Time-delay systems: an overview of some recent advances and open problems☆
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 . 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:where is an appropriate polynomial without any zero in the right half-plane . 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: ).
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:
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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.
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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.
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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/).
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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.