In this paper, the problem of active clock distribution network synchronization is considered. The network is made of identical oscillators interconnected through a distributed array of phase-locked-loops (PLLs). The problem of the PLL network design is reformulated, from a control theory point of view, as a control law design for a distributed multi-agent system. Inspired by the decentralized control law design methodology using the dissipativity input-output approach, the particular topology

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... f interconnected subsystems is exploited to solve the problem by applying a convex optimization approach involving simple Linear Matrix Inequality (LMI) constraints. After choosing the dissipativity properties which is satisfied by the interconnection matrix, the constraints are transformed into an H ∞ norm constraint on a particular transfer function that must be fulfilled for global stability. Additional constraints on inputs and outputs are introduced in order to ensure the desired performance specifications during the H ∞ control design procedure. I. INTRODUCTION HE active clock distribution network ( Fig.1) can be used as an alternative way to distribute the clock in synchronous many-core microprocessor systems. It has a large number of advantages in terms of perturbation rejection, robustness properties and power consumption [1] [2] [3] [4] [5] [6] . In these systems, synchronization is crucial to ensure a correct system operation. Additionally, it is required that the generated clock signal is slightly sensitive to the external perturbations: temperature, noise and power, i.e. it possesses some signal purity properties. The network is generally composed of identical, independent voltage controlled oscillators (VCOs represented with dots in the right part of Fig. 1 ) that are spatially separated and connected in a two dimensional regular grid with phase detectors (PDs, represented with rectangles in Fig 1) which are located in between adjacent VCOs in order to compensate the clock propagation delays; and identical filters (for sake of cleanness, represented by F only in the left part of Fig 1) . In general, each oscillator can be followed by a frequency divider (1/d in the left part of Fig. 1 ) in order to distribute lower frequency clock signal throughout the network. The overall network is modelled as an interconnection of multiple input phase locked loop (PLL) nodes represented in the left part of Fig.1 with m i shared phase detectors. In general, PLLs can be digital (All Digital PLLs) [4, 7] . Depending on the application, one or several reference inputs can be introduced in the network. These reference oscillators must be synchronized and serve to enforce a common network clock frequency. Dynamically speaking, the described system has a much more complex behaviour than conventional clock distribution network. It is actually an automatic control system made of several feedback loops and caution must be taken to ensure stability of the overall system. Due to its complexity, classical microelectronics methods and tools fail to analyze and efficiently design this class of systems. In this paper, we illustrate how to apply control theory tools in order to ensure appropriate working operation of the whole system. Methods for designing a stand-alone PLL are well-known in the field of microelectronics [8, 9] . From a control system point of view, a PLL represents an LTI feedback dynamical system in the phase domain and one can design quite easily a controller ensuring good stability margins and performance requirements using standard control theory tools [8] [9] [10] [11] . However, there is no guarantee for the global network that it convergences to the "synchronous state" even though each node is identical and properly designed to ensure the local convergence on an average input signal. There is no guarantee neither that the performance will not be degraded by the network. These aspects of global network inter-connection and nodes coupling are very important and must be taken into account during the system design procedure. One of the most important results in understanding of global network dynamic or, more-generally, behavior of networked multi-agent systems is the work [12] where the authors give a necessary and sufficient stability condition for such LTI network. They use a transformation of the Laplacian interconnection matrix allowing the expression of the global stability test in the form of simultaneous stability tests of N independent systems that have different feedback gains. Since these gains are defined by Laplacian eigenvalues (in general complex), one may apply in straight manner the Nyquist stability test of the global network. The paper [12] gave rise to a large number of various papers that use the same transformation to specify the stability test based on variety of approaches such as passivity, ℒ 2 gain [13-18], more general IQC characterization [19, 20] . These papers deal with different set of additional problems such as nonlinear interconnections, effects of delay propagation and structure switching, but all of them are tools allowing only the stability analysis. However, there are two extensions to the design of a control law based on the result of [12] .