In this section the yaw-rate reference-tracking problem is solved by front wheel braking.
First, the D-K-W iteration method is applied and the resulting uncertainty model is an-alyzed. Then, the performance of the controller in closed-loop experiments is discussed.
6.6.1 D-K-W iteration design results
Five experiments are performed on the HFVS atv= 12m/s and also the responses of the nominal modelGnare computed for the same inputs. The data defining the constraints
6.6. DESIGN RESULTS
0 20 40 60 80 100 120 140 160
0 0.05 0.1 0.15 0.2 0.25
yaw−rate peak−signals [rad/s]
time [s]
0 20 40 60 80 100 120 140 160
0 0.02 0.04 0.06 0.08 0.1
steering angle peak−signals [deg] time [s]
Figure 6.15: Peak-signal plot of yaw-rate and steering angle responses. Nonlinear system with nonzero road-slope (solid), with zero road-slope (dashed), linear model (dotted), v= 12m/s
10−4 10−3 10−2 10−1 100 101 102
−100
−80
−60
−40
−20 0 20
uncertainty on yaw−rate
frequency [rad/s]
10−4 10−3 10−2 10−1 100 101 102
−100
−80
−60
−40
−20 0
uncertainty on steering angle
frequency [rad/s]
Figure 6.16: Perturbation (dotted) and disturbance (solid) errors in the frequency-domain, v= 12m/s.
on the uncertainty model are selected from the discrete Fourier-transformation of the nominal model error (y−Gnu) and control input u = ∆p in a frequency range of [0.04, 20]rad/s. The weighting functions W∆and Wdare initialized with the minimal disturbance necessary to have a consistent uncertainty model with the data of the 5 experiments. The weighting functions are shown in Figure 6.19 by dashed lines. Then during the D-K-W-iteration, Wd increased and the W∆ of the perturbation decreased.
Finally they arrived at the solid lines of the figures. The thick points denote the upper bounds v1k and v2k,k= 1, ..., nω.
One might think that we got back the real distribution of the uncertainty indicated by Figure 6.15 and Figure 6.16: large disturbance and small perturbation. Then we should get the converse atv= 28m/s, where the effect of perturbation seems to be more significant than that of disturbances, see Figure 6.17 and Figure 6.18. It is emphasized that the information of the real distribution is not contained in the data. The resulting weighting functions fit the robust performance criterion and the data set is only a
CONTROL
0 20 40 60 80 100 120 140 160
0 0.05 0.1 0.15 0.2 0.25
yaw−rate peak−signals [rad/s]
time [s]
0 20 40 60 80 100 120 140 160
0 0.02 0.04 0.06 0.08 0.1
steering angle peak−signals [deg] time [s]
Figure 6.17: Peak-signal plot of yaw-rate and steering angle responses. Nonlinear system with nonzero road-slope (solid), with zero road-slope (dashed), linear model (dotted), v= 28m/s
10−4 10−3 10−2 10−1 100 101 102
−100
−80
−60
−40
−20 0 20
uncertainty on yaw−rate
frequency [rad/s]
10−4 10−3 10−2 10−1 100 101 102
−100
−80
−60
−40
−20 0
uncertainty on steering angle
frequency [rad/s]
Figure 6.18: Modeling (dotted) and disturbance (solid) errors in the frequency-domain, v= 28m/s.
constraint for having an unfalsified uncertainty model. To confirm the discussion the synthesis is repeated for another set of experiments at aboutv= 28m/s. In contrast to the real situation the disturbance is again dominant in the resulting uncertainty model, see Figure 6.20.
At the end of this section the achieved criterion values γ are presented in Table 6.2. For comparison a standard D-K iteration is performed with the initial weighting functions of the D-K-W iteration. By tuning the uncertainty model, considerable im-provement in robust performance is achieved. Sinceγ <1, the closed-loop system with the real plant (HFVS) is likely to be stable (it depends on the quality of the data).
Closed-loop simulation is presented in the next section.
6.6. DESIGN RESULTS
10−4 10−3 10−2 10−1 100 101 102
−100
−50 0 50
Magnitude [dB]
Disturbance weight 1
10−4 10−3 10−2 10−1 100 101 102
−100
−50 0 50
Magnitude [dB]
Disturbance weight 2
frequency [rad/sec]
(a) Weighting function for the additive distur-bances
10−4 10−3 10−2 10−1 100 101 102
−200
−150
−100
−50 0 50
Magnitude [dB]
Perturbation weight
frequency [rad/sec]
(b) Weighting function for the input-multiplicative perturbation
Figure 6.19: Weighting functions of the two additive and the input-multiplicative per-turbations. The initial (dashed) and the result after D-K-W-iteration (solid). Nominal model is at v= 12m/s.
10−4 10−3 10−2 10−1 100 101 102
−40
−20 0 20
Magnitude [dB]
Disturbance weight 1
10−4 10−3 10−2 10−1 100 101 102
−60
−40
−20 0
Magnitude [dB]
Disturbance weight 2
frequency [rad/sec]
(a) Weighting functions for the additive dis-turbances
10−4 10−3 10−2 10−1 100 101 102
−60
−50
−40
−30
−20
−10 0 10
Magnitude [dB]
Perturbation weight
frequency [rad/sec]
(b) Weighting function for the input-multiplicative perturbation
Figure 6.20: Weighting functions of the two additive and the input-multiplicative per-turbations. The initial (dashed) and the result after D-K-W-iteration (solid). Nominal model is at v= 28m/s
Table 6.2: The achieved γ values of the D-K and D-K-W iterations on low and high speed.
v=12 m/s v=28 m/s
DK 2.049 2.503
DKW 0.843 0.496
CONTROL
6.6.2 Closed-loop simulation
The two controllers designed by D-K and D-K-W iteration, respectively, are compared.
The former controller used the uncertainty model with the assumption of small distur-bances and large perturbations, as could be expected based on physical or engineering considerations. The other controller is the result of the D-K-W iteration with large disturbance and small perturbation models. The velocity is kept at about v = 12m/s by driving the rear wheels. The lateral road-slope is varying as in Figure 6.21.
0 10 20 30 40 50 60
−2
−1 0 1 2
lateral road−slope [deg]
time [s]
Figure 6.21: Disturbance ϕR to the closed-loop experiments
0 10 20 30 40 50 60
−5 0 5
pressure difference [bar]
time [s]
0 10 20 30 40 50 60
−0.2
−0.1 0 0.1 0.2
yaw−rate [rad/s]
time [s]
0 10 20 30 40 50 60
0 0.1 0.2 0.3 0.4
performance ||z(t)||2
time [s]
Figure 6.22: Closed-loop simulation at v = 12 m/s. From top to bottom: front wheel pressure difference ∆p; yaw-rate reference (solid) and yaw-rate measurements; perfor-mancekzk2. D-K: dashed, D-K-W iteration: dotted. The areas of the dotted boxes are enlarged in Figure 6.23.
The control input, the yaw-rate reference-tracking and the measured performance kzk22 =
Z t 0
Wt(rref −ψ)˙ Wu∆p
T
Wt(rref −ψ)˙ Wu∆p
dt
are plotted in Figure 6.22. In order to see the difference better some typical portions of the signals are enlarged in Figure 6.23. It can be seen that the control input ∆p is
6.6. DESIGN RESULTS
0 0.5 1 1.5 2
−5
−4
−3
−2
−1 0
pressure difference [bar]
time [s]
0 0.5 1 1.5 2
0 0.05 0.1 0.15 0.2
yaw−rate [rad/s]
time [s]
6 8 10 12 14 16
0.196 0.198 0.2 0.202
yaw−rate [rad/s]
time [s]
Figure 6.23: Closed-loop simulation at v = 12 m/s. Enlarged areas of Figure 6.22.
Solid line: yaw-rate reference rref, dashed: D-K, dotted: D-K-W
−1200 −100 −80 −60 −40 −20 0 20 40 60 80
100 200 300 400 500 600
x position [m]
y position [m]
Figure 6.24: Closed-loop simulation at v = 12 m/s. Trajectory of the vehicle. Solid line: reference path, dashed: D-K, dotted: D-K-W
moderated and, in addition, tracking is improved due to the tuning of the weighting functions W∆ and Wd. The path of the vehicle on the road is shown in Figure 6.24.
CONTROL
The position error with controller KDKW,3 decreased to the half or third of that with controller KDK. Closed-loop experiment data with controller KDKW,3 not invalidates the model, so the final controller can be accepted.
It can also be concluded that the strategy of braking only the front wheels is con-firmed: a satisfactory steering performance (yaw-rate of0.2rad/s) can be achieved with moderate braking effort (below5bar).