Time-delay systems: Analysis and control using the lambert W function

Time-Delay Systems: Analysis and Control Using the Lambert W Function

Volumn , Issue , 2010, Pages 1-142

  Yi, Sun ^a   Nelson, Patrick W ^a   Ulsoy, A Galip ^a  

^a UNIVERSITY OF MICHIGAN   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

DIFFERENTIAL EQUATIONS; TIME DELAY; TIMING CIRCUITS;

ACHIEVABLE PERFORMANCE; ANALYSIS AND CONTROLS; DELAY DIFFERENTIAL EQUATIONS; DESIGN CONTROLLERS; LAMBERT W FUNCTION; NOVEL METHODOLOGY; SYSTEM PROPERTY; TIME-DELAY SYSTEMS;

DELAY CONTROL SYSTEMS;

EID: 84967566763     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1142/7759     Document Type: Book

References (162)

1. Adams, B. M., Banks, H. T., Davidian, M., Kwon, H. D., Tran, H. T., Wynne, S. N. and Rosenberg, E. S. (2005). HIV dynamics: Modeling, data analysis, and optimal treatment protocols, Journal of Computational and Applied Mathematics 184, 1, pp. 10-49.
2. Adams, B. M., Banks, H. T., Kwon, H. D. and Tran, H. T. (2004). Dynamic multidrug therapies for HIV: Optimal and STI control approaches, Mathe-matical Biosciences and Engineering 1, 2, pp. 223-241.
3. Asl, F. M. and Ulsoy, A. G. (2003). Analysis of a system of linear delay differential equations, Journal of Dynamic Systems, Measurement and Control 125, 2, pp. 215-223.
4. Asl, F. M. and Ulsoy, A. G. (2007). Closure to "Discussion of 'analysis of a system of linear delay differential equations' ", Journal of Dynamic Systems Measurement and Control 129, 1, pp. 123-123.
5. Banks, H. T. and Bortz, D. M. (2005). A parameter sensitivity methodology in the context of hiv delay equation models, Journal of Mathematical Biology50, pp. 607-625.
6. Banks, H. T. and Manitius, A. (1975). Projection series for retarded functional differental equations with applications to optimal control problems, Journal of Differential Equatios 18, 2, pp. 296-332.
7. Beddington, J. R. and May, R. M. (1975). Time delays are not necessarily desta-bilizing, Mathematical Biosciences 27, 1-2, pp. 109-117.
8. Bellman, R. E. and Cooke, K. L. (1963). Differential-Difference Equations (Aca-demic Press, New York).
9. Bengea, S. C., Li, X. Q. and DeCarlo, R. A. (2004). Combined controller-observer design for uncertain time delay systems with application to engine idle speed control, Journal of Dynamic Systems Measurement and Control 126, 4, pp. 772-780.
10. Bhat, K. P. M. and Koivo, H. N. (1976a). Modal characterizations of controlla-bility and observability in time delay systems, IEEE Transactions on Au-tomatic Control 21, 2, pp. 292-293.
11. Bhat, K. P. M. and Koivo, H. N. (1976b). An observer theory for time delay systems, IEEE Transactions on Automatic Control 21, 2, pp. 266-269.
12. Bortz, D. M. and Nelson, P. W. (2004). Sensitivity analysis of a nonlinear lumped parameter model of HIV infection dynamics, Bulletin of Mathematical Bi-ology 66, 5, pp. 1009-1026.
13. Bortz, D. M. and Nelson, P. W. (2006). Model selection and mixed-effects mod-eling of HIV infection dynamics, Bulletin of Mathematical Biology 68, 8, pp. 2005-2025.
14. Brethe, D. and Loiseau, J. J. (1998). An effective algorithm for finite spectrum as-signment of single-input systems with delays, Mathematics and Computers in Simulation 45, 3-4, pp. 339-348.
15. Buckalo, A. F. (1968). Explicit conditions for controllability of linear systems with time lag, IEEE Transactions on Automatic Control 13, pp. 193-195.
16. Chen, C. T. (1984). Linear System Theory and Design (Holt, Rinehart and Winston, New York).
17. Chen, S. G., Ulsoy, A. G. and Koren, Y. (1997). Computational stability analysis of chatter in turning, Journal of Dynamic Systems, Measurement and, Control 119, 4, pp. 457-460.
18. Chen, Y. and Moore, K. L. (2002a). Analytical stability bound for a class of delayed fractional-order dynamic systems, Nonlinear Dynamics 29, 1-4, pp. 191-200.
19. Chen, Y. and Moore, K. L. (2002b). Analytical stability bound for delayed second-order systems with repeating poles using Lambert function W, Automatica38, 5, pp. 891-895.
20. Cheng, Y. and Hwang, C. (2006). Use of the Lambert W function for time-domain analysis of feedback fractional delay systems, IEE Proceedings: ControlTheory and Applications 153, 2, pp. 167-174.
21. Choudhury, A. K. (1972a). Algebraic and transfer-function criteria of fixed-time controllability of delay-differential systems, International Journal of Control 16, 6, pp. 1073-1082.
22. Choudhury, A. K. (1972b). Necessary and sufficient conditions of pointwise com-pleteness of linear time-invariant delay-differential systems, Internation Journal of Control 16, 6, pp. 1083-1100.
23. Ciupe, M. S., Bivort, B. L., Bortz, D. M. and Nelson, P. W. (2006). Estimat-ing kinetic parameters from HIV primary infection data through the eyes of three different mathematical models, Mathematical Biosciences 200, 1, pp. 1-27.
24. Ciupe, S. M., Ribeiro, R. M., Nelson, P. W., Dusheiko, G. and Perelson, A. S. (2007). The role of cells refractory to productive infection in acute hepatitis B viral dynamics, Proceedings of The National Academy of Sciences of the United States of America 104, 12, pp. 5050-5055.
25. Cooke, K. L. and Grossman, Z. (1982). Discrete delay, distributed delay and stability switches, Journal of Mathematical Analysis and Applications 86, 2, pp. 592-627.
26. Corless, R. M., Gonnet, G. H., Hare, D. E. G., Jeffrey, D. J. and Knuth, D. E. (1996). On the Lambert W function, Advances in Computational Mathe-matics 5, 4, pp. 329-359.
27. D'Amato, R. M., D'Aquila, R. T. and Wein, L. M. (2000). Management of an-tiretroviral therapy for HIV infection: Analyzing when to change therapy, Management Science 46, 9, pp. 1200-1213.
28. Darouach, M. (2001). Linear functional observers for systems with delays in state variables, IEEE Transactions on Automatic Control 46, 3, pp. 491-496.
29. de la Sen, M. (2005). On pole-placement controllers for linear time-delay systems with commensurate point delays, Mathematical Problems in Engineering, 1, pp. 123-140.
30. Delfour, M. C. and Mitter, S. K. (1972). Controllability, observability and optimal feedback control of affine hereditary differential systems, SIAM Journal of Control 10, pp. 298-328.
31. Dixit, N. M. and Perelson, A. S. (2004). Complex patterns of viral load decay un-der antiretroviral therapy: influence of pharmacokinetics and intracellular delay, Journal of Theoretical Biology 226, 1, pp. 95-109.
32. Engelborghs, K., Dambrine, M. and Roose, D. (2001). Limitations of a class of stabilization methods for delay systems, IEEE Transactions on Automatic Control 46, 2, pp. 336-339.
33. Fattouh, A., Sename, O. and Dion, J. M. (1998). H-infinity observer design for time-delay systems, in Proceedings ofthe 37th IEEE Conference on Decision and Control, Tampa FL, Dec.1998, pp. 4545-4546.
34. Fliess, M., Marquez, R. and Mounier, H. (2002). An extension of predictive control, PID regulators and smith predictors to some linear delay systems, International Journal of Control 75, 10, pp. 728-743.
35. Fofana, M. S. (2003). Delay dynamical systems and applications to nonlinear machine-tool chatter, Chaos Solitons & Fractals 17, pp. 731-747.
36. Forde, J. and Nelson, P. (2004). Applications of sturm sequences to bifurcation analysis of delay differential equation models, Journal of Mathematical Analysis and Applications 300, 2, pp. 273-284.
37. Franklin, G. F., Powell, J. D. and Emami-Naeini, A. (2005). Feedback Control of Dynamic Systems (Pearson Prentice Hall, Upper Saddle River, NJ).
38. Frost, M. G. (1982). Controllability, observability and the transfer function matrix for a delay-differential system, Internation Journal of Control 35, 1, pp. 175-182.
39. Gabasov, R., Zhevnyak, R. M., Kirillova, F. M. and Kopeikina, T. B. (1972). Relative observability of linear systems. 2. observations under constantly acting perturbations, Automation and Remote Control 33, 10, pp. 1563-1573.
40. Glizer, V. Y. (2004). Observability of singularly perturbed linear time-dependent differential systems with small delay, Journal of Dynamical and Control Systems 10, 3, pp. 329-363.
41. Gorecki, H., Fuksa, S., Grabowski, P. and Korytowski, A. (1989). Analysis and, Synthesis of Time Delay Systems (John Wiley & Sons, New York).
42. Gu, K. and Niculescu, S. I. (2006). Stability analysis of time-delay systems: A Lyapunov approach, Advanced Topics in Control Systems Theory, Lecture Notes in Control and Information Sciences 328, pp. 139-170.
43. Hale, J. K. and Lunel, S. M. V. (1993). Introduction to Functional Differential Equations (Springer-Verlag, New York).
44. Herz, A. V. M., Bonhoeffer, S., Anderson, R. M., May, R. M. and Nowak, M. A. (1996). Viral dynamics in vivo: Limitations on estimates of intracellular delay and virus decay, Proceedings of the National Academy of Sciences of The United States of America 93, 14, pp. 7247-7251.
45. Ho, D. D., Neumann, A. U., Perelson, A. S., Chen, W., Leonard, J. M. and Markowitz, M. (1995). Rapid turnover of plasma virions and CD4 lym-phocites in HIV-1 infection, Nature 373, 6510, pp. 123-126.
46. Holford, S. and Agathoklis, P. (1996). Use of model reduction techniques for designing IIR filters with linear phase in the passband, IEEE Transactions on Signal Processing 44, 10, pp. 2396-2404.
47. Hovel, P. and Scholl, E. (2005). Control of unstable steady states by time-delayed feedback methods, Physical Review E 72, 4, pp. 1-7.
48. Hrissagis, K. and Kosmidou, O. I. (1998). Delay-dependent robust stability con-ditions and decay estimates for systems with input delays, Kybernetika 34, 6, pp. 681-691.
49. Hu, G. and Davison, E. J. (2003). Real stability radii of linear time-invariant time-delay systems, Systems and Control Letters 50, 3, pp. 209-219.
50. Huang, Y. X., Rosenkranz, S. L. and Wu, H. L. (2003). Modeling HIV dynamics and antiviral response with consideration of time-varying drug exposures, adherence and phenotypic sensitivity, Mathematical Biosciences 184, 2, pp. 165-186.
51. Hwang, C. and Cheng, Y. (2005). A note on the use of the Lambert W function in the stability analysis of time-delay systems, Automatica 41, 11, pp. 1979-1985.
52. Jankovic, M. (2001). Control design for a diesel engine model with time delay, in Proceedings ofthe 40th IEEE Conference on Decision and Control, Orlando, FL, Dec. 2001, pp. 117-122.
53. Jankovic, M. and Kolmanovsky, I. (2000). Constructive Lyapunov control design for turbocharged diesel engines, IEEE Transactions on Control Systems Technology 8, 2, pp. 288-299.
54. Jankovic, M. and Kolmanovsky, I. (2009). Developments in Control of Time-Delay Systems for Automotive Powertrain Applications, Delay Differential Equations: Recent Advances and New Directions (Springer).
55. Jarlebring, E. and Damm, T. (2007). The Lambert W function and the spectrum of some multidimensional time-delay systems, Automatica 43, 12, pp. 2124-2128.
56. Kalmar-Nagy, T., Stepan, G. and Moon, F. C. (2001). Subcritical hopf bifurcation in the delay equation model for machine tool vibarations, Nonlinear Dynamics 26, pp. 121-142.
57. Kawabata, K. and Mori, T. (2009). Feedback enlargement of stability radius by nondifferentiable optimization, Electrical Engineering in Japan 166, 3, pp. 55-61.
58. Kaya, I. (2004). IMC based automatic tuning method for PID controllers in a Smith predictor configuration, Computers & Chemical Engineering 28, 3, pp. 281-290.
59. Kirillova, F. M. and Churakova, S. V. (1967). The controllability problem for linear systems with aftereffect, Differential Equations 3, pp. 221-225.
60. Kirschner, D. and Webb, G. F. (1996). A model for treatment strategy in the chemotherapy of AIDS, Bulletin of Mathematical Biology 58, 2, pp. 367-390.
61. Kopeikina, T. B. (1998). Relative observability of linear nonstationary singularly perturbed delay systems, Differential Equations 34, 1, pp. 22-28.
62. Kuang, Y. (1993). Delay Differential Equations: with Applications in Population Dynamics (Academic Press, Cambridge, Mass.).
63. Lee, E. B. and Olbrot, A. (1981). Observability and related structural results for linear hereditary systems, International Journal of Control 34, 6, pp. 1061-1078.
64. Lee, T. N. and Dianat, S. (1981). Stability of time-delay systems, IEEE Trans-actions on Automatic Control 26, 4, pp. 951-953.
65. Leyvaramos, J. and Pearson, A. E. (1995). An asymptotic modal observer for linear autonomous time-lag systems, IEEE Transactions on Automatic Con-trol 40, 7, pp. 1291-1294.
66. Li, X. and deSouza, C. E. (1998). Output feedback stabilization of linear timedelay systems, in L. Dugard and E. I. Verriest (eds.), Stability and Control of Time-delay Systems (Springer, New York), pp. 241-258.
67. Li, X., Ji, J. C., Hansen, C. H. and Tan, C. (2006). The response of a Duffing-van der Pol oscillator under delayed feedback control, Journal of Sound and Vibration 291, 3-5, pp. 644-655.
68. Lien, C. H., Sun, Y. J. and Hsieh, J. G. (1999). Global stabilizability for a class of uncertain systems with multiple time-varying delays via linear control, International Journal of Control 72, 10, pp. 904-910.
69. Liu, P. L. (2003). Exponential stability for linear time-delay systems with delay dependence, Journal of the Franklin Institute 340, 6-7, pp. 481-488.
70. Lunel, S. M. V. (1989). Exponential Type Calculus for Linear Delay Equations (Centrum voor Wiskunde en Informatica, Amsterdam, The Netherlands).
71. Maccari, A. (2001). The response of a parametrically excited van der Pol oscillator to a time delay state feedback, Nonlinear Dynamics 26, 2, pp. 105-119.
72. Maccari, A. (2003). Vibration control for the primary resonance of the van der Pol oscillator by a time delay state feedback, International Journal of Non-Linear Mechanics 38, 1, pp. 123-131.
73. Mahmoud, M. S. (2000). Robust Control and Filtering for Time-Delay Systems(Marcel Dekker, New York).
74. Mahmoud, M. S. and Ismail, A. (2005). New results on delay-dependent control of time-delay systems, IEEE Transactions on Automatic Control 50, 1, pp. 95-100.
75. Malek-Zavarei, M. and Jamshidi, M. (1987). Time-delay Systems: Analysis, Op-timization, and Applications (Elsevier Science Pub., New York, U.S.A.).
76. Manitius, A. and Olbrot, A. W. (1979). Finite spectrum assignment problem for systems with delays, IEEE Transactions on Automatic Control 24, 4, pp. 541-553.
77. Mao, W. J. and Chu, J. (2006). D-stability for linear continuous-time systems with multiple time delays, Automatica 42, 9, pp. 1589-1592.
78. Marino, S., Beretta, E. and Kirschner, D. E. (2007). The role of delays in innate and adaptive immunity to intracellular bacterial infection, Mathematical Biosciences and Engineering 4, 2, pp. 261-286.
79. Martinez-Picado, J., Negredo, E., Ruiz, L., Shintani, A., Fumaz, C. R., Zala, C., Domingo, P., Vilaro, J., Llibre, J. M., Viciana, P., Hertogs, K., Boucher, C., D'Aquila, R. T. and Clotet, B. (2003). Alternation of antiretroviral drug regimens for HIV infection-a randomized, controlled trial, Annals of Interna, l Medicine 139, 2, pp. 81-89.
80. Merritt, H. E. (1965). Theory of self-excited machine-tool chatter-contribution to machine-too chatter research-1, Journal of Engineering for Industry 87, 4, pp. 447-454.
81. Michiels, W., Engelborghs, K., Vansevenant, P. and Roose, D. (2002). Continuous pole placement for delay equations, Automatica 38, 5, pp. 747-761.
82. Michiels, W. and Roose, D. (2003). An eigenvalue based approach for the robust stabilization of linear time-delay systems, International Journal of Control76, 7, pp. 678-686.
83. Minis, I. E., Magrab, E. B. and Pandelidis, I. O. (1990). Improved methods for the prediction of chatter in turning, part 3. a generalized linear theory, Journal of Engineering for Industry 112, 1, pp. 28-35.
84. Mittler, J. E., Sulzer, B., Neumann, A. U. and Perelson, A. S. (1998). Influence of delayed viral production on viral dynamics in HIV-1 infected patients, Mathematical Biosciences 152, 2, pp. 143-163.
85. Moore, B. C. (1981). Principal component analysis in linear systems: controllabil-ity, observability, and model reduction, IEEE Transactions on Automatic Control 26, 1, pp. 17-32.
86. Mounier, H. (1998). Algebraic interpretation of the spectral controllability of a linear delay system, Forum Mathematicum 10, 1, pp. 39-58.
87. Nayfeh, A., Chin, C., Pratt, J. and Moon, F. C. (1997). Application of perturbation methods to tool chatter dynamics, in F. C. Moon (ed.), Dynamics and Chaos in Manufacturing Processes (Wiley, New York), pp. 193-213.
88. Nelson, P. W., Mittler, J. E. and Perelson, A. S. (2001). Effect of drug efficacy and the eclipse phase of the viral life cycle on estimates of HIV viral dynamic parameters, Journal of Acquired Immune Deficiency Syndromes 26, 5, pp. 405-412.
89. Nelson, P. W., Murray, J. D. and Perelson, A. S. (2000). A model of HIV-1 pathogenesis that includes an intracellular delay, Mathematical Biosciences163, 2, pp. 201-215.
90. Nelson, P. W. and Perelson, A. S. (2002). Mathematical analysis of delay differ-ential equation models of HIV-1 infection, Mathematical Biosciences 179, 1, pp. 73-94.
91. Niculescu, S. I. (1998). H-infinity memoryless control with an alpha-stability con-straint for time-delay systems: An LMI approach, IEEE Transactions on Automatic Control 43, 5, pp. 739-743.
92. Niculescu, S. I. (2001). Delay Effects on Stability: a Robust Control Approach(Springer, New York).
93. Niculescu, S. I. and Annaswamy, A. M. (2003). An adaptive Smith-controller for time-delay systems with relative degree n<=2, Systems & Control Letters 49, 5, pp. 347-358.
94. Nowak, M. A., Bonhoeffer, S., Shaw, G. M. and May, R. M. (1997). Anti-viral drug treatment: Dynamics of resistance in free virus and infected cell pop-ulations, Journal of Theoretical Biology 184, 2, pp. 205-219.
95. Olbrot, A. W. (1972). On controllability of linear systems with time delays in control, IEEE Transactions on Automatic Control 17, 5, pp. 664-666.
96. Olgac, N., Elmali, H., Hosek, M. and Renzulli, M. (1997). Active vibration control of distributed systems using delayed resonator with acceleration feedback, Journal of Dynamic Systems, Measurement and Control 119, 3, pp. 380389.
97. Olgac, N., Ergenc, A. and Sipahi, R. (2005). "Delay scheduling": A new concept for stabilization in multiple delay systems, Journal of Vibration and Control 11, 9, pp. 1159-1172.
98. Olgac, N. and Sipahi, R. (2002). An exact method for the stability analysis of time-delayed linear time-invariant (LTI) systems, IEEE Transactions on Automatic Control 47, 5, pp. 793-797.
99. Olgac, N. and Sipahi, R. (2005). A unique methodology for chatter stability mapping in simultaneous machining, Journal of Manufacturing Science and, Engineering 127, 4, pp. 791-800.
100. Optiz, H. and Bernardi, F. (1970). Investigation and calculation of the chatter behavior of lathes and milling machines, Annlas of CIRP 18, pp. 335-343.
101. Pearson, A. E. and Fiagbedzi, Y. A. (1989). An obsever for time lag systems, IEEE Transactions on Automatic Control 34, 7, pp. 775-777.
102. Pease, M. C. (1965). Methods of Matrix Algebra (Academic Press, New York).
103. Perelson, A. S. (2002). Modelling viral and immune system dynamics, Nature Reviews Immunology 2, 1.
104. Perelson, A. S. and Nelson, P. W. (1999). Mathematical analysis of HIV-1 dynamics in VIVO, SIAM Review 41, 1, pp. 3-44.
105. Perelson, A. S., Neumann, A. U., Markowitz, M., Leonard, J. M. and Ho, D. D. (1996). HIV-1 dynamics in vivo: Virion clearance rate, infected cell life-span, and viral generation time, Science 271, 5255, pp. 1582-1586.
106. Pila, A. W., Shaked, U. and de Souza, C. E. (1999). H-infinity filtering for continuous-time linear systems with delay, IEEE Transactions on Auto-matic Control 44, 7, pp. 1412-1417.
107. Postlethwaite, I. and Foo, Y. K. (1985). Robustness with simultaneous pole and zero movement across the j-omega-axis, Automatica 21, 4, pp. 433-443.
108. Pyragas, K. (1992). Continuous control of chaos by self-controlling feedback, Physics Letters A 170, 6, pp. 421-428.
109. Qiu, L., Bernhardsson, B., Rantzer, A., Davison, E. J., Young, P. M. and Doyle, J. C. (1995). A formula for computation of the real stability radius, Auto-matica 31, 6, pp. 879-890.
110. Radjavi, H. and Rosenthal, P. (2000). Simultaneous Triangularization (Springer, New York).
111. Reddy, D. V. R., Sen, A. and Johnston, G. L. (2000). Dynamics of a limit cycle oscillator under time delayed linear and nonlinear feedbacks, Physica D: Nonlinear Phenomena 144, 3, pp. 335-357.
112. Richard, J. P. (2003). Time-delay systems: an overview of some recent advances and open problems, Automatica 39, 10, pp. 1667-1694.
113. Rong, L. B., Feng, Z. L. and Perelson, A. S. (2007). Emergence of HIV-1 drug resistance during antiretroviral treatment, Bulletin of Mathematical Biology69, pp. 2027-2060.
114. Rosenwasser, Y. and Yusupov, R. (2000). Sensitivity of Automatic Control Sys-tems (CRC Press, Boca Raton, Fla.).
115. Salamon, D. (1980). Observers and duality between observation and state feedback for time-delay systems, IEEE Transactions on Automatic Control 25, 6, pp. 1187-1192.
116. Shafiei, Z. and Shenton, A. T. (1997). Frequency-domain design of PID controllers for stable and unstable systems with time delay, Automatica 33, 12, pp. 2223-2232.
117. Sharifi, F. J., Beheshti, M. T. H., Ganjefar, S. and Momeni, H. (2003). Behavior of Smith predictor in teleoperation systems with modeling and delay time errors, in Proceedings of 2003 IEEE Conference on Control Applications, Maui, HI, Dec. 2003, pp. 1176-1180.
118. Shinozaki, H. and Mori, T. (2006). Robust stability analysis of linear time-delay systems by Lambert W function: Some extreme point results, Automatica42, 10, pp. 1791-1799.
119. Silva, G. J. and Datta, A. B. S. P. (2001). Controller design via Pade approximation can lead to instability, in Proceedings of the 40th IEEE Conference on Decision and Control, Orlando, FL, Dec. 2001, pp. 4733-4737.
120. Sipahi, R. and Olgac, N. (2003a). Active vibration suppression with time delayed feedback, Journal of Vibration and Acoustics 125, 3, pp. 384-388.
121. Sipahi, R. and Olgac, N. (2003b). Degenerate cases in using the direct method, Journal of Dynamic Systems Measurement and Control 125, 2, pp. 194201.
122. Smith, O. (1957). Closer control of loops with dead time, Chemical Engineering Progress 53, pp. 217-219.
123. Stepan, G. (1989). Retarded Dynamical Systems: Stability and Characteristic Functions (Wiley, New York).
124. Stepan, G. and Insperger, T. (2006). Stability of time-periodic and delayed sys-tems-a route to act-and-wait control, Annual reviews in control 30, 2, pp. 159-168.
125. Stepan, G. and Moon, F. C. (1997). Delay differential equation models for machine tool chatter, in F. C. Moon (ed.), Dynamics and, Chaos in Manufacturing Processes (Wiley, New York), pp. 165-191.
126. Suh, B. S. and Yang, J. H. (2005). A tuning of PID regulators via LQR approach, Journal of Chemical Engineering of Japan 38, 5, pp. 344-356.
127. Thowsen, A. (1977). On pointwise degeneracy, controllability and minimal time control of linear dynamical systems with delays, International Journal of Control 25, 3, pp. 345-360.
128. Tlusty, J. (2000). Manufacturing Processes and Equipment (Prentice Hall, New Jersey).
129. Tobias, S. A. (1965). Machine-tool Vibration (Wiley, New York).
130. Trinh, H. (1999). Linear functional state observer for time-delay systems, International Journal of Control 72, 18, pp. 1642-1658.
131. Tsoi, A. C. and Gregson, M. J. (1978). Recent advances in the algebraic system theory of delay differential equations, in M. J. Gregson (ed.), Recent Theo-retical Developments in Control (Academic Press, New York), pp. 67-127.
132. Van Assche, V., Dambrine, M., Lafay, J. F. and Richard, J. P. (1999). Some problems arising in the implementation of distributed-delay control laws, in Proceedings of 38th IEEE Conference on Decision and Control, Phoenix, AZ, Dec. 1999, pp. 4668-4672.
133. Vandevenne, H. F. (1972). Controllability and stabilizability properties of delay systems, in Proceedings of the 1972 IEEE Conference on Decision Con-trol and 11th Symposium on Adaptive Processes, Phoenix, AZ, Dec. 1999, pp. 370-377.
134. Verriest, E. I. and Kailath, T. (1983). On generalized balanced realizations, IEEE Transcations on Automatic Control 28, 8, pp. 833-844.
135. Wang, Q., Lee, H. T. and Tan, K. K. (1995). Automatic tuning of finite spectrum assignment controllers for delay systems, Automatica 31, 3, pp. 477-482.
136. Wang, Q. G., Liu, M. and Hang, C. C. (2007). Approximate pole placement with dominance for continuous delay systems by PID controllers, Canadian Journal of Chemical Engineering 85, 4, pp. 549-557.
137. Wang, Z. H. and Hu, H. Y. (2007). Robust stability of time-delay systems with uncertain parameters, IUTAM Symposium on Dynamics and Control of Nonlinear Systems with Uncertainty 2, pp. 363-372.
138. Wang, Z. H. and Hu, H. Y. (2008). Calculation of the rightmost characteristic root of retarded time-delay systems via Lambert W function, Journal of Sound and Vibration 318, 4-5, pp. 757-767.
139. Weiss, L. (1967). On the controllability of delay-differential equations, SIAM Journal on Control and Optimization 5, 4, pp. 575-587.
140. Weiss, L. (1970). An algebraic criterion for controllability of linear systems with time delay, IEEE Transactions on Automatic Control 15, 4, pp. 443-444.
141. Wirkus, S. and Rand, R. (2002). The dynamics of two coupled van der pol oscil-lators with delay coupling, Nonlinear Dynamics 30, 3, pp. 205-221.
142. Wodarz, D. and Lloyd, A. L. (2004). Immune responses and the emergence of drug-resistant virus strains in vivo, Proceedings of the Royal Society of London Series B-Biological Sciences 271, pp. 1101-1109.
143. Wodarz, D. and Nowak, M. A. (2000). HIV therapy: Managing resistance, Pro-ceedings of the National Academy of Sciences of the United States of Amer-ica 97, 15, pp. 8193-8195.
144. Wu, H. L., Huang, Y. X., Acosta, P., Rosenkranz, S. L., Kuritzkes, D. R., Eron, J. J., Perelson, A. S. and Gerber, J. G. (2005). Modeling long-term HIV dynamics and antiretroviral response-effects of drug potency, pharma-cokinetics, adherence, and drug resistance, Journal of Acquired Immune Deficiency Syndromes 39, 3, pp. 272-283.
145. Xu, J. and Chung, K. W. (2003). Effects of time delayed position feedback on a van der pol-duffing oscillator, Physica D: Nonlinear Phenomena 180, 1-2, pp. 17-39.
146. Yi, S., Nelson, P. W. and Ulsoy, A. G. (2006a). Chatter stability analysis us-ing the matrix lambert function and bifurcation analysis, in Proceedings of the 2006 ASME International Conference on Manufacturing Science and Engineering, Ypsilanti, MI, Sept. 2006, MSEC 2006-21131.
147. Yi, S., Nelson, P. W. and Ulsoy, A. G. (2007a). Controllability and observability of systems of linear delay differential equations via the matrix Lambert W function, in Proceedings of2007 American Control Conference, New York, NY, July 2007, pp. 5631-5636.
148. Yi, S., Nelson, P. W. and Ulsoy, A. G. (2007b). Delay differential equations via the matrix Lambert W function and bifurcation analysis: Application to machine tool chatter, Mathematical Biosciences and, Engineering 4, 2, pp. 355-368.
149. Yi, S., Nelson, P. W. and Ulsoy, A. G. (2007c). Feedback control via eigenvalue assignment for time delayed systems using the Lambert W function, in Proceedings of the ASME 2007 IDETC, Las Vegas, NV, Sept. 2007, DETC 2007-35711.
150. Yi, S., Nelson, P. W. and Ulsoy, A. G. (2007d). Survey on analysis of time delayed systems via the Lambert W function, Dynamics of Continuous, Discrete and Impulsive Systems (Series A) 14, S2, pp. 296-301.
151. Yi, S., Nelson, P. W. and Ulsoy, A. G. (2008a). Controllability and observability of systems of linear delay differential equations via the matrix Lambert W function, IEEE Transactions on Automatic Control 53, 3, pp. 854-860.
152. Yi, S., Nelson, P. W. and Ulsoy, A. G. (2008b). Eigenvalues and sensitivity analysis for a model of HIV-1 pathogenesis with an intracellular delay, in Pro-ceedings of 2008 ASME Dynamic Systems and Control Conference, Ann Arbor, MI, Oct. 2008, DSCC2008-2408.
153. Yi, S., Nelson, P. W. and Ulsoy, A. G. (2008c). Robust control and time-domain specifications for systems for delay differential equations via eigenvalue as-signment, in Proceedings of2008 American Control Conference, Seattle, WA, June 2008, pp. 4298-4933.
154. Yi, S., Nelson, P. W. and Ulsoy, A. G. (2009). Design of observer-based feedback control for time-delay systems with application to automotive powertrain control, in Proceedings of 2009 ASME Dynamic Systems and Control Conference, Hollywood, CA, Oct. 2009, DSCC2009-2590.
155. Yi, S., Nelson, P. W. and Ulsoy, A. G. (2010a). Design of observer-based feedback control for time-delay systems with application to automotive powertrain control, Journal of Franklin Institute 347, 1, pp. 358-376.
156. Yi, S., Nelson, P. W. and Ulsoy, A. G. (2010b). Eigenvalue assignment via the Lambert W function for control for time-delay systems, Journal of Vibration and Control (published online, doi:10.1177/1077546309341102).
157. Yi, S., Nelson, P. W. and Ulsoy, A. G. (2010c). Robust control and time-domain specifications for systems for delay differential equations via eigenvalue as-signment, Journal of Dynamic Systems, Measurement and Control 132, 3, (7 pages, doi:10.1115/1.4001339).
158. Yi, S. and Ulsoy, A. G. (2006). Solution of a system of linear delay differential equations using the matrix Lambert function, in Proceedings of 2006 American Control Conference, Minneapolis, MN, 2006 June, pp. 2433-2438.
159. Yi, S., Ulsoy, A. G. and Nelson, P. W. (2006b). Solution of systems of linear delay differential equations via Laplace transformation, in Proceedings of 45th IEEE Conference on Decision and Control, San Diego, CA, DEC. 2006, pp. 2535-2540.
160. Yu, X. G. (2008). An LMI approach to robust Hœ filtering for uncertain systems with time-varying distributed delays, Journal of the Franklin Institute 345, pp. 877-890.
161. Zafer, N. (2007). Discussion: "Analysis of a system of linear delay differential equations", Journal of Dynamic Systems Measurement and Control 129, 1, pp. 121-122.
162. Zhong, Q. C. (2006). Robust Control of Time-Delay Systems (Springer, London).