
−20 −15 −10 −5 0 5 10
−15
−10
−5 0 5 10 15
M1
Real
Imag
−8 −6 −4 −2 0 2 4 6 8
−8
−6
−4
−2 0 2 4 6 8
M2
Real
Imag
−60 −50 −40 −30 −20 −10 0 10
−8
−6
−4
−2 0 2 4 6 8
M3
Real
Imag
−8 −6 −4 −2 0 2 4 6 8
−8
−6
−4
−2 0 2 4 6 8
M4
Real
Imag
Figure 6.5: Structure validation: the excitation bandwidth and pole locations ofA−LC fromv= 8m/s to20m/s
∆p, the yaw rate r and the steering angle δ are plotted. The outputs of the predictor (solid lines) fit the measurements well-markedly. In many control synthesis problems the role of the filter L(θ) is redesigned, only the A(θ, v), B(θ, v) and C(θ, v) matrices of the predictor are kept. The outputs of the predictor without the filter (L(θ) = 0) are plotted with dashed lines. This approach of omitting L after identification fits to the prediction error identification paradigm (separation principle [48, 60]): first try to model as well as possible with a full order model then reduce the model for any (e.g.
control) purposes.
6.2.3 Simplified nominal model Chosen nominal model M2 is characterized by
ψ¨ = (−p1l
v +p2v) ˙ψ+p1δ,
¨δ = (p3l
v +p4v) ˙ψ−p3δ+p5δ˙+p6∆p,
6.2. MODELLING FOR STEER-BY-BRAKE CONTROL
0 1 2 3 4 5 6 7 8
−1 0 1
pressure,∆p
M2 v=8.05 m/s
0 1 2 3 4 5 6 7 8
−0.1
−0.05 0 0.05 0.1
yaw−rate, r
0 1 2 3 4 5 6 7 8
−0.04
−0.02 0 0.02 0.04
steering angle, δm
time [s]
0 1 2 3 4 5 6 7 8
−1 0 1
pressure,∆p M2 v=17.49 m/s
0 1 2 3 4 5 6 7 8
−0.15
−0.1
−0.05 0 0.05 0.1
yaw−rate, r
0 1 2 3 4 5 6 7 8
−0.02 0 0.02
steering angle, δm
time [s]
Figure 6.6: Validation of the nominal model in time domain. Measurements - dotted, predictor - solid, with filter gainL= 0 - dashed.
where parameterspiare defined in Table 6.1. Due to the crude dynamical simplifications and linearizations parameter identification is necessary. The result is also shown in the table.
Table 6.1: Identified parameters of the nominal model p1= cJflf
z = 22.92 p2 =−mlf(cJrzlrcr−cl flf) =−0.0602 p3= nRKJ cf
s = 66.43 p4 =−mlfJnsRKcrlcf =−0.4794 p5=−kJss =−3.255 p6 =−JCspTref frs =−0.3603
We can measureψ˙ andδ by standard sensors available on commercial vehicles. Let the nominal model be denoted by Gn: y = Gnu, with y = [ ˙ψ δ]T and u = ∆p. It appears in state-space form as
Gn :
ψ¨
δ˙ δ¨
=
−p1l
v +p2v p1 0
0 0 1
p3l
v +p4v −p3 p5
ψ˙
δ δ˙
+
0 0 p6
∆p, y= ψ˙
δ
.
6.2.4 Uncertainty model structure
The next task in this section is to define the uncertainty model which describes the difference between the nominal model and the real system. This can be performed by defining a set of models so that the set should contain both the nominal and the unknown real system.
Recall that uncertainty model contains both neglected dynamics (model perturba-tion) and the description of unknown outer signals. Assuming stable systems, the goal of uncertainty model construction is to describe the input-output behavior of the real
CONTROL
system. This could be achieved by assuming a large enough outer signal on the output of the nominal model without assumption of neglected dynamics, y =Gnu+d, how-ever structured descriptions of uncertainty generally lead to better performance of the closed-loop. Concerning model perturbation the nominal model assumes a flat vehicle model neglecting all vertical dynamics and wheel dynamics. It simplifies yaw dynamics and the steering system. Furthermore the system is linearized. Concerning unknown outer signals, the effect of the lateral slope of the road should be taken into account.
During the experiments the roll angle ϕR of the road varies causing the vehicle to skid sidewards and turns round the vertical axis, i.e., turns across the road. The reason for this cornering is the acting of different side-forces at the front and rear owing to the different wheel loads. The evolving yaw moment also turns the steering system, thus amplifying the cornering.
All these effects can be described by the uncertainty model
y=Gn(1 + ∆)u+d (6.3)
where ∆ is called input-multiplicative perturbation, which is a stable unknown but bounded system. The outer disturbances are denoted byd, which is a bounded energy unknown signal.