92 CHAPTER 7 Connected cruise controllers andV(h)is a range policy function (e.g. constant time-gap spacing policy). The derivative ofV(h) at the uniform equilibrium flow(h∗, v∗)equals toV0(h∗) =κ+ ˜κ(where1/V0(h∗)is the time-gap).
The parameters of the linearized system areκ,α, βandτ, where the additive uncertainty of these parameters (˜κ,α,˜ β,˜ τ) denote the static uncertainty of the parameters of the human-driven vehicle.˜ The transfer function between the velocity fluctuations of the follower and leader vehicles is written as
T(s) =
(κ+ ˜κ)(α+ ˜α) + (β+ ˜β)s
e−sτ1−sϑ(s)˜ 1 +sϑ(s)˜ s2+ (κ+ ˜κ)(α+ ˜α) + (α+ ˜α+β+ ˜β)s
e−sτ1−sϑ(s)˜ 1 +sϑ(s)˜ ,
where the uncertainty of the time delay is modeled by the Rekasius substitution with restriction to s = iω, moreover ω is the frequency and ϑ(iω) =˜ ω−1tan(0.5ω˜τ) is a frequency-dependent real parameter. The follower vehicle is robust string stable if|T(iω)| <1, ω > 0for all allowed uncertainties. Robust string stability with respect to uncertain parameters in the follower’s model can be analyzed by constructing the M-∆uncertain interconnection structure and analyzing the structured singular values, where (s= iωand in case of the Rekasius substitution0≤ω < π/|τ˜|)
M(s) =
M1,1(s) M1,2(s) M2,1(s) M2,2(s)
,
∆(s) = diag[˜κ,α,˜ β,˜ ϑ(s), δ˜ c], δc ∈C,|δc|<1, moreover
M1,1(s) = 1
D(s)
−αe−sτ −e−sτ −e−sτ −2 s2+βse−sτ −(κ+s)e−sτ −(κ+s)e−sτ 2(κ+s)
−αse−sτ −se−sτ −se−sτ 2s
αs3e−sτ s3e−sτ s3e−sτ −s3+s κα+s(α+β) e−sτ
,
M1,2(s) = 1 D(s)
s+αe−sτ κs−βse−sτ s2+αse−sτ (καs2+βs3)e−sτ
,
M2,1(s) = 1 D(s)
αse−sτ se−sτ se−sτ −2s (κα+βs)e−sτ , M2,2(s) = (κα+βs)e−sτ
D(s) , D(s) = s2+ κα+s(α+β)
e−sτ. Related publications: [179], [180].
APPENDIX A
Computation of the structured singular value
The structured singular value was originally introduced for complex diagonal perturbations in [30], and more detailed mathematical investigations were provided in [181, 134]. Real structured perturbations and mixed perturbation problems were investigated more in details in [182, 135, 175].
It is important to mention that a very similar approach (based on geometric considerations) to calculate the stability margins was introduced in [183, 184].
The definition ofµdepends upon the underlying block structureXK of allowed uncertainties
∆. Here, we follow the notations and definition provided by [175]. Let us consider a matrix M ∈ Cn×n, and three non-negative integers mr,mcandmC withm = mr+mc+mC ≤ n, the block structure is anm-tuple of positive integerski ∈Z+is written as
K={k1, . . . , kmr, kmr+1, . . . , kmr+mc, kmr+mc+1, . . . , kmr+mc+mC}, Σmi=1ki =n, (A.1) and define the set of all allowed perturbation matrices as
XK={∆= diag δr1Ik1, . . . , δmrrIkmr, δ1cIkmr+1, . . . , δcmcIkmr+mc,∆C1, . . . ,∆CmC :
δri ∈R, δci ∈C,∆Ci ∈Ckmr+mc+i×kmr+mc+i}. (A.2) This block structure is able to represent repeated real scalar (δir), repeated complex scalar (δic) and full block complex perturbations (∆Ci ).
Letσ(¯ ·) denote the largest singular value of(·). Then theµ-value ofM is the inverse of the smallestσ(∆)¯ ,∆∈ XK, such that1is an element of the spectrum of the matrix product∆M(see [21, 185]). An equivalent definition says that theµ-value of matrixMis the inverse of the smallest perturbation such thatdet(I−∆M) = 0, i.e.,
µ(M) :=
∆min∈XK
¯
σ(∆) : det (I−M∆) = 0 −1
0, if no∆∈ XKmakesI−M∆singular.
(A.3)
Note that the larger theµ-value, the less robust the system. Also, it can be shown from the definition that for anyα∈ C,µ(αM) =|α|µ(M). However,µ:Cn×n →Ris not a norm, since it does not satisfy the triangle inequality [134].
The computation of the structured singular value is not straightforward, and in general only upper and lower bounds can be determined. It can be shown, that very conservative bounds for complex perturbations can be calculated as
ρ(M)≤µ(M)≤σ(M),¯ (A.4)
where ρ(·)is the spectral radius. Lower bound is exact if ∆ = δc1I, δ1c ∈ Cis a repeated scalar perturbation, and upper bound is exact if∆∈ Cn×nis a full block perturbation. A review on the complex structured singular values and their computation is given by [134].
93
94 Appendix A Computation of the structured singular value The bounds provided by the formulas (A.4) are easily computable, however, rarely sufficient when accurate calculations are required, since the gap between upper and lower bounds can be arbitrarily large [181]. Tighter bounds can be calculated by introducing transformations that affect ρandσ, but does not affect¯ µ. These bounds in case of complex perturbations are reformulated as
max
U∈UK
ρ(UM)≤µ(M)≤ inf
D∈DK
¯
σ(DMD−1) =νC(M), (A.5) where
UK={U∈ XK : U∗U=I}, (A.6)
DK={0<D=D∗ :D∆=∆D, for all∆∈ XK}, (A.7) for details see [30, 181]. The upper bound can be reformulated as a mathematically convex optimization problem that makes the computation of the upper bound feasible. The reformulated bound as a linear matrix inequality (LMI) is written as
νC(M) = inf
ν>0 D∈DK
{ν :M∗DM< ν2D}. (A.8) In comparison, the lower bound can be found by maximizing the spectral radius of a scaled matrix, however, the problem is not convex and global maximum is not guaranteed to be found.
The computation of real structured singular values is more time consuming, similarly to the complex case, the upper bound can be computed by
νR(M) = inf
ν>0 D∈DK
G∈GK
{ν:M∗DM+ i(GM−M∗G)< ν2D}, (A.9)
where
GK ={G=G∗ :G∆=∆∗G, for all∆∈ XK}, (A.10) andGis block-diagonal with zero block for complex uncertainties. For the detailed structures, see [175].
In the present work the computation of the structured singular value is carried out in Matlab environment, using the built-inmussvfunction included in theµ-Analysis and Synthesis Toolbox developed by [137].
APPENDIX B
Structured singular value and weighted maximum norm
The calculation of the structured singular values is time consuming, even if the upper bound is reformulated into a mathematically convex problem, see Appendix A. Computational effort can be reduced if simpler, but more conservative upper bounds are calculated. Following the results of [21], we give an upper approximation ofµfor complex perturbation structures.
Let us consider a matrix M ∈ Cn×n and the set of perturbations X = {∆ ∈ Cn×n : ∆ = diag(∆C1, . . . ,∆Cl )}, then theµ-value ofMis defined as
µ(M) :=
∆min∈X
¯
σ(∆) : det (I−M∆) = 0 −1
0, if no∆∈ X makesI−M∆singular. (B.1) Another alternative expression follows from the definition
µ(M) = max
∆∈X
¯ σ(∆)≤1
ρ M∆
, (B.2)
where the proof uses the property that for anyα∈C,µ(αM) =|α|µ(M).
The computation of the complex structured singular value requires the solution of a convex optimization problem (see Appendix A) that can be time-consuming for large-scale matrices. In order to speed up the calculation, [21] provided an alternative approach for complex perturbations in order to bound the structured singular value from above. Let us replace the perturbation measure
¯
σin definition (B.1), and introduce a weighted maximum norm instead as k∆kmax:= max
j,k R−1jk|∆jk|, (B.3)
where Rjk ≥ 0 are the elements of the non-negative weight matrixR ∈ Rn×n. In this case the following equivalence holds for∆
k∆kmax≤1 ⇔ |∆jk| ≤Rjk, for all j, k. (B.4) Then an upper bound ofµ-values can be given by the relations [21]
µ(M) = max
∆∈X k∆kmax≤1
ρ(M∆)≤ρ MR
, (B.5)
whereMis a non-negative matrix with elementsMjk =|Mjk|. Using this approach the structured singular value can be bounded from above using the spectral radius. To prove the inequality, the
95
96 Appendix B Structured singular value and weighted maximum norm following properties can be used
P1:|[M∆]jk|=| Xn
i=1
Mji∆ik| ≤ Xn
i=1
|Mji||∆ik|= [M ∆]jk, (B.6) P2:if A1,A2 ∈Rn×n, and0≤[A1]jk ≤[A2]jk, thenρ(A1)≤ρ(A2), (B.7) P3:ρ
A11 · · · A1n ... ... ...
An1 · · · Ann
≤ρ
|A11| · · · |A1n| ... ... ...
|An1| · · · |Ann|
. (B.8)
Detailed derivations, theorems and proofs related to the topic can be found in [21]. An important note is that the inequality in (B.5) is actually an equality ifMis a diagonal matrix.
References
[1] T. Insperger and G. Stepan. Semi-Discretization for Time-Delay Systems: Stability and Engineering Applications. Applied Mathematical Sciences. Springer New York, 2011.
[2] O. J. M. Smith. Closer control of loops with dead time. Chem. Eng. Prog., 53(5):217–219, 1957.
[3] G. Stépán. Delay, nonlinear oscillations and shimmying wheels.IUTAM Symposium on New Applica-tions of Nonlinear and Chaotic Dynamics in Mechanics, pages 373–386, Dordrecht, 1999. Springer Netherlands.
[4] D. Takács and G. Stépán. Micro-shimmy of towed structures in experimentally uncharted unstable parameter domain.Vehicle System Dynamics, 50(11):1613–1630, 2012.
[5] Y. Altintas. Manufacturing Automation: Metal Cutting Mechanics, Machine Tool Vibrations, and CNC Design. Cambridge University Press, 2 edition, 2012.
[6] G. Stépán.Retarded Dynamical Systems: Stability and Characteristic Functions. Longman Scientific
& Technical, 1989.
[7] G. Orosz and G. Stépán. Subcritical hopf bifurcations in a car-following model with reaction-time delay. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 462(2073):2643–2670, 2006.
[8] T. Insperger and J. Milton. Sensory uncertainty and stick balancing at the fingertip. Biological Cybernetics, 108(1):85–101, 2014.
[9] J. K. Hale. Theory of Functional Differential Equations. Springer, 1977.
[10] J. K. Hale and S. M. V. Lunel.Introduction to Functional Differential Equations. Springer New York, 1993.
[11] O. Diekmann, S. A. Gils, S. M. Lunel, and H.-O. Walther. Delay Equations: Functional-, Complex-, and Nonlinear Analysis. Springer, 1995.
[12] W. Michiels and S.-I. Niculescu. Stability and Stabilization of Time-Delay Systems. Society for Industrial and Applied Mathematics, 2007.
[13] H. Smith. An Introduction to Delay Differential Equations with Applications to the Life Sciences. Springer-Verlag, 2011.
[14] Z. Dombovari. Local and Global Dynamics of Cutting Processes. PhD thesis, Budapest University of Technology and Economics, 2012.
[15] D. Lehotzky. Numerical methods for the stability and stabilizability analysis of delayed dynamical systems. PhD thesis, Budapest University of Technology and Economics, 2016.
[16] D. Breda, S. Maset, and R. Vermiglio.Linearized Stability of Delay Differential Equations. Springer-Verlag GmbH, 2014.
[17] M. Farkas. Periodic Motions. Springer-Verlag GmbH, 1994.
[18] L. Trefethen and M. Embree. Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators. Princeton University Press, 2005.
[19] L. Qiu, B. Bernhardsson, A. Rantzer, E. Davison, P. Young, and J. Doyle. A formula for computation of the real stability radius. Automatica, 31(6):879–890, 1995.
[20] J. Demmel. A counterexample for two conjectures about stability. IEEE Transactions on Automatic Control, 32(4):340–342, 1987.
97
98 References [21] M. Karow, D. Hinrichsen, and A. J. Pritchard. Interconnected systems with uncertain couplings:
Explicit formulae for mu-values, spectral value sets, and stability radii.SIAM Journal on Control and Optimization, 45(3):856–884, 2006.
[22] F. Borgioli, W. Michiels, and N. Guglielmi. Characterizing and computing real pseudospectral abscissa for nonlinear eigenvalue problems. Technical Report TW Reports 668, 2016.
[23] D. Hinrichsen and A. J. Pritchard. Stability radius for structured perturbations and the algebraic Riccati equation. Systems & Control Letters, 8(2):105 – 113, 1986.
[24] D. Hinrichsen and B. Kelb. Spectral value sets: A graphical tool for robustness analysis. Systems &
Control Letters, 21(2):127 – 136, 1993.
[25] T. Wagenknecht, W. Michiels, and K. Green. Structured pseudospectra for nonlinear eigenvalue problems. Journal of Computational and Applied Mathematics, 212(2):245 – 259, 2008.
[26] D. Hinrichsen and A. J. Pritchard. Stability radii of linear systems. Systems & Control Letters, 7(1):1–10, 1986.
[27] W. Michiels and D. Roose. An eigenvalue based approach for the robust stabilization of linear time-delay systems. International Journal of Control, 76(7):678–686, 2003.
[28] K. Green and T. Wagenknecht. Pseudospectra and delay differential equations. Journal of Computa-tional and Applied Mathematics, 196(2):567 – 578, 2006.
[29] W. Michiels, K. Green, T. Wagenknecht, and S.-I. Niculescu. Pseudospectra and stability radii for analytic matrix functions with application to time-delay systems.Linear Algebra and its Applications, 418(1):315 – 335, 2006.
[30] J. C. Doyle. Analysis of feedback systems with structured uncertainties.IEE Proceedings D - Control Theory and Applications, 129(6):242–250, 1982.
[31] Z. J. Palmor. Time-delay compensation: Smith predictor and its modifications.The Control Handbook, 2000.
[32] W. Michiels and S.-I. Niculescu. On the delay sensitivity of Smith predictors. International Journal of Systems Science, 34(8-9):543–551, 2003.
[33] M. Krstic. Delay compensation for nonlinear, adaptive, and PDE systems. Birkhauser, Boston, 2009.
[34] Z. Palmor. Stability properties of Smith dead-time compensator controllers. International Journal of Control, 32(6):937–949, 1980.
[35] D. H. Owens and A. Raya. Robust stability of Smith predictor controllers for time-delay systems.IEE Proceedings D - Control Theory and Applications, 129(6):298–304, 1982.
[36] K. Yamanaka and E. Shimemura. Effects of mismatched Smith controller on stability in systems with time-delay. Automatica, 23(6):787 – 791, 1987.
[37] W. Feng. On practical stability of linear multivariable feedback systems with time-delays.Automatica, 27(2):389 – 394, 1991.
[38] K. Gu. A review of some subtleties of practical relevance for time-delay systems of neutral type.
ISRN Applied Mathematics, 2012.
[39] T. Erneux and T. Kalmár-Nagy. Nonlinear stability of a delayed feedback controlled container crane.
Journal of Vibration and Control, 13(5):603–616, 2007.
[40] M. Morari and E. Zafiriou. Robust process control. Prentice Hall, Englewood Cliffs, NJ, 1989.
[41] K. J. Astrom, C. C. Hang, and B. C. Lim. A new Smith predictor for controlling a process with an integrator and long dead-time. IEEE Transactions on Automatic Control, 39(2):343–345, 1994.
[42] Q.-C. Zhong and G. Weiss. A unified Smith predictor based on the spectral decomposition of the plant. International Journal of Control, 77(15):1362–1371, 2004.
References 99 [43] A. Manitius and A. Olbrot. Finite spectrum assignment problem for systems with delays. IEEE
Transactions on Automatic Control, 24(4):541–552, 1979.
[44] Q. G. Wang, T. H. Lee, and K. K. Tan. Finite spectrum assignment for time delay systems. Springer, London, 1999.
[45] P. J. Gawthrop and E. Ronco. Predictive pole-placement control with linear models. Automatica, 38(3):421 – 432, 2002.
[46] M. Jankovic. Forwarding, backstepping, and finite spectrum assignment for time delay systems.
Automatica, 45(1):2 – 9, 2009.
[47] L. Mirkin and N. Raskin. Every stabilizing dead-time controller has an observer-predictor-based structure. Automatica, 39(10):1747 – 1754, 2003.
[48] Q. Zhong. Robust Control of Time-delay Systems. Springer London, 2006.
[49] S. Mondie, M. Dambrine, and O. Santos. Approximation of control laws with distributed delays: A necessary condition for stability.Kybernetika, 38(5):541–551, 2002.
[50] W. Michiels, S. Mondie, and D. Roose. Robust stabilization of time-delay systems with distributed delay control laws: Necessary and sufficient conditions for a safe implementation. Technical Report TW Report 363, Department of Computer Science, Katholieke Universiteit Leuven, Belgium, 2003.
[51] T. G. Molnar and T. Insperger. On the robust stabilizability of unstable systems with feedback delay by finite spectrum assignment.Journal of Vibration and Control, 22(3):649–661, 2016.
[52] D. Hajdu and T. Insperger. Demonstration of the sensitivity of the Smith predictor to parameter uncertainties using stability diagrams. International Journal of Dynamics and Control, 4(4):384–
392, 2016.
[53] D. Hajdu and T. Insperger. A smith-prediktor időtartománybeli vizsgálata. A Gépipari Tudományos Egyesület Műszaki Folyóirata, 3:12–15, 2013.
[54] D. Hajdu and T. Insperger. Time domain analysis of the Smith predictor in case of parameter uncertainties: A case study.International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, 7B:V07BT10A060, 2013.
[55] S. G. Leveille, D. P. Kiel, R. N. Jones, A. Roman, M. T. Hannan, F. A. Sorond, H. G. Kang, E. J.
Samelson, M. Gagnon, M. Freeman, and L. A. Lipsitz. The MOBILIZE Boston Study: Design and methods of a prospective cohort study of novel risk factors for falls in an older population. BMC Geriatrics, 8:16, 2008.
[56] F. Moss and J. Milton. Balancing the unbalanced. Nature, 425:911–912, 2003.
[57] J. T. Bingham, J. T. Choi, and L. H. Ting. Stability in a frontal plane model of balance requires coupled changes to postural configuration and neural feedback control. Journal of Neurophysiology, 106(1):437–448, 2011.
[58] A. D. Goodworth and R. J. Peterka. Influence of stance width on frontal plane postural dynamics and coordination in human balance control.Journal of Neurophysiology, 104(2):1103–1118, 2010.
[59] A. D. Goodworth and R. J. Peterka. Sensorimotor integration for multisegmental frontal plane balance control in humans. Journal of Neurophysiology, 107(1):12–28, 2012.
[60] D. B. Lockhart and L. H. Ting. Optimal sensorimotor transformations for balance. Nature Neuro-science, 10(10):1329–1336, 2007.
[61] T. D. J. Welch and L. H. Ting. A feedback model explains the differential scaling of human postural responses to perturbation acceleration and velocity.Journal of Neurophysiology, 101(6):3294–3309, 2009.
[62] T. Insperger, J. Milton, and G. Stépán. Acceleration feedback improves balancing against reflex delay.
Journal of The Royal Society Interface, 10(79), 2013.
100 References [63] P. Gawthrop, I. Loram, M. Lakie, and H. Gollee. Intermittent control: a computational theory of
human control. Biological Cybernetics, 104(1-2):31–51, 2011.
[64] P. Gawthrop, I. Loram, H. Gollee, and M. Lakie. Intermittent control models of human standing:
similarities and differences. Biological Cybernetics, 108(2):159–168, 2014.
[65] A. Bottaro, Y. Yasutake, T. Nomura, M. Casadio, and P. Morasso. Bounded stability of the quiet standing posture: An intermittent control model. Human Movement Science, 27(3):473–495, 2008.
[66] Y. Asai, Y. Tasaka, K. Nomura, T. Nomura, M. Casidio, and P. Morasso. A model of postural control in quiet standing: Robust compensation of delay–induced instability using intermittent activation of feedback control. PLoS ONE, 4:e6169, 2009.
[67] Y. Suzuki, T. Nomura, M. Casadio, and P. Morasso. Intermittent control with ankle, hip, and mixed strategies during quiet standing: A theoretical proposal based on a double inverted pendulum model.
Journal of Theoretical Biology, 310:55–79, 2012.
[68] B. Mehta and S. Schaal. Forward models in visuomotor control. Journal of Neurophysiology, 88(2):942–953, 2002.
[69] T. Insperger. Act-and-wait concept for continuous-time control systems with feedback delay. IEEE Transactions on Control Systems Technology, 14(5):974–977, 2006.
[70] F. Valero-Cuevas, H. Hoffmann, M. Kurse, J. Kutch, and E. Theodorou. Computational models for neuromuscular function. IEEE Reviews in Biomedical Engineering, 2:110–135, 2009.
[71] J. L. Cabrera and J. G. Milton. On-off intermittency in a human balancing task. Physical Review Letters, 89(15), 2002.
[72] J. G. Milton, T. Ohira, J. L. Cabrera, R. M. Fraiser, J. B. Gyorffy, F. K. Ruiz, M. A. Strauss, E. C.
Balch, P. J. Marin, and J. L. Alexander. Balancing with vibration: A prelude for "Drift and Act"
balance control. PLoS ONE, 4(10):1–11, 2009.
[73] I. D. Loram, C. N. Maganaris, and M. Lakie. Human postural sway results from frequent, ballistic bias impulses by soleus and gastrocnemius. The Journal of Physiology, 564(1):295–311, 2005.
[74] F. J. Valero-Cuevas, N. Smaby, M. Venkadesan, M. Peterson, and T. Wright. The strength-dexterity test as a measure of dynamic pinch performance. Journal of Biomechanics, 36(2):265–270, 2003.
[75] P. Paoletti and L. Mahadevan. Balancing on tightropes and slacklines. Journal of The Royal Society Interface, 9(74):2097–2108, 2012.
[76] J. T. Bingham and L. H. Ting. Stability radius as a method for comparing the dynamics of neurome-chanical systems. IEEE Transactions on Neural Systems and Rehabilitation Engineering, 21(5):840–
848, 2013.
[77] J. Massion and M. H. Woollacott. Posture and equilibrium. Clinical Disorders of Balance, Posture and Gait, pages 1–18, New York, 1996. Oxford University Press.
[78] S. M. Henry, J. Fung, and F. B. Horak. Effect of stance width on multidirectional postural responses.
Journal of Neurophysiology, 85(2):559–570, 2001.
[79] G. Stepan. Delay effects in the human sensory system during balancing. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 367(1891):1195–1212, 2009.
[80] L. Moreau and E. Sontag. Balancing at the border of instability. Physical Review E, 68(2):020901, 2003.
[81] K. Costello, S. Matrangola, and M. Madigan. Independent effects of adding weight and inertia on balance during quiet standing. BioMedical Engineering OnLine, 11(1):20, 2012.
[82] W. Horsthemke and R. Lefever. Noise-induced transitions: Theory and applications in Physics, Chemistry and Biology. Springer, Berlin, 1984.
References 101 [83] D. Hajdu, J. Milton, and T. Insperger. Extension of stability radius to neuromechanical systems with structured real perturbations.IEEE Transactions on Neural Systems and Rehabilitation Engineering, 24(11):1235–1242, 2016.
[84] J. Munoa, X. Beudaert, Z. Dombovari, Y. Altintas, E. Budak, C. Brecher, and G. Stepan. Chatter suppression techniques in metal cutting.CIRP Annals, 65(2):785 – 808, 2016.
[85] G. Duncan, M. Kurdi, T. L. Schmitz, and J. P. Snyder. Uncertainty propagation for selected analytical milling stability limit analyses.34th North American Manufacturing Research Conference, 2006.
[86] X. Zhang, L. Zhu, D. Zhang, H. Ding, and Y. Xiong. Numerical robust optimization of spindle speed for milling process with uncertainties.International Journal of Machine Tools and Manufacture, 61:9 – 19, 2012.
[87] Y. Liu, L.-l. Meng, K. Liu, and Y.-m. Zhang. Chatter reliability of milling system based on first-order second-moment method. The International Journal of Advanced Manufacturing Technology, 87(1):801–809, 2016.
[88] X. Huang, M. Hu, Y. Zhang, and C. Lv. Probabilistic analysis of chatter stability in turning. The International Journal of Advanced Manufacturing Technology, 87(9):3225–3232, 2016.
[89] Y. Liu, T. xiang Li, K. Liu, and Y. min Zhang. Chatter reliability prediction of turning process system with uncertainties. Mechanical Systems and Signal Processing, 66-67:232 – 247, 2016.
[90] S. Hu, X. Huang, Y. Zhang, and C. Lv. Reliability analysis of the chatter stability during milling using a neural network. International Journal of Aerospace Engineering, 2016.
[91] M. Löser and K. Großmann. Influence of parameter uncertainties on the computation of stability lobe diagrams. Procedia CIRP, 46:460 – 463, 2016.
[92] M. Löser, A. Otto, S. Ihlenfeldt, and G. Radons. Chatter prediction for uncertain parameters.Advances in Manufacturing, 6(3):319–333, 2018.
[93] G. Totis. RCPM - A new method for robust chatter prediction in milling. International Journal of Machine Tools and Manufacture, 49(3):273 – 284, 2009.
[94] N. D. Sims, G. Manson, and B. Mann. Fuzzy stability analysis of regenerative chatter in milling.
Journal of Sound and Vibration, 329(8):1025 – 1041, 2010.
[95] D. Hamann, N.-P. Walz, A. Fischer, M. Hanss, and P. Eberhard. Fuzzy arithmetical stability analysis of uncertain machining systems. Mechanical Systems and Signal Processing, 98:534 – 547, 2018.
[96] X. Huang, Y. Zhang, and C. Lv. Probabilistic analysis of dynamic stability for milling process.
Nonlinear Dynamics, 86(3):2105–2114, 2016.
[97] S. S. Park and Y. M. Qin. Robust regenerative chatter stability in machine tools. The International Journal of Advanced Manufacturing Technology, 33(3):389–402, 2007.
[98] E. Graham, M. Mehrpouya, and S. Park. Robust prediction of chatter stability in milling based on the analytical chatter stability.Journal of Manufacturing Processes, 15(4):508 – 517, 2013.
[99] E. Graham, M. Mehrpouya, R. Nagamune, and S. Park. Robust prediction of chatter stability in micro milling comparing edge theorem and LMI.CIRP Journal of Manufacturing Science and Technology, 7(1):29 – 39, 2014.
[100] S. A. Tobias. Machine-tool Vibration. Blackie, Glasgow, 1965.
[101] J. Tlusty and L. Spacek. Self-excited vibrations on machine tools. Nakl. CSAV, Prague. in Czech., 1954.
[102] T. Insperger, G. Stépán, and J. Turi. State-dependent delay in regenerative turning processes.Nonlinear Dynamics, 47(1):275–283, 2007.
[103] G. Stepan, Z. Dombovari, and J. Muñoa. Identification of cutting force characteristics based on chatter experiments.CIRP Annals, 60(1):113–116, 2011.
102 References [104] D. Bachrathy and G. Stépán. Bisection method in higher dimensions and the efficiency number.
Periodica Polytechnica Mechanical Engineering, 56(2):81–86, 2012.
[105] Y. Ding, L. Zhu, X. Zhang, and H. Ding. A full-discretization method for prediction of milling stability. International Journal of Machine Tools and Manufacture, 50(5):502 – 509, 2010.
[106] X. J. Zhang, C. H. Xiong, Y. Ding, M. J. Feng, and Y. L. Xiong. Milling stability analysis with simultaneously considering the structural mode coupling effect and regenerative effect. International Journal of Machine Tools and Manufacture, 53(1):127 – 140, 2012.
[107] E. A. Butcher, O. A. Bobrenkov, E. Bueler, and P. Nindujarla. Analysis of milling stability by the chebyshev collocation method: Algorithm and optimal stable immersion levels.J. Comput. Nonlinear Dynam, 4(3):031003, 2009.
[108] G. Totis, P. Albertelli, M. Sortino, and M. Monno. Efficient evaluation of process stability in milling with spindle speed variation by using the chebyshev collocation method. Journal of Sound and Vibration, 333(3):646 – 668, 2014.
[109] F. A. Khasawneh and B. P. Mann. A spectral element approach for the stability of delay systems.
International Journal for Numerical Methods in Engineering, 87(6):566–592, 2011.
[110] G. Stepan, J. Munoa, T. Insperger, M. Surico, D. Bachrathy, and Z. Dombovari. Cylindrical milling tools: Comparative real case study for process stability. CIRP Annals, 63(1):385 – 388, 2014.
[111] G. Hu and E. J. Davison. Real stability radii of linear time-invariant time-delay systems. Systems &
Control Letters, 50(3):209–219, 2003.
[112] Z. Mao and M. Todd. Statistical modeling of frequency response function estimation for uncertainty quantification. Mechanical Systems and Signal Processing, 38(2):333–345, 2013.
[113] H. S. Kim and T. L. Schmitz. Bivariate uncertainty analysis for impact testing.Measurement Science and Technology, 18(11):3565–3571, 2007.
[114] D. Hajdu, T. Insperger, and G. Stepan. Robust stability of machining operations in case of uncertain frequency response functions. Procedia CIRP, 46:151 – 154, 2016.
[115] D. Hajdu, T. Insperger, and G. Stepan. Robust stability analysis of machining operations. The International Journal of Advanced Manufacturing Technology, 88(1):45–54, 2017.
[116] N. D. Sims. Vibration absorbers for chatter suppression: A new analytical tuning methodology.
Journal of Sound and Vibration, 301(3):592 – 607, 2007.
[117] D. J. Segalman and E. A. Butcher. Suppression of regenerative chatter via impedance modulation.
Journal of Vibration and Control, 6(2):243–256, 2000.
[118] G. Aguirre, M. Gorostiaga, T. Porchez, and J. Munoa. Self-tuning dynamic vibration absorber for machine tool chatter suppression. 28th Annual Meeting of the American Society for Precission Engineering (ASPE), St. Paul, Minnesota, United States, 2013.
[119] N. van Dijk, N. van de Wouw, H. Nijmeijer, R. Faassen, E. Doppenberg, and J. Oosterling. Real-time detection and control of machine tool chatter in high speed milling. 2nd International Conference Innovative Cutting Processes and Smart Machining, 2008.
[120] N. van de Wouw, N. van Dijk, and H. Nijmeijer. Pyragas-type feedback control for chatter mitigation in high-speed milling. IFAC-PapersOnLine, 48(12):334–339, 2015.
[121] N. van Dijk, N. van de Wouw, E. Doppenberg, H. Oosterling, and H. Nijmeijer. Chatter control in the high-speed milling process usingµ-synthesis. American Control Conference, pages 6121–6126, 2010.
[122] D. Lehotzky and T. Insperger. Stability of turning processes subjected to digital PD control.Periodica Polytechnica Mechanical Engineering, 56(1):33 – 42, 2012.
[123] D. Bachrathy and G. Stepan. Improved prediction of stability lobes with extended multi frequency solution. CIRP Annals, 62(1):411 – 414, 2013.
References 103 [124] D. Hajdu, T. Insperger, and G. Stepan. Robust controller design for turning operations based on
measured frequency response functions. IFAC-PapersOnLine, 50(1):7103 – 7108, 2017.
[125] M. H. Kurdi, T. L. Schmitz, R. T. Haftka, and B. P. Mann. A numerical study of uncertainty in stability and surface location error in high-speed milling.Manufacturing Engineering and Materials Handling, Parts A and B, pages 387–395, 2005.
[126] M. T. Loukil, V. Gagnol, and T.-P. Le. Reliability evaluation of machining stability prediction. The International Journal of Advanced Manufacturing Technology, 93(1):337–345, 2017.
[127] E. Budak and Y. Altintas. Analytical prediction of chatter stability in milling – Part I: General formulation.J. Dyn. Sys., Meas., Control, 120(1):22–30, 1998.
[128] Z. Dombovari and G. Stepan. The effect of helix angle variation on milling stability. Journal of Manufacturing Science and Engineering, 134(5), 2012.
[129] Y. Altintas and E. Budak. Analytical prediction of stability lobes in milling. CIRP Annals, 44(1):357 – 362, 1995.
[130] D. J. Ewins. Modal Testing: Theory, Practice and Applications. John Wiley & Sons, 2009.
[131] M. Zatarain and Z. Dombovari. Stability analysis of milling with irregular pitch tools by the implicit subspace iteration method.International Journal of Dynamics and Control, 2(1):26 – 34, 2014.
[132] V. Sellmeier and B. Denkena. Stable islands in the stability chart of milling processes due to unequal tooth pitch. International Journal of Machine Tools and Manufacture, 51(2):152 – 164, 2011.
[133] G. Stepan, D. Hajdu, A. Iglesias, D. Takacs, and Z. Dombovari. Ultimate capability of variable pitch milling cutters. CIRP Annals, 2018.
[134] A. Packard and J. Doyle. The complex structured singular value. Automatica, 29(1):71 – 109, 1993.
[135] M. K. H. Fan, A. L. Tits, and J. C. Doyle. Robustness in the presence of mixed parametric uncertainty and unmodeled dynamics. IEEE Transactions on Automatic Control, 36(1):25–38, 1991.
[136] F. R. Gantmacher. The theory of matrices, volume 2. Chelsea Pub. Co., New York, 1959.
[137] G. J. Balas, J. C. Doyle, K. Glover, A. Packard, and R. Smith. µ-analysis and synthesis toolbox.
MUSYN Inc. and The MathWorks, Natick MA, 1993.
[138] D. Hajdu, T. Insperger, D. Bachrathy, and G. Stepan. Prediction of robust stability boundaries for milling operations with extended multi-frequency solution and structured singular values.Journal of Manufacturing Processes, 30:281 – 289, 2017.
[139] D. Hajdu, T. Insperger, and G. Stepan. Quantification of uncertainty in machining operations based on probabilistic and robust approaches.Procedia CIRP, 77:82–85, 2018.
[140] P. Labuhn and W. Chundrlik. Adaptive cruise control, 1995.
[141] P. Barber, G. Engelman, P. King, and M. Richardson. Adaptive cruise control system and methodology, including control of inter-vehicle spacing, 2009.
[142] V. Milanés, J. Alonso, L. Bouraoui, and J. Ploeg. Cooperative maneuvering in close environments among cybercars and dual-mode cars. IEEE Transactions on Intelligent Transportation Systems, 12(1):15–24, 2011.
[143] M. Wang, W. Daamen, S. P. Hoogendoorn, and B. van Arem. Rolling horizon control framework for driver assistance systems. Part II: Cooperative sensing and cooperative control. Transportation Research Part C: Emerging Technologies, 40:290–311, 2014.
[144] J. Ploeg, N. van de Wouw, and H. Nijmeijer. Lpstring stability of cascaded systems: Application to vehicle platooning.IEEE Transactions on Control Systems Technology, 22(2):786–793, 2014.
[145] V. Milanés and S. E. Shladover. Modeling cooperative and autonomous adaptive cruise control dynamic responses using experimental data.Transportation Research Part C: Emerging Technologies, 48:285–300, 2014.
104 References [146] B. van Arem, C. J. G. van Driel, and R. Visser. The impact of cooperative adaptive cruise control on traffic-flow characteristics. IEEE Transactions on Intelligent Transportation Systems, 7(4):429–436, 2006.
[147] J. Ploeg, D. P. Shukla, N. van de Wouw, and H. Nijmeijer. Controller synthesis for string stability of vehicle platoons. IEEE Transactions on Intelligent Transportation Systems, 15(2):854–865, 2014.
[148] J. Lioris, R. Pedarsani, F. Y. Tascikaraoglu, and P. Varaiya. Platoons of connected vehicles can double throughput in urban roads. Transportation Research Part C: Emerging Technologies, 77:292–305, 2017.
[149] S. E. Li, R. Li, J. Wang, X. Hu, B. Cheng, and K. Li. Stabilizing periodic control of automated vehicle platoon with minimized fuel consumption. IEEE Transactions on Transportation Electrification, 3(1):259–271, 2017.
[150] G. Orosz. Connected cruise control: modelling, delay effects, and nonlinear behaviour. Vehicle System Dynamics, 54(8):1147–1176, 2016.
[151] G. Orosz, J. I. Ge, C. R. He, S. S. Avedisov, W. B. Qin, and L. Zhang. Seeing beyond the line of site – controlling connected automated vehicles. Mechanical Engineering, 139(12):S8–S12, 2017.
[152] L. Zhang and G. Orosz. Motif-based design for connected vehicle systems in presence of heterogeneous connectivity structures and time delays. IEEE Transactions on Intelligent Transportation Systems, 17(6):1638–1651, 2016.
[153] J. I. Ge and G. Orosz. Dynamics of connected vehicle systems with delayed acceleration feedback.
Transportation Research Part C: Emerging Technologies, 46:46–64, 2014.
[154] S. S. Avedisov and G. Orosz. Analysis of connected vehicle networks using network-based perturbation techniques. Nonlinear Dynamics, 89(3):1651–1672, 2017.
[155] J. I. Ge, G. Orosz, D. Hajdu, T. Insperger, and J. Moehlis. To Delay or Not to Delay—Stability of Connected Cruise Control, pages 263–282. Springer International Publishing, Cham, 2017.
[156] P. Seiler and R. Sengupta. AnH∞approach to networked control. IEEE Transactions on Automatic Control, 50(3):356–364, 2005.
[157] J. P. Maschuw, G. C. Keßler, and D. Abel. LMI-based control of vehicle platoons for robust longitudinal guidance. IFAC Proceedings Volumes, 41(2):12111–12116, 2008.
[158] S. E. Li, F. Gao, K. Li, L. Y. Wang, K. You, and D. Cao. Robust longitudinal control of multi-vehicle systems–A distributed H-infinity method. IEEE Transactions on Intelligent Transportation Systems, 19(9):2779 – 2788, 2018.
[159] Y. Zheng, S. E. Li, K. Li, and W. Ren. Platooning of connected vehicles with undirected topologies:
Robustness analysis and distributed H-infinity controller synthesis. IEEE Transactions on Intelligent Transportation Systems, 19(5):1353 – 1364, 2018.
[160] F. Gao, S. E. Li, Y. Zheng, and D. Kum. Robust control of heterogeneous vehicular platoon with uncertain dynamics and communication delay. IET Intelligent Transport Systems, 10(7):503–513, 2016.
[161] A. Stotsky, C. Chien, and P. Ioannou. Robust platoon-stable controller design for autonomous intelligent vehicles. Proceedings of 33rd IEEE Conference on Decision and Control. IEEE, 1994.
[162] L. Zhang, J. Sun, and G. Orosz. Hierarchical design of connected cruise control in the presence of in-formation delays and uncertain vehicle dynamics.IEEE Transactions on Control Systems Technology, 26(1):139–150, 2018.
[163] M. di Bernardo, A. Salvi, and S. Santini. Distributed consensus strategy for platooning of vehicles in the presence of time-varying heterogeneous communication delays. IEEE Transactions on Intelligent Transportation Systems, 16(1):102–112, 2015.
References 105 [164] W. B. Qin and G. Orosz. Scalable stability analysis on large connected vehicle systems subject to stochastic communication delays.Transportation Research Part C: Emerging Technologies, 83:39–60, 2017.
[165] Y. Zhou, S. Ahn, M. Chitturi, and D. A. Noyce. Rolling horizon stochastic optimal control strategy for ACC and CACC under uncertainty. Transportation Research Part C: Emerging Technologies, 83:61–76, 2017.
[166] B. Besselink and K. H. Johansson. String stability and a delay-based spacing policy for vehicle platoons subject to disturbances. IEEE Transactions on Automatic Control, 62(9):4376–4391, 2017.
[167] J. I. Ge, S. S. Avedisov, C. R. He, W. B. Qin, M. Sadeghpour, and G. Orosz. Experimental validation of connected automated vehicle design among human-driven vehicles.Transportation Research Part C: Emerging Technologies, 91:335–352, 2018.
[168] J. I. Ge and G. Orosz. Connected cruise control among human-driven vehicles: Experiment-based parameter estimation and optimal control design. Transportation Research Part C: Emerging Tech-nologies, 95:445–459, 2018.
[169] A. Salvi, S. Santini, and A. S. Valente. Design, analysis and performance evaluation of a third order distributed protocol for platooning in the presence of time-varying delays and switching topologies.
Transportation Research Part C: Emerging Technologies, 80:360–383, 2017.
[170] N. Bekiaris-Liberis, C. Roncoli, and M. Papageorgiou. Predictor-based adaptive cruise control design.
IEEE Transactions on Intelligent Transportation Systems, 19(10):3181–3195, 2018.
[171] T. G. Molnar, W. B. Qin, T. Insperger, and G. Orosz. Application of predictor feedback to compensate time delays in connected cruise control. IEEE Transactions on Intelligent Transportation Systems, 19(2):545–559, 2018.
[172] Y. Zheng, S. E. Li, J. Wang, D. Cao, and K. Li. Stability and scalability of homogeneous vehicular platoon: Study on the influence of information flow topologies. IEEE Transactions on Intelligent Transportation Systems, 17(1):14–26, 2016.
[173] P. Seiler, A. Pant, and K. Hedrick. Disturbance propagation in vehicle strings. IEEE Transactions on Automatic Control, 49(10):1835–1841, 2004.
[174] C. Scherer. Theory of Robust Control. TU Delft, 2001.
[175] P. M. Young, M. P. Newlin, and J. C. Doyle. µanalysis with real parametric uncertainty.Proceedings of the 30th IEEE Conference on Decision and Control, 2:1251–1256, 1991.
[176] N. Olgac and R. Sipahi. An exact method for the stability analysis of time-delayed linear time-invariant (LTI) systems. IEEE Transactions on Automatic Control, 47(5):793–797, 2002.
[177] Z.-Q. Wang, P. Lundström, and S. Skogestad. Representation of uncertain time delays in the H∞
framework.International Journal of Control, 59(3):627–638, 1994.
[178] Dedicated Short Range Communications (DSRC) Message Set Dictionary Set. SAE J2735SET_201603, SAE International, 2016.
[179] D. Hajdu, J. I. Ge, T. Insperger, and G. Orosz. Stability, Control and Application of Time-delay Systems, chapter Robust stability of connected cruise controllers. 2019.
[180] D. Hajdu, J. I. Ge, T. Insperger, and G. Orosz. Robust design of connected cruise control among human-driven vehicles.IEEE Transactions on Intelligent Transportation Systems, 2019.
[181] A. Packard, M. K. H. Fan, and J. Doyle. A power method for the structured singular value.Proceedings of the 27th IEEE Conference on Decision and Control, pages 2132–2137 vol.3, 1988.
[182] P. M. Young and J. C. Doyle. Computation of µ with real and complex uncertainties. 29th IEEE Conference on Decision and Control, 3:1230–1235, 1990.
[183] M. Safonov. Stability margins of diagonally perturbed multivariable feedback systems. 20th IEEE Conference on Decision and Control including the Symposium on Adaptive Processes. IEEE, 1981.