Elsevier

Automatica

Volume 39, Issue 10, October 2003, Pages 1667-1694
Automatica

Time-delay systems: an overview of some recent advances and open problems

https://doi.org/10.1016/S0005-1098(03)00167-5Get rights and content

Abstract

After presenting some motivations for the study of time-delay system, this paper recalls modifications (models, stability, structure) arising from the presence of the delay phenomenon. A brief overview of some control approaches is then provided, the sliding mode and time-delay controls in particular. Lastly, some open problems are discussed: the constructive use of the delayed inputs, the digital implementation of distributed delays, the control via the delay, and the handling of information related to the delay value.

Section snippets

Some motivations for investigating TDS

Time-delay systems (shortly, TDS) are also called systems with aftereffect or dead-time, hereditary systems, equations with deviating argument or differential-difference equations. They belong to the class of functional differential equations (FDEs) which are infinite dimensional, as opposed to ordinary differential equations (ODEs). It is not the place here to recall the great number of monographs devoted to this field of active research (at least 30 English-language books since 1963); the

Forward and backward solutions of retarded differential equations

A classical hypothesis in the modelling of physical processes is to assume that the future behavior of the deterministic system can be summed up in its present state only. In the case of ODEs, the state is a n-vector x(t) moving in Euclidean space Rn. Now, if one has to take into account an irreducible influence of the past, leading to the introduction of a deviated time-argument, then the state cannot anymore be a vector x(t) defined at a discrete value of time t. For instance, the simple

Rational approximations

The main interest for approximating a delay with some rational approximation lies in the hope of treating an infinite-dimensional system like a finite-dimensional one. Approximating delays by means of rational transfer functions (“shift-operator” techniques) generally involves the truncation of some infinite series, and can be achieved via the following approximation:e−hsp(−hs)p(hs),where p∈R[hs] is an appropriate polynomial without any zero in the right half-plane Res⩾0. Let us mention in

Modelling in connection with stability studies

This section presents some links between the stability analysis of delay systems and the way one transforms their state-space representation.

A brief overview

Since the Smith “posicast control” (Smith, 1957) and predictor (Smith, 1959) in the late 1950s, the control via or of delay systems has been widely considered. Many of the techniques were based on approximation methods, which are not necessarily convenient when significant uncertainties—including delay variations—are involved. During the last 10 years, many studies were devoted to the control of TDS: Table 3, provides some references. Of course, if the control can be studied with some success

Using the delayed inputs

As mentioned previously, many results have been published about the control of systems with state delays but without input or output delays. They lead to “memoryless controllers”, which means control laws of the form u(t)=Kx(t), or to more general controllers with memory that include, nevertheless, an instantaneous feedback term (for example: u(t)=Kx(t)+∑iKix(t−hi)).

But a much more difficult and challenging question is to control a process without instantaneous measurement access to state

Conclusion

Faced with the wide number of results connected with delay systems, we hope that this overview has provided some enlightenment to the matter. To conclude, let us stress some of the main lines:

  • Delay systems constitute a good compromise between the too simple models with finite dimension and the great complexity of PDEs. The behavior features and the structural characteristics of delay systems are particular enough to justify specific techniques.

  • In what concerns robust stability, the main

Acknowledgements

In 1993, several teams in France initiated a cooperation network on delay systems, supported by French CNRS (“GdR Automatique”) since 1999. For their fruitful discussions, the author would like to acknowledge all colleagues related to this group, as well as those involved in the current 2002–2004 NSF-CNRS program (managed by S.I. Niculescu, Fr. and K. Gu, USA). The author is also grateful to the referees for helping to complete the scope of this survey. Special and warm thanks are addressed to

Jean-Pierre Richard is Professor at the Ecole Centrale de Lille, France (French “Grande Ecole”). His major research fields are delay systems, stabilization and control of continuous systems (linear/nonlinear), with applications to Transportations and Sciences & Technologies of Information and Communication. He is heading the team “Nonlinear and Delay Systems” (http://syner.free.fr/) of the LAIL (Lab. of Aut. control & Computer Sc. for Industry, CNRS UMR 8021).

Born in 1956, he obtained his D.Sc.

References (271)

  • J.H. Ge et al.

    Robust H state feedback control for linear systems with state delay and parameter uncertainty

    Automatica

    (1996)
  • F. Gouaisbaut et al.

    Robust control of systems with variable delayA sliding mode control design via LMIs

    System and Control Letters

    (2002)
  • A. Goubet-Bartholomeus et al.

    Stability of perturbed systems with time-varying delay

    Systems and Control Letters

    (1997)
  • Abdallah, C., Birdwell, J. D., Chiasson, J., Chupryna, V., Tang, Z., & Wang, T. (2001). Load balancing instabilities...
  • Abdallah, C., & Chiasson, J. (2001). Stability of communication networks in the presence of delays. In Third IFAC...
  • Abdallah, G., Dorato, P., Benitez-Read, J., & Byrne, R. (1993). Delayed positive feedback can stabilize oscillatory...
  • W. Aernouts et al.

    Delayed control of a Moore-Greitzer axial compressor model

    International Journal of Bifurcation and Chaos

    (2000)
  • Aggoune, W. (1999). Contribution à la Stabilisation de Systèmes Non Linéaires: Application aux Systèmes Non Réguliers...
  • Akian, M., Bliman, P. A., & Sorine, M. (1998). P.I. control of nonlinear oscillations for a system with delay. Research...
  • Al-Amer, S. H., & Al-Sunni, F. M. (2000). Approximation of time-delay systems. In ACC’00 (American control conference),...
  • Z. Artstein

    Linear systems with delayed controlsA reduction

    IEEE Transactions on Automatic Control

    (1982)
  • K.J. Åström et al.

    A new Smith predictor for controlling a process with an integrator and long deadtime

    IEEE Transactions on Automatic Control

    (1994)
  • S.P. Banks

    Nonlinear delay systems, Lie algebras and Lyapunov transformations

    IMA Journal of Mathematical Control and Information

    (2002)
  • A. Bartholoméüs et al.

    Bounded domains and constrained control of linear time-delays systems

    JESA, European Journal of Automatic Systems

    (1997)
  • C. Battle et al.

    On the approximation of delay elements by feedback

    Automatica

    (2000)
  • A. Beghi et al.

    Approximating delay elements by feedback

    IEEE Transactions on Circuits and Systems

    (1997)
  • Belkoura, L., Dambrine, M., Richard, J.-P., & Orlov, Y. (1998). Sliding mode on-line identification of delay systems....
  • Belkoura, L., Richard, J. P., & Orlov, Y. (2000). Identifiability of linear time delay systems. In Second IFAC workshop...
  • Bellen, A., & Zennaro, M. (2001). A free step-size implementation of second order stable methods for neutral delay...
  • R. Bellman et al.

    Differential difference equations

    (1963)
  • Biberovic, E., Iftar, A., & Ozbay, H. (2001). A solution to the robust flow control problem for networks with multiple...
  • C. Bonnet et al.

    Stabilization of fractional exponential systems including delays

    Kybernetika

    (2001)
  • C. Bonnet et al.

    Robust control and tracking of a delay system with discontinuous nonlinearity in the feedback

    International Journal of Control

    (1999)
  • C. Bonnet et al.

    Robust stabilization of a delay system with saturating actuator or sensor

    International Journal of Robust and Nonlinear Control

    (2000)
  • Borne, P., Dambrine, M., Perruquetti, W., & Richard, J. P. (2002). Vector Lyapunov functions: Nonlinear, time-varying,...
  • Boukas, E. K., & Liu, Z. K. (2002). Deterministic and stochastic time-delay systems. Control engineering. Basel:...
  • Brethé, D. (1997). Contribution à l'Etude de la Stabilisation des Systèmes Linéaires à Retards. IRCCyN, University of...
  • L. Bushnell

    Editorial: Networks and control

    IEEE Control System Magazine

    (2001)
  • C.I. Byrnes et al.

    A several complex variables approach to feedback stabilization of linear neutral delay-differential systems

    Mathematical Systems Theory

    (1984)
  • Y.Y. Cao et al.

    H control of uncertain markovian jump systems with time delay

    IEEE Transactions on Automatic Control

    (2000)
  • P.H. Chang et al.

    Time delay observerA robust observer for nonlinear plants

    ASME Journal of Dynamic Systems Measurement and Control

    (1997)
  • Chang, P. H., & Park, S. H. (1998). The enhanced time delay observer for nonlinear systems. In 37th IEEE CDC’98...
  • J. Chen et al.

    Frequency sweeping tests for stability independent of delay

    IEEE Transactions on Automatic Control

    (1995)
  • E. Cheres et al.

    Stabilization of uncertain dynamic systems including state delay

    IEEE Transactions on Automatic Control

    (1989)
  • Choi, H. H. (1999). An LMI approach to sliding mode control design for a class of uncertain time delay systems. In...
  • S.B. Choi et al.

    An observer-based controller design method for improving air/fuel characterization of spark ignition engines

    IEEE Transactions on Control Systems Technology

    (1998)
  • G. Conte et al.

    The disturbance decoupling problem for systems over a ring

    SIAM Journal on Control and Optimization

    (1995)
  • G. Conte et al.

    Non-interacting control problems for delay-differential systems via systems over rings

    JESA, European Journal on Automatic Systems

    (1997)
  • Conte, G., & Perdon, A. M. (1998). Systems over rings: Theory and applications. In First IFAC workshop on linear time...
  • Dambrine, M., Gouaisbaut, F., Perruquetti, W., & Richard, J.-P. (1998). Robustness of sliding mode control under delays...
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    Jean-Pierre Richard is Professor at the Ecole Centrale de Lille, France (French “Grande Ecole”). His major research fields are delay systems, stabilization and control of continuous systems (linear/nonlinear), with applications to Transportations and Sciences & Technologies of Information and Communication. He is heading the team “Nonlinear and Delay Systems” (http://syner.free.fr/) of the LAIL (Lab. of Aut. control & Computer Sc. for Industry, CNRS UMR 8021).

    Born in 1956, he obtained his D.Sc. in Physical Sciences in 1984, Ph.D. in Automatic Control in 1981, Dipl. Eng. and DBA in Electronics in 1979. He is a Member of the IEEE (Senior), of the Russian Academy of Nonlinear Sciences, of the IFAC TC2.2 Linear Systems and of the editorial board of the International Journal of Systems Science.

    He is President of the GRAISyHM (Research Group in Integrated Automation and Man-Machine Systems, 220 researchers from 10 labs of Region Nord—Pas de Calais, France) and Director of the doctoral researches training in automatic control, University of Lille and Ecole Centrale de Lille. He is in charge of several programs and research networks (CNRS and French Ministry of Research) and belongs to the Advisory Committee of the IEEE biennial conference CIFA, “Conférence Internationale Francophone d'Automatique” (http://cifa2004.ec-lille.fr/).

    A preliminary version of this paper was presented in Second IFAC Workshop on Linear Time Delay Systems (Richard, 2000). This paper was recommended for publication in revised form by Editor Manfred Morari.

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