Zeros of networked systems with time-invariant interconnections☆
Introduction
Recent developments of enabling technologies such as communication systems, cheap computation equipment and sensor platforms have given great impetus to the creation of networked systems. Thus, this area has attracted significant attention worldwide and researchers have studied networked systems from different perspectives (see e.g. Olfati-Saber & Murray, 2002, Sinopoli, Sharp, Schenato, Schaffert, & Sastry, 2003, Tanner, Jadbabaie, & Pappas, 2003 and Zamani & Lin, 2009). In particular, in view of the recent chain of events (Falliere et al., 2011, Gorman, 2009, Rid, 2012), the issues of security and cyber threats to the networked systems have gained growing attention. This paper uses system theoretic approaches to deal with problems involved with the security of networks.
Recent works have shown that control theory can be used as an effective tool to detect and mitigate the effects of cyber attacks on the networked systems; see for example Amin, Cárdenas, and Sastry (2009), Cárdenas et al. (2011), Gupta, Langbort, and Basar (2010), Mo et al. (2012), Sridhar, Hahn, and Govindarasu (2012), Teixeira, Shames, Sandberg, and Johansson (2012) and the references listed therein. The authors of Teixeira et al. (2012) have introduced the concept of zero-dynamics attacks and shown how attackers can use knowledge of networks’ zeros to produce control commands such that they are not detected as security threats.1 They have further shown that zeros of networks provide valuable information relevant to detecting cyber attacks. The authors in Teixeira et al. (2012) were more concerned with mitigating such attacks and did not provide a detailed discussion about zeros of the networked systems. In addition to this, even though various aspects of the dynamics of networked systems have been extensively studied in the literature, see e.g. Fax and Murray (2004), Olfati-Saber, Fax, and Murray (2007) and Ren, Beard, and Atkins (2007), to the authors’ best knowledge, the zeros of networked systems have not been studied in any detail except in Zamani, Helmke, and Anderson (2013). The current paper establishes a link between the problem of zero-dynamics attacks and the analysis of zeros that has been recorded in Zamani et al. (2013). Furthermore, several new results are introduced in the current paper compared to its preliminary conference version including the provision of proofs of certain results which were not part of the conference version.
This paper examines the zeros of networked systems in more depth. Our focus is on networks of finite-dimensional linear discrete-time dynamical systems that arise through static interconnections of a finite number of such systems. Such models arise naturally in applications of linear networked systems, e.g. for cyclic pursuit (Marshall, Broucke, & Francis, 2004); shortening flows in image processing (Bruckstein, Sapiro, & Shaked, 1995), or for the discretization of partial differential equations (Brockett & Willems, 1974).
Our ultimate goal is to analyze the zeros of networked systems with periodic, or more generally time-varying interconnection topology. An important tool for this analysis is blocking or lifting technique for networks with time-invariant interconnections. Note that blocking of linear time-invariant systems is useful if not standard in design of controllers for linear periodic systems as shown by Chen and Francis (1995) and Khargonekar, Poolla, and Tannenbaum (1985). The authors of Bolzern, Colaneri, and Scattolini (1986) and Grasselli and Longhi (1988) have examined zeros of blocked systems obtained from blocking of time-invariant systems. Their works have been extended in Chen, Anderson, Deistler, and Filler (2012) and Zamani, Chen, Anderson, Deistler, and Filler (2011). However, these earlier contributions do not take any underlying network structure into consideration. In this paper, we introduce some results that provide a first step in that direction.
It is worthwhile noting the blocking technique has been used in the networked systems literature for both control and identification purposes. For instance, the authors in Haber and Verhaegen (2014) have exploited this technique to identify the system parameters in a networked system via the subspace approach. The same set of authors have employed the blocking technique to study moving horizon estimation problem for networked systems (Haber & Verhaegen, 2013). In Montestruque and Antsaklis (2006) the authors have utilized the blocking technique to provide a sufficient and necessary condition for stability of a class of networked systems with communication bandwidth limitation. A similar problem has been addressed in Garcia and Antsaklis (2010) using the blocking.
The structure of this paper is as follows. First, in Section 2 we introduce state-space and higher order polynomial system models for time-invariant networks of linear systems. A central result used is the strict system equivalence between these different system representations. Moreover, we completely characterize both finite and infinite zeros of arbitrary heterogeneous networks. For homogeneous networks of identical SISO systems more explicit results are provided in Section 3. Homogeneous networks with a circulant coupling topology are studied as well. In Section 4, a relation between the transfer function of the blocked system and the transfer function of the associated unblocked system is explained. We then relate the zeros of blocked networked systems to those of the corresponding unblocked systems, generalizing work in Chen et al. (2012), Zamani, Anderson, Helmke, and Chen (2013) and Zamani et al. (2011). Finally, Section 5 provides the concluding remarks.
Section snippets
Problem statement and preliminaries
We consider networks of linear systems, coupled through constant interconnection parameters. Each agent is assumed to have the state-space representation as a linear discrete-time system Here, , and are the associated system matrices. We assume that each system is reachable and observable and that the agents are interconnected by static coupling laws with , and
Zeros of homogeneous networks
The preceding result has a nice simplification in the case of homogeneous networks of SISO agents, i.e. where the node systems are single input single output systems with identical transfer function. Let us define the interconnection transfer function as
The next theorem relates the zeros of the system (5) to those of the interconnection dynamics3
Zeros of blocked networked systems
The technique of blocking or lifting a signal is well-known in systems and control (Chen & Francis, 1995) and signal processing (Vaidyanathan, 1993). In systems theory, this method has been mostly exploited to transform linear discrete-time periodic systems into linear time-invariant systems in order to apply the well-developed tools for linear time-invariant systems; see Bittanti and Colaneri (2009) and the literature therein. As we mentioned in the introduction section, the blocking technique
Conclusions
In this paper, we explored the zeros of networks of linear systems. It was assumed that the interaction topology is time-invariant. The zeros were characterized for both homogeneous and heterogeneous networks. In particular, it was shown that for homogeneous networks with full rank direct feedthrough matrix, the finite zeros of the whole network are exactly the preimages of interconnection dynamics zeros under the inverse of an agent transfer function. We then discussed the condition under
Mohsen Zamani was born in Shiraz, Iran. He did both his undergraduate and graduate studies in Electrical Engineering at the Shiraz University of Technology in 2007 and the National University of Singapore in 2009, respectively. He obtained his Ph.D. from the Australian National University in 2014. He then joined School of Electrical Engineering and Computer Science, University of Newcastle, Australia as a post-doctoral researcher. His research interests include control and estimation for
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Mohsen Zamani was born in Shiraz, Iran. He did both his undergraduate and graduate studies in Electrical Engineering at the Shiraz University of Technology in 2007 and the National University of Singapore in 2009, respectively. He obtained his Ph.D. from the Australian National University in 2014. He then joined School of Electrical Engineering and Computer Science, University of Newcastle, Australia as a post-doctoral researcher. His research interests include control and estimation for cyber–physical systems and econometric modeling.
Uwe Helmke studied mathematics and physics at the University of Bremen in Germany (Master 1979, Ph.D. 1983). Between 1983 and 1995 he was a member of the real algebraic geometry group of M. Knebusch at the university of Regensburg. Since 1995 he is Full Professor in Mathematics at the University of Würzburg, where holds the chair in dynamical systems and control theory. He is also the founding director of the Interdisciplinary Research Center for Science and Technology (IFZM) at the University of Würzburg. His current research interest are in the application of differential geometric methods to distributed computing control and the analysis of networks of systems. He is a member of the Bavarian Academy of Sciences and a Fellow of the IEEE.
Brian D. O. Anderson was educated at Sydney and Stanford Universities in mathematics and electrical engineering. He is a Distinguished Professor at the Australian National University and Distinguished Researcher in National ICT Australia. He received the IFAC Quazza Medal in 1999 and an Automatica best paper prize, as well as a number of other awards and medals from IEEE, IEE, the Institution of Engineers, Australia and the Australian Academy of Science. He is a Fellow of the Australian Academy of Science, the Australian Academy of Technological Sciences and Engineering, the Royal Society, and a foreign member of the US National Academy of Engineering. He holds honorary doctorates from a number of universities, including Université Catholique de Louvain, Belgium, and ETH, Zürich. He held various IFAC offices including the presidency (1990–1993) and was also president of the Australian Academy of Science (1998–2002). His current research interests are in distributed control, sensor networks and econometric modeling. He holds awards from the Australian and Japanese governments.
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The material in this paper was partially presented at the 52nd IEEE Conference on Decision and Control, December 10–13, 2013, Florence, Italy. This paper was recommended for publication in revised form by Associate Editor Constantino M. Lagoa under the direction of Editor Richard Middleton.