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Global Stabilization of Planar Systems with Input Delay and Saturation

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Truncated Predictor Feedback for Time-Delay Systems

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Abstract

This chapter investigates the problem of global stabilization of planar linear systems with both actuator saturation and delay. For a double integrator system, two families of TPF-based linear feedback solutions to the problem are proposed. Both of these families of solutions are parameterized in a single-parameter γ. The first solution is delay dependent in the sense that the delay information is explicitly used in the controller design, while the second solution is delay independent as the delay information is not directly used in the feedback. For an oscillator system, we also propose a delay-dependent TPF-based controller. Explicit ranges of the value of γ for all the three cases are provided to guarantee the global stability of the closed-loop system.

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References

  1. Aeyels D, Willems JL (1992) Pole assignment for linear time-invariant systems by periodic memoryless output feedback. Automatica 28:1159–1168

    MATH  MathSciNet  Google Scholar 

  2. Anderson BDO, Moore JB (1969) New results in linear system stability. SIAM J Control 7(3):398–414

    MATH  MathSciNet  Google Scholar 

  3. Anderson BDO, Moore JB (1990) Optimal control: linear quadratic methods. Prentice Hall, Englewood Cliffs

    MATH  Google Scholar 

  4. Artstein Z (1982) Linear systems with delayed controls: a reduction. IEEE Trans Autom Control 27(4):869–879

    MATH  MathSciNet  Google Scholar 

  5. Bando M, Ichikawa A (2012) Graphical generation of periodic robits of Tschauner-Hempel equations. AIAA J Guid Control Dyn 35(3):1002–1007

    Google Scholar 

  6. Barabanov NE (2005) Periodic linear systems with delays: stability and H ∞ control problem. In: The 17th IMACS world congress, paper T5-I-01-0041, Paris

    Google Scholar 

  7. Barbalat I (1959) Système d’équations différentielle d’oscillations non linéaires. Rev Roum Math Pures Appl 4(2):267–270

    MATH  MathSciNet  Google Scholar 

  8. Bekiaris-Liberis N, Krstic M (2010) Delay-adaptive feedback for linear feedforward systems. Syst Control Lett 59(5):277–283

    MATH  MathSciNet  Google Scholar 

  9. Bekiaris-Liberis N, Krstic M (2010) Stabilization of linear strict-feedback systems with delayed integrators. Automatica 46(11):1902–1910

    MATH  MathSciNet  Google Scholar 

  10. Bekiaris-Liberis N, Krstic M (2011) Lyapunov stability of linear predictor feedback for distributed input delays. IEEE Trans Autom Control 56(3):655–660

    MathSciNet  Google Scholar 

  11. Bhat KPM, Koivo HN (1976) An observer theory for time-delay systems. IEEE Trans Autom Control 21(2):266–269

    MATH  Google Scholar 

  12. Bittanti S, Colaneri P (2008) Periodic systems: filtering and control. Springer, Berlin/London

    Google Scholar 

  13. Bresch-Pietri D, Krstic M (2010) Delay-adaptive predictor feedback for systems with unknown long actuator delay. IEEE Trans Autom Control 55(9):2106–2112

    MathSciNet  Google Scholar 

  14. Bueler E, Butcher E (2002) Stability of periodic linear delay-differential equations and the Chebyshev approximation of fundamental solutions. Preprint

    Google Scholar 

  15. Butcher E, Mann B (2009) Stability analysis and control of linear periodic delayed systems using Chebyshev and temporal finite element methods. In: Balachandran B, Kalma-Nagy T, Gilsinn DE (eds) Delay differential equations: recent advances and new directions. Springer, New York/London, pp 93–129

    Google Scholar 

  16. Cao YY, Lam J, Sun YX (1998) Static output feedback stabilization: an ILMI approach. Automatica 34(12):1641–1645

    MATH  Google Scholar 

  17. Cao YY, Lin Z, Hu T (2002) Stability analysis of linear time-delay systems subject to input saturation. IEEE Trans Circuits Syst I Fundam Theory Appl 49(2):233–240

    MathSciNet  Google Scholar 

  18. Carter TE (1998) State transition matrices for terminal rendezvous studies: brief survey and new example. AIAA J Guid Control Dyn 21(1):148–155

    MATH  Google Scholar 

  19. Carter TE, Pardis CJ (1996) Optimal power-limited rendezvous with upper and lower bounds on thrust. AIAA J Guid Control Dyn 19(5):1124–1133

    MATH  Google Scholar 

  20. Chen GR, Duan ZS (2008) Network synchroni- zability analysis: a graph-theoretic approach. Chaos Interdiscip J Nonlinear Sci 18(3):037102-1–037102-10

    Google Scholar 

  21. Chen J, Latchman HA (1995) Frequency sweeping tests for stability independent of delay. IEEE Trans Autom Control 40(9):1640–1645

    MATH  MathSciNet  Google Scholar 

  22. Chen WH, Zheng WX (2007) Delay-dependent robust stabilization for uncertain neutral systems with distributed delays. Automatica 43(1):95–104

    MATH  MathSciNet  Google Scholar 

  23. Chen J, Gu G, Nett CN (1995) A new method for computing delay margins for stability of linear delay systems. Syst Control Lett 26(2):107–117

    MATH  MathSciNet  Google Scholar 

  24. Chen B-S, Wang S-S, Lu R-C (1988) Stabilization of time-delay systems containing saturating actuators. Int J Control 47:867–881

    MATH  MathSciNet  Google Scholar 

  25. Chen J, Fu P, Niculescu S-I, Guan Z (2010) An eigenvalue perturbation approach to stability analysis, part I: eigenvalue series of matrix operators. SIAM J Control Optim 48(8): 5564–5582

    MATH  MathSciNet  Google Scholar 

  26. Choi H-L, Lim J-T (2010) Output feedback regulation of a chain of integrators with an unknown time-varying delay in the input. IEEE Trans Autom Control 55(1):263–268

    MathSciNet  Google Scholar 

  27. Clohessy WH, Wiltshire RS (1960) Terminal guidance system for satellite rendezvous. J Aeronaut Sci 27(9):653–658

    MATH  Google Scholar 

  28. Collado J, Lozano R, Ailon A (1999) Semi-global stabilization of discrete-time systems with bounded inputs using a periodic controller. Syst Control Lett 36:267–275

    MATH  MathSciNet  Google Scholar 

  29. Cong S (2013) On exponential stability conditions of linear neutral stochastic differential systems with time-varying delay. Int J Robust Nonlinear Control 23(11):1265–1276

    MATH  MathSciNet  Google Scholar 

  30. Cui NG, Zhou B, Duan G-R (2013) Output feedback elliptical orbital rendezvous via state-dependent Riccati differential equations. IET Control Theory Appl 7(10):1429–1436

    MathSciNet  Google Scholar 

  31. Duan Z, Wang J, Chen G, Huang L (2008) Stability analysis and decentralized control of a class of complex dynamical networks. Automatica 44(4):1028–1035

    MATH  MathSciNet  Google Scholar 

  32. Engelborghs K, Luzyanina T, Samaey G (2001) DDE-BIFTOOL v.2.00: a Matlab package for bifurcation analysis of delay differential equation. Dept. Comput. Sci., K.U. Leuven, T.W. Rep. 330

    Google Scholar 

  33. El Haoussi F, Tissir EH (2007) An LMI-based approach for robust stabilization of time delay systems containing saturating actuators. IMA J Math Control Inf 24(3):347–356

    MATH  MathSciNet  Google Scholar 

  34. Elaydi S, Zhang S (1994) Stability and periodicity of difference equations with finite delay. Funkc Ekvacio 37(3):401–413

    MATH  MathSciNet  Google Scholar 

  35. Fang H, Lin Z (2006) A further result on global stabilization of oscillators with bounded delayed input. IEEE Trans Autom Control 51(1):121–128

    MathSciNet  Google Scholar 

  36. Fax J, Murray R (2004) Information flow and cooperative control of vehicle formations. IEEE Trans Autom Control 49(9):1465–1476

    MathSciNet  Google Scholar 

  37. Foias C, Ozbay H, Tannenbaum A (1996) Robust control of infinite dimensional systems: frequency domain methods. Lecture notes in control and information sciences. Springer, Berlin

    Google Scholar 

  38. Fridman E, Shaked U (2005) Stability and guaranteed cost control of uncertain discrete delay systems. Int J Control 78(4):235–246

    MATH  MathSciNet  Google Scholar 

  39. Fridman E, Pila A, Shaked U (2003) Regional stabilization and H ∞ control of time-delay systems with saturating actuators. Int J Robust Nonlinear Control 13(9):885–907

    MATH  MathSciNet  Google Scholar 

  40. Fridman E, Shaked U, Liu K (2009) New conditions for delay-derivative-dependent stability. Automatica 45(11):2723–2727

    MATH  MathSciNet  Google Scholar 

  41. Furukawa T, Shimemura E (1983) Stabilizability conditions by memolyless feedback for linear systems with time-delay. Int J Control 37(3):553–565

    MATH  MathSciNet  Google Scholar 

  42. Gary NK, Mann BP, Kim NH, Kurdi MH (2007) Stability of a time-delayed system with parametric excitation. J Dyn Syst Measure Control 129:125–135

    Google Scholar 

  43. Goebel G, Munz U (2010) Allgower F, Stabilization of linear systems with distributed input delay. In: American control conference, Baltimore, 30 June–02 July 2010, pp 5800–5805

    Google Scholar 

  44. Goldberg RR (1964) Methods of real analysis. Blaisdell, Waltham

    MATH  Google Scholar 

  45. Gronwall TH (1919) Note on the derivatives with respect to a parameter of the solutions of a system of differential equations. Ann Math 20(4):292–296

    MATH  MathSciNet  Google Scholar 

  46. Gu G (1989) Lee EB, Stability testing of time delay systems. Automatica 25:777–780

    MATH  Google Scholar 

  47. Gu K (2000) An integral inequality in the stability problem of time-delay systems. In: Proceedings of the 39th IEEE conference on decision and control, Sydney, Dec 2000, pp 2805–2810

    Google Scholar 

  48. Gu K (2003) An improved stability criterion for systems with distributed delays. Int J Robust Nonlinear Control 13:819–831

    MATH  Google Scholar 

  49. Gu K (2012) A review of some subtleties of practical relevance for time-delay systems of neutral type. ISRN Appl Math 2012:Article ID 725783, 46 p. doi:10.5402/2012/725783

    Google Scholar 

  50. Gu K, Han QL, Luo ACJ, Niculescu SI (2001) Discretized Lyapunov functional for systems with distributed delay and piecewise constant coefficients. Int J Control 74(7):737–744

    MATH  MathSciNet  Google Scholar 

  51. Gu K Kharitonov VL, Chen J (2003) Stability of time-delay systems. Springer, Berlin

    MATH  Google Scholar 

  52. Guelman M, Aleshin M, Optimal bounded low-thrust rendezvous with fixed terminal-approach direction. AIAA J Guid Control Dyn 24(2):378–385

    Google Scholar 

  53. Halanay A (1966) Differential equations: stability, oscillations, time lags. Academic, New York/London

    MATH  Google Scholar 

  54. Hale JK (1977) Theory of functional differential equations. Applied mathematical sciences, vol 3. Springer, New York

    Google Scholar 

  55. Han Q (2005) Absolute stability of time-delay systems with sector-bounded nonlinearity. Automatica 41(12):2171–2176

    MATH  MathSciNet  Google Scholar 

  56. Haurani A, Michalska HH, Boulet B (2004) Robust output feedback stabilization of uncertain time-varying state-delayed systems with saturating actuators. Int J Control 77(4):399–414

    MATH  MathSciNet  Google Scholar 

  57. He Y, Wang QG, Lin C, Wu M (2007) Delay-range-dependent stability for systems with time-varying delay. Automatica 43(2):371–376

    MATH  MathSciNet  Google Scholar 

  58. Hong Y, Chen G, Bushnell L (2008) Distributed observers design for leader-following control of multi-agent networks. Automatica 44(3):846–850

    MATH  MathSciNet  Google Scholar 

  59. Hu T, Lin Z (2001) Control systems with actuator saturation: analysis and design. Birkhäuser, Boston

    Google Scholar 

  60. Huang L (2003) The theory foundation of stability and robustness. Science Press, Beijing (in Chinese)

    Google Scholar 

  61. Ichikawa A (2006) Null controllability with vanishing energy for infinite dimensional periodic systems. In: Proceedings of the seventeenth international symposium on mathematical theory of networks and systems, Kyoto, pp 638–642

    Google Scholar 

  62. Ichikawa A (2008) Null controllability with vanishing energy for discrete-time systems. Syst Control Lett 57:34–38

    MATH  MathSciNet  Google Scholar 

  63. Inalhan G, Tillerson M, How JP (2002) Relative dynamics and control of spacecraft formations in eccentric orbits. J Guid Control Dyn 25(1):48–59

    Google Scholar 

  64. Insperger T, Stepan G (2011) Semi-discretization for time-delay systems: stability and engineering applications. Springer, New York/Dordrecht/Heidelberg/London

    Google Scholar 

  65. Jankovic M (2009) Forwarding, backstepping, and finite spectrum assignment for time delay systems. Automatica 45(1):2–9

    MATH  MathSciNet  Google Scholar 

  66. Kailath T (1980) Linear systems. Prentice-Hall, Englewood Cliffs

    MATH  Google Scholar 

  67. Kaliora G, Astolfi A (2004) Nonlinear control of feedforward systems with bounded signals. IEEE Trans Autom Control 49(11):1975–1990

    MathSciNet  Google Scholar 

  68. Kalman RE (1960) Contributions to the theory of optimal control. Boletin De la Sociedad Matematica Mexicana 5(2):102–119

    MathSciNet  Google Scholar 

  69. Karafyllis I (2011) Stabilization by means of approximate predictors for systems with delayed input. SIAM J Control Optim 49(3):1100–1123

    MATH  MathSciNet  Google Scholar 

  70. Karafyllis I, Krstic M, Predictor-based output feedback for nonlinear delay systems. Available at: http://arxiv.org/abs/1108.4499

  71. Karlgaard CD (2006) Robust rendezvous navigation in elliptical orbit. J Guid Control Dyn 29(2):495–499

    Google Scholar 

  72. Khargonekar PP, Poola K, Tannenbaum A (1985) Robust control of linear time-invariant plants using periodic compensation. IEEE Trans Autom Control 30:1088–1096

    MATH  Google Scholar 

  73. Klamka J (1977) Absolute controllability of linear systems with time-variable delays in control. Int J Control 26:57–63

    MATH  MathSciNet  Google Scholar 

  74. Klamka J (1982) Observer for linear feedback control of systems with distributed delays in controls and outputs. Syst Control Lett 1(5):326–331

    MATH  MathSciNet  Google Scholar 

  75. Kleinman DL (1976) On the linear regulator problem and the matrix Riccati equation. Report, Electronic Systems Laboratory, Department of Electrical Engineering, Massachusetts Institute of Technology

    Google Scholar 

  76. Kleinman DL, Athans M (1982) The design of suboptimal linear time-varying equations, IEEE Trans Autom Control 27:461–463

    Google Scholar 

  77. Kluever CA (1999) Feedback control for spacecraft rendezvous and docking. AIAA J Guid Control Dyn 22(4):609–611

    Google Scholar 

  78. Kojima A, Uchida K, Shimemura E, Ishijima S (1994) Robust stabilization of a system with delays in control. IEEE Trans Autom Control 39(8):1694–1698

    MATH  MathSciNet  Google Scholar 

  79. Kolmanovskii VB, Richard JP (1997) Stability of some systems with distributed delays. JESA 31:971–982. Special issue on Analysis and Control of Time-Delay Systems

    Google Scholar 

  80. Krstic M (2008) Lyapunov tools for predictor feedbacks for delay systems: inverse optimality and robustness to delay mismatch. Automatica 44:2930–2935

    MATH  MathSciNet  Google Scholar 

  81. Krstic M (2010) Lyapunov stability of linear predictor feedback for time-varying input delay. IEEE Trans Autom Control 55(2):554–559

    MathSciNet  Google Scholar 

  82. Krstic M (2010) Compensation of infinite-dimensional actuator and sensor dynamics. IEEE Control Syst Mag 30(1):22–41

    MathSciNet  Google Scholar 

  83. Khalil HK (2002) Nonlinear systems. Prentice Hall, Upper Saddle River

    MATH  Google Scholar 

  84. Kharitonov VL, Melchior-Aguliar D (2000) On delay-dependent stability conditions. Syst Control Lett 40(1):71–76

    MATH  Google Scholar 

  85. Kucera V (1973) A review of the matrix Riccati equation. Kybernetika 9(1):42–61

    MATH  MathSciNet  Google Scholar 

  86. Kwakernaak H, Sivan R (1973) Linear optimal control systems. Wiley-Interscience, New York

    Google Scholar 

  87. Lam J, Gao H, Wang C (2007) Stability analysis for continuous systems with two additive time-varying delay components. Syst Control Lett 56(1):16–24

    MATH  MathSciNet  Google Scholar 

  88. Lampe BP, Rosenwasser EN (2011) Stability investigation for linear periodic time-delayed systems using Fredholm theory. Autom Remote Control 72(1):38–60

    MATH  MathSciNet  Google Scholar 

  89. Letyagina ON, Zhabko AP (2009) Robust stability analysis of linear periodic systems with time delay. Int J Mod Phys A 24(5):893–907

    MATH  Google Scholar 

  90. Leyva-Ramos J, Pearson AE (2000) Output feedback stabilizing controller for time-delay systems. Automatica 36(4):613–617

    MATH  MathSciNet  Google Scholar 

  91. Li Y-L (2011) Sums of a class of power series. Stud Coll Math 14(3):11–12

    MATH  Google Scholar 

  92. Li Z, Duan Z, Chen G, Huang L (2010) Consensus of multiagent systems and synchronization of complex networks: a unified viewpoint. IEEE Trans Circuits Syst I Regul Pap 57(1): 213–224

    MathSciNet  Google Scholar 

  93. Lin Z (2007) On asymptotic stabilizability of discrete-time linear systems with delayed input. Commun Inf Syst 7(3):227–264

    MATH  MathSciNet  Google Scholar 

  94. Lin Z, Fang H (2007) On asymptotic stabilizability of linear systems with delayed input. IEEE Trans Autom Control 52(6):998–1013

    MathSciNet  Google Scholar 

  95. Lin Z, Saberi A (1993) Semi-global exponential stabilization of linear systems subject to ‘input saturation’ via linear feedbacks. Syst Control Lett 21(3):225–239

    MATH  MathSciNet  Google Scholar 

  96. Lin P, Jia Y, Du J, Yuan S (2007) Distributed consensus control for second-order agents with fixed topology and time delay. In: 2007 Chinese control conference, Hunan, pp 577–581

    Google Scholar 

  97. Liu B, Marquez HJ (2007) Razumikhin-type stability theorems for discrete-time delay systems. Automatica 43(7):1219–1225

    MATH  MathSciNet  Google Scholar 

  98. Liu W, Chitour Y, Sontag E (1996) On finite-gain stabilizability of linear systems subject to input saturation. SIAM J Control Optim 34(4):1190–1219

    MATH  MathSciNet  Google Scholar 

  99. Liu X, Lu W, Chen T (2010) Consensus of multi-agent systems with unbounded time-varying delays. IEEE Trans Autom Control 55(10):2396–2401

    MathSciNet  Google Scholar 

  100. Long X, Insperger T, Balachandran B (2009) Systems with periodic coefficients and periodically varying delays: semi-discretization-based stability analysis. In: Balachandran B, Kalma-Nagy T, Gilsinn DE (eds) Delay differential equations: recent advances and new directions. Springer, New York/London, pp 131–153

    Google Scholar 

  101. Lozano R, Sanchez A, Salazar-Cruz S, Fantoni I, Torres J (2006) Discrete-time stabilization of integrators in cascade: real-time stabilization of a mini-rotorcraft. In: Proceedings of the 45th IEEE conference on decision & control, San Diego pp 6265–6270

    Google Scholar 

  102. Lv J, Chen G, di Bernardo M (2010) On some recent advances in synchronization and control of complex networks. In: Proceedings of the 2010 IEEE international symposium on circuits and systems, Paris, pp 3773–3776

    Google Scholar 

  103. Lv L, Duan G, Zhou B (2010) Parametric pole assignment and robust pole assignment for discrete-time linear periodic systems. SIAM J Control Optim 48(6):3975–3996

    MATH  MathSciNet  Google Scholar 

  104. MacDonald N (1978) Time lags in biological models. Lecture notes biomath, vol 27. Springer, New York

    Google Scholar 

  105. Manitius AZ, Olbrot AW (1979) Finite spectrum assignment for systems with delays. IEEE Trans Autom Control 24(4):541–553

    MATH  MathSciNet  Google Scholar 

  106. Marchand N, Hably A (2005) Global stabilization of multiple integrators with bounded control. Automatica 41(12):2147–2152

    MATH  MathSciNet  Google Scholar 

  107. Mazenc F, Mondié S, Niculescu S-I (2003) Global asymptotic stabilization for chains of integrators with a delay in the input. IEEE Trans Autom Control 48(1):57–63

    Google Scholar 

  108. Mazenc F, Mondié S, Francisco R (2004) Global asymptotic stabilization of feedforward systems with delay in the input. IEEE Trans Autom Control 49(5):844–850

    Google Scholar 

  109. Mazenc F, Mondié S, Niculescu S-I (2004) Global stabilization of oscillators with bounded delayed input. Syst Control Lett 53(5):415–422

    MATH  Google Scholar 

  110. Mazenc F, Niculescu S, Krstic M (2012) Lyapunov–Krasovskii functionals and application to input delay compensation for linear time-invariant systems. Automatica 48(7):1317–1323

    MATH  MathSciNet  Google Scholar 

  111. Mengali G, Quarta AA (2005) Fuel-optimal, power-limited rendezvous with variable thruster efficiency. J Guid Control Dyn 28(6):1194–1199

    Google Scholar 

  112. Michiels W (2002) Stability and stabilization of time-delay systems. Ph.D. thesis, Katholieke Universiteit Leuven, Leuven

    Google Scholar 

  113. Michiels W, Niculescu S-I (2007) Characterization of delay-independent stability and delay interference phenomena. SIAM J Control Optim 45(6):2138–2155

    MATH  MathSciNet  Google Scholar 

  114. Michiels W, Niculescu S-I (2007) Stability and stabilization of time-delay systems: an eigenvalue-based approach. SIAM, Singapore

    Google Scholar 

  115. Michiels W, Roose D (2001) Global stabilization of multiple integrators with time delay and input constraints. In: Proceedings of the third IFAC workshop on time-delay systems, Santa Fe, pp 266–271

    Google Scholar 

  116. Michiels W, Engelborghs K, Vansevenant P, Roose D (2002) Continuous pole placement for delay equations. Automatica 38(5):747–761

    MATH  MathSciNet  Google Scholar 

  117. Michiels W, Sepulchre R, Roose D (2001) Stability of perturbed delay differential equations and stabilization of nonlinear cascade systems. SIAM J Control Optim 40(3):661–680

    MATH  MathSciNet  Google Scholar 

  118. Mirkin L (2004) On the approximation of distributed-delay control laws. Syst Control Lett 51(5):331–342

    MATH  MathSciNet  Google Scholar 

  119. Mondie S, Michiels W (2003) Finite spectrum assignment of unstable time-delay systems with a safe implementation. IEEE Trans Autom Control 48(12):2207–2212

    MathSciNet  Google Scholar 

  120. Morarescu CI, Niculescu SI, Michiels W (2007) Asymptotic stability of some distributed delay systems: an algebraic approach. Int J Tomogr Stat 7(7):128–133

    MathSciNet  Google Scholar 

  121. Molinari BP (1977) The time-invariant linear-quadratic optimal control problem. Automatica 13(4):347–357

    MATH  Google Scholar 

  122. Munz U, Papachristodoulou A, Allgower F (2011) Consensus in multi-agent systems with coupling delays and switching topology. IEEE Trans Autom Control 56(12):2976–2982

    MathSciNet  Google Scholar 

  123. Munz U, Papachristodoulou A, Allgower F (2012) Delay robustness in non-identical multi-agent systems. IEEE Trans Autom Control 57(6):1597–1603

    MathSciNet  Google Scholar 

  124. Niculescu S-I (2001) Delay effects on stability: a robust control approach. Springer, Berlin

    Google Scholar 

  125. Niculescu S-I, Gu K (2004) Advances in time-delay systems. Lecture notes in computational science and engineering, vol 38. Springer, Berlin/New York

    Google Scholar 

  126. Olbrot A (1972) On controllability of linear systems with time delays in control. IEEE Trans Autom Control 17(5):664–666

    MATH  MathSciNet  Google Scholar 

  127. Olbrot AW (1973) Algebraic criteria of controllability to zero function for linear constant time-lag systems. Control and Cybern 2:59–77

    MATH  Google Scholar 

  128. Olbrot AW (1978) Stabilizability, detectability, and spectrum assignment for linear autonomous systems with general time delays. IEEE Trans Autom Control 23(5):887–890

    MATH  MathSciNet  Google Scholar 

  129. Olfati-Saber R, Murray R (2004) Consensus problems in networks of agents with switching topology and time-delays. IEEE Trans Autom Control 49(9):1520–1533

    MathSciNet  Google Scholar 

  130. Ozbay H, Bonnet C, Clairambault J (2008) Stability analysis of systems with distributed delays and applications to hematopoietic cell maturation dynamics. In: Proceedings of the 47th IEEE conference on decision and control, Cancún, pp 2050–2055

    Google Scholar 

  131. Pardis CJ, Carter TE (1995) Optimal power-limited rendezvous with thrust saturation. AIAA J Guid Control Dyn 18(5):1145–1150

    Google Scholar 

  132. Polites ME (1999) Technology of automated rendezvous and capture in space. J Guid Control Dyn 36(2):280–291

    Google Scholar 

  133. Priola E, Zabczyk J (2003) Null controllability with vanishing energy. SIAM J Control Optim 42(3):1013–1032

    MATH  MathSciNet  Google Scholar 

  134. Rao VG, Berstein DS (2001) Naive control of the double integrator. IEEE Trans Control Syst Mag 21(5):86–97

    Google Scholar 

  135. Ren W (2007) Formation keeping and attitude alignment for multiple spacecraft through local interactions. AIAA J Guid Control Dyn 30(2):633–638

    Google Scholar 

  136. Ren W, Cao Y (2011) Distributed coordination of multi-agent networks. Communications and control engineering series. Springer, London

    Google Scholar 

  137. Ren W, Beard R, Atkins E (2007) Information consensus in multi-vehicle cooperative control. IEEE Control Syst Mag 27(2):71–82

    Google Scholar 

  138. Richard JP (2003) Time-delay systems:an overview of some recent advances and open problems. Automatica 39(10):1667–1694

    MATH  MathSciNet  Google Scholar 

  139. Ruan S, Wei J (2003) On the zeros of transcendental functions with applications to stability of delay differential equations with two delays. Dyn Contin Discret Impuls Syst A Math Anal 10:863–874

    MATH  MathSciNet  Google Scholar 

  140. Rudy C, Olgac N (2011) An exact method for the stability analysis of linear consensus protocols with time delay. IEEE Trans Autom Control 56(7):1734–1740

    Google Scholar 

  141. Rugh WJ (1996) Linear systems theory, 2nd edn. Prentice Hall, Upper Saddle River

    Google Scholar 

  142. Scharf DP, Hadaegh FY, Ploen SR (2004) A survey of spacecraft formation flying: guidance and control (part II): control. In: Proceedings of the 2004 American control conference, Boston, pp 2976–2985

    Google Scholar 

  143. Seo JH, Shim H, Back J (2009) Consensus of high-order linear systems using dynamic output feedback compensator: low gain approach. Automatica 45:2659–2664

    MATH  MathSciNet  Google Scholar 

  144. Sheng J, Elbeyli O, Sun JQ (2004) Stability and optimal feedback controls for time-delayed linear periodic systems. AIAA J 42(5):908–911

    Google Scholar 

  145. Shi T, Su HY, Chu J (2010) On stability and stabilisation for uncertain stochastic systems with time-delay and actuator saturation. Int J Syst Sci 41(5):501–509

    MATH  MathSciNet  Google Scholar 

  146. Shibata M, Ichikawa A (2007) Orbital rendezvous and flyaround based null controllability with vanishing energy. AIAA J Guid Control Dyn 30(4):934–945

    Google Scholar 

  147. Shustin E, Fridman L, Fridman E, Castanos F (2008) Robust semiglobal stabilization of the second order system by relay feedback with an uncertain variable time delay. SIAM J Control Optim 47(1):196–217

    MATH  MathSciNet  Google Scholar 

  148. Son NK, Su NV (1986) Linear periodic control systems: controllability with restrained controls. Appl Math Optim 14(1):173–185

    MATH  MathSciNet  Google Scholar 

  149. Sontag ED (1984) An algebraic approach to bounded controllability of linear systems. Int J Control 39(1):181–188

    MATH  MathSciNet  Google Scholar 

  150. Su Y, Huang J (2012) Stability of a class of linear switching systems with applications to two consensus problems. IEEE Trans Autom Control 57(6):1420–1430

    MathSciNet  Google Scholar 

  151. Sun YJ (2002) Global stabilizability of uncertain systems with time-varying delays via dynamic observer-based output feedback. Linear Algebra Appl 353:91–105

    MATH  MathSciNet  Google Scholar 

  152. Sussmann HJ, Sontag ED, Yang Y (1994) A general result on the stabilization of linear systems using bounded controls. IEEE Trans Autom Control 39(12):2411–2425

    MATH  MathSciNet  Google Scholar 

  153. Tarbouriech S, da Silva JMG (2000) Synthesis of controllers for continuous-time delay systems with saturating controls via LMI’s. IEEE Trans Autom Control 45(1):105–111

    MATH  Google Scholar 

  154. Teel AR (1992) Global stabilization and restricted tracking for multiple integrators with bounded controls. Syst Control Lett 18(3):165–171

    MATH  MathSciNet  Google Scholar 

  155. Teel AR (1998) Connections between Razumikhin-type theorems and the ISS nonlinear small gain theorem. IEEE Trans Autom Control 43(7):960–964

    MATH  MathSciNet  Google Scholar 

  156. Tian Y, Liu C (2009) Robust consensus of multi-agent systems with diverse input delays and asymmetric interconnection perturbations. Automatica 45:1347–1353

    MATH  MathSciNet  Google Scholar 

  157. Tian Y, Zhang Y (2012) High-order consensus of heterogeneous multi-agent systems with unknown communication delays. Automatica 48(6):1205–1212

    MATH  MathSciNet  Google Scholar 

  158. Tschauner J, Hempel P (1965) Rendezvous zu einem in elliptischer Bahn umlaufenden Ziel. Astronaut Acta 11(2):104–109

    MATH  Google Scholar 

  159. Tyan F, Bernstein DS (1999) Global stabilization of systems containing a double integrator using a saturated linear controller. Int J Robust Nonlinear Control 9:1143–1156

    MATH  MathSciNet  Google Scholar 

  160. Van Assche V, Dambrine M, Lafay JF, Richard JP (1999) Some problems arising in the implementation of distributed-delay control laws. In: Proceedings of the 38th IEEE conference on decision and control, Arizona, pp 4668–4672

    Google Scholar 

  161. Varga A (2005) On solving periodic differential matrix equations with applications to periodic system norms computation. In: Proceedings of the 44th IEEE conference on decision and control, Seville, pp 6545–6550

    Google Scholar 

  162. Watanabe K, Ito M (1981) An observer for linear feedback control laws of multivariable systems with multiple delays in the controls and outputs. Syst Control Lett 1(1):54–59

    MATH  Google Scholar 

  163. Wang X, Saberi A, Stoorvogel AA, Grip HF, Yang T (2013) Consensus in the network with uniform constant communication delay. Automatica 49(8):2461–2467

    MathSciNet  Google Scholar 

  164. Willems JC (1971) Least squares stationary optimal control and the algebraic Riccati equations. IEEE Trans Autom Control 16(6):621–634

    MathSciNet  Google Scholar 

  165. Woodbury MA (1949) The stability of out-input matrices. University of Chicago Press, Chicago, p 93

    Google Scholar 

  166. Wu M, He Y, She J-H (2004) New delay-dependent stability criteria and stabilizing method for neutral systems. IEEE Trans Autom Control 49(12):2266–2271

    MathSciNet  Google Scholar 

  167. Xu S, Lam J (2008) A survey of linear matrix inequality techniques in stability analysis of delay systems. Int J Syst Sci 39(12):1095–1113

    MATH  MathSciNet  Google Scholar 

  168. Xu S, Lam J, Yang C (2001) Quadratic stability and stabilization of uncertain linear discrete-time systems with state delay. Syst Control Lett 43(2):77–84

    MATH  MathSciNet  Google Scholar 

  169. Xu S, Lam J, Mao X (2007) Delay-dependent \(\mathcal{H}_{\infty }\) control and filtering for uncertain Markovian jump systems with time-varying delays. IEEE Trans Circuits Syst I Regul Pap 54(9):2070–2077

    MathSciNet  Google Scholar 

  170. Yakoubi K, Chitour Y (2006) Linear systems subject to input saturation and time delay: finite-gain \({\mathcal{L}}^{p}\)-stabilization. SIAM J Control Optim 45(3):1084–1115

    MATH  MathSciNet  Google Scholar 

  171. Yakoubi K, Chitour Y (2007) Linear systems subject to input saturation and time delay: global asymptotic stabilization. IEEE Trans Autom Control 52(5):874–879

    MathSciNet  Google Scholar 

  172. Yamanaka K, Ankersen F (2002) New state transition matrix for relative motion on an arbitrary elliptical orbit. AIAA J Guid Control Dyn 25(1):60–66

    Google Scholar 

  173. Yang Y, Sontag ED, Sussmann HJ (1997) Global stabilization of linear discrete-time systems with bounded feedback. Syst Control Lett 30(5):273–281

    MATH  MathSciNet  Google Scholar 

  174. Ye H, Wang H, Wang H (2007) Stabilization of a PVTOL aircraft and an inertia wheel pendulum using saturation technique. IEEE Trans Control Syst Technol 15(6):1143–1150

    MathSciNet  Google Scholar 

  175. You K, Xie L (2011) Coordination of discrete-time multi-agent systems via relative output feedback. Int J Robust Nonlinear Control 21(13):1587–1605

    MATH  MathSciNet  Google Scholar 

  176. You K, Xie L (2011) Network topology and communication data rate for consensusability of discrete-time multi-agent systems. IEEE Trans Autom Control 56(10):2262–2275

    MathSciNet  Google Scholar 

  177. Yu W, Chen G, Cao M (2010) Some necessary and sufficient conditions for second-order consensus in multi-agent dynamical systems. Automatica 46:1089–1095

    MATH  MathSciNet  Google Scholar 

  178. Yu W, Chen G, Cao M (2010) Distributed leader-follower flocking control for multi-agent dynamical systems with time-varying velocities. Syst Control Lett 59(9):543–552

    MATH  MathSciNet  Google Scholar 

  179. Zaccarian L, Nesic D, Teel AR (2005) \(\mathcal{L}_{2}\) anti-windup for linear dead-time systems. Syst Control Lett 54(12):1205–1217

    MATH  MathSciNet  Google Scholar 

  180. Zhang S, Chen M-P (1998) A new Razumikhin theorem for delay difference equations. Comput Math Appl 36(10):405–412

    MATH  MathSciNet  Google Scholar 

  181. Zhang X, Sun J (2013, to appear) A note on the stability of linear dynamical systems with time delay. J Vib Control. doi:10.1177/1077546312473319

    Google Scholar 

  182. Zhang H, Zhang D, Xie L (2004) An innovation approach to \(\mathcal{H}_{\infty }\) prediction with applications to systems with delayed measurements. Automatica 40(7):1253–1261

    MATH  MathSciNet  Google Scholar 

  183. Zhong QC (2004) On distributed delay in linear control laws-part I: discrete-delay implementations. IEEE Trans Autom Control 49(11):2074–2080

    Google Scholar 

  184. Zhou B, Li Z, Lin Z, Reduced-order observer based output feedback control of time-delay systems, A personal communication

    Google Scholar 

  185. Zhou B, Consensus of delayed multi-agent systems by reduced-order observer based truncated predictor feedback protocols. Submitted for publication

    Google Scholar 

  186. Zhou B, Consensus of discrete-time linear multi-agent systems with large input and communication delays. Submitted for publication

    Google Scholar 

  187. Zhou B, Observer based output feedback control of discrete-time linear systems with input and output delays. International Journal of Control, to appear.

    Google Scholar 

  188. Zhou B, Input delay compensation of linear systems with both state and input delays by nested prediction. Automatica. http://dx.doi.org/10.1016/j.automatica.2014.03.010.

  189. Zhou B (2010) Parametric Lyapunov approach to the design of control systems with saturation nonlinearity and its applications. PhD thesis, Harbin Institute of Technology, Harbin

    Google Scholar 

  190. Zhou B (2013) Analysis and design of discrete-time linear systems with nested actuator saturations. Syst Control Lett 62(10):871–879

    MATH  Google Scholar 

  191. Zhou B, Duan G-R (2009) On analytical approximation of the maximal invariant ellipsoids for linear systems with bounded controls. IEEE Trans Autom Control 54(2):346–353

    MathSciNet  Google Scholar 

  192. Zhou B, Duan G-R (2012) Periodic Lyapunov equation based approaches to the stabilization of continuous-time periodic linear systems. IEEE Trans Autom Control 57(8):2139–2146

    MathSciNet  Google Scholar 

  193. Zhou B, Li Z-Y, Truncated predictor feedback for periodic linear systems with input delays with applications to the elliptical spacecraft rendezvous. To be presented at the 26th Chinese Control and Decision Conference (2014 CCDC) to be held on May 31-June 2, 2014 in Changsha, China.

    Google Scholar 

  194. Zhou B, Lin Z, Truncated predictor feedback stabilization of polynomially unstable linear systems with multiple time-varying input delays. IEEE Trans Autom Control. doi:10.1109/TAC.2013.2297188

    Google Scholar 

  195. Zhou B, Lin Z (2011) Parametric Lyapunov equation approach to stabilization of discrete-time systems with input delay and saturation. IEEE Trans Circuits Syst I Regul Pap 58(11): 2741–2754

    MathSciNet  Google Scholar 

  196. Zhou B, Lin Z (2013) Consensus of high-order multi-agent systems with large input and communication delays. Automatica. Vol. 50, No. 2, pp. 452–464, 2014.

    MathSciNet  Google Scholar 

  197. Zhou B, Li Z-Y, Lin Z, On higher-order truncated predictor feedback for linear systems with input delay. Int J Robust Nonlinear Control. doi:10.1002/rnc.3012

    Google Scholar 

  198. Zhou B, Xu C, Duan G (to appear) Distributed and truncated reduced-order observer based output feedback consensus of multi-agent systems. IEEE Trans Autom Control. doi:10.1109/TAC.2014.2301573

    Google Scholar 

  199. Zhou B, Duan G-R, Li Z-Y (2008) On improving transient performance in global control of multiple integrators system by bounded feedback. Syst Control Lett 57(10):867–875

    MATH  MathSciNet  Google Scholar 

  200. Zhou B, Duan G-R, Lin Z (2008) A parametric Lyapunov equation approach to the design of low gain feedback. IEEE Trans Autom Control 53(6):1548–1554

    MathSciNet  Google Scholar 

  201. Zhou B, Lin Z, Duan G-R (2009) A parametric Lyapunov equation approach to low gain feedback design for discrete-time systems. Automatica 45(1):238–244

    MATH  MathSciNet  Google Scholar 

  202. Zhou B, Lin Z, Duan G-R (2009) Properties of the parametric Lyapunov equation based low-gain design with applications in stabilization of time-delay systems. IEEE Trans Autom Control 54(7):1698–1704

    MathSciNet  Google Scholar 

  203. Zhou B, Li Z-Y, Duan G-R, Wang Y (2009) Optimal pole assignment for discrete-time systems via Stein equations. IET Control Theory Appl 3(8):983–994

    MathSciNet  Google Scholar 

  204. Zhou B, Lin Z, Duan G-R (2010) Global and semi-global stabilization of linear systems with multiple delays and saturations in the input. SIAM J Control Optim 48(8):5294–5332

    MATH  MathSciNet  Google Scholar 

  205. Zhou B, Lin Z, Duan G-R (2010) Stabilization of linear systems with input delay and saturation–a parametric Lyapunov equation approach. Int J Robust Nonlinear Control 20(13):1502–1519

    MATH  MathSciNet  Google Scholar 

  206. Zhou B, Duan G-R, Lin Z (2010) Global stabilization of the double integrator system with saturation and delay in the input. IEEE Trans Circuits Syst I Regul Pap 57(6):1371–1383

    MathSciNet  Google Scholar 

  207. Zhou B, Duan G-R, Lin Z (2010) Approximation and monotonicity of the maximal invariant ellipsoid for discrete-time systems by bounded controls. IEEE Trans Autom Control 55(2):440–446

    MathSciNet  Google Scholar 

  208. Zhou B, Lam J, Duan G-R (2010) Full delayed state feedback pole assignment of discrete-time time-delay systems. Optim Control Appl Methods 31(2):155–169

    MATH  MathSciNet  Google Scholar 

  209. Zhou B, Lam J, Lin Z, Duan G-R (2010) Global stabilization and restricted tracking with bounded feedback for multiple oscillator systems. Syst Control Lett 59(7):414–422

    MATH  MathSciNet  Google Scholar 

  210. Zhou B, Duan G-R, Lin Z (2011) A parametric periodic Lyapunov equation with application in semi-global stabilization of discrete-time periodic systems subject to actuator saturation. Automatica 47(2):316–325

    MATH  MathSciNet  Google Scholar 

  211. Zhou B, Lin Z, Duan G-R (2011) L ∞ and L 2 low gain feedback: their properties, characterizations and applications in constrained control. IEEE Trans Autom Control 56(5):1030–1045

    MathSciNet  Google Scholar 

  212. Zhou B, Lin Z, Duan G-R (2011) Lyapunov differential equation approach to elliptical orbital rendezvous with constrained controls. AIAA J Guid Control Dyn 34(2):345–358

    Google Scholar 

  213. Zhou B, Lin Z, Duan G-R (2011) Stabilization of a class of linear systems with input delay and the zero distribution of their characteristic equations. IEEE Trans Circuits Syst I Regul Pap 58(2):388–401

    MathSciNet  Google Scholar 

  214. Zhou B, Lin Z, Duan G-R (2011) A truncated prediction approach to stabilization of linear systems with long time-varying input delay. In: Proceedings of the 50th IEEE conference on decision and control and European control conference, Orlando, 12–15 Dec 2011, pp 4146–4151

    Google Scholar 

  215. Zhou B, Lin Z, Duan G-R (2012) Truncated predictor feedback for linear systems with long time-varying input delays. Automatica 48(10):2387–2399

    MATH  MathSciNet  Google Scholar 

  216. Zhou B, Lin Z, Lam J (2012) Semi-global stabilization of linear time-delay systems with control energy constraint. Automatica 48(4):694–698

    MATH  MathSciNet  Google Scholar 

  217. Zhou B, Cui NG, Duan G-R (2012) Circular orbital rendezvous with actuator saturation and delay: a parametric Lyapunov equation approach. IET Control Theory Appl 6(9):1281–1287

    MathSciNet  Google Scholar 

  218. Zhou B, Li Z-Y, Lin Z (2012) Observer based output feedback control of linear systems with multiple input and output delays. In: Proceedings of the 51st IEEE conference on decision and control, Maui, 10–13 Dec 2012, pp 2370–2375

    Google Scholar 

  219. Zhou B, Gao H, Lin Z, Duan G-R (2012) Stabilization of linear systems with distributed input delay and input saturation. Automatica 48(5):712–724

    MATH  MathSciNet  Google Scholar 

  220. Zhou B, Li Z-Y, Zheng WX, Duan G-R (2012) Stabilization of some linear systems with both state and input delays. Syst Control Lett 61(10):898–998

    MathSciNet  Google Scholar 

  221. Zhou B, Cai G-B, Duan G-R (2013) Stabilization of time-varying linear systems via Lyapunov differential equations. Int J Control 86(2):332–347

    MATH  MathSciNet  Google Scholar 

  222. Zhou B, Hou M, Duan G-R (2013) L ∞ and L 2 semi-global stabilisation of continuous-time periodic linear systems with bounded controls. Int J Control 86(4):709–720

    MATH  MathSciNet  Google Scholar 

  223. Zhou B, Li Z-Y, Lin Z (2013) Stabilization of discrete-time systems with multiple actuator delays and saturations. IEEE Trans Circuits Syst I Regul Pap 60(2):389–400

    MathSciNet  Google Scholar 

  224. Zhou B, Li Z-Y, Lin Z (2013) Observer based output feedback control of linear systems with input and output delays. Automatica 49(7):2039–2052

    MathSciNet  Google Scholar 

  225. Zhou B, Li Z-Y, Lin Z (2013) Discrete-time l ∞ and l 2 norm vanishment and low gain feedback with their applications in constrained control. Automatica 49(1):111–123

    MATH  MathSciNet  Google Scholar 

  226. Zhou B, Sang Q, Lin Z (2013) Sampled-data periodic predictor feedback for linear systems with input delay. In: 2013 Chinese control conference, Xi’an, 26–28 July 2013, pp 1409–1414

    Google Scholar 

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    Bin Zhou

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Zhou, B. (2014). Global Stabilization of Planar Systems with Input Delay and Saturation. In: Truncated Predictor Feedback for Time-Delay Systems. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54206-0_6

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