the position of the vehicle is measured on-line, the curvature of the path near the vehicle is known as a function of time. Hence, the following differential equation can be formulated for the desired orientation of the vehicle
ψ˙ref(t) = 1 R(t)v(t)
where ψref and v denote, respectively, desired orientation and actual forward speed of the vehicle. In the pictured scene of Figure 6.7, the redesign of the path is necessary at timet2 because of the increased lateral error of the trajectory-tracking. In order to get a realizable trajectory,ψ˙ref(t)should be driven through a low-pass filter, Wψ, so we get
rref :=Wψ
v R
for the yaw-rate reference (see also Figure 6.1 for the control strategy).
6.4 Robust control design based on D-K-W-iteration
The development of the D-K-W iteration was motivated by control problems like the steer-by-brake problem. The necessity of safe and robust design implies the need for simple low-order models and the appropriate modelling of the nominal model error. We have freedom in creating uncertainty models that cannot be invalidated by the data, however, our choice strongly influences the uncertainty model and so the controller.
The freedom consists of selecting the structure of the uncertainty model (for example input-multiplicative perturbation and additive disturbance model on the output) and choosing the weighting functions that normalize the blocks of perturbation and the disturbance. The former choice remains the matter of engineering insight. We propose, in general, to test a few simple structures. The latter is designed by D-K-W-iteration together with the controller. The weighting functions are set to tightly over-bound the unfalsified model set and tuned to minimize the same criterion that the control design minimizes.
In the steer-by-brake problem of this section, D-K-W iteration of Chapter 4 is adopted for uncertainty modelling and control design, with some modifications. In Chapter 4 skew µ synthesis was applied to guarantee stability of control for the actu-ally unfalsified model set. Recall that skew µsynthesis guarantees robust performance level µ∆a(M) for perturbations ∆∈BH∞. When stability margin is high in a control problem, i.e., µ∆(M11) << 1, then a robust performance level µ∆a(M) >1 might al-low stable experiments and standard µtools (available in Matlab) are appropriate for implementing the algorithm of Chapter 4. In this case the criterion for control design and uncertainty modelling is the standard (not scaled) µ upper-bound, denoted by γ, guaranteeing robust performance level γ for perturbations ∆ ∈ γ1BH∞. When γ is near to 1, the µand skew µcriteria coincide. In this chapter the standard µtools are applied.
Suppose there are input/output measurements in the frequency-domain, (u, y) ∈ EN. Once the uncertainty model is designed in variables θlk, stable and invertible
CONTROL
weighting functions have to be designed for over-bounding. They should be chosen of low order to keep the controller’s order low. For example
W∆,i(jω) = A∆,ijω+z∆,i
jω+p∆,i, i= 1, . . . , τ Wd,i(jω) = Ad,i(jω+zd,i1)(jω+zd,i2)
(jω+pd,i1)(jω+pd,i2), i= 1, . . . , nw.
where parametersA∗,z∗,p∗ are determined so that the inequalities are tightly satisfied.
This over-bounding method is faster and simpler to implement than the one presented in Chapter 4.
In the steer-by-brake problem the uncertain system model (6.3) consists of an input-multiplicative perturbation and additive disturbance that is formulated according to the notations of Figure 4.1 by
U =
0 0 1 Gn I2 Gn
The next task is to augment the uncertain system in order to specify the aims of the robust control. The performance input wp is constituted by the disturbance dand the known reference signal r. Specifically in the steer-by-brake problem, the control specifications are the following. Let the yaw-rate reference be tracked up to 1 rad/s bandwidth; the control signal must not contain significant components beyond 1 rad/s;
and let the control be robust against uncertainties. To this end, Figure 4.3 turns Figure 6.8, where blocks ∆,W∆, Gn and Wd constitute the uncertainty model whose output y= [ ˙ψ δ]T is assumed to be equal to the measured output by consistency condition (4.4).
Block[0 1]selects the yaw-rate. The yaw-rate reference signal is normalized by weighting function WC, which is computed by over-bounding a typical yaw-rate reference signal in magnitude in the frequency-domain, see Figure 6.9a, thus, rref = WCrref,0 with rref,0 ∈ BL2. The performance signal is chosen to consists of the tracking error and the control input. It is required that the low frequency components (below1 rad/s) of the yaw-rate reference be followed. For good tracking in steady state, a high gain ofWt
is required on zero frequency, for that reason a second order filter Wt= 0.25(s+50)(s+1)22 is chosen with crossover at about1rad/s. The maximal applicable brake pressure is10bar, but this value counts as very strong braking and should be avoided in order to spare the brake-lining. After closed-loop simulations, holding the limit must be confirmed.
The control input is penalized beyond 1 rad/s by Wu = 0.0008(s+0.2)(s+50)22 in order to avoid high-frequency dynamics of the controller. The resulting weighting functions are plotted in Figure 6.9b. Thus, the performance channel is defined as wp = [dT rref,0]T and zp = [z1 z2]T.
The augmented plant mapping[wu|d1d2 rref,0|∆p]T 7−→[zu|z1 z2|rref ψ δ]˙ T can be
6.4. ROBUST CONTROL DESIGN BASED ON D-K-W-ITERATION
W∆
Gn Wu ∆
WC K
Wd
Wt
-?
--+? +?
?
z1
d
- -- +
[0 1]
-rref,0
-6
- 6
- y
rref
- ψ˙ z2
+
-uK = ∆p -zu wu wu0
d0
Figure 6.8: Control specification: the closed-loop system with normalizing weighting functions.
10−4 10−3 10−2 10−1 100 101 102
−90
−80
−70
−60
−50
−40
−30
−20
−10 0 10
magnitude of yaw−rate and WC, [dB]
frequency [rad/s]
(a)
10−3 10−2 10−1 100 101 102 103
−80
−60
−40
−20 0 20 40
magnitude of Wt and Wu, [dB]
frequency [rad/s]
(b)
Figure 6.9: (a) The magnitude of DFT of a typical realizable yaw-rate reference signal (solid line) and the magnitude of its normalizing weighting function |WC(jω)| (dashed line). (b) The magnitude of Wu (solid line) and Wt(dashed line).
described as
G0 =
0 0 0 0 1
−WtGn1 −Wt/√
3 0 WtWC/√
3 −WtGn1
0 0 0 0 Wu
0 0 0 WC/√
3 0
Gn1 1/√
3 0 0 Gn1
Gn2 0 1/√
3 0 Gn2
where Gn =
Gn1 Gn2
and 1/√
3 is incorporated in order to normalize input vector [dT rref,0]T.
The optimization problem based onµ analysis can be formulated as follows infD inf
K inf
W∆,Wd
γ
CONTROL
subject to (4.6), (4.7) and sup
ω
¯
σ(DLFL(G0, K)D−1R diag(W∆, Wd, I))< γ,
whereD, DL, DRare defined in Section 3.2 withτ = 1. Since the uncertainty structure is additive, the optimization can be performed as a series of convex programs as presented in Section 4.5. The only difference is that skew µscaling diag{Inw, γk2Ind+nr} in (4.17) must be replaced byγk2.
If γ can not be minimized below one, the performance specifications should be loosened, otherwise robust performance and even robust stability can not be guaranteed.
The resulting controller depends on the quality of the experimental data. Pure excitation of the system might result in overly optimistic uncertainty model and, finally, a destabilizing controller. Therefore D-K-W algorithm is worth using in an iteration of simulation (closed-loop test of the controller), data collection and application of the algorithm on the extended data set (see Chapter 4). If none of the new data sets invalidates the uncertain model and the robust performance is satisfied (γ ≤ 1), it can increase our confidence that the model set contains the real system and the controller performs well on the real system. On the other hand, based on some a priori information, lower limits of the magnitudes of weighting functions can always be added to the constraints in order to enforce some cautiousness of the controller.