the last iteration steps of a given number. During the iteration only the stable controller with improved performance is stored. For notational brevity index N is omitted.
Recall that normalized dynamic uncertainty ∆u∈ S∆u is characterized by dynamic multiplier Πu in (5.2), and set of scheduling parameters S∆s by multiplier Ps ∈ SP
defined in (3.6).
Step B1. Controller synthesis
Compute generalized plant GaD including scalings and weighting functions. Then, a nominal LPV controller is designed according to Section 5.3.2. Criterionγp is minimized subject to synthesis inequalities (5.7).
Step B2. Optimization of the uncertainty model
This step is introduced in Section 5.4. Conditions (5.11) can be rewritten as
"
QDVIk2 +SDMD0VIk2 +VIk2MD0∗ SDT VIk2MD0∗ R
1 2
DT T R
1 2
DMD0VIk2 −T2
#
<0, (5.12)
by congruent transformation with diag{I, T}, whereT = diag{Inzu+nzs+nzc, γ
1
p2Izp}, in order to obtain an LMI in γp as well.
For each frequency point ωk, minimize γp =γk subject to (5.12) and (4.15) in the variables γk,v2ik and Θlk,i= 1, ..., τ +nd,l= 1, ..., N.
Then, point-wise bounds vik are over-bounded by stable minimal-phase transfer function. The orders of the transfer functions are increased until the criterion with the fitted weighting functions is below εγk (ε > 1). This is a similar procedure to that in Step B2 in Section 4.4.
Step B3. Skew µ analysis by finding the scalings
By applying Schur complement on the term with γp−1, analysis inequality (5.5) can be transformed to an LMI in variables d2ik,i = 1, ..., τ, and γp. For each frequency point γp = γk can be minimized. Then, point-wise scalings dik are fitted in magnitude by stable, stable invertible transfer functions where the selection of order and the weighting of the fitting criterion are similar to those in the previous step.
Remark. 5.1 The implications of Theorem 4.2 hold also for the presented method, since the applied controller synthesis procedure corresponds to skew µ synthesis, as it has been shown in Section 5.2.
5.6 Case study: Active steering for vehicle stability en-hancement - LPV case
In order to illustrate Algorithm 2, the example of Section 4.6 is continued by designing an LPV (velocity-scheduled) controller for the steering problem.
FOR LPV SYSTEMS
W∆
Gn
∆u
Wu
K Wref
ψ˙ref
Wd
Wt
u=Td
- -Tdriver
Wr
wu
∆s
+
?
+?
?
z1 d
-+
z2
-uK=Tc
6
y= ˙ψ
+
-- ? r
+ yK
-zu wu0
d0
G¯n
?
∆c
-- ?
-Figure 5.5: Specification of the closed-loop. z1 and z2 are performance outputs, FU(Gn,∆s)is the LPV nominal model, and G¯n is an LTI nominal model.
The true vehicle is the same as in Section 4.6. The nominal model is described by (4.18), however, it is renamed toGn(v)expressing the velocity-dependence of the linear model.
The control specification is plotted in Figure 5.5. It differs from Figure 4.8 in the un-certainty model. Since the iterative unun-certainty modelling and control design algorithm applies for LTI (speed independent) uncertainty models, the input-multiplicative per-turbation∆u cannot excite nominal LPV modelGn(v). Instead, it excites a fixed-speed single-track modelG¯n=Gn(v0), for example at an average speedv0. The LPV nominal model is reformulated by an LFT, in Figure 5.5, as Gn(v) =FU(Gn,∆s). Blocks∆u,
∆s, W∆, Gn, G¯n and Wd constitute the uncertain LPV system. The LPV feedback controller isFL(K,∆c).
By redrawing the system of Figure 5.5 we can arrive to Figure 5.1. The input-output data set for the uncertainty modeling consists of u = δ and y = ˙ψ measurements of three experiments, where the adhesion coefficient is 1, 0.7 and 0.4. The side-wind and vehicle velocities are plotted in Figure 5.6. The yaw-rate measurement for someTdriver
input with Tc = 0 is plotted in Figure 5.7 by dash-dot lines. At this point, all input parameters of the algorithm have been constructed.
The initial uncertainty model is set according to the following scene. 10% for the neglected dynamics is assumed: W∆0 := 0.1. The minimal necessary disturbance weight-ing functions are designed as in Section 4.6 for satisfyweight-ing the consistency constraints for the initial data set. The controller designed by the iterative algorithm is compared to the controller designed for the initial model which is denoted byK0. After the first D-K-W iteration the size of the perturbation W∆ increased significantly, specially at the higher frequency range while that of the disturbance decreased. Then, the three closed-loop experiments are carried out by the controller. The new data invalidated the model and the iteration was continued with the extended data set resulting the controller K2. After this second iteration the algorithm finished.
The reference-tracking of D-K controller K0 and D-K-W controllers K1 and K2
5.6. CASE STUDY: ACTIVE STEERING FOR VEHICLE STABILITY ENHANCEMENT - LPV CASE
0 10 20 30 40 50 60 70 80 90
0 5 10 15 20 25 30
time [s]
velocity [m/s]
Side−wind speed and vehicle velocity
vwind v
Figure 5.6: Side-wind speed (solid) and forward velocity (dashed).
are illustrated in Figure 5.7. It can be seen that both disturbance attenuation and steady state tracking performance (see also Figures 5.8-5.9) have been improved due to the algorithm. With the initial controller the achieved robust performance level is γ = 0.1060, after the first D-K-W iteration it is γ = 0.0544 due to the shaping of W, and then, with additional data, the final controller achieved γ = 0.0905. This implies that the robust performance has been improved.
Remark. 5.2 When comparing the results with the LTI case (Figures 4.11(b,d) and Figures 5.7, 5.8), it can be seen that LPV controllers achieved weaker performance. The reason is that the true uncertainty is very large in this example and its embedding by the uncertainty model in the LTI case can easily describe the effects of speed variation as well. The additional constraints imposed by the LPV design allow weaker performance.
0 10 20 30 40 50 60 70 80 90
−0.8
−0.6
−0.4
−0.2 0 0.2 0.4 0.6 0.8
time [s]
yaw−rate [rad/s]
Reference tracking, µ=0.4
reference K=0 DK controller DKW final
Figure 5.7: Simulation results with µ= 0.4. Yaw-rate referenceψ˙ref (solid), yaw-rate measurement ψ˙ without controller (dash-dot), with initial controller K0 (dotted), with final controller K2 (bold dashed).
FOR LPV SYSTEMS
0 10 20 30 40 50 60 70 80 90
0 0.5 1 1.5 2 2.5 3
time [s]
tracking error norm
Tracking performance
DK DKW 1 DKW 2
Figure 5.8: Evaluation of the norm of the reference tracking error, kψ˙ref(t)−ψ(t)˙ k2, during the simulation with µ = 0.4. The norm with K0 (dotted), with K1 (dashed), withK2 (solid).
0 10 20 30 40 50 60 70 80 90
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
time [s]
Control energy
Actuator performance
DK DKW 1 DKW 2
Figure 5.9: Evaluation of the norm of control input, kuKk2, during the simulation with µ= 0.4. The norm with K0 (dotted), with K1 (dashed), with K2 (solid).
5.7. SUMMARY AND THESIS 2
5.7 Summary and Thesis 2
In this chapter, an iterative uncertainty modelling and robust control design algorithm (Algorithm 2) is elaborated for LPV systems. Since LFT scheduling parameter depen-dence enables robust quadratic performance analysis to be performed in both time- and frequency-domain, LPV LFT is the appropriate system class for which a common crite-rion can be defined for both the time-domain control design and the frequency-domain uncertainty modelling methods.
By using IQCs a more general uncertainty and system class can be handled as com-pared to (skew)µanalysis tools. In Section 5.2 the relation between the two approaches is exhibited. The derived identity in Lemma 5.1 provides multipliers for the IQC char-acterization of structured dynamic uncertainty in a unit ball and guaranteed induced L2-gain of systems with a given levelγs.
In Section 5.3 analysis and synthesis inequalities are derived for uncertain LPV systems with LFT dependence on scheduling parameters and dynamic uncertainty. Due to the particular structure of dynamic multiplier, it can be factorized and the factors are utilized for scaling the nominal augmented LPV plant. Thus, a two step LMI-based iterative algorithm, similar to D-K iteration in (skew) µ synthesis, can be applied for robust LPV controller synthesis.
An optimization problem based on LMIs is derived for tuning additive, structured uncertainty models. Robust quadratic performance criterion of special structure, (5.10), is formulated as an LMI optimization problem in terms of the parameters of the unfal-sified uncertainty models, see Lemma 5.2.
Due to the skew µ like characterization of the performance problem, the advan-tageous properties presented in Lemma 4.1 and Theorem 4.2 hold also for Algorithm 2:
• If the LPV model is consistent with the data of all experiments performed in closed-loop with the LPV controller, then an upper-bound of the performance is guaranteed, even if the true system and disturbance bounds are not known. The upper-bound is calculated based on the LPV model.
• Performance degradation beyond the guaranteed level in a new experiment implies the falsification of the model, which enforces the update and re-tuning of both the uncertainty model and the controller.
The contribution of this chapter can be summarized in the following thesis:
Thesis 2
An iterative uncertainty modelling and robust control design algorithm is elaborated for LPV models with LFT dependence on scheduling parameters, structured dynamic uncer-tainty and disturbances. Based on time- and frequency-domain IQCs, guaranteed robust quadratic performance level is minimized by searching for an unfalsified uncertainty model and a robust LPV controller. For additive uncertainty structures, the algorithm is formulated as a series of LMI problems. Due to the advantageous properties of the
FOR LPV SYSTEMS
algorithm, unstable experiments are handled and improvement of guaranteed robust per-formance on the true, unknown system is ensured.
Own publications related to the chapter and thesis: [RLB07, MKD+09, RGB09].
Chapter 6
Vehicle safety enhancement by steer-by-brake control
The topic is motivated by the research into emergency situations of heavy trucks when the driver becomes incapable of controlling the vehicle due to e.g. lipothymy or drowsi-ness. The need for keeping the vehicle on the road and drive it to a safe position emerges due to increased traffic density. In the most common vehicles where no electronic steer-ing system is available but the braksteer-ing is controlled by onboard computers, the only way to automate or assist steering is the use of the electronic brake system, the application of individual or unilateral wheel brakes.
There are many papers concerning different approaches and aims that improve steer-ing by the braksteer-ing systems. In [4] active steersteer-ing and individual wheel braksteer-ing are discussed from a yaw and roll control point of view. The advantage of individual wheel braking is that it is implementable with less hardware effort since the actuators and wheel speed sensors are available in the existing anti-lock braking systems. A method for unilateral braking for rollover prevention can be found in [51], for preventing unintended lane departure in [71, 100, 104].
Figure 6.1: Block scheme of the control concept. The topic of this paper covers the diagram with the solid lines.
In this section a yaw-rate trajectory-tracking problem is solved by using the elec-tronic brake system. The concept of the control algorithm is presented in Figure 6.1.
It is assumed that the vehicle is equipped with sensors, such as a camera, GPS or radar, providing information about the environment and the position of the vehicle. It
CONTROL
is assumed that there is a monitoring and path planning system (MPPS) that detects the emergency situation. A decision can be made based on for example detecting lane avoidance without turn signal. When this occurs, MPPS defines a path along the road and switches on the emergency controller that steers, slows up and halts the vehicle unless the controller is switched off by some driver action. Such monitoring systems are treated in many papers, e.g. [19, 66, 71, 72, 84, 93, 114]. The desired path can be spec-ified in terms of the curvature of the trajectory, thus, by taking the vehicle speed into account, a yaw-rate reference signal can be defined for a trajectory-tracking controller.
Based on the yaw-rate and steering wheel angle measurements, the proposed con-troller affects the front wheel brakes by small pressures in order to turn the steering system which turns the vehicle. In case of heavy trucks this approach has an advantage over unilateral braking1: in frequent cases of uneven load distribution between front and rear axles, the rear brake has practically no effect on the lateral dynamics. The performance aims are specified in terms of yaw-rate tracking error and control energy.
Both must be bounded for all occurrences of the model uncertainty, such as neglected and unknown dynamics of the vehicle, unknown adhesion characteristics and the lat-eral slope of the road. Because of strict safety demands on vehicle control, the H∞/µ design is chosen, which guarantees robust performance against bounded uncertainty of the nominal model.
This problem setup suits well the D-K-W iteration algorithm. The advantage of this method is that uncertainty is modelled automatically: choose a simple uncertainty structure, then consider the set of all models in this structure that are not invalidated by measurement data; and this set is parameterized and tuned in order to achieve robust performance specifications. The resulting joint algorithm simplifies the design, reduces the number of design parameters and tailors modelling criterion to the final goal, i.e.
closed-loop robust performance.
In Section 6.1 the experimental environment is presented. The model used for control design is derived in Section 6.2. The concept of trajectory generation is discussed in Section 6.3. Section 6.4 presents details about the control design and uncertainty modelling by using D-K-W iteration.
6.1 Experimental conditions
The modelling and control synthesis are based on measurement data which are generated by the following experimental conditions. A High-Fidelity Vehicle Simulator (HFVS) program was implemented under Matlab/Simulink, which is a relatively good approx-imation of the MAN truck shown in Figure 6.2a. Some physical parameters (lengths, masses, inertias) of the HFVS match the parameters of the MAN truck, while other pa-rameters (cumulative damping and spring coefficients) are results of identification based on real-life experiments. The HFVS model contains all important features that may be relevant in the yaw motion control problem: the 17-degree of freedom (DoF) nonlinear Matlab/Simulink model contains the dynamics of the suspension system with vertical wheel center motion (4DoF), the roll, pitch, heave motions of the sprung mass (3DoF),
1unilateral braking means the braking of both front and rear wheels on one side