Brief paperStability and reliable data reconstruction of uncertain dynamic systems over finite capacity channels☆
Introduction
Recent development in wireless communication and electronics has given birth to micro-electro-mechanical systems (MEMSs) which are small in size and communicate in short distances. These tiny embedded systems, in general, consist of sensors, a data processor, a communication unit and an actuator. They are densely deployed either inside the phenomenon or very close to it. These embedded systems collaborate with each other by exchanging control and observation signals via wireless links. However, due to the limited power of embedded components, the transmission is subject to limited capacity and noise.
In the above applications, the encoders, decoders and controllers must be designed for real-time communication and control when the communication is via limited capacity and noisy communication channels. References Charalambous and Farhadi (2008), Elia (2004), Farhadi and Charalambous (2008), Li and Baillieul (2004), Liberzon and Hespanha (2005), Martins, Dahleh, and Elia (2006), Malyavej and Savkin (2005), Matveev and Savkin (2007), Nair and Evans (2004), Nair, Evans, Mareels, and Moran (2004), Savkin and Petersen (2003), Tatikonda, Sahai, and Mitter (2004) and Yuksel and Basar (2007) are representative although not exhaustive of the recent activity addressing the above questions. They present necessary and sufficient conditions for the stability and reliable data reconstruction of dynamic systems. However, most of these publications are concerned with cases when the dynamic model and communication channel are known. In practice, uncertain dynamic systems and channels are more realistic representations of the actual problems. Only in few publications (e.g., Martins et al., 2006, Matveev and Savkin, 2007) uncertain dynamic systems are considered, in which the uncertain dynamic systems are subject to uniformly bounded disturbances. This excludes dynamic systems which are subject to deterministic or stochastic disturbances of finite energy or power, which are often dealt with using minimax techniques.
This paper addresses control over limited capacity for a class of dynamic systems described by a relative entropy constraint. Such an uncertainty description is a generalization of the sum quadratic uncertainty description considered in Moheimani, Savkin, and Petersen (1995) and Petersen and James (1996). The sum quadratic uncertainty description includes the uniformly bounded uncertainty description as a special case. Consequently, this paper complements the results of Farhadi and Charalambous (2008) by presenting an encoder, a decoder and a controller for uniform reliable data reconstruction and robust stability of an uncertain dynamic system subject to the relative entropy constraint, when it is controlled over AWGN channels. It also complements the results of Farhadi and Charalambous (2008) by calculating the robust entropy rate using stochastic dynamic programming.
The paper is organized as follows. In Section 2, the problem formulation is presented. In Section 3 we summarize the main results of Farhadi and Charalambous (2008) and we calculate the robust entropy rate using stochastic dynamic programming. Then, in Section 4, an encoder, a decoder and a robust controller are presented for uniform reliable data reconstruction and robust stability via AWGN channels. Here, it is shown that the necessary condition presented in Farhadi and Charalambous (2008) is tight. Proofs are given in the Appendix.
Section snippets
Problem formulation
Throughout we adopt the following notations. A sequence of random vectors (R.V.s) with length is denoted by for . The density function associated with the R.V. is denoted by . The conditional density function of the R.V. given R.V. is denoted by . The joint density function of the R.V.s and is denoted by . The natural logarithm is denoted by . We denote by the space of all matrices , and by the identity matrix on
Robust entropy rate — Necessary condition for reconstructability
As mentioned earlier, this paper particularly complements the results of Farhadi and Charalambous (2008) by presenting an encoder, a decoder and a controller for uniform reliable data reconstruction and robust stability over AWGN channels. Therefore, in the following section, we recall the main results of Farhadi and Charalambous (2008). Moreover, here we identify a connection between or robust control techniques and the design of an encoder and a decoder.
Let the directed information from
Encoder, decoder and stabilizing controller
In this section we present an encoder, a decoder and a robust controller for uniform mean-square reconstructability and robust stability, as described by Definition 2.1, Definition 2.2.
Consider the control/communication system of Fig. 1, where the information source is the observation sequence of the uncertain controlled dynamic system described by the relative entropy constraint (1) with the nominal system (2). For this system, the encoder and decoder are a linear encoder and a linear decoder,
Conclusion
This paper complements the results of Farhadi and Charalambous (2008) by showing that the necessary condition presented there can be tight. It also complements Matveev and Savkin (2007) and Martins et al. (2006) by considering a relative entropy uncertainty description. In this paper a connection between the existence of an encoder and a decoder (for reliable data reconstruction) and the problem was also established. The encoding, decoding and stabilizing schemes presented in this paper can
Alireza Farhadi was born in Tehran, Iran. He finished his elementary and post secondary education in Iran and received his M.S. degree in Electrical Engineering from Iran University of Science and Technology, Tehran, Iran, in 2000. He then continued his studies at the University of Ottawa, Ontario, Canada, as a Ph.D. student (2002–2007) and received his Ph.D. degree in Electrical Engineering. After graduation he worked at the University of Ottawa as a post-doctoral fellow (2008–2009). He is
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Alireza Farhadi was born in Tehran, Iran. He finished his elementary and post secondary education in Iran and received his M.S. degree in Electrical Engineering from Iran University of Science and Technology, Tehran, Iran, in 2000. He then continued his studies at the University of Ottawa, Ontario, Canada, as a Ph.D. student (2002–2007) and received his Ph.D. degree in Electrical Engineering. After graduation he worked at the University of Ottawa as a post-doctoral fellow (2008–2009). He is currently working in INRIA (the French national institute for research in computer science and control), Grenoble, France. His research interests lie in the intersection of control and communication. He has published several journal articles on this subject.
Charalambos D. Charalambous received his Electrical Engineering B.S. degree in 1987, his M.E. degree in 1988, and his Ph.D. in 1992, all from the Department of Electrical Engineering, Old Dominion University, Virginia, USA. In 2003 he joined the Department of Electrical and Computer Engineering, University of Cyprus, where he served as Associate Dean of the School of Engineering till 2009. He was an Associate Professor at University of Ottawa, School of Information Technology and Engineering, from 1999 to 2003. He served on the faculty of McGill University, Department of Electrical and Computer Engineering, as a non-tenure faculty member, from 1995 to 1999. From 1993 to 1995 he was a post-doctoral fellow at Idaho State University, Engineering Department. He is currently an associate editor of the Systems and Control Letters and IEEE Communications Letters, and from 2002 to 2004 he served as an Associate Editor of the IEEE Transactions on Automatic Control. He was a member of the Canadian Centers of Excellence through MITACS (the mathematics of information technology and complex systems), from 1998 to 2001. In 2001 he received the Premier’s Research Excellence Award of the Ontario Province of Canada.
Charalambous’ research group ICCCSystemS, Information, Communication and Control of Complex Systems, is interested in theoretical and technological developments concerning large-scale distributed communication and control systems and networks in science and engineering. These include the theory and applications of stochastic processes and systems subject to uncertainty, communication and control systems and networks, large deviations, information theory, robustness and their connections to statistical mechanics.
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The research leading to these results has received funding from the European Community’s Seventh Framework Programme (FP7/2007-2013) under grant agreement no. INFSO-ICT-223844 and the Cyprus Research Promotion Foundation under the project ARTEMIS. The material in this paper was partially presented in the Proceedings of the 2008 American Control Conference. This paper was recommended for publication in revised form by Associate Editor George Yin under the direction of Editor Ian R. Petersen. The authors would like to thank Professor N.U. Ahmed for many helpful discussions.