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Analysis and Control of Deterministic and Stochastic Dynamical Systems with Time Delay

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Complex Systems

Part of the book series: Nonlinear Physical Science ((NPS,volume 0))

Abstract

This chapter presents a comprehensive summary of recent advances in the analysis and control of time-delayed deterministic and stochastic systems. The studies of numerical methods for time-delayed systems in the mathematics literature are reviewed including a discussion of the abstract Cauchy problem for delayed differential equations. Several numerical methods for computing the response of and designing controls for time-delayed systems are presented. These include semidiscretization, continuous time approximation, lowpass filter based continuous time approximation, and continuous time approximation with Chebyshev nodes. A 1arge number of examples are presented including optimal feedback gain design, stability domains in the feedback gain space of linear time-invariant and periodic systems, optimal control, Lyapunov stability, supervisory control of systems with uncertain time delay, moment stability, Fokker-Planck-Kolmogorov equation and reliability formulation of stochastic systems.

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References

  • Ali M.S., Hou Z.K. and Noori M.N., 1998, Stability and performance of feedback control systems with time delays, Computers and Structures, 66(2–3), 241–248.

    Article  MATH  Google Scholar 

  • Atay F.M., 2002, Delayed-feedback control of oscillations in non-linear planar systerns, International Journal of Control, 75, 297–304.

    Article  MATH  MathSciNet  Google Scholar 

  • Bellen A. and Maset S., 2000, Numerical solution of constant coefficient linear delay differential equations as abstract Cauchy problems, Numerische Mathematik; 84, 351–374.

    Article  MATH  MathSciNet  Google Scholar 

  • Bellen A. and Zennaro M., 2003, Numerical Methods for Delay Differential Equations, Oxford University Press, New York.

    Book  MATH  Google Scholar 

  • Breda D., Maset S. and Vermiglio R., 2004, Computing the characteristic roots for delay differential equations, IMA Journal of Numerical Analysis, 24, 1–19.

    Article  MATH  MathSciNet  Google Scholar 

  • Breda D., Maset S. and Vermigho R., 2005, Pseudospectral differencing methods for characteristic roots of delay differential equations, SIAM Journal Science Computing, 27, 482–495.

    Article  MATH  Google Scholar 

  • Buckwar E., 2000, Introduction to the numerical analysis of stochastic delay differential equations, Journal of Computational and Applied Mathematics, 125, 297–307.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  • Butcher E. and Bobrenkov O., 2009, The chebyshev spectral continuous time approximation for periodic delay differential equations, In: Proceedings of ASME 2009 International Design Engineering Technical Conferences (IDETC) and Computers and Information in Engineering Conference (CIE), San Diego, California.

    Google Scholar 

  • Cai G. and Huang J., 2002, Optimal control method with time delay in control, Journal of Sound and Vibration, 251, 383–394.

    Article  ADS  MathSciNet  Google Scholar 

  • Cai G.P., Huang J.Z. and Yang S.X., 2003, An optimal control method for linear systems with time delay, Computers & structures, 81, 1539–1546.

    Article  MathSciNet  Google Scholar 

  • Camacho E.F., Bordons C., 1999, Model Predictive Control, Springer, New York.

    Google Scholar 

  • Cao Y.Y., Lin Z.L. and Hu T.S., 2002, Stability analysis of linear time-delay systerns subject to input saturation, IEEE Transactions on Circuits and Systems I-Fundamental Theory and Applications, 49, 233–240.

    Article  MathSciNet  Google Scholar 

  • Carnahan B., Luther H.A. and Wilkes J.O., 1969, Applied Numerical Methods, John Wiley and Sons, New York.

    MATH  Google Scholar 

  • Chen W.H., Guan Z.H. and Lu X.M., 2004, Delay-dependent output feedback guaranteed cost control for uncertain time-delay systems, Automatica, 40, 1263–1268.

    Article  MATH  MathSciNet  Google Scholar 

  • Deshmukh V., Ma H. and Butchern E.A., 2006, Optimal control of parametrically excited linear delay differential systems via Chebyshev polynomials, Optimal Control Applications and Methods, 27, 123–136.

    Article  MathSciNet  Google Scholar 

  • Deshmukh V., Butcher E.A. and Bueler E., 2008, Dimensional reduction of nonlinear delay differential equations with periodic coefficients using Chebyshev spectral collocation, Nonlinear Dynamics, 52.

    Google Scholar 

  • Dumont G.A., Elnaggar A. and Elshafei A., 1993, Adaptive predictive control of systems with time-varying time delay, International Journal of Adaptive Control and Signal Processing, 7, 91–101.

    Article  MATH  Google Scholar 

  • Elbeyli O. and Sun J.Q., 2004, On the semi-discretization method for feedback control design of linear systems with time delay, Journal of Sound and Vibration, 273, 429–440.

    Article  ADS  MathSciNet  Google Scholar 

  • Engelborghs K. and Roose D., 2002, On stability of LMS methods and characteristic roots of delay differential equations, IMA Journal of Numerical Analysis, 40, 629–650.

    MATH  MathSciNet  Google Scholar 

  • Fan K.K., Chen J.D., Lien C.H. and Hsieh J.G., 2002, Delay-dependent stability criterion for neutral time-delay systems via linear matrix inequality approach, Journal of Mathematical Analysis and Applications, 273, 580–589.

    Article  MATH  MathSciNet  Google Scholar 

  • Filipovic D., Olgac N., 1998, Torsional delayed resonator with velocity feedback, IEEE/ASME Transactions on Mechatronics, 3, 67–72.

    Article  Google Scholar 

  • Frank T.D. and Beek P.J., 2001, Stationary solutions of linear stochastic delay differential equations: Applications to biological systems, Physical Review E Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics, 64, 219, 171/1-12.

    Article  Google Scholar 

  • Franklin G.F., Powell J.D. and Emami-Naeini A., 1986, Feedback Control of Dynamic Systems, Addison-Wesley, Reading, Massachusetts.

    MATH  Google Scholar 

  • Franklin G.F., Powell J.D. and Workman M.L., 1998, Digital Control of Dynamic Systems, Addison Wesley Longman, Inc., Berkeley, California.

    Google Scholar 

  • Fridman E. and Orlov Y., 2009, Exponential stability of linear distributed parameter systems with time-varying delays, Automatica, 45, 194–201.

    Article  MATH  MathSciNet  Google Scholar 

  • Fu Y., Tian Z. and Shi S., 2003, State feedback stabilization for a class of stochastic time-delay nonlinear systems, IEEE Transactions on Automatic Control, 48, 282–286.

    Article  MathSciNet  Google Scholar 

  • Fujii H.A., Ichiki W., Suda S.I. and Watanabe T.R., 2000, Chaos analysis on librational control of gravity-gradient satellite in elliptic orbit, Journal of Guidance, Control, and Dynamics, 23, 145–146.

    Article  Google Scholar 

  • Gao H.J., Chen T.W. and Lam J., 2008, A new delay system approach to network-based control, Automatica, 44, 39–52.

    Article  MATH  MathSciNet  Google Scholar 

  • Garg N.K., Mann B.P., Kim N.H. and Kurdi M.H., 2007, Stability of a time-delayed system with parametric excitation, Journal of Dynamic Systems, Measurement, and Control, 129, 125–135.

    Article  Google Scholar 

  • Golub G.H. and Loan C.F.V., 1983, Matrix Computations, The Johns Hopkins University Press, Baltimore, Maryland.

    MATH  Google Scholar 

  • Gu K. and Niculescu S.I., 2003, Survey on recent results in the stability and control of time-delay systems, Journal of Dynamic Systems, Measurement, and Control, 125, 158–165.

    Article  Google Scholar 

  • Guillouzic S., L’Heureux I. and Longtin A., 1999, Small delay approximation of stochastic delay differential equations, Physical Review E Statistical Physics; Plasmas, Fluids, and Related Interdisciplinary Topics, 59, 3970–3982.

    Article  Google Scholar 

  • Ha C. and Ly D.L., 1996, Sampled-data system with computation time delay: optimal w-synthesis method, Journal of Guidance, Control, and Dynamics, 19, 584–591.

    Article  MATH  Google Scholar 

  • Han Q.L., 2009, A discrete delay decomposition approach to stability of linear retarded and neutral systems, Automatica, 45, 517–524.

    Article  MATH  Google Scholar 

  • He Y., Wang Q.G., Lin C. and Wu M., 2007, Delay-range-dependent stability for systems with time-varying delay, Automatica, 43, 371–376.

    Article  MATH  MathSciNet  Google Scholar 

  • Hespanha J.P., Liberzon D. and Morse A.S., 1999, Logic-based switching control of a nonholomic system with parametric modeling uncertainty, Systems & Control Letters, 38, 167–177.

    Article  MATH  MathSciNet  Google Scholar 

  • Hespanha J.P., Liberzon D. and Morse A.S, 2003, Hysteresis-based switching algorithms for supervisory control of uncertain systems, Automatica, 39, 263–272.

    Article  MATH  MathSciNet  Google Scholar 

  • Hu X.B. and Chen W.H., 2004, Model predictive control for constrained systems with uncertain state-delays, International Journal of Robust and Nonlinear Control, 14, 1421–1432.

    Article  MATH  MathSciNet  Google Scholar 

  • Insperger T. and Stepan G., 2001, Semi-discretization of delayed dynamical systerns, In: Proceedings of ASME 2001 Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Pittsburgh, Pennsylvania.

    Google Scholar 

  • Insperger T. and Stepan G., 2002, Semi-discretization method for delayed systems, International Journal for Numerical Methods in Engineering, 55, 503–518.

    Article  MATH  MathSciNet  Google Scholar 

  • Ivanescu D., Dion J.M., Dugard L. and Niculescu S.I., 2000, Dynamical compensation for time-delay systems: An LMI approach, International Journal of Robust and Nonlinear Control, 10, 611–628.

    Article  MATH  MathSciNet  Google Scholar 

  • Ji G. and Luo Q., 2006, Iterative learning control for uncertain time-delay systems. Dynamics of Continuous Discrete and Impulsive Systems — Series A Mathematical Analysis, 13, 1300–1306.

    MathSciNet  Google Scholar 

  • Jiang X.F. and Han Q.L., 2008, New stability criteria for linear systems with interval time-varying delay, Automatica, 44, 2680–2685.

    Article  MATH  MathSciNet  Google Scholar 

  • Kalmar-Nagy T., 2005, A novel method for efficient numerical stability analysis delay-differential equations, In: Proceedings of American Control Conference, Portland, Oregon, pp 2823–2826.

    Google Scholar 

  • Kao C.Y. and Rantzer A, 2007, Stability analysis of systems with uncertain timevarying delays, Automatica, 43, 959–970

    Article  MATH  MathSciNet  Google Scholar 

  • Kapila V. and Haddad W.M., 1999, Robust stabilization for systems with parametric uncertainty and time delay, Journal of the Franklin Institute, 336, 473–480.

    Article  MATH  MathSciNet  Google Scholar 

  • Kim J.H., 2008, Improved ellipsoidal bound of reachable sets for time-delayed linear systems with disturbances, Automatica, 44, 2940–2943.

    Article  MATH  Google Scholar 

  • Klein E.J. and Ramirez W.F., 2001, State controllability and optimal regulator control of time-delayed systems, International Journal of Control, 74, 281–289.

    Article  MATH  MathSciNet  Google Scholar 

  • Kolmanovskii V.B. and Richard J.P., 1999, Stability of some linear systems with delays, IEEE Transactions on Automatic Control, 44, 984–989.

    Article  MATH  MathSciNet  Google Scholar 

  • Koto T., 2004, Method of lines approximations of delay differential equations, Computers and Mathematics with Applications, 48, 45–59.

    Article  MATH  MathSciNet  Google Scholar 

  • Koto T., 2009, Stability of implicit-explicit linear multistep methods for ordinary and delay differential equations, Frontiers of Mathematics in China, 4, 113–129.

    Article  MATH  MathSciNet  Google Scholar 

  • Kuchler U. and Platen E., 2002, Weak discrete time approximation of stochastic differential equations with time delay, Mathematics and Computers in Simulation, 59, 497–507.

    Article  MathSciNet  Google Scholar 

  • Kwon O.M., Park J.H. and Lee S.M., 2008, On delay-dependent robust stability of uncertain neutral systems with interval time-varying delays, Applied Mathematics and Computation, 203, 843–853.

    Article  MATH  MathSciNet  Google Scholar 

  • Kwon W. and Pearson A., 1980, Feedback stabilization of linear systems with delayed control. IEEE Transactions on Automatic Control, 25, 266–269.

    Article  MATH  MathSciNet  Google Scholar 

  • Kwon W.H., Lee G.W. and Kim S.W., 1990, Performance improvement using time delays in multivariable controller design, International Journal of Control, 52, 1455–1473.

    Article  MATH  Google Scholar 

  • Leugering G., 2000, On the semi-discretization of optimal control problems for networks of elastic strings: global optimality systems and domain decomposition, Journal of Computational and Applied Mathematics, 120, 133–157.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  • Lewis F.L. and Syrmos V.L., 1995, Optimal Control, John Wiley and Sons, New York.

    Google Scholar 

  • Li Z., Ye L. and Liu Y., 1989, Unconditional stability of discrete systems with any time delay, Advances in Modelling and Simulation, 17, 11–18.

    Google Scholar 

  • Lin C.L., Chen C.H. and Huang H.C., 2008, Stabilizing control of networks with uncertain time varying communication delays, Control Engineering Practice, 16, 56–66.

    Article  Google Scholar 

  • Lin Y.K. and Cai G.Q., 1995, Probabilistic Structural Dynamics — Advanced Theory and Applications, McGraw-Hill, New York.

    Google Scholar 

  • Ma H., Butcher E.A. and Bueler E., 2003, Chebyshev expansion of linear and piecewise linear dynamic systems with time delay and periodic coefficients under control excitations, Journal of Dynamic Systems, Measurement, and Control, 125, 236–243.

    Article  Google Scholar 

  • Ma H., Deshmukh V., Butcher E.A. and Averina V., 2005, Delayed state feedback and chaos control for time periodic systems via a symbolic approach, Communications in Nonlinear Science and Numerical Simulation, 10, 479–497.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  • Maset S, 2003, Numerical solution of retarded functional differential equations as abstract Cauchy problems, Journal of Computational and Applied Mathematics; 161, 259–282.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  • Miller D.E. and Davison D.E., 2005, Stabilization in the presence of an uncertain arbitrarily large delay, IEEE Transactions on Automatic Control, 50, 1074–1089.

    Article  MathSciNet  Google Scholar 

  • Morse A.S., 1996, Supervisory control of families of linear set-point controllers — Part 1: Exact matching, IEEE Transactions on Automatic Control, 41, 1413–1431.

    Article  MATH  MathSciNet  Google Scholar 

  • Morse A.S., 1997, Supervisory control of families of linear set-point controllers — Part 2: Robustness, IEEE Transactions on Automatic Control, 42, 1500–1515.

    Article  MATH  MathSciNet  Google Scholar 

  • Niculescu S.I., Verriest E.I., Dugard L. and Dion J.M., 1998, Stability of linear systems with delayed state: A guided tour. In: Proceedings of the IFAC Workshop: Linear Time Delay Systems, Grenoble, France, pp 31–38.

    Google Scholar 

  • Nohmi M. and Matsumoto K., 2002, Teleoperation of a truss structure by force command in ets-vii robotics mission, AIAA Journal, 40, 334–339.

    Article  ADS  Google Scholar 

  • Normey-Rico O.E. and Camacho E.F., 1999, Robustness effects of a prefilter in a smith predictor-based generalized predictive controller, IEE Proceedings: Control Theory and Applications, 146, 179–185.

    Article  Google Scholar 

  • de Oliveira MC and Geromel JC, 2004, Synthesis of non-rational controllers for linear delay systems, Automatica, 40, 171–188.

    Article  MATH  Google Scholar 

  • Pfeiffer B.M. and Marquardt W., 1996, Symbolic semi-discretization of partial differential equation systems, Mathematics and Computers in Simulation, 42, 617–628.

    Article  MATH  MathSciNet  Google Scholar 

  • Pinto O.C. and Goncalves P.B., 2002, Control of structures with cubic and quadratic non-linearities with time delay consideration, Journal of the Brazilian Society of Mechanical Sciences, 24, 99–104.

    Article  Google Scholar 

  • Rawlings J.B., 2000, Tutorial overview of model predictive control, IEEE Control. Systems Magazine, 20, 38–52.

    Article  Google Scholar 

  • Shanmugathasan N. and Johnston R.D., 1988, Exploitation of time delays for improved process control, International Journal of Control, 48, 1137–1152.

    Article  MATH  Google Scholar 

  • Shao H.Y., 2008, Improved delay-dependent stability criteria for systems with a delay varying in a range, Automatica, 44, 3215–3218.

    Article  MATH  Google Scholar 

  • Sheng J. and Sun J.Q., 2005, Feedback controls and optimal gain design of delayed periodic linear systems, Journal of Vibration and Control, 11, 277–294.

    Article  MATH  MathSciNet  Google Scholar 

  • Sheng J., Elbeyli O. and Sun J.Q., 2004, Stability and optimal feedback controls for time-delayed linear periodic systems, AIAA Journal, 42, 908–911.

    Article  ADS  Google Scholar 

  • Singh T., 1995, Fuel/time optimal control of the benchmark problem, Journal of Guidance, Control, and Dynamics, 18, 1225–1231.

    Article  MATH  Google Scholar 

  • Slotine J.J.E. and Li W., 1991, Applied Nonlinear Control, Prentice Hall, New Jersey.

    MATH  Google Scholar 

  • Smith O.J.M., 1957, Closer control of loops with dead time, Chemical Engineering Progress, 53, 217–219.

    Google Scholar 

  • Stepan G., 1998, Delay-differential equation models for machine tool chatter, In: Moon FC (ed.) Dynamics and Chaos in Manufacturing Processes, Wiley, New York, pp 165–192.

    Google Scholar 

  • Suh I.H. and Bien Z., 1979, Proportional minus delay controller, IEEE Transactions; on Automatic Control, 24, 370–372.

    Article  MATH  MathSciNet  Google Scholar 

  • Sun I.H. and Bien Z., 1980, Use of time-delay actions in the controller design, IEE Transactions on Automatic Control, 25, 600–603.

    Article  Google Scholar 

  • Sun J.Q., 2006, Stochastic Dynamics and Control, Elsevier Science, Ltd., Oxford, U.K.

    Google Scholar 

  • Sun J.Q., 2009, A method of continuous time approximation of delayed dynamical systems, Communications in Nonlinear Science and Numerical Simulation, 14, 998–1007.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  • Tallman G.H. and Smith O.J.M., 1958, Analog study of dead-beat posicast control, IEEE Transactions on Automatic Control, 3, 14–21.

    Google Scholar 

  • Vijta M., 2000, Some remarks on the Padé-approximations, In: Proceedings of the 3rd TEMPUS-INTCOM Symposium, Veszpré., Hungary, pp 1–6

    Google Scholar 

  • Wu H. and Mizukami K., 1995, Robust stability criteria for dynamical systems including delayed perturbations, IEEE Transactions on Automatic Control, 40, 487–490.

    Article  MATH  MathSciNet  Google Scholar 

  • Xia Y.Q. and Jia Y.M., 2003, Robust control of state delayed systems with polytopic type uncertainties via parameter-dependent lyapunov functionals, Systems & Control Letters, 50, 183–193.

    Article  MATH  MathSciNet  Google Scholar 

  • Xiao H. and Liu Y, 1994, The stability of linear time-varying discrete systems with tIme-delay, Journal of Mathematical Analysis and Applications, 188, 66–77.

    Article  MATH  MathSciNet  Google Scholar 

  • Yang B. and Wu X., 1998, Modal expansion of structural systems with time delays, AIAA Journal, 36, 2218–2224.

    Article  ADS  Google Scholar 

  • Yue D., Tian E., Zhang Y. and Peng C., 2009, Delay-distribution-dependent robust stability of uncertain systems with time-varying delay, International Journal of Robust and Nonlinear Control, 19, 377–393.

    Article  MATH  MathSciNet  Google Scholar 

  • Zhang X.P., Tsiotras P. and Knospe C., 2002, Stability analysis of lpv time-delayed systems, International Journal of Control, 75, 538–558.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Engineering, University of California, Merced, CA, 9533, USA

    Jian-Qiao Sun & Bo Song

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  1. Jian-Qiao Sun
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  2. Bo Song
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Editors and Affiliations

  1. Department of Mechanical and Industrial Engineering, Southern Illinois University, Edwardsville, Edwardsville, IL, 62026-1805, USA

    Albert C. J. Luo

  2. University of California, Merced, 5200 N. Lake Road, P.O. Box 2039, Merced, CA, 95344, USA

    Jian-Qiao Sun

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Sun, JQ., Song, B. (2011). Analysis and Control of Deterministic and Stochastic Dynamical Systems with Time Delay. In: Luo, A.C.J., Sun, JQ. (eds) Complex Systems. Nonlinear Physical Science, vol 0. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-17593-0_4

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