Brief paperA dynamic output feedback controller for NCS based on delay estimates☆
Introduction
Network controlled systems (NCSs) are frequently encountered in practice for widespread fields of applications due to their suitable and flexible structure (Antsaklis and Baillieul, 2007, Åström and Wittenmark, 1997). Nevertheless, some aspects of NCSs, such as time-varying delays, quantization and drop-out, imply that the stability analysis and control design are becoming fundamentally more difficult to investigate (Donkers, Heemels, van de Wouw, & Hetel, 2011). They lead to a rich literature in automatic control, particularly in what concerns the stability aspects of systems with time-varying delays (Cloosterman et al., 2009, Dritsas and Tzes, 2009, Hespanha et al., 2007, Walsh et al., 2002, Zhang et al., 2001).
In order to achieve stability requirements, controller synthesis methodologies have been proposed. Among of them the use of Lyapunov functions is the most popular for stability analysis (Cloosterman et al., 2009, Walsh et al., 2002) and control synthesis (Cloosterman et al., 2010). Robust controllers that are independent of the time-varying delays may be designed to stabilize an NCS but such strategies are generally a source of conservatism. In order to reduce such a conservatism for an NCS for which the current time-varying delay is available, controllers depending on the time-varying delays may be proposed.
Recently, output dynamic feedback design for NCSs has been proposed using linear matrix inequalities (LMIs), based on the design procedure introduced in Scherer, Gahinet, and Chilali (1997) or on the elimination lemma. The output dynamic feedback may be independent with respect to the time-varying delay (Dritsas and Tzes, 2007, Fujioka, 2008, Park, 2004), or time-varying delay dependent: (Tai & Uchida, 2007) considers two delays (back and forth), (Moraes, Castelan, & Moreno, submitted for publication) a global time-varying delay less than the sampling period and (Melin, Jungers, Daafouz, & Iung, 2011) a global and bounded time-varying delay multiple of the sampling period.
Knowing the value of the current time-varying delay is in practice difficult, not to say impossible. Nevertheless, tools allowing one to obtain a time-varying delay estimate are provided in the literature (for more details see Carter, 1981, Richard, 2003). The goal of this paper is not to enrich this mature literature, but to make the best use of the information conveyed by the time-varying delay estimate in the stabilizing controller design. Such an idea was investigated in Hetel, Daafouz, Richard, and Jungers (2011) for designing a state-feedback controller dependent on this estimate by considering an additional uncertain term in the exponential uncertainty.
This note aims at extending the results in Hetel et al. (2011) by relaxing the assumption of the state availability and by proposing a dynamic output feedback controller dependent on an estimate of the time-varying delay. The main result is allowed by considering the exponential uncertainty related to the estimate of the time-varying delay decomposed into a polytopic term and an uncertain term. Via Petersen’s lemma and changes of variables similar to the one proposed by Scherer, we obtain sufficient conditions formulated as LMIs leading to the controller design ensuring the closed-loop stability of the NCS.
The note is organized as follows. In Section 2, the problem of dynamic output feedback controller design dependent on an estimate of the time-varying delay is formulated. Some preliminaries are proposed in Section 3 to allow the proof of the main result presented in Section 4. A numerical illustration is given in Section 5 to highlight the efficiency of our result compared with the literature, before some concluding remarks in Section 6.
Notation. and denote respectively the transpose and the symmetric block in a symmetric matrix. denotes the diagonal matrix composed by matrices and . The Hermitian operator is given by , for any square matrix . denotes the maximum singular value of the matrix . Matrices and are respectively the identity matrix of size and the null matrix of size . The notation and will be used. The symbol denotes an irrelevant block in a matrix.
Section snippets
Problem description
Consider the following continuous-time system: where and are respectively the continuous-time state, input and output of the system. The state is sampled with a sampling period to obtain the sampling states at time , (). Due to the NCS structure, the continuous-time input applied to the system is delayed by a global time-varying delay which is assumed to be unknown and to verify ; that is,
Preliminaries
By defining , we can decompose into an uncertain polynomial parameter dependent matrix and an uncertain bounded remainder : . Let us introduce the following lemma, proven in Jungers, Hetel, and Daafouz (2010).
Lemma 6 The uncertain polynomial parameter dependent matrix , with , can be rewritten as a convex polytope with vertices, with : , with and
Designing the dynamic output feedback controller
Inequality (13) is not linear with respect to the gain matrices of the controller, but via a change of variables, originally proposed in a linear time invariant (LTI) case in Scherer et al. (1997), this is reformulated into LMIs.
Proposition 10 Let us consider the predefined system (8) and a scalar verifying inequality (12). Assume that there exist matrices , matrices , , matrices and and symmetric positive-definite matrices
Illustration
In this section, we consider an example coming from Cloosterman et al. (2010) and revisited in Hetel et al. (2011), for which . It has been pointed out that the design of a delay-independent state-feedback controller by applying the design method proposed by Cloosterman et al. (2010) failed with ; however, the design of a state-feedback controller dependent on an estimate of the delay with succeeded via the method provided in Hetel
Conclusion
The design of a dynamic output feedback controller for a network controlled system based on the knowledge of a time-varying delay estimate has been studied in this note. As a preliminary result, the exponential uncertainty is rewritten as the sum of a polytopic term and an uncertain bounded term. With the help of Petersen’s lemma and a change of variables inspired by Scherer, a sufficient condition formulated as LMIs has been provided to design a dynamic output feedback controller stabilizing
Acknowledgments
E.B. Castelan has financial support from CNPq Brazil. This work was partially supported by the project PICS CNRS no 5284, by CAPES-COFECUB Project no 701/11 and by ANR project ArHyCo, Programme “Systèmes Embarqués et Grandes Infrastructures” — ARPEGE, contract number ANR-2008 SEGI 004 01-30011459 and finally by the European Community’s Seventh Framework Programme (FP7/2007–2013) under grant agreement no 257462: HYCON2 Network of Excellence “Highly-Complex and Networked Control Systems”.
Marc Jungers was born in Semur-En-Auxois, France, in 1978. He entered the Ecole Normale Supérieure of Cachan (ENS Cachan, France) in 1999. He received his “Agrégation” in Applied Physical Science in 2002, his Master’s degree in automatic control in 2003 from ENS Cachan and University Paris Sud, (Orsay, France) and his Ph.D. degree from ENS Cachan in September 2006. From 2003 to 2007 he was with Laboratory SATIE and Electrical Engineering Department in ENS Cachan, as a Ph.D. student and then as
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Marc Jungers was born in Semur-En-Auxois, France, in 1978. He entered the Ecole Normale Supérieure of Cachan (ENS Cachan, France) in 1999. He received his “Agrégation” in Applied Physical Science in 2002, his Master’s degree in automatic control in 2003 from ENS Cachan and University Paris Sud, (Orsay, France) and his Ph.D. degree from ENS Cachan in September 2006. From 2003 to 2007 he was with Laboratory SATIE and Electrical Engineering Department in ENS Cachan, as a Ph.D. student and then as an assistant professor. In October 2007 he joined the CNRS (National Center for Scientific Research) and the CRAN (Nancy, France) to develop research activities in control theory. His research interests include games theory, robust control, non-convex optimization, coupled Riccati equations and switched systems.
Eugênio B. Castelan was born in Criciúma (S.C.), Brazil. He received his Electrical Engineering degree, in 1982, and his M.Sc. degree, in 1985, both from UFSC, Brazil, and his Doctoral degree, in 1992, from Paul Sabatier University, France. In 1993, he joined the Department of Automation and Systems at UFSC, Brazil, where he has been developing his teaching and research activities. In 2003, he spent a year at LAAS du CNRS, France, as an invited researcher in the Group MAC. He was the chair of the Master and Doctorate Graduate Program on Automation and Systems Engineering at UFSC, from 2007 until 2010, and vice-chair from 2010 until 2012. His main research interests are on constrained control systems, control theory and control applications to mechatronics systems.
Vitor M. Moraes was born in Ijuí, Brazil. He received his Electrical Engineering degree in 2007 from UFMT, Brazil, and his M.Sc. degree in Automation and Systems Engineering in 2012 from UFSC, Brazil. He is currently a Doctoral student in Automation and Systems Engineering at UFSC, Brazil. His research interests are mainly networked control systems, nonlinear systems and robust control.
Ubirajara F. Moreno was born in S. Bernardop do Campo, Brazil, in 1971. He graduated in Industrial Electrical Engineering from the Federal Technological University of Paraná (1994), obtained his MS degree in Electrical Engineering from the Federal University of Santa Catarina (1997) and his Ph.D. degree in Electrical Engineering from Universidade Estadual de Campinas (2001). He is currently a professor at the Federal University of Santa Catarina. He has experience in Automation and Systems, working on the following topics: networked control systems, nonlinear systems, chaotic oscillators, mechatronic systems, monitoring systems and engineering education.
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The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Wei Ren under the direction of Editor Frank Allgöwer.
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