FOR LTI SYSTEMS
4.6 Case study: Active steering for vehicle stability
4.6. CASE STUDY: ACTIVE STEERING FOR VEHICLE STABILITY ENHANCEMENT - LTI CASE u=Td and output y= ˙ψ. The system matrices are given by
A=
−cfmv+cr −1 +crlrmv−c2flf mvcf 0 0 0 0 0
crlr−cflf
Jz −crl
2 r+cfl2f Jzv
cflf
Jz 0 0 0 0 0
0 0 0 1 0 0 0 0
nRKcf
Jsl
Kslǫ
Jsl+nRKJcflf
slv −Ksl+nJslRKcf −BJslsl NKLslJsl 0 0 0
0 0 0 0 0 1 0 0
0 −MbǫR MbR 0 −RpaMR−NLbMR −MBRR MaR 0
0 0 0 0 0 0 0 1
0 0 0 0 JswCSRp JswkmRp −JCswS −Jkswm
Bd=
wβ wr
06×1
, Bu = 07×1
1 cJsw
, C =
0 1 01×6
, Dd= 0, Du = 0 The state and input variables are defined in Table 4.1 and the physical parameters in
Figure 4.6: Single-track model
Figure 4.7: Neglected dynamics: a stock steering system.
Table 4.2.
Adhesion coefficient µ and vehicle velocity v are constant uncertain parameters which define setST: in each iteration of the algorithm, Step C consists of three exper-iments with (µ, v)∈ {(0.2,10),(1,30),(1,10)}.
FOR LTI SYSTEMS
Table 4.1: State and input variables β vehicle side-slip angle
ψ˙ yaw-rate (measured output) δ steering angle
∆R rack position
δm steering wheel angle
vwind side-wind velocity (disturbance)
Td applied torque on the steering wheel (control input)
Table 4.2: Physical parameters of the vehicle model
lf= 2.3444m,lr= 2.8056m COG-front and COG-rear axis distances
m= 14200kg Mass of vehicle
cf =µ2.5201 106,cr=µ1.9144 106 Front and rear cornering stiffness
Jz= 123780Nm/(rad/sec2) Inertia of vehicle around the vertical axis Jsw = 0.034Nm/(rad/sec2) Inertia of steering wheel
Ksc= 42700N mrad,Bsc= 0.36042rad/s)N m Steering column stiffness and damping Ktb= 83N mrad,Btb= 0.0rad/s)N m Torsion bar stiffness and damping
MR= 6 kg Mass of rack
Rp= 0.007367m Radius of pinion BR= 352Nm/(rad/s) Rack damping
NL= 0.11816m Steering linkage ration
Ksl= 14878N mrad,Bsl = 160rad/s)N m Steering linkage stiffness and damping
ǫ= 0.01 Roll steer coefficient
ηF = 0.985,ηβ= 0.985 Efficiency of forwd./backwd. torque transmission ηps= 0.95 Efficiency of power steering torque transmission
Gps= 30N/rad Power steering gain
Jsl = 6.1Nm/(rad/sec2) Cumulative inertia of steering linkage and tire nRK = 0.0061m Total tire trail length
c= 0.0039 Normalizing factor such thatδ(∞)≈Td(∞) wβ= 0.0012,wr= 8.1435∗10−5 Wind force factors on slip and yaw-rate Cs= KKsc∗Ktb
sc+Ktb,km=KBsc∗Ktb
sc+Ktb
a=ηFCs
Rp+ηpsKsc∗Gps
Ksc+Ktb, b= 2ηβ Ksl
RRNL
µ= 1(nominal) 0.2, 1, 1 (test) Adhesion coefficient in experiment 1,2,3 v= 15(nominal) 10, 30, 10 m/s (test) Absolute velocity in experiment 1,2,3
The steering system dynamics is considered as unknown and the linearized single-track model is used for control design. The nominal model is given by
Gn :
β˙ ψ¨ ynom
=
−cfmv+cr −1 +crlrmv−c2flf
cf mv crlr−cflf
Jz −crl
2 r+cflf2
Jzv
cflf Jz
0 1 0
β ψ˙ Td
(4.18)
Based on Table 4.2, note that the input is scaled so that Td=δ in the nominal model.
4.6. CASE STUDY: ACTIVE STEERING FOR VEHICLE STABILITY ENHANCEMENT - LTI CASE
W∆
Gn
∆
Wu K
Wref ψ˙ref
Wd
Wt u=Td
-?
--+? +?
?
z1
d
-+
-z2
-Tdriver
uK =Tc 6
y= ˙ψ
+ Wr
?
-+
yK
-zu wu wu0
d0 r
-Figure 4.8: Control specification: the closed-loop system with normalizing weighting functions.
The nominal value of uncertain adhesion coefficient µ= 1 is built in cornering stiffness parameters (cf, cr) and the nominal velocity is v = 15 m/s. The neglected steering system dynamics acts on the steering angle input of the nominal model, for this reason a natural choice for the uncertainty structure is input-multiplicative perturbation ∆0 =
T
Gn−1that belongs to setS∆u={∆∈ RH∞}. The effect of the side-wind is modelled by an additive disturbance on the output. Thus, the system describing the structure of the uncertainty will be U =
0 0 1 Gn 1 Gn
.
The goal of the control is given by the specification of the closed-loop system in Figure 4.8. Blocks ∆, W∆, Gn and Wd constitute the uncertain system. The torque of the driver on the steering wheel Tdriver drives reference model Wref = (s+5)75 2, which represents the required behavior of the vehicle. Block Wr normalizes physical signal Tdriver. Signal ψ˙ref is to be followed by applying an additional torque Tc generated by feedback controller K. Tracking error ψ˙ref −ψ˙ is penalized by low-pass filter Wt = 0.025s+0.001s+3 in order to achieve good tracking at steady state and at low frequencies.
Control signal uK = Tc is penalized by high-pass filter Wu = 1.4(s+0.04)(s+10)22 in order to avoid high-frequency components.
Augmented plant mapping[wu0 |d0 r |uK]T 7−→[zu|z1 z2|yK]T is given by
G0 =
0 0 Wr/√
2 1
−WtGn −Wt/√
2 Wt(Wref −Gn)Wr/√
2 −WtGn
0 0 0 Wu
−Gn −1/√
2 (Wref −Gn)Wr/√
2 −Gn
,
where term √
2is incorporated in order to normalize input vector [d r]T.
Initial data setE1 is generated by three open-loop experiments associated with the three systems inST. In all experiments (open- and closed-loop) the same driver torque Tdriver, plotted in Figure 4.11(c) (dash-dot), and the same side-wind velocity, plotted in Figure 4.9, will be applied. A sampling time of 0.01s is used and the frequency domain data is derived by discrete Fourier-transformation (DFT).
FOR LTI SYSTEMS
0 10 20 30 40 50 60 70 80 90
0 5 10 15 20
time [s]
side wind velocity [m/s]
Disturbance on the vehicle
vwind
Figure 4.9: Side-wind velocity in the experiments.
10−2 100 102
−50
−40
−30
−20
−10 0 10 20
frequency, rad/s
magnitude, dB
Disturbance weighting function |Wd|
true sup|d|
DK DKW 1 DKW 2 DKW 3
(a)
10−2 100 102
−50
−40
−30
−20
−10 0 10
frequency, rad/s
magnitude, dB
Perturbation weighting function |W∆|
true sup|∆|
DK DKW 1 DKW 2 DKW 3
(b)
Figure 4.10: Disturbance (a) and perturbation (b) weighting functions. True uncer-tainty (o), initial (DK), and subsequent iterations DKW1,2,3. (Data at ω1 = 0 is plotted at 100ω2 = 7∗10−4 rad/s)
Given the control specifications, the first task is to chose initial weighting functions for the uncertainty. In this numerical example we can exactly calculate the boundaries of the multiplicative perturbation set as|W∆,true(jω)|= maxT ∈ST |GTn(jω)(jω)−1|. Similarly, the maximal magnitude of the true additive disturbance can be calculated. These boundaries are plotted in Figure 4.10 by small ’o’ markers. It can be seen from Figure 4.10(b) that the size of the perturbation is greater than -6dB=0.5012 (minimum at ω = 0), i.e. the nominal model is very inaccurate. In the real problems, however, the boundary of the uncertainty set is not known. Assume, the initial weighting functions in the uncertainty model are set according to the following scene: Based on engineering judgement constant 60% is assumed for the neglected dynamics, i.e. W∆0 := 0.6, which is a quite good estimation - a bit overestimated at low frequencies and a bit underestimated at higher frequencies. In order to start from a correct (consistent) model the following procedure is applied for the determination of the disturbance weighting function. Using the model validation constraints of Section 4.2 the upper-bound of the disturbance weighting function is minimized for each data point in E1 (open-loop data)
minθlk αlk
4.6. CASE STUDY: ACTIVE STEERING FOR VEHICLE STABILITY ENHANCEMENT - LTI CASE
subject to
0.6 ≥ |wu0,l(jωk, θlk)|
|zul(jωk)| αlk ≥ |d0,l(jωk, θlk)|
for l = 1, ..., N and k = 1, ..., nω. Values αlk are, then, over-bounded by weighting function Wd0 which is shown in Figure 4.10 by dotted line.
The controller designed by Algorithm 1 is initialized by the controller designed by the skew D-K iteration (steps B1 and B3 in Algorithm 1) for the initial model withWd0 and W∆0. This controller is denoted by KDK. The evaluation of the uncertainty weighting functions during Algorithm 1 is presented in Figure 4.10. After the first application of D-K-W iteration the size of perturbation decreases significantly to W∆(Θ1), while that of the disturbance increases to Wd(Θ1) (dashed lines in Figure 4.10). This means that the weighting functions got farther from the true weighting functions: the neglected dynamics is modelled partly as an effect of disturbance. The resulting controller is denoted by KDKW,1. Then, the three closed-loop simulations withKDKW,1 are carried out. Step D reveals that the new data invalidates the model. Thus, the iteration continues. After three iterations the analysis shows that data E4 are consistent with the resulting uncertainty model, therefore, the algorithm stops.
The reference-tracking and the control signals ofKDKand the last controllerKDKW,3 are plotted in Figures 4.11(a)-4.11(c). Among the three plants in ST the one with (µ, v) = (0.2,10)is the worst-case showing the largest oscillations1. This plant is desta-bilized by initial controller KDK: at about t= 43s the yaw-rate attains the predefined saturation level and uK =Tc is set to zero2, meanwhile the yaw-rate converges to the open-loop case. Att= 53scontrollerKDK is switched on again and the yaw-rate starts to oscillate. Figure 4.11(b) show the yaw-rate of the final DKW controller for both the worst-case plant with (µ, v) = (0.2,10) and for the case with (µ, v) = (1,30). Despite the large uncertainty the final controller achieved good tracking performance for all plants in the set.
The L2-norm of the tracking error, kψ˙ref(t)−ψ(t)˙ k2, is also computed during the simulations. The norm is plotted in Figure 4.11(d), showing the superiority of the DKW controllers over KDK. Regarding each controller, Table 4.3 shows the achieved robust performance levelγ¯s(K)and the satisfaction of the consistency conditions based on the closed-loop simulation data.
Table 4.3: Performance levels of the controllers KDK KDKW,1 KDKW,2 KDKW,3
¯
γs(K) 0.2234 0.1595 0.2035 0.1980
consistency × × × X
1Only the experiments with parameter setting(µ, v) = (0.2,10)are plotted in Figures 4.11(a)-4.11(c)
2Applied steering torqueTdbecame equal to driver torqueTdriverin periodt∈[43s,53s], see Figure 4.11(c).
FOR LTI SYSTEMS
Figure 4.10 show that the weighting functions represent the size and shape of a fictive uncertainty set that serves for a mathematical description (or rather inclusion) of the true system. They values depend on the control specification, not on the true system.
It can be seen that uncertainty modelling improved the guaranteed performance.
Controller KDKW,1 after the first iteration is the best performing controller, however, it is designed based on an invalid model. In the subsequent iterations the robustness of the controllers is increased. The final controller is consistent with all measurements.
0 20 40 60 80
−2
−1.5
−1
−0.5 0 0.5 1 1.5
time, s
yaw−rate, rad/s
Reference tracking with K=0 and with K
DK, (µ,v)=(0.2,10)
rref
open−loop KDK
(a)
0 10 20 30 40
−1
−0.8
−0.6
−0.4
−0.2 0 0.2 0.4 0.6
time, s
yaw−rate, rad/s
Reference tracking with the final KDKW
rref
KDKW,3, (µ,v)=(0.2,10) KDKW,3, (µ,v)=(1,30)
(b)
0 20 40 60 80
−0.8
−0.6
−0.4
−0.2 0 0.2 0.4
time, s
Steering torque, Nm
Control signals Td with (µ,v)=(0.2,10)
Tdriver KDK KDKW,3
(c)
0 20 40 60 80
0 1 2 3 4 5
time, s
Norm of the tracking errors
Tracking performance of the controllers KDK
KDKW,1 KDKW,2 KDKW,3
(d)
Figure 4.11: Simulation results (a) - yaw-rate referenceψ˙ref (solid), open-loop response (dash-dot), closed-loop response withKDK(dotted). (b) - yaw-rate with final controller KDKW,3 in two experiments. (c) - control signals: driver torque Tdriver (dash-dot), Td
with KDK (dotted), with KDKW,3 (solid). (d) - evaluation of reference tracking error kψ˙ref(t)−ψ(t)˙ k2 (sum of the three experiments)
4.7. SUMMARY AND THESIS 1
4.7 Summary and Thesis 1
In this chapter, a design algorithm for weighting functions characterizing structured uncertainty is developed. It is based on frequency-domain model validation results by Smith and Doyle [123], but consistency conditions are utilized in an optimization problem as constraints where the objective is to tune the weighting functions in order to minimize a control performance criterion. This weight optimization problem is placed into an iterative experimenting, unfalsification and robust control design framework.
No a priori information is assumed on the true system that generates experiment data. It is only assumed that the true system is stable and in every experiment it can be modelled by a given LTI nominal model with LFT interconnection to structured dynamic perturbations and disturbances. The objective is to minimize the robust per-formance of the true controlled system by designing both the weighting functions and the controller.
It is shown that if the model is consistent with all experiment data that can be obtained with a given controller, then an upper-bound of the true performance can be computed based on the model (Lemma 4.1). The lemma suggests the idea of using closed-loop experiments to falsify/unfalsify models, and thus improve the consistency of the model. In fact, one cannot do more than just improving robustness when no a priori information is available on the bounds of uncertainty. Guarantee cannot be given for stability and performance.
It is proved, furthermore, in Theorem 4.2 that, when using a control synthesis method developed for minimizing skew µcriterion, then the following statements hold:
• if a model is not falsified by a new experiment, then the actual performance of the control is, as expected, below the guaranteed level
• if the realized performance in the new experiment exceeds the guaranteed level, then the model is necessarily falsified
Based on these implications astability uncertain (defined by [130]) iterative scheme can be proposed where instability is detected in time and measurement data is utilized for improving robustness of the controller.
The proposed scheme is summarized in Algorithm 1 where details of skewµ-synthesis and weighting function design are elaborated. Finally, a class of uncertainty structures are defined which turns the steps of the algorithm into convex problems which can be solved based on LMIs.
The contribution of this chapter can be summarized in the following thesis:
Thesis 1
An iterative algorithm has been elaborated for LTI models with the aim of shaping struc-tured uncertainty models and designing robust controllers based on measurement data.
The algorithm handles unstable experiments ensuring safe improvement of guaranteed robust performance on the true, unknown system. For additive uncertainty structures, the algorithm is formulated as a series of LMI problems. The steps of skew µ synthesis
FOR LTI SYSTEMS
has been elaborated as part of the algorithm. It has been proved that performance degra-dation beyond the guaranteed level implies falsification of the actual model, which forces the continuation of the algorithm.
Own publications related to the chapter and thesis: [RB05b, RB06b, RB06a, RGSB08, Röd09, RBar, RG10].
Chapter 5
Uncertainty modelling and robust control design for LPV systems
The class of LPV systems has become more important in the last two decades due to its modelling power for nonlinear/time-varying systems and the quick development in the related control theory. Many analysis and synthesis results have been successfully inherited from the LTI system’s theory, as it is shown e.g. by the unified framework of Chapter 3, and the adapted popular control methods: LQG control and Kalman-filtering [134], [RBL02], fault detection and isolation [11], model predictive control [105], and so on. This line is continued in the chapter: iterative uncertainty modelling and robust control design method is elaborated for LPV systems with LFT scheduling parameter dependence and structured dynamic uncertainty.
The design of robust LPV controllers based on intuitive choice of weighting func-tions is an even more time-consuming and laborious iterative task than in case of LTI controllers. The surplus labour is due to the increased computational burden of con-troller synthesis and the additional design parameters related to LPV modelling (e.g.
choice of scheduling variables and its polytope). The automated and control oriented design of uncertainty weighting functions may drastically decrease the design effort while improving robust performance.
In the iterative algorithm presented in Chapter 4, a frequency-domain consistency constraint is applied for the reason of computational advantages. The linearity of nom-inal model U23 is not utilized in the uncertainty modelling step of Algorithm 1, even the nominal model does not play a role there. In fact, the nominal model can be of any class of systems. One important property we require of the algorithm is that criterion for uncertainty modelling and control design are the same. This means that the system class of the nominal model must be such that control criterion (robust performance) can be characterized in the frequency-domain. This requirement is fulfilled in case of LPV systems with LFT parametrization of the scheduling variables. The modelling and the analysis are carried out in the frequency-domain and the LPV control design is performed in the time-domain.
Analysis and synthesis of LPV systems are carried out via multipliers and IQCs. The relation between µanalysis and IQCs is formulated in Section 5.2. Controller synthesis method for uncertain LPV systems with LFT parameter dependence is discussed in
FOR LPV SYSTEMS
Section 5.3. The uncertainty modelling criterion and constraints are presented in Section 5.4. The joint uncertainty modelling and control design algorithm for LPV systems is summarized in Section 5.5. The efficiency of the method is demonstrated on the vehicle steering problem in Section 5.6.
5.1 Problem formulation
The true plant to be controlled and which generates the data is described by (4.1).
Suppose that in finite time experiments its input-output behavior can be described with the help of a fictive disturbance d0 ∈ Ln2d and by an LFT of a given LPV model and a structured dynamic uncertainty ∆u0 in the form
zu
ˆ y
=
U11 U12 U13
U21 U22 U23(∆s)
| {z }
U
wu0
d0 u
, wu0 = ∆u0zu, (5.1)
where U23(∆s) is an ny×nu LPV nominal model with LFT dependence on scheduling parameter∆s∈ S∆s, whereS∆s is defined by (3.31). Other blocksUij,ij 6= 23belong to RH∞ and describe the interconnection structure of the nominal model and structured dynamic uncertainty, ∆u0∈ S∆u, whereS∆u is defined by (4.3). Normalizing weighting functions, W∆, Wd, are stable LTI systems such that ∆u0 = W∆∆u, ∆u ∈BS∆u and d0 =Wdd,d∈BL2.
Suppose that the input-output data of system T in the lth experiment are given in the frequency domain as El. The data containing N experiments are denoted by EN =SN
l=1El. The consistency definition in Section 4.1 still holds with the modification that transfer functionU(jωk)does not exists,Ubeing an LPV system, but model output ˆ
y can be transformed to the frequency-domain by DFT.
The control objective is to minimize (4.5), where controllerK =K(∆s)is allowed to depend on the on-line measurable scheduling parameter via LFT. An upper-bound can be calculated for ¯γ(K(∆s)) based on a consistent model in the same way as described in Chapter 5. To this end the control performance is specified on the model, as well, by augmenting U, as described in Section 4.1, and so, the system configuration plotted in Figure 5.1 is obtained.
It is also supposed that assumptions 4.1 and 4.2 hold. The assumptions allow the existence of unfalsified uncertainty models and the establishment of consistency based on generalized plant G0. We can summarize the problem statement as follows.
Problem 2. Given an unknown system T, a plant model U and control specification defined by G0. Provide an iterative uncertainty model unfalsification and robust con-trol design procedure that 1. constructs a consistent model (U, W∆, Wd) for the true plant, based on closed-loop experiments, and 2. designs an LPV controllerK(∆s) that minimizes objective (4.5).