Skip to main content
SpringerLink
Log in

Basics and Preliminaries of Time-Delay Systems

  • Chapter
  • First Online:
Dynamic Systems with Time Delays: Stability and Control

  • 974 Accesses

Abstract

In this chapter, we review the basic characteristics and properties of dynamic systems with time delays, and revisit remarkable research achievements made by many researchers during the last decades. In particular, we explore remarkable mathematical equations, inequalities, and lemmas which are widely used in system analysis to deal with state variables including time delays that are essentially encountered when the stability and control problems of dynamic systems and networks with time delays have been considered. We will also look at the results of Lyapunov function selection for the stability analysis of dynamic systems because the Lyapunov stability analysis method is almost the only alternative for solving the stability and control problems. Finally, we end this chapter by introducing the main issues related to stabilization of the time delay system.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 79.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 99.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 129.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Hale JK (1977) Theory of functional differential equations. Springer, New York

    Book  MATH  Google Scholar 

  2. Hale JK, Verduyn Lunel SM (1993) Introduction to functional differential equations. Springer, New York

    Book  MATH  Google Scholar 

  3. Kolmanovskii VB, Nosov VR (1986) Stability of functional differential equations. Academic, London

    Google Scholar 

  4. Kolmanovskii V, Myshkis A (1992) Applied theory of functional differential equations. Kluwer Academic Publishers, Dordrecht

    Book  MATH  Google Scholar 

  5. Kharitonov VL (2013) Time-delay systems: Lyapunov functionals and matrices. Birkhauser, Basel

    Book  MATH  Google Scholar 

  6. Gu K, Niculescu SI (2003) Survey on recent results in the stability and control of time-delay systems. J Dyn Syst Meas Control 125:158–165

    Article  Google Scholar 

  7. Kharitonov V (1998) Robust stability analysis of time delay systems: a survey. In: Proceedings of fourth IFAC conference on system structure and control, pp 1–12

    Article  Google Scholar 

  8. Kolmanovskii V, Niculescu SI, Gu K (1999) Delay effects on stability: a survey. In: Proceeding of 38th IEEE conference on decision and control, pp 1993–1998

    Google Scholar 

  9. Richard JP (2003) Time-delay systems: an overview of some recent advances and open problems. Automatica 39:1667–1694

    Article  MathSciNet  MATH  Google Scholar 

  10. Fridman E, Shaked U (2003) Delay systems. Int J Robust Nonlinear Control 13:791–937

    Article  MATH  Google Scholar 

  11. Niculescu SI, Richard JP (2002) Analysis and design of delay and propagation systems. IMA J Math Control Inf 19:1–227

    Article  MathSciNet  Google Scholar 

  12. Niculescu SI (2001) Delay effects on stability. Lecture notes in control and information sciences, vol 269

    Google Scholar 

  13. Niculescu SI, Verriest EI, Dugard L, Dion JM (1997) Stability and robust stability of time-delay systems: a guided tour. Lecture notes in control and information sciences, vol 228

    Google Scholar 

  14. Gorelik G (1939) To the theory of feedback with delay. J Tech Phys 9:450–454

    Google Scholar 

  15. Hutchinson GE (1948) Circular causal systems in ecology. Ann N Y Acad Sci 50:221–246

    Article  Google Scholar 

  16. Malek-Zaverei M, Jamshidi M (1987) Time-delay systems: analysis, optimization and applications. North-Holland, Amsterdam

    MATH  Google Scholar 

  17. Fridman E, Shaked U (2005) Stability and guaranteed cost control of uncertain discrete delay systems. Int J Control 78:235–246

    Article  MathSciNet  MATH  Google Scholar 

  18. Gao H, Chen T (2007) New results on stability of discrete-time systems with time-varying state delay. IEEE Trans Autom Control 52:328–334

    Article  MathSciNet  MATH  Google Scholar 

  19. Zhang B, Xu S, Zou Y (2008) Improved stability criterion and its applications in delayed controller design for discrete-time systems. Automatica 44:2963–2967

    Article  MathSciNet  MATH  Google Scholar 

  20. He Y, Wu M, Liu GP, She JH (2008) Output feedback stabilization for a discrete-time system with a time-varying delay. IEEE Trans Autom Control 53:2372–2377

    Article  MathSciNet  MATH  Google Scholar 

  21. Yue D, Tian E, Zhang Y (2009) A piecewise analysis method to stability analysis of linear continuous/discrete systems with time-varying delay. Int J Robust Nonlinear Control 19:1493–1518

    Article  MathSciNet  MATH  Google Scholar 

  22. Meng X, Lam J, Du B, Gao H (2010) A delay-partitioning approach to the stability analysis of discrete-time systems. Automatica 46:610–614

    Article  MathSciNet  MATH  Google Scholar 

  23. Shao H, Han QL (2011) New stability criteria for linear discrete-time systems with interval-like time-varying delays. IEEE Trans Autom Control 56:619–625

    Article  MathSciNet  MATH  Google Scholar 

  24. Ramakrishnan K, Ray G (2013) Robust stability criteria for a class of uncertain discrete-time systems with time-varying delay. Appl Math Model 37:1468–1479

    Article  MathSciNet  MATH  Google Scholar 

  25. Peng C (2012) Improved delay-dependent stabilisation criteria for discrete systems with a new finite sum inequality. IET Control Theory Appl 6:448–453

    Article  MathSciNet  Google Scholar 

  26. Lee SM, Kwon OM, Park JH (2012) Regional asymptotic stability analysis for discrete-time delayed systems with saturation nonlinearity. Nonlinear Dyn 67:885–892

    Article  MathSciNet  MATH  Google Scholar 

  27. Kwon OM, Park MJ, Park JH, Lee SM, Cha EJ (2015) Improved delay-partitioning approach to robust stability analysis for discrete-time systems with time-varying delays and randomly occurring parameter uncertainties. Optim Control Appl Methods 36:496–511

    Article  MathSciNet  MATH  Google Scholar 

  28. Seuret A, Gouaisbaut F, Fridman E (2015) Stability of discrete-time systems with time-varying delays via a novel summation inequality. IEEE Trans Autom Control 60:2740–2745

    Article  MathSciNet  MATH  Google Scholar 

  29. Castanos F, Estrada E, Mondie S, Ramirez A (2015) Passivity-based PI control of first-order systems with I/O communication delays: a complete sigma-stability analysis. arXiv:150701146

  30. Lakshmanan S, Senthilkumar T, Balasubramaniam P (2011) Improved results on robust stability of neutral systems with mixed time-varying delays and nonlinear perturbations. Appl Math Model 35:5355–5368

    Article  MathSciNet  MATH  Google Scholar 

  31. Mohajerpoor R, Lakshmanan S, Abdi H, Rakkiyappan R, Nahavandi S, Park JH (2017) Improved delay-dependent stability criteria for neutral systems with mixed interval time-varying delays and nonlinear disturbances. J Frankl Inst 354:1169–1194

    Article  MathSciNet  MATH  Google Scholar 

  32. Li LM (1988) Stability of linear neutral delay-differential systems. Bull Aust Math Soc 38:339–344

    Article  MathSciNet  MATH  Google Scholar 

  33. Hu GD, Hu GD (1996) Some simple stability criteria of neutral delay-differential systems. Appl Math Comput 80:257–271

    MathSciNet  MATH  Google Scholar 

  34. Verriest EI, Niculescu SI (1997) Delay-independent stability of linear neutral systems: a Riccati equation approach. In: Proceedings of 1997 European control conference, pp 3632–3636

    Google Scholar 

  35. Park JH, Won S (1999) A note on stability of neutral delay-differential systems. J Frankl Inst 336:543–548

    Article  MathSciNet  MATH  Google Scholar 

  36. Chen J (1995) On computing the maximal delay intervals for stability of linear delay systems. IEEE Trans Autom Control 40:1087–1093

    Article  MathSciNet  MATH  Google Scholar 

  37. Hui GD, Hu GD (1997) Simple criteria for stability of neutral systems with multiple delays. Int J Syst Sci 28:1325–1328

    Article  MATH  Google Scholar 

  38. Park JH, Won S (1999) Asymptotic stability of neutral systems with multiple delays. J Optim Theory Appl 103:183–200

    Article  MathSciNet  MATH  Google Scholar 

  39. Yue D, Won S, Kwon OM (2003) Delay dependent stability of neutral systems with time delay: an LMI approach. IEE Proc Control Theory Appl 150:23–27

    Article  Google Scholar 

  40. Park JH, Won S (2000) Stability analysis for neutral delay-differential systems. J Frankl Inst 337:1–9

    Article  MathSciNet  MATH  Google Scholar 

  41. Park JH, Won S (2001) A note on stability analysis of neutral systems with multiple time-delays. Int J Syst Sci 32:409–412

    Article  MathSciNet  MATH  Google Scholar 

  42. Fridman E (2001) New Lyapunov-Krasovskii functionals for stability of linear retarded and neutral type systems. Syst Control Lett 43:309–319

    Article  MathSciNet  MATH  Google Scholar 

  43. Chen JD, Lien CH, Fan KK (2001) Criteria for asymptotic stability of a class of neutral systems via a LMI approach. IEE Proc Control Theory Appl 148:442–447

    Article  Google Scholar 

  44. Park JH (2001) A new delay-dependent criterion for neutral systems with multiple delays. J Comput Appl Math 136:177–184

    Article  MathSciNet  MATH  Google Scholar 

  45. Park JH (2002) Stability criterion for neutral differential systems with mixed multiple time-varying delay arguments. Math Comput Simul 59:401–412

    Article  MathSciNet  MATH  Google Scholar 

  46. Han QL (2002) Robust stability of uncertain delay-differential systems of neutral type. Automatica 38:719–723

    Article  MathSciNet  MATH  Google Scholar 

  47. Park JH (2003) Robust guaranteed cost control for uncertain linear differential systems of neutral type. Appl Math Comput 140:523–535

    MathSciNet  MATH  Google Scholar 

  48. Han QL (2004) On robust stability of neutral systems with time-varying discrete delay and norm-bounded uncertainty. Automatica 40:1087–1092

    Article  MathSciNet  MATH  Google Scholar 

  49. He Y, Wu M, She JH, Liu GP (2004) Delay-dependent robust stability criteria for uncertain neutral systems with mixed delays. Syst Control Lett 51:57–65

    Article  MathSciNet  MATH  Google Scholar 

  50. Wu M, He Y, She JH (2004) New delay-dependent stability criteria and stabilizing method for neutral systems. IEEE Trans Autom Control 49:2266–2271

    Article  MathSciNet  MATH  Google Scholar 

  51. Park JH (2005) LMI optimization approach to asymptotic stability of certain neutral delay differential equation. Appl Math Comput 160:355–361

    MathSciNet  MATH  Google Scholar 

  52. Xu S, Lam J, Zou Y (2005) Further results on delay-dependent robust stability conditions of uncertain neutral systems. Int J Robust Nonlinear Control 15:233–246

    Article  MathSciNet  MATH  Google Scholar 

  53. Kharitonov V, Mondie S, Collado J (2005) Exponential estimates for neutral time-delay systems: an LMI approach. IEEE Trans Autom Control 50:666–670

    Article  MathSciNet  MATH  Google Scholar 

  54. Lien CH (2005) Delay-dependent stability criteria for uncertain neutral systems with multiple time-varying delays via LMI approach. IEE Proc Control Theory Appl 152:707–714

    Article  Google Scholar 

  55. Liu XG, Wu M, Martin R, Tang ML (2007) Delay-dependent stability analysis for uncertain neutral systems with time-varying delays. Math Comput Simul 75:15–27

    Article  MathSciNet  MATH  Google Scholar 

  56. Chen WH, Zheng WX (2007) Delay-dependent robust stabilization for uncertain neutral systems with distributed delays. Automatica 43:95–104

    Article  MathSciNet  MATH  Google Scholar 

  57. Zhang J, Shi P, Qiu J (2008) Robust stability criteria for uncertain neutral system with time delay and nonlinear uncertainties. Chaos Solitons Fractals 38:160–167

    Article  MathSciNet  MATH  Google Scholar 

  58. Sun J, Liu GP, Chen J (2009) Delay-dependent stability and stabilization of neutral time-delay systems. Int J Robust Nonlinear Control 19:1364–1375

    Article  MathSciNet  MATH  Google Scholar 

  59. Rakkiyappan R, Balasubramaniam P (2008) LMI conditions for global asymptotic stability results for neutral-type neural networks with distributed time delays. Appl Math Comput 204:317–324

    MathSciNet  MATH  Google Scholar 

  60. Kwon OM, Park JH, Lee SM (2008) On stability criteria for uncertain delay-differential systems of neutral type with time-varying delays. Appl Math Comput 197:864–873

    MathSciNet  MATH  Google Scholar 

  61. Kwon OM, Park JH (2008) On improved delay-dependent stability criterion of certain neutral differential equations. Appl Math Comput 199:385–391

    MathSciNet  MATH  Google Scholar 

  62. Li M, Liu L (2009) A delay-dependent stability criterion for linear neutral delay systems. J Frankl Inst 346:33–37

    Article  MathSciNet  MATH  Google Scholar 

  63. Kwon OM, Park JH, Lee SM (2009) Augmented Lyapunov functional approach to stability of uncertain neutral systems with time-varying delays. Appl Math Comput 207:202–212

    MathSciNet  MATH  Google Scholar 

  64. Wu M, He Y, She JH (2010) Stability analysis and robust control of time-delay systems. Springer, Beijing

    Book  MATH  Google Scholar 

  65. Han QL (2009) A discrete delay decomposition approach to stability of linear retarded and neutral systems. Automatica 45:517–524

    Article  MathSciNet  MATH  Google Scholar 

  66. Wu M, He Y, She JH, Liu GP (2004) Delay-dependent stability criteria for robust stability of time-varying delay systems. Automatica 40:1435–1439

    Article  MathSciNet  MATH  Google Scholar 

  67. Fridman E, Shaked U (2003) Delay-dependent stability and \(\mathscr {H}_\infty \) control: constant and time-varying delays. Int J Control 76:48–60

    Article  MathSciNet  MATH  Google Scholar 

  68. Gu K, Niculescu SI (2000) Additional dynamics in transformed time delay systems. IEEE Trans Autom Control 45:572–575

    Article  MathSciNet  MATH  Google Scholar 

  69. Gu K, Niculescu SI (2001) Further remarks on additional dynamics in various model transformations of linear delay systems. IEEE Trans Autom Control 46:497–500

    Article  MathSciNet  MATH  Google Scholar 

  70. Park PG (1999) A delay-dependent stability criterion for systems with uncertain linear state-delayed systems. IEEE Trans Autom Control 35:876–877

    Article  MATH  Google Scholar 

  71. Moon YS, Park PG, Kwon WH, Lee YS (2001) Delay-dependent robust stabilization of uncertain state-delayed systems. Int J Control 74:1447–1455

    Article  MathSciNet  MATH  Google Scholar 

  72. Fridman E, Shaked U (2002) An improved stabilization method for linear time-delay systems. IEEE Trans Autom Control 47:1931–1937

    Article  MathSciNet  MATH  Google Scholar 

  73. Gu K (1999) Discretized Lyapunov functional for uncertain systems with multiple time-delay. Int J Control 72:1436–1445

    Article  MathSciNet  MATH  Google Scholar 

  74. Kwon OM, Park JH, Lee SM (2010) An improved delay-dependent criterion for asymptotic stability of uncertain dynamic systems with time-varying delays. J Optim Theory Appl 145:343–353

    Article  MathSciNet  MATH  Google Scholar 

  75. Kwon OM, Park MJ, Lee SM, Park JH, Cha EJ (2012) New delay-partitioning approaches to stability criteria for uncertain neutral systems with time-varying delays. J Frankl Inst 349:2799–2823

    Article  MathSciNet  MATH  Google Scholar 

  76. Ko JW, Park PG (2011) Delay-dependent stability criteria for systems with asymmetric bounds on delay derivative. J Frankl Inst 348:2674–2688

    Article  MathSciNet  MATH  Google Scholar 

  77. He Y, Wang QG, Lin C, Wu M (2005) Augmented Lyapunov functional and delay-dependent stability criteria for neutral systems. Int J Robust Nonlinear Control 15:923–933

    Article  MathSciNet  MATH  Google Scholar 

  78. Park MJ, Kwon OM, Park JH, Lee SM (2011) A new augmented Lyapunov-Krasovskii functional approach for stability of linear systems with time-varying delays. Appl Math Comput 217:7197–7209

    MathSciNet  MATH  Google Scholar 

  79. Kwon OM, Park MJ, Park JH, Lee SM, Cha EJ (2013) Analysis on robust \(\mathscr {H}_\infty \) performance and stability for linear systems with interval time-varying state delays via some new augmented Lyapunov-Krasovskii functional. Appl Math Comput 224:108–122

    MathSciNet  MATH  Google Scholar 

  80. Park JH, Kwon OM (2005) Novel stability criterion of time delay systems with nonlinear uncertainties. Appl Math Lett 18:683–688

    Article  MathSciNet  MATH  Google Scholar 

  81. Parlakci MNA (2006) Robust stability of uncertain time-varying state-delayed systems. IEE Proc Control Theory Appl 153:469–477

    Article  MathSciNet  Google Scholar 

  82. Suplin V, Fridman E, Shaked U (2006) \(\mathscr {H}_\infty \) control of linear uncertain time-delay systems-a projection approach. IEEE Trans Autom Control 51:680–685

    Article  MathSciNet  MATH  Google Scholar 

  83. Ariba Y, Gouaisbaut F (2007) Delay-dependent stability analysis of linear systems with time-varying delay. In: Proceedings of the 46th IEEE conference on decision and control, pp 2053–2058

    Google Scholar 

  84. Kwon OM, Lee SM, Park JH (2011) On the reachable set bounding of uncertain dynamic systems with time-varying delays and disturbances. Inf Sci 181:3735–3748

    Article  MathSciNet  MATH  Google Scholar 

  85. Sun J, Liu GP, Chen J, Rees D (2010) Improved delay-range-dependent stability criteria for linear systems with time-varying delays. Automatica 46:466–470

    Article  MathSciNet  MATH  Google Scholar 

  86. Sakthivel R, Mathiyalagan K, Marshal Anthoni S (2012) Robust stability and control for uncertain neutral time delay systems. Int J Control 85:373–383

    Article  MathSciNet  MATH  Google Scholar 

  87. Lee WI, Lee SY, Park PG (2014) Improved criteria on robust stability and \(\mathscr {H}_\infty \) performance for linear systems with interval time-varying delays via new triple integral functionals. Appl Math Comput 243:570–577

    MathSciNet  MATH  Google Scholar 

  88. Kim JH (2011) Note on stability of linear systems with time-varying delay. Automatica 47:2118–2121

    Article  MathSciNet  MATH  Google Scholar 

  89. Fang M, Park JH (2013) A multiple integral approach to stability of neutral time-delay systems. Appl Math Comput 224:714–718

    MathSciNet  MATH  Google Scholar 

  90. Gu K, Kharitonov VK, Chen J (2003) Stability of time-delay systems. Birkhauser, Boston

    Book  MATH  Google Scholar 

  91. Liu K, Suplin V, Fridman E (2011) Stability of linear systems with general sawtooth delay. IMA J Math Control Inf 27:419–436

    Article  MathSciNet  MATH  Google Scholar 

  92. Seuret A, Gouaisbaut F (2013) Wirtinger-based integral inequality: application to time-delay systems. Automatica 49:2860–2866

    Article  MathSciNet  MATH  Google Scholar 

  93. Zheng M, Li K, Fei M (2014) Comments on “Wirtinger-based integral inequality: application to time-delay systems [Automatica 49 (2013) 2860–2866]”. Automatica 50:300–301

    Article  MathSciNet  Google Scholar 

  94. Park MJ, Kwon OM, Park JH, Lee SM, Cha EJ (2015) Stability of time-delay systems via Wirtinger-based double integral inequality. Automatica 55:204–208

    Article  MathSciNet  MATH  Google Scholar 

  95. Mohajerpoor R, Lakshmanan S, Abdi H, Rakkiappan R, Nahavandi S, Shi P (2018) New delay-range-dependent stability criteria for interval-time-varying delay systems via Wirtinger-based inequalities. Int J Robust Nonlinear Control 28:661–677

    Article  MathSciNet  MATH  Google Scholar 

  96. Skelton RE, Iwasaki T, Grigoradis KM (1997) A unified algebraic approach to linear control design. Taylor & Francis

    Google Scholar 

  97. Zeng HB, He Y, Wu M, She J (2015) Free-matrix-based integral inequality for stability analysis of systems with time-varying delay. IEEE Trans Autom Control 60:2768–2772

    Article  MathSciNet  MATH  Google Scholar 

  98. Lee TH, Park JH, Xu S (2017) Relaxed conditions for stability of time-varying delay systems. Automatica 75:11–15

    Article  MathSciNet  MATH  Google Scholar 

  99. Seuret A, Gouaisbaut F (2015) Hierarchy of LMI conditions for the stability analysis of time-delay systems. Syst Control Lett 81:1–7

    Article  MathSciNet  MATH  Google Scholar 

  100. Park PG, Lee WI, Lee SY (2015) Auxiliary function-based integral inequalities for quadratic functions and their applications to time-delay systems. J Frankl Inst 352:1378–1396

    Article  MathSciNet  MATH  Google Scholar 

  101. Chen J, Xu S, Chen W, Zhang B, Ma Q, Zou Y (2016) Two general integral inequalities and their applications to stability analysis for systems with time-varying delay. Int J Robust Nonlinear Control 26:4088–4103

    Article  MathSciNet  MATH  Google Scholar 

  102. Zhang XM, Han QL, Seuret A, Gouaisbaut F (2017) An improved reciprocally convex inequality and an augmented Lyapunov-Krasovskii functional for stability of linear systems with time-varying delay. Automatica 84:221–226

    Article  MathSciNet  MATH  Google Scholar 

  103. Seuret A, Gouaisbaut F (2016) Delay-dependent reciprocally convex combination lemma. Rapport LAAS n16006

    Google Scholar 

  104. Park PG, Ko J, Jeong J (2011) Reciprocally convex approach to stability of systems with time-varying delays. Automatica 47:235–238

    Article  MathSciNet  MATH  Google Scholar 

  105. Kim JH (2016) Further improvement of Jensen inequality and application to stability of time-delayed systems. Automatica 64:121–125

    Article  MathSciNet  MATH  Google Scholar 

  106. Shao H (2009) New delay-dependent stability criteria for systems with interval delay. Automatica 45:744–749

    Article  MathSciNet  MATH  Google Scholar 

  107. Park PG, Ko JW, Jeong CK (2011) Reciprocally convex approach to stability of systems with time-varying delays. Automatica 47:235–238

    Article  MathSciNet  MATH  Google Scholar 

  108. Lee WI, Park PG (2014) Second-order reciprocally convex approach to stability of systems with interval time-varying delays. Appl Math Comput 229:245–253

    MathSciNet  MATH  Google Scholar 

  109. Lee TH, Park JH (2017) A novel Lyapunov functional for stability of time-varying delay systems via matrix-refined-function. Automatica 80:239–242

    Article  MathSciNet  MATH  Google Scholar 

  110. Kwon OM, Park MJ, Park JH, Lee SM, Cha EJ (2014) Improved results on stability of linear systems with time-varying delays via Wirtinger-based integral inequality. J Frankl Inst 351:5386–5398

    Article  MathSciNet  MATH  Google Scholar 

  111. Chen J, Park JH, Xu S (2018) Stability analysis of continuous-time systems with time-varying delay using new Lyapunov-Krasovskii functionals. J Frankl Inst 355:5957–5967

    Article  MathSciNet  MATH  Google Scholar 

  112. Smith OJM (1957) Posicast control of damped oscillatory systems. Proc IRE 45:1249–1255

    Article  Google Scholar 

  113. Zhong QC (2006) Robust control of time-delay systems. Springer, London

    MATH  Google Scholar 

  114. Leva A, Maffezzoni C, Scattolini R (1994) Self-tuning PI-PID regulators for stable systems with varying delay. Automatica 30:1171–1183

    Article  MathSciNet  MATH  Google Scholar 

  115. Shafiei Z, Shenton AT (1994) Tuning of PID-type controllers for stable and unstable systems with time delay. Automatica 30:1609–1615

    Article  MathSciNet  MATH  Google Scholar 

  116. Poulin E, Pomerleau A (1996) PID tuning for integrating and unstable processes. IEE Proc Control Theory Appl 143:429–435

    Article  MATH  Google Scholar 

  117. Astrom KJ, Hagglund T (2001) The future of PID control. Control Eng Pract 9:1163–1175

    Article  Google Scholar 

  118. Ge M, Chiu MS, Wang QG (2002) Robust PID controller design via LMI approach. J Process Control 12:3–13

    Article  Google Scholar 

  119. Silva GJ, Datta A, Bhattacharyya SP (2002) New results on the synthesis of PID controllers. IEEE Trans Autom Control 47:241–252

    Article  MathSciNet  MATH  Google Scholar 

  120. Park JH (2001) Robust stabilization for dynamic systems with multiple time-varying delays and nonlinear uncertainties. J Optim Theory Appl 108:155–174

    Article  MathSciNet  MATH  Google Scholar 

  121. Park JH, Jung HY (2003) On the exponential stability of a class of nonlinear systems including delayed perturbations. J Comput Appl Math 159:467–471

    Article  MathSciNet  MATH  Google Scholar 

  122. Park JH (2004) On dynamic output feedback guaranteed cost control of uncertain discrete-delay systems: an LMI optimization approach. J Optim Theory Appl 121:147–162

    Article  MathSciNet  MATH  Google Scholar 

  123. Fridman E, Shaked U (2003) Delay dependent stability and \(\mathscr {H}_\infty \) control: constant and time-varying delays. Int J Control 76:48–60

    Article  MathSciNet  MATH  Google Scholar 

  124. Mahmoud MS, Ismail A (2005) New results on delay-dependent control of time-delay systems. IEEE Trans Autom Control 50:95–100

    Article  MathSciNet  MATH  Google Scholar 

  125. Zhang XM, Wu M, She JH, He Y (2005) Delay-dependent stabilization of linear systems with time-varying state and input delays. Automatica 41:1405–1412

    Article  MathSciNet  MATH  Google Scholar 

  126. Michiels W, Van Assche V, Niculescu SI (2005) Stabilization of time-delay systems with a controlled time-varying delay and applications. IEEE Trans Autom Control 50:493–504

    Article  MathSciNet  MATH  Google Scholar 

  127. Parlakli MNA (2006) Improved robust stability criteria and design of robust stabilizing controller for uncertain linear time-delay systems. Int J Robust Nonlinear Control 16:599–636

    Article  MathSciNet  Google Scholar 

  128. Zhong R, Yang Z (2006) Delay-dependent robust control of descriptor systems with time delay. Asian J Control 8:36–44

    Article  MathSciNet  Google Scholar 

  129. Lin Z, Fang H (2007) On asymptotic stabilizability of linear systems with delayed input. IEEE Trans Autom Control 52:998–1013

    Article  MathSciNet  MATH  Google Scholar 

  130. Xu S, Lam J (2008) A survey of linear matrix inequality techniques in stability analysis of delay systems. Int J Syst Sci 39:1095–1113

    Article  MathSciNet  MATH  Google Scholar 

  131. Hien LV, Phat VN (2009) Exponential stability and stabilization of a class of uncertain linear time-delay systems. J Frankl Inst 346:611–625

    Article  MathSciNet  MATH  Google Scholar 

  132. Choi HH, Chung MJ (1996) Observer-based \(\mathscr {H}_\infty \) controller design for state delayed linear systems. Automatica 32:1073–1075

    Article  MathSciNet  MATH  Google Scholar 

  133. Park JH (2004) On the design of observer-based controller of linear neutral delay-differential systems. Appl Math Comput 150:195–202

    MathSciNet  MATH  Google Scholar 

  134. Kwon OM, Park JH, Lee SM, Won SC (2006) LMI optimization approach to observer-based controller design of uncertain time-delay systems via delayed feedback. J Optim Theory Appl 128:103–117

    Article  MathSciNet  MATH  Google Scholar 

  135. Hua C, Li F, Guan X (2006) Observer-based adaptive control for uncertain time-delay systems. Inf Sci 176:201–214

    Article  MathSciNet  MATH  Google Scholar 

  136. Chen JD (2007) Robust output observer-based control of neutral uncertain systems with discrete and distributed time delays: LMI optimization approach. Chaos Solitons Fractals 34:1254–1264

    Article  MathSciNet  MATH  Google Scholar 

  137. Karimi HR (2008) Observer-based mixed \(\mathscr {H}_2/\mathscr {H}_\infty \) control design for linear systems with time-varying delays: an LMI approach. Int J Control Autom Syst 6:1–14

    Google Scholar 

  138. Majeed R, Ahmad S, Rehan M (2015) Delay-range-dependent observer-based control of nonlinear systems under input and output time-delays. Appl Math Comput 262:145–159

    MathSciNet  MATH  Google Scholar 

  139. Zhou J, Park JH, Ma Q (2016) Non-fragile observer-based \(\mathscr {H}_\infty \) control for stochastic time-delay systems. Appl Math Comput 291:69–83

    MathSciNet  MATH  Google Scholar 

  140. Liu Q, Zhou B (2017) Extended observer based feedback control of linear systems with both state and input delays. J Frankl Inst 354:8232–8255

    Article  MathSciNet  MATH  Google Scholar 

  141. Eskandari N, Dumont GA, Wang ZJ (2017) An observer/predictor-based model of the user for attaining situation awareness. IEEE Trans Hum Mach Syst 46:279–290

    Article  Google Scholar 

  142. Mohajerpoor R, Abdi H, Nahavandi S (2016) A new algorithm to design minimal multi-functional observers for linear systems. Asian J Control 18:842–857

    Article  MathSciNet  MATH  Google Scholar 

  143. Fernando TL, Trinh HM, Jennings L (2010) Functional observability and the design of minimum order linear functional observers. IEEE Trans Autom Control 55:1268–1273

    Article  MathSciNet  MATH  Google Scholar 

  144. Mohajerpoor R, Lakshmanan S, Abdi H, Nahavandi S, Park JH (2018) Delay-dependent functional observer design for linear systems with unknown time-varying state delays. IEEE Trans Cybern 48:2036–2048

    Article  Google Scholar 

  145. Chang SSL, Peng TKC (1972) Adaptive guaranteed cost control of systems with uncertain parameters. IEEE Trans Autom Control 17:474–483

    Article  MathSciNet  MATH  Google Scholar 

  146. Yu L, Chu J (1999) An LMI approach to guaranteed cost control of linear uncertain time-delay systems. Automatica 35:1155–1159

    Article  MathSciNet  MATH  Google Scholar 

  147. Esfahani SH, Petersen IR (2000) An LMI approach to output feedback guaranteed cost control for uncertain time-delay systems. Int J Robust Nonlinear Control 10:157–174

    Article  MathSciNet  MATH  Google Scholar 

  148. Yu L, Gao F (2001) Optimal guaranteed cost control of discrete-time uncertain systems with both state and input delays. J Frankl Inst 338:101–110

    Article  MathSciNet  MATH  Google Scholar 

  149. Chen WH, Guan ZH, Lu X (2003) Delay-dependent guaranteed cost control for uncertain discrete-time systems with delay. IEE Proc Control Theory Appl 150:412–416

    Article  Google Scholar 

  150. Shi P, Boukas EK, Shi Y, Agarwal RK (2003) Optimal guaranteed cost control of uncertain discrete time-delay systems. J Comput Appl Math 157:435–451

    Article  MathSciNet  MATH  Google Scholar 

  151. Park JH, Jung HY (2004) On the design of nonfragile guaranteed cost controller for a class of uncertain dynamic systems with state delays. Appl Math Comput 150:245–257

    MathSciNet  MATH  Google Scholar 

  152. Chen WH, Guan ZH, Lu X (2004) Delay-dependent output feedback guaranteed cost control for uncertain time-delay systems. Automatica 40:1263–1268

    Article  MathSciNet  MATH  Google Scholar 

  153. Park JH (2004) Delay-dependent guaranteed cost stabilization criterion for neutral-delay-differential systems: matrix inequality approach. Comput Math Appl 47:1507–1515

    Article  MathSciNet  MATH  Google Scholar 

  154. Park JH (2005) Delay-dependent criterion for guaranteed cost control of neutral delay systems. J Optim Theory Appl 124:491–502

    Article  MathSciNet  MATH  Google Scholar 

  155. Lien CH (2007) Delay-dependent and delay-independent guaranteed cost control for uncertain neutral systems with time-varying delays via LMI approach. Chaos Solitons Fractals 33:1017–1027

    Article  MathSciNet  MATH  Google Scholar 

  156. Fernando TL, Phat VN, Trinh HM (2013) Output feedback guaranteed cost control of uncertain linear discrete systems with interval time-varying delays. Appl Math Model 37:1580–1589

    Article  MathSciNet  MATH  Google Scholar 

  157. Keel L, Bhattacharyya S (1997) Robust, fragile, or optimal. IEEE Trans Autom Control 42:1098–1105

    Article  MathSciNet  MATH  Google Scholar 

  158. Dorato P (1998) Non-fragile controller design: an overview. In: Proceedings of American control conference, pp 2829–2831

    Google Scholar 

  159. Xu S, Lam J, Wang J, Yang GH (2004) Non-fragile positive real control for uncertain linear neutral delay systems. Syst Control Lett 52:59–74

    Article  MathSciNet  MATH  Google Scholar 

  160. Park JH (2004) Robust non-fragile control for uncertain discrete-delay large-scale systems with a class of controller gain variations. Appl Math Comput 149:147–164

    MathSciNet  MATH  Google Scholar 

  161. Yue D, Lam J (2005) Non-fragile guaranteed cost control for uncertain descriptor systems with time-varying state and input delays. Optim Control Appl Methods 26:85–105

    Article  MathSciNet  Google Scholar 

  162. Xie N, Tang GY (2006) Delay-dependent nonfragile guaranteed cost control for nonlinear time-delay systems. Nonlinear Anal Theory Methods Appl 64:2084–2097

    Article  MathSciNet  MATH  Google Scholar 

  163. Lien CH, Yu KW (2007) Nonfragile control for uncertain neutral systems with time-varying delays via the LMI optimization approach. IEEE Trans Syst Man Cybern Part B (Cybernetics) 37:493–499

    Article  Google Scholar 

  164. Chen JD, Yang CD, Lien CH, Horng JH (2008) New delay-dependent non-fragile \(\mathscr {H}_\infty \) observer-based control for continuous time-delay systems. Inf Sci 178:4699–4706

    Article  MathSciNet  MATH  Google Scholar 

  165. Liu L, Han Z, Li W (2010) \(\mathscr {H}_\infty \) non-fragile observer-based sliding mode control for uncertain time-delay systems. J Frankl Inst 347:567–576

    Article  MathSciNet  MATH  Google Scholar 

  166. Park JH (2004) Design of dynamic output feedback controller for a class of neutral systems with discrete and distributed delay. IEE Proc Control Theory Appl 151:610–614

    Article  Google Scholar 

  167. Park JH (2005) Convex optimization approach to dynamic output feedback control for delay differential systems of neutral type. J Optim Theory Appl 127:411–423

    Article  MathSciNet  MATH  Google Scholar 

  168. Li L, Jia Y (2009) Non-fragile dynamic output feedback control for linear systems with time-varying delay. IET Control Theory Appl 3:995–1005

    Article  MathSciNet  Google Scholar 

  169. Thuan MV, Phat VN, Trinh HM (2012) Dynamic output feedback guaranteed cost control for linear systems with interval time-varying delays in states and outputs. Appl Math Comput 218:10697–10707

    MathSciNet  MATH  Google Scholar 

  170. Lee TH, Park JH (2018) New methods of fuzzy sampled-data control for stabilization of chaotic systems. IEEE Trans Syst Man Cybern Syst 48:2026–2034

    Article  Google Scholar 

  171. Lee TH, Park JH (2017) Stability analysis of sampled-data systems via free-matrix-based time-dependent discontinuous Lyapunov approach. IEEE Trans Autom Control 62:3653–3657

    Article  MathSciNet  MATH  Google Scholar 

  172. Wen S, Huang T, Yu X, Chen MZQ, Zeng Z (2016) Aperiodic sampled-data sliding-mode control of fuzzy systems with communication delays via the event-triggered method. IEEE Transactions on Fuzzy Systems 24:1048–1057

    Article  Google Scholar 

  173. Wang B, Cheng J, Al-Barakati A, Fardoun HM (2017) A mismatched membership function approach to sampled-data stabilization for T-S fuzzy systems with time-varying delayed signals. Signal Process 140:161–170

    Article  Google Scholar 

  174. Ge C, Shi Y, Park JH, Hua C (2019) Robust \(\mathscr {H}_\infty \) stabilization for T-S fuzzy systems with time-varying delays and memory sampled-data control. Appl Math Comput 346:500–512

    MathSciNet  Google Scholar 

  175. Wu ZG, Shi P, Su H, Chu J (2013) Sampled-data synchronization of chaotic Lur’e systems with time delays. IEEE Trans Neural Netw Learn Syst 24:410–421

    Article  Google Scholar 

  176. Hua C, Ge C, Guan X (2015) Synchronization of chaotic Lur’e systems with time delays using sampled-data control. IEEE Trans Neural Netw Learn Syst 26:1214–1221

    Article  MathSciNet  Google Scholar 

  177. Shi K, Liu X, Zhu H, Zhong S, Liu Y, Yin C (2016) Novel integral inequality approach on master-slave synchronization of chaotic delayed Lur’e systems with sampled-data feedback control. Nonlinear Dyn 83:1259–1274

    Article  MathSciNet  MATH  Google Scholar 

  178. Wen G, Duan Z, Yu W, Chen G (2013) Consensus of multi-agent systems with nonlinear dynamics and sampled-data information: a delayed-input approach. Int J Robust Nonlinear Control 23:602–619

    Article  MathSciNet  MATH  Google Scholar 

  179. Wu ZG, Shi P, Su H, Chu J (2013) Stochastic synchronization of Markovian jump neural networks with time-varying delay using sampled data. IEEE Trans Cybern 43:1796–1806

    Article  Google Scholar 

  180. Lee TH, Park JH, Kwon OM, Lee SM (2013) Stochastic sampled-data control for state estimation of time-varying delayed neural networks. Neural Netw 46:99–108

    Article  MATH  Google Scholar 

  181. Rakkiyappan R, Sakthivel N, Cao J (2015) Stochastic sampled-data control for synchronization of complex dynamical networks with control packet loss and additive time-varying delays. Neural Netw 66:46–63

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Electrical Engineering, Yeungnam University, Gyeongsan-si, Korea (Republic of)

    Ju H. Park

  2. Division of Electronic Engineering, Chonbuk National University, Jeonju, Korea (Republic of)

    Tae H. Lee

  3. Control and Computer Engineering, North China Electric Power University, Beijing, China

    Yajuan Liu

  4. School of Electrical Engineering and Automation, Jiangsu Normal University, Xuzhou, China

    Jun Chen

Authors
  1. Ju H. Park
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tae H. Lee
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yajuan Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jun Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ju H. Park .

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Singapore Pte Ltd.

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Park, J., Lee, T.H., Liu, Y., Chen, J. (2019). Basics and Preliminaries of Time-Delay Systems. In: Dynamic Systems with Time Delays: Stability and Control. Springer, Singapore. https://doi.org/10.1007/978-981-13-9254-2_2

Download citation

Publish with us

Policies and ethics

Search

Navigation