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

Time-Delay Systems Delay Differential Equations Lambert W Function Stability Controllability Observability Feedback Control Eigenvalue Assignment State Observer Chatter in Machining Human Immunodeficiency Virus Diesel Engine Control Robust Control Time-Domain Specification

[1]  M. Malek-Zavarei,et al.  Time-Delay Systems: Analysis, Optimization and Applications , 1987 .

[2]  H T Banks,et al.  A parameter sensitivity methodology in the context of HIV delay equation models , 2005, Journal of mathematical biology.

[3]  Gaston H. Gonnet,et al.  On the LambertW function , 1996, Adv. Comput. Math..

[4]  Manuel De la Sen Parte.  On pole-placement controllers for linear time-delay systems with commensurate point delays , 2005 .

[5]  Sun Yi,et al.  Chatter Stability Analysis Using the Matrix Lambert Function and Bifurcation Analysis , 2006 .

[6]  Stephen Holford,et al.  The Use of Techniques for Model Reduction Designing IIR Filters with Linear Phase in the Passband , 1996 .

[7]  H. Koivo,et al.  An observer theory for time delay systems , 1976 .

[8]  Guangdi Hu,et al.  Real stability radii of linear time-invariant time-delay systems , 2003, Syst. Control. Lett..

[9]  S. Niculescu.  Delay Effects on Stability: A Robust Control Approach , 2001 .

[10]  M. Fofana.  Delay dynamical systems and applications to nonlinear machine-tool chatter , 2003 .

[11]  Sjoerd Verduyn Lunel.  Exponential type calculus for linear delay equations , 1989 .

[12]  Chi-Tsong Chen,et al.  Linear System Theory and Design , 1995 .

[13]  A. K. Choudhury.  Necessary and sufficient conditions of pointwise completeness of linear time-invariant delay-differential systems† , 1972 .

[14]  B. Moore.  Principal component analysis in linear systems: Controllability, observability, and model reduction , 1981 .

[15]  Ravi N. Banavar,et al.  Output Feedback Stabilization of a Mobile Robot , 2006 .

[16]  Sun Yi,et al.  Design of observer-based feedback control for time-delay systems with application to automotive powertrain control , 2010, J. Frankl. Inst..

[17]  R. Bellman,et al.  Differential-Difference Equations , 1967 .

[18]  A. Perelson,et al.  HIV-1 Dynamics in Vivo: Virion Clearance Rate, Infected Cell Life-Span, and Viral Generation Time , 1996, Science.

[19]  Denise Kirschner,et al.  A model for treatment strategy in the chemotherapy of AIDS , 1996 .

[20]  A. Olbrot,et al.  Finite spectrum assignment problem for systems with delays , 1979 .

[21]  David S. Goodsell,et al.  Modeling HIV , 1999, SIGGRAPH '99.

[22]  A. Perelson.  Modelling viral and immune system dynamics , 2002, Nature Reviews Immunology.

[23]  Chyi Hwang,et al.  A note on the use of the Lambert W function in the stability analysis of time-delay systems , 2005, Autom..

[24]  Edward J. Davison,et al.  A formula for computation of the real stability radius , 1995, Autom..

[25]  M. E. Deich,et al.  INVESTIGATION AND CALCULATION OF AXIAL-TURBINE STAGES , 1967 .

[26]  Anuradha M. Annaswamy,et al.  An adaptive Smith-controller for time-delay systems with relative degree n*≤2 , 2003, Syst. Control. Lett..

[27]  M. C. Pease,et al.  Methods of Matrix Algebra , 1965 .

[28]  Tamás Kalmár-Nagy,et al.  Subcritical Hopf Bifurcation in the Delay Equation Model for Machine Tool Vibrations , 2001 .

[29]  A. Buckalo.  Explicit conditions for controllability of linear systems with time lag , 1968 .

[30]  G. Stépán.  Retarded dynamical systems : stability and characteristic functions , 1989 .

[31]  C. Hwang,et al.  Use of the Lambert W function for time-domain analysis of feedback fractional delay systems , 2006 .

[32]  Nejat Olgac,et al.  “Delay Scheduling”: A New Concept for Stabilization in Multiple Delay Systems , 2005 .

[33]  Hulin Wu,et al.  Modeling HIV dynamics and antiviral response with consideration of time-varying drug exposures, adherence and phenotypic sensitivity. , 2003, Mathematical biosciences.

[34]  A. Mcguire,et al.  What is it to be a model? , 2000, HEPAC Health Economics in Prevention and Care.

[35]  Jiří Tlustý,et al.  Manufacturing processes and equipment , 1999 .

[36]  M A Nowak,et al.  Viral dynamics in vivo: limitations on estimates of intracellular delay and virus decay. , 1996, Proceedings of the National Academy of Sciences of the United States of America.

[37]  Robert M. May,et al.  Time delays are not necessarily destabilizing , 1975 .

[38]  H. Trinh.  Linear functional state observer for time-delay systems , 1999 .

[39]  A. T. Shenton,et al.  Frequency-domain design of pid controllers for stable and unstable systems with time delay , 1997, Autom..

[40]  Michel C. Delfour,et al.  Controllability, Observability and Optimal Feedback Control of Affine Hereditary Differential Systems , 1972 .

[41]  A. Galip Ulsoy,et al.  Analysis of a System of Linear Delay Differential Equations , 2003 .

[42]  Takehiro Mori,et al.  Robust stability analysis of linear time-delay systems by Lambert W function: Some extreme point results , 2006, Autom..

[43]  Kestutis Pyragas.  Continuous control of chaos by self-controlling feedback , 1992 .

[44]  Rifat Sipahi,et al.  A Unique Methodology for Chatter Stability Mapping in Simultaneous Machining , 2005 .

[45]  Sun Yi,et al.  Controllability and Observability of Systems of Linear Delay Differential Equations via the Matrix Lambert W Function , 2007, 2007 American Control Conference.

[46]  Stephen Wirkus,et al.  The Dynamics of Two Coupled van der Pol Oscillators with Delay Coupling , 2002 .

[47]  Sun Yi,et al.  Solution of a system of linear delay differential equations using the matrix Lambert function , 2006, 2006 American Control Conference.

[48]  Herman Vandevenne.  Controllability and stabilizability properties of delay systems , 1972, CDC 1972.

[49]  Ilya Kolmanovsky,et al.  Developments in Control of Time-Delay Systems for Automotive Powertrain Applications , 2009 .

[50]  H. Koivo,et al.  Modal characterizations of controllability and observability in time delay systems , 1976 .

[51]  Sun Yi,et al.  Solution of Systems of Linear Delay Differential Equations via Laplace Transformation , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[52]  Jian Chu,et al.  I-stability for linear continuous-time systems with multiple time delays , 2006, Autom..

[53]  Raymond A. DeCarlo,et al.  Combined Controller-Observer Design for Uncertain Time Delay Systems With Application to Engine Idle Speed Control , 2004 .

[54]  Dirk Roose,et al.  Limitations of a class of stabilization methods for delay systems , 2001, IEEE Trans. Autom. Control..

[55]  Tobias Damm,et al.  The Lambert W function and the spectrum of some multidimensional time-delay systems , 2007, Autom..

[56]  L. Weiss.  An algebraic criterion for controllability of linear systems with time delay , 1970 .

[57]  Taeyoung Lee,et al.  Stability of time-delay systems , 1981 .

[58]  Olga I. Kosmidou,et al.  Delay-dependent robust stability conditions and decay estimates for systems with input delays , 1998, Kybernetika.

[59]  E. B. Magrab,et al.  Improved Methods for the Prediction of Chatter in Turning, Part 3: A Generalized Linear Theory , 1990 .

[60]  A. K. Choudhury.  Algebraic and transfer-function criteria of fixed-time controllability of delay-differential systems. , 1972 .

[61]  A. Maccari.  Vibration Control for the Primary Resonance of a Cantilever Beam by a Time Delay State Feedback , 2003 .

[62]  A. Olbrot.  On controllability of linear systems with time delays in control , 1972 .

[63]  K. Moore,et al.  Analytical Stability Bound for a Class of Delayed Fractional-Order Dynamic Systems , 2001, Proceedings of the 40th IEEE Conference on Decision and Control (Cat. No.01CH37228).

[64]  A. Olbrot,et al.  Observability and related structural results for linear hereditary systems , 1981 .

[65]  T. Kailath,et al.  On generalized balanced realizations , 1980, 1980 19th IEEE Conference on Decision and Control including the Symposium on Adaptive Processes.

[66]  M. DE LA SEN,et al.  Pole placement controllers for linear time-delay systems with commensurate point delays , 2004 .

[67]  Harvey Thomas Banks,et al.  Projection series for retarded functional differential equations with applications to optimal control problems , 1975 .

[68]  Alan S. Perelson,et al.  Effect of drug efficacy and the eclipse phase of the viral life cycle on estimates of HIV viral dynamic parameters. , 2001 .

[69]  V. Van Assche,et al.  Some problems arising in the implementation of distributed-delay control laws , 1999, Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No.99CH36304).

[70]  Francis C. Moon,et al.  Dynamics and chaos in manufacturing processes , 1998 .

[71]  A. Galip Ulsoy,et al.  Closure to "Discussion of 'Analysis of a System of Linear Delay Differential Equations' " (2007, ASME J. Dyn. Syst., Meas., Control, 129, pp. 121-122) , 2007 .

[72]  Magdi S. Mahmoud,et al.  New results on delay-dependent control of time-delay systems , 2005, IEEE Transactions on Automatic Control.

[73]  A. Perelson,et al.  A model of HIV-1 pathogenesis that includes an intracellular delay. , 2000, Mathematical biosciences.

[74]  M. G. Frost.  Controllability, observability and the transfer function matrix for a delay-differential system , 1982 .

[75]  D M Bortz,et al.  Estimating kinetic parameters from HIV primary infection data through the eyes of three different mathematical models. , 2006, Mathematical biosciences.

[76]  Sun Yi,et al.  EIGENVALUES AND SENSITIVITY ANALYSIS FOR A MODEL OF HIV-1 PATHOGENESIS WITH AN INTRACELLULAR DELAY , 2008 .

[77]  M A Nowak,et al.  HIV therapy: managing resistance. , 2000, Proceedings of the National Academy of Sciences of the United States of America.

[78]  O Smith,et al.  CLOSER CONTROL OF LOOPS WITH DEAD TIME , 1957 .

[79]  A. Galip Ulsoy,et al.  Eigenvalue Assignment via the Lambert W Function for Control of Time-delay Systems , 2010 .

[80]  A. Perelson,et al.  Influence of delayed viral production on viral dynamics in HIV-1 infected patients. , 1998, Mathematical biosciences.

[81]  Colin H. Hansen,et al.  The response of a Duffing-van der Pol oscillator under delayed feedback control , 2006 .

[82]  Ian Postlethwaite,et al.  Robustness with simultaneous pole and zero movement across the jω-axis , 1985, Autom..

[83]  K. Cooke,et al.  Discrete delay, distributed delay and stability switches , 1982 .

[84]  S. Niculescu.  H∞ memoryless control with an α-stability constraint for time-delay systems: an LMI approach , 1998, IEEE Trans. Autom. Control..

[85]  B. Adams,et al.  HIV dynamics: Modeling, data analysis, and optimal treatment protocols , 2005 .

[86]  Wim Michiels,et al.  An eigenvalue based approach for the robust stabilization of linear time-delay systems , 2003 .

[87]  Martin Hosek,et al.  Active Vibration Control of Distributed Systems Using Delayed Resonator With Acceleration Feedback , 1997 .

[88]  Patrick W. Nelson,et al.  Applications of Sturm sequences to bifurcation analysis of delay differential equation models , 2004 .

[89]  Hulin Wu,et al.  Modeling Long-Term HIV Dynamics and Antiretroviral Response: Effects of Drug Potency, Pharmacokinetics, Adherence, and Drug Resistance , 2005, Journal of acquired immune deficiency syndromes.

[90]  Thomas Kailath,et al.  On generalized balanced realizations , 1980 .

[91]  Alan S. Perelson,et al.  Emergence of HIV-1 Drug Resistance During Antiretroviral Treatment , 2007, Bulletin of mathematical biology.

[92]  Henryk Górecki,et al.  Analysis and Synthesis of Time Delay Systems , 1989 .

[93]  E Schöll,et al.  Control of unstable steady states by time-delayed feedback methods. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[94]  Chang-Chieh Hang,et al.  Approximate pole placement with dominance for continuous delay systems by pid controllers , 2008 .

[95]  Wim Michiels,et al.  Continuous pole placement for delay equations , 2002, Autom..

[96]  Lawrence M. Wein,et al.  Management of Antiretroviral Therapy for HIV Infection: Analyzing When to Change Therapy , 2000 .

[97]  Jesus Leyva-Ramos,et al.  An asymptotic modal observer for linear autonomous time lag systems , 1995, IEEE Trans. Autom. Control..

[98]  Rifat Sipahi,et al.  An exact method for the stability analysis of time-delayed linear time-invariant (LTI) systems , 2002, IEEE Trans. Autom. Control..

[99]  M. Hautus.  Controllability and stabilizability of sampled systems , 1972 .

[100]  V. Shaygannejad.  In response , 2009, Journal of research in medical sciences : the official journal of Isfahan University of Medical Sciences.

[101]  Chang-Hua Lien,et al.  Global stabilizability for a class of uncertain systems with multiple time-varying delays via linear control , 1999 .

[102]  Attilio Maccari.  Vibration control for the primary resonance of the van der Pol oscillator by a time delay state feedback , 2003 .

[103]  A. Galip Ulsoy,et al.  Delay differential equations via the matrix Lambert W function and bifurcation analysis: application to machine tool chatter. , 2007, Mathematical biosciences and engineering : MBE.

[104]  S. A. Tobias.  Machine-tool vibration , 1965 .

[105]  M. Mahmoud.  Robust Control and Filtering for Time-Delay Systems , 2000 .

[106]  E. Rosenwasser,et al.  Sensitivity of Automatic Control Systems , 1999 .

[107]  Dominik Wodarz,et al.  Immune responses and the emergence of drug–resistant virus strains in vivo , 2004, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[108]  Pin-Lin Liu,et al.  Exponential stability for linear time-delay systems with delay dependence , 2003, J. Frankl. Inst..

[109]  Xin-Gang Yu,et al.  An LMI approach to robust H∞ filtering for uncertain systems with time-varying distributed delays , 2008, J. Frankl. Inst..

[110]  Jack K. Hale,et al.  Introduction to Functional Differential Equations , 1993, Applied Mathematical Sciences.

[111]  Matematik.  Necessary and Sufficient Condition , 2010 .

[112]  Gene F. Franklin,et al.  Feedback Control of Dynamic Systems , 1986 .

[113]  S. Niculescu,et al.  Stability analysis of time-delay systems : A lyapunov approach , 2006 .

[114]  Sun Yi,et al.  Robust control and time-domain specifications for systems of delay differential equations via eigenvalue assignment , 2008, 2008 American Control Conference.

[115]  A. Perelson,et al.  Complex patterns of viral load decay under antiretroviral therapy: influence of pharmacokinetics and intracellular delay. , 2004, Journal of theoretical biology.

[116]  Takehiro Mori,et al.  Feedback enlargement of stability radius by nondifferentiable optimization , 2009 .

[117]  Carlos E. de Souza,et al.  Output feedback stabilization of linear time-delay systems , 1997, 1997 European Control Conference (ECC).

[118]  Richard Bellman,et al.  Differential-Difference Equations , 1967 .

[119]  Uri Shaked,et al.  𝒽∞ Filtering for Continuous-time Linear Systems with Delay , 1999, IEEE Trans. Autom. Control..

[120]  A. Galip Ulsoy,et al.  Computational stability analysis of chatter in turning , 1997 .

[121]  Alan S. Perelson,et al.  Mathematical Analysis of HIV-1 Dynamics in Vivo , 1999, SIAM Rev..

[122]  Gilbert G. Walter.  Delay-Differential Equation Models for Fisheries , 1973 .

[123]  Hugues Mounier,et al.  Algebraic interpretations of the spectral controllability of a linear delay system , 1998 .

[124]  Kwok-wai Chung,et al.  Effects of time delayed position feedback on a van der Pol–Duffing oscillator , 2003 .

[125]  P. D. Roberts,et al.  Recent Theoretical Developments in Control , 1979 .

[126]  Ji Hoon Yang,et al.  A Tuning of PID Regulators via LQR Approach , 2005 .

[127]  D. Salamon.  Observers and duality between observation and state feedback for time delay systems , 1980 .

[128]  D. V. Reddy,et al.  Dynamics of a limit cycle oscillator under time delayed linear and nonlinear feedbacks , 1999, chao-dyn/9908005.

[129]  D. M. Bortz,et al.  Sensitivity analysis of a nonlinear lumped parameter model of HIV infection dynamics , 2004, Bulletin of mathematical biology.

[130]  Erik I. Verriest,et al.  Stability and Control of Time-delay Systems , 1998 .

[131]  Tamás Kalmár-Nagy,et al.  Delay differential equations : recent advances and new directions , 2009 .

[132]  D. Kirschner,et al.  A model for treatment strategy in the chemotherapy of AIDS. , 1996, Bulletin of mathematical biology.

[133]  Mrdjan J. Jankovic,et al.  Control design for a diesel engine model with time delay , 2001, Proceedings of the 40th IEEE Conference on Decision and Control (Cat. No.01CH37228).

[134]  Gábor Stépán,et al.  Stability of time-periodic and delayed systems - a route to act-and-wait control , 2006, Annu. Rev. Control..

[135]  Nejat Olgac,et al.  Degenerate Cases in Using the Direct Method , 2003 .

[136]  B. Adams,et al.  Dynamic multidrug therapies for hiv: optimal and sti control approaches. , 2004, Mathematical biosciences and engineering : MBE.

[137]  Z. Wang,et al.  Calculation of the rightmost characteristic root of retarded time-delay systems via Lambert W function , 2008 .

[138]  A. Thowsen.  On pointwise degeneracy, controllability and minimal time control of linear dynamical systems with delays , 1977 .

[139]  Nejat Olgac,et al.  Active Vibration Suppression With Time Delayed Feedback , 2003 .

[140]  Sun Yi,et al.  FEEDBACK CONTROL VIA EIGENVALUE ASSIGNMENT FOR TIME DELAYED SYSTEMS USING THE LAMBERT W FUNCTION , 2007 .

[141]  Qing-Chang Zhong,et al.  Robust Control of Time-delay Systems , 2006 .

[142]  A. Maccari,et al.  The Response of a Parametrically Excited van der Pol Oscillator to a Time Delay State Feedback , 2001 .

[143]  S. Ganjefar,et al.  Behavior of Smith predictor in teleoperation systems with modeling and delay time errors , 2003, Proceedings of 2003 IEEE Conference on Control Applications, 2003. CCA 2003..

[144]  M A Nowak,et al.  Anti-viral drug treatment: dynamics of resistance in free virus and infected cell populations. , 1997, Journal of theoretical biology.

[145]  Ibrahim Kaya,et al.  IMC based automatic tuning method for PID controllers in a Smith predictor configuration , 2004, Comput. Chem. Eng..

[146]  Geoffrey Dusheiko,et al.  The role of cells refractory to productive infection in acute hepatitis B viral dynamics , 2007, Proceedings of the National Academy of Sciences.

[147]  Mrdjan Jankovic,et al.  Constructive Lyapunov control design for turbocharged diesel engines , 2000, IEEE Trans. Control. Syst. Technol..

[148]  Kok Kiong Tan,et al.  Automatic tuning of finite spectrum assignment controllers for delay systems , 1995, Autom..

[149]  Shankar P. Bhattacharyya,et al.  Controller design via Pade approximation can lead to instability , 2001, Proceedings of the 40th IEEE Conference on Decision and Control (Cat. No.01CH37228).

[150]  Patrick W Nelson,et al.  Mathematical analysis of delay differential equation models of HIV-1 infection. , 2002, Mathematical biosciences.

[151]  J. Loiseau,et al.  An effective algorithm for finite spectrum assignment of single-input systems with delays , 1998 .

[152]  Mohamed Darouach,et al.  Linear functional observers for systems with delays in state variables , 2001, IEEE Trans. Autom. Control..

[153]  Lidia Ruiz,et al.  Alternation of Antiretroviral Drug Regimens for HIV Infection , 2003, Annals of Internal Medicine.

[154]  D M Bortz,et al.  Model Selection and Mixed-Effects Modeling of HIV Infection Dynamics , 2006, Bulletin of mathematical biology.

[155]  Valery Y. Glizer.  Observability of Singularly Perturbed Linear Time-Dependent Differential Systems with Small Delay , 2004 .

[156]  A. Pearson,et al.  An observer for time lag systems , 1989 .

[157]  Jean-Pierre Richard,et al.  Time-delay systems: an overview of some recent advances and open problems , 2003, Autom..

[158]  Z. Wang,et al.  Robust Stability of Time-Delay Systems with Uncertain Parameters , 2007 .

[159]  Simeone Marino,et al.  The role of delays in innate and adaptive immunity to intracellular bacterial infection. , 2007, Mathematical biosciences and engineering : MBE.

[160]  Kevin L. Moore,et al.  Analytical stability bound for delayed second-order systems with repeating poles using Lambert function W , 2002, Autom..

[161]  H. E. Merritt.  Theory of Self-Excited Machine-Tool Chatter: Contribution to Machine-Tool Chatter Research—1 , 1965 .

[162]  Richard Marquez,et al.  An extension of predictive control, PID regulators and Smith predictors to some linear delay systems , 2002 .