A velocity scheduled model is derived by applying the following simplifying assumptions to the nonlinear model.
1. First order Taylor-series approximation around β = 0 and δ = 0. If these angles are small it follows that α is also small
2. The adhesion function µ(s) is approximated in the region of small slip values, since the braking of one wheel is performed with small brake pressure: µ(s)≈µs¯ 3. Due to the applied small braking, the yaw moment generated by he brake forces
are neglected
4. In case of braking or on approximately constant velocitymax(1,cos(α)λ)≈1can be written
5. The wheel loads and adhesion are symmetric on the left and right sides, i.e.
Fzf
2 =Fz11=Fz12, F2zr =Fz21=Fz22,µ¯,µ¯f = ¯µ11= ¯µ12,µ¯r = ¯µ21= ¯µ22 6. |rlf| ≪(v− |rlw2|)2
7. (rl2w)2 ≪v2
8. vR11+vR12 ≈2v,vR21+vR22≈2v,vR21≈vR22 The lateral forces can be written as
Fyf := Fy11+Fy12=cfαf, Fyr := Fy21+Fy22=crαr,
where cornering stiffness parameters cf = ¯µ(Fz11+Fz12) and cr = ¯µ(Fz21+Fz22) are introduced. For the average front and rear slip angles we get
αf = δ−β−lfr v , αr = −β+lrr
v .
B.4. DERIVATION OF THE SIMPLIFIED MODEL
Accordingly, the yaw dynamics reduces to the linearized single-track model β˙ = −cf +cr
mv β+ (−1 +crlr−cflf
mv2 ) ˙ψ+ cf
mvδ ψ¨ = crlr−cflf
Jz
β−crlr2+cflf2
Jzv ψ˙+cflf Jz
δ
The above results are standard and can be found in any vehicle dynamics textbooks.
The following derivation is dedicated to the differential wheel model and the steering model of the steer-by-brake problem [Röd03].
Assuming symmetric driving forces the aligning torque can be written as
Ms = rs(Fx,12−Fx,11) +nRKFyf (B.13) where nRK :=nR+nK. According to the linearized slip model, the braking forces are FBij=−Fxij=−µs¯ ijFz. Let’s introduce the brake force difference
∆FB = Fx,12−Fx,11= ¯µFzf
2 (s11−s12)
where ∆s := s11−s12 is the slip difference at the front wheels. The next task is to formulate ∆s. It can be assumed that steering angles, and so wheel slip angles, are small during steer-by-brake maneuvers and the lateral slips can be neglected, i.e.
s1j =sL,1j = 1−λ1jcosα1j. It follows that
∆s ≈ (1−λ11cosα11)−(1−λ12cosα12)≈(λ12−λ11) cosαf ≈ vR12
vw12 −vR11
vw11 With assumption|rlf| ≪(v− |rl2w|)2 and from (B.7)-(B.8), wheel center velocities vw =|¯vw|can be approximated as
vw11 ≈ v+rlw
2 cosβ+rlfsinβ ≈v+rlw
2 vw12 ≈ v−rlw
2 cosβ+rlfsinβ ≈v−rlw 2 which results in
∆s ≈ vR12
v−rlw2 − vR11 v+rl2w
= v(vR12−vR11) +rlw2(vR12+vR11) v2−(rlw2)2
≈ −∆vR
v +lw
vr,
where differential wheel speed∆vR:=vR11−vR12is introduced. The last approximation follows from assumptions (rlw2)2 ≪ v2 and vR11+vR12 ≈ 2v. Inserting the resulted formulas into (B.13), we obtain for the aligning torque
Ms = rsµ¯Fzf
2 (−∆vR
v +lw
v r) +nRKcf(δ−β−lfr v )
Differential wheel dynamics can be characterized by
∆ ˙vR = v˙R11−v˙R12= (Fx12−Fx11)
| {z }
∆FB
ref f2 Jw
+ (p12−p11)CpTref f Jw
= cf
2∆sr2ef f
Jw −∆pCpTref f Jw
(B.14) where applied brake pressure difference ∆p = p11−p12 is introduced and µ¯F2zf is re-placed by c2f.
Finally, the three dynamic components are unified and, by assuming released hand-wheelTd= 0 andδm =δ, we obtain the velocity dependent linear model
M0 : x˙ =A0x+B0u y=C0x
where x= [β r δδ˙∆vR]T,u= ∆p,y= [r δ∆vR]T and the state-space matrices are
A0=
a11
v −1 +av122 av13 0 0 a21 a22
v a23 0 0
0 0 0 1 0
a41 a42
v a43 a44 a45
v
0 av52 0 0 av55
, B0=
0 0 0 0 b1
, C0=
0 1 0 0 0 0 0 1 0 0 0 0 0 0 1
,(B.15)
with constant parameters
a11=−cfm+cr, a12=crlr−mcflf, a13= cmf, a21=crlrJ−zcflf, a22=−crl
2 r+cfl2f
Jz , a23=cJfzlf, a41= nRKJcf
s , a42=nRKcflf−J0.5rscflw
s ,
a43=−nRKJscf, a44=−kJss, a45= 0.5rJscf
s , a52=0.5cfr
2 ef flw
Jw , a55=−0.5cfr
2 ef f
Jw b1=−CpTJrwef f.
Note that measurement y is available in the ESP data packets.
OWN PUBLICATIONS CITED
Own publications cited
[MKD+09] A. Marcos, M. Kerr, G. De Zaiacomo, L.F. Penín, Z. Szabó, G. Rödönyi, and J. Bokor. Application of LPV/LFT modeling and data-based validation to a re-entry vehicle. In AIAA Guidance, Navigation and Control Conference, Chicago, US, 2009.
[RB04a] G. Rödönyi and J. Bokor. Identification of LPV steering models of a truck.
5th International PhD Workshop on Systems and Control, a Young Gener-ation Viewpoint, Balatonfüred, Hungary, ISBN 963 311 359 8, page on CD, 2004.
[RB04b] G. Rödönyi and J. Bokor. Identification of LPV vehicle models for steering control involving asymmetric front wheel braking. 6th IFAC Symposium on Nonlinear Control Systems, NOLCOS-2004, Stuttgart, Germany, pages 663–668, 2004.
[RB05a] G. Rödönyi and J. Bokor. Identification of an LPV vehicle model based on experimental data for brake-steering control. 16th IFAC World Congress, Prague, page on CD, 2005.
[RB05b] G. Rödönyi and J. Bokor. Uncertainty identification for a nominal LPV vehicle model based on experimental data. 44th IEEE Conference on Deci-sion and Control and European Control Conference CDC-ECC’05, Seville, Spain, pages 2682–2687, 2005.
[RB06a] G. Rödönyi and J. Bokor. Integrated uncertainty model identification and robust control synthesis for linear time-invariant systems. 14th Mediter-ranean Conference on Control and Automation, Ancona, Italy, pages WM4–
6, 2006.
[RB06b] G. Rödönyi and J. Bokor. A joint structured complex uncertainty identifica-tion and mu-synthesis algorithm. Proceedings of the 2006 IEEE Conference on Control Applications, Munich, Germany, pages 2927–2932, 2006.
[RBL02] G. Rödönyi, J. Bokor, and B. Lantos. LQG control of LPV systems with parameter-dependent lyapunov function. 10th Mediterranean Conference on Control and Automation, Lisbon, Portugal, pages WP5–2, 417.pdf on CD, 8 pages, 2002.
[RBar] G. Rödönyi and J. Bokor. Uncertainty remodeling for robust control of linear time-invariant plants. Periodica Politechnica, Elect. Eng, to appear.
[Röd03] G. Rödönyi. Vehicle models for steering control. Technical report, Sys-tem and Control Laboratory, Computer and Automation Research Institute, SCL-4-2003, 2003.
[Röd07] G. Rödönyi. A dynamic model of a heavy truck. Technical report, Sys-tem and Control Laboratory, Computer and Automation Research Institute, SCL-1-2007, 2007.
[Röd09] G. Rödönyi. Iterative design of uncertainty model and robust controller based on experiment data. European Control Conference, ECC’09, Bu-dapest, Hungary, pages 802–807, 2009.
[RG10] G. Rödönyi and P. Gáspár. Iterative design of structured uncertainty models and robust controllers based on closed-loop data. Accepted in IET Control Theory and Applications, 2010.
[RGB09] G. Rödönyi, P. Gáspár, and J. Bokor. Robust LPV controller synthesis with uncertainty modelling. 17th Mediterranean Conference on Control and Automation, Thessaloniki, Greece, pages 815–820, 2009.
[RGB10] G. Rödönyi, P. Gáspár, and J. Bokor. The emergency steering of a heavy truck by front-wheel braking. Appears in International Journal of Heavy Vehicle Systems, 2010.
[RGSB08] G. Rödönyi, P. Gáspár, Z. Szabó, and J. Bokor. Uncertainty remodeling for robust control of linear time-invariant plants. 16th Mediterranean Confer-ence on Control and Automation, Ajaccio, France, pages 232–237, 2008.
[RLB07] G. Rödönyi, B. Lantos, and J. Bokor. Uncertainty modeling and robust control of LTI systems based on integral quadratic constraints. 11th Inter-national Conference on Intelligent Engineering Systems, INES07, Budapest, Hungary, pages 207–212, 2007.
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