Brief paperSynchronization in networks of identical linear systems☆
Introduction
In recent years, consensus, coordination and synchronization problems have been popular subjects in systems and control, motivated by many applications in physics, biology, and engineering. These problems arise in multi-agent systems with the collective objective of reaching agreement about some variables of interest.
In the consensus literature, the emphasis is on the communication constraints rather than on the individual dynamics: the agents exchange information according to a communication graph that is not necessarily complete, nor even symmetric or time-invariant, but, in the absence of communication, the agreement variables usually have no dynamics. It is the exchange of information only that determines the time-evolution of the variables, aiming at asymptotic synchronization to a common value. The convergence of such consensus algorithms has attracted much attention in recent years. It only requires a weak form of connectivity for the communication graph (Jadbabaie et al., 2003, Moreau, 2005, Moreau, 2004, Olfati-Saber and Murray, 2004).
In the synchronization literature, the emphasis is on the individual dynamics rather than on the communication limitations: the communication graph is often assumed to be complete (or all-to-all), but in the absence of communication, the time-evolution of the systems’ variables can be oscillatory or even chaotic. The system dynamics can be modified through the information exchange, and, as in the consensus problem, the goal of the interconnection is to reach synchronization to a common solution of the individual dynamics (Hale, 1996, Pham and Slotine, 2007, Pogromsky, 1998, Stan and Sepulchre, 2007).
Coordination problems encountered in the engineering world can often be rephrased as consensus or synchronization problems in which both the individual dynamics and the limited communication aspects play an important role. Designing interconnection control laws that can ensure synchronization of relevant variables is therefore a control problem that has attracted quite some attention in recent years (Nair and Leonard, 2008, Sarlette et al., 2007, Scardovi et al., 2008, Scardovi et al., 2007, Sepulchre et al., 2008).
The present paper deals with a fairly general solution of the synchronization problem in the linear case. Assuming identical individual agents dynamics each described by the linear state-space model , the main result is the construction of a dynamic output feedback controller that ensures exponential synchronization to a solution of the linear system under the following assumptions: (i) has no exponentially unstable modes, (ii) is stabilizable and is detectable, and (iii) the communication graph is uniformly connected. The result can be interpreted as a generalization of classical consensus algorithms, studied recently, corresponding to the particular case (Moreau, 2005, Moreau, 2004). The generalization includes the non-trivial examples of synchronizing harmonic oscillators and chains of integrators. In turn, these models, are applicable to distributed coordination of interconnected mobile systems. Models of vehicles often require to take into account second-order dynamics that can be feedback linearized as double integrators. Furthermore clocks synchronization problems have been reduced to synchronization of double integrator models (Carli, Chiuso, Schenato, & Zampieri, 2008).
The proposed dynamic controller structure proposed in this paper differs from the static diffusive coupling often considered in the synchronization literature, which requires more stringent assumptions on the communication graph. For instance, the results in the recent papers (Carli et al., 2008, Ren, 2008, Tuna, 2008) on synchronization of harmonic oscillators and double integrators, assume a time-invariant communication topology. The idea of designing dynamic controllers for synchronization of networked systems was recently proposed in Scardovi et al. (2007) and has been applied to stabilize planar and three-dimensional collective motion (Scardovi et al., 2008, Sepulchre et al., 2008).
The paper also summarizes several sufficient conditions for synchronization by static diffusive coupling and illustrates on simple examples that synchronization may fail under diffusive coupling when the stronger assumptions on the communication graph are not satisfied.
The paper is organized as follows. In Section 2 the notation used throughout the paper is summarized, some preliminary results are reviewed and the synchronization problem is introduced and defined. In Section 3 the main result is presented. In Section 4 we derive the discrete-time counterpart of the main result. Finally, in Section 5, two-dimensional examples are reported to illustrate the role of the proposed dynamic controller in situations where static diffusive coupling fails to achieve synchronization.
Section snippets
Preliminaries
Throughout the paper we will use the following notation. Given vectors we indicate with the stacking of the vectors, i.e. . We denote with the -dimensional diagonal matrix and we define . Given two matrices and we denote their Kronecker product with . For notational convenience, we use the convention and . For a comprehensive list of properties of the Kronecker product the reader is referred to (Horn & Johnson, 1994).
Main result
Before stating the main result we introduce a preliminary Lemma that is a direct extension of Theorem 1 when, in (5), and are nonsingular (square) matrices.
Lemma 1 Consider the linear systems(5). Letandbenonsingular matrices and assume that all the eigenvalues ofbelong to the imaginary axis. Assume that the communication graphis uniformly connected and the corresponding Laplacian matrixpiecewise continuous and bounded. Then the control law
Discussion and extensions
For the sake of completeness, in this section we extend the obtained results to address synchronization of discrete-time linear systems. Then we briefly discuss the existing results (providing also a simple extension) about synchronization under (static) diffusive output coupling.
Examples
The conditions of Theorem 4 are only sufficient conditions for exponential synchronization under diffusive coupling. We provide two simple examples to illustrate that these conditions are not far from being necessary when considering time-varying and directed graphs and that the internal model of the dynamic controller (7) plays an important role in such situations.
Example 1 (Synchronization of Harmonic Oscillators) Consider a group of harmonic oscillators for ,
Conclusions
In this paper the problem of synchronizing a network of identical linear systems described by the state-space model has been addressed. A dynamic controller ensuring exponential convergence of the solutions to a synchronized solution of the decoupled systems is provided assuming that (i) has no exponentially unstable modes, (ii) is stabilizable and is detectable, and (iii) the communication graph is uniformly connected. Stronger conditions are shown to be sufficient (and,
Luca Scardovi received his engineering degree and Ph.D. degree in Electronic and Computer Engineering from the University of Genoa, Italy, in 2001 and 2005 respectively. In 2005 he was Adjunct Professor at the University of Salento, Italy. He held research associate positions at the Department of Electrical Engineering and Computer Science at the University of Liège, Belgium (2005–2007) and at the Department of Mechanical and Aerospace Engineering at Princeton University, USA (2007–2009). He is
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Luca Scardovi received his engineering degree and Ph.D. degree in Electronic and Computer Engineering from the University of Genoa, Italy, in 2001 and 2005 respectively. In 2005 he was Adjunct Professor at the University of Salento, Italy. He held research associate positions at the Department of Electrical Engineering and Computer Science at the University of Liège, Belgium (2005–2007) and at the Department of Mechanical and Aerospace Engineering at Princeton University, USA (2007–2009). He is currently an Assistant Professor in the Department of Electrical Engineering and Information Technology at the Technische Universität München (TUM), Germany. His research interests focus on dynamical systems with special emphasis in analysis and control of complex networked systems.
Rodolphe Sepulchre is Professor in the Department Electrical Engineering and Computer Science, Université de Liège. He received his engineering degree and Ph.D. degree in applied mathematics from the University of Louvain, Belgium, in 1990 and 1994 respectively. He held research or teaching positions at the University of California, Santa Barbara (1994–1996), Princeton (2002–2003), and Louvain (2001–2009). His research interests include nonlinear control systems, dynamical systems, and optimization on manifolds. He is the recipient of the 2008 IEEE Antonio Ruberti prize.
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The work is supported in part by ONR grants N00014–02–1–0826 and N00014–04–1–0534. The material in this paper was partially presented at 47th IEEE Conference on Decision and Control. This paper was recommended for publication in revised form by Associate Editor Ben M. Chen under the direction of Editor Ian R. Petersen. This paper presents research results of the Belgian Network DYSCO (Dynamical Systems, Control, and Optimization), funded by the Interuniversity Attraction Poles Programme, initiated by the Belgian State, Science Policy Office. The scientific responsibility rests with its authors.