Linear Approach for Synchronous State Stability in Fully Connected PLL Networks

Synchronization is an essential feature for the use of digital systems in telecommunication networks, integrated circuits, and manufacturing automation. Formerly, master-slave MS architectures, with precise master clock generators sending signals to phase-locked loops PLLs working as slave oscillators, were considered the best solution. Nowadays, the development of wireless networks with dynamical connectivity and the increase of the size and the operation frequency of integrated circuits suggest that the distribution of clock signals could be more efficient if distributed solutions with fully connected oscillators are used. Here, fully connected networks with second-order PLLs as nodes are considered. In previous work, how the synchronous state frequency for this type of network depends on the node parameters and delays was studied and an expression for the long-term frequency was derived Piqueira, 2006 . Here, by taking the first term of the Taylor series expansion for the dynamical system description, it is shown that for a generic network with N nodes, the synchronous state is locally asymptotically stable.


Introduction
Digital engineering technologies for communications, control, and computation require reliable clock distribution systems to guarantee the correct temporal order in the information processing by the several parts of a spatially distributed system 1-4 .Synchronization network is the general denomination of the part of the whole system responsible for this temporal order and the several possible solutions for its design are presented in 5 .
Originally, master-slave architectures were used to distribute a precise clock signal generated by a master node to the other points of the systems where PLLs regenerate the phase and frequency information 1, 5 .

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The evolution of the telecommunication services to wireless and dynamical networks hasshown the inadequacy of centralized clock distribution structures in these cases, motivating the study of fully connected architectures to generate reference signals with the phase-locked loops operating as nodes of the synchronization networks 1, 5-7 .
Besides telecommunication networks, the main fields for the application of fully connected systems are time signal distribution in digital electronic circuits 2, 7-9 and wireless sensor networks 10 .Another very important application of networks is the implementation of oscillatory neural-computing devices, where the vector of phase-differences amongst a group of synchronized oscillators is associated with some memory information 11, 12 .In this work, a fully connected N-node PLL network is studied, starting with a review of the model for a single node and the derivation of the synchronous state frequency 13 .Then, the dynamical equation for the phase differences between nodes is presented providing a model for the whole N-node network.
As it was shown in 13, 14 , this kind of network presents a synchronous state that is reachable for any possible combination of node parameters.Consequently, the linear approximation gives practical hints about the lock-in operation of the network 15 .Then, considering the first term of the Taylor series development around the synchronous state 16 , it is shown that the synchronous state is locally asymptotically stable for any number N of nodes.

Nodes in a fully connected architecture and synchronous state frequency
In this section, the nodes of the fully connected network are analyzed, considering that they are second-order phase-locked loops modeled as the classical analog version 17 .The results shown in 13 are summarized and the expression of synchronous state frequency is derived.This expression will be used to determine the stability conditions.PLL nodes are closed loops composed of a phase detector PD , a low-pass filter F , and a voltage-controlled oscillator VCO 17 .The double frequency jitter in the output of the phase detector is neglected 18-20 .
The N nodes are geographically distributed and mutually coupled by using two-way connections.Each i-node exchanges clock signals with all the j-nodes such that j / i.Each VCO belonging to a PLL is described by its free-running frequency ω i and by its phase φ i t .The whole network has its dynamics described by phase errors and frequency errors defined by Δφ ji t and Δ φji t , respectively.Time delays τ ji corresponding to the propagation of signals from the node j VCO output to the node i PD input are considered, as shown in Figure 1, containing the model of the signal processed in node 1.
It can be noticed that in an N-node fully connected PLL network, the nodes need N − 1 phase detectors 21 where the local VCO output signal is multiplied by the delayed signals sent by the VCO of the other nodes.As there is no integrated circuit with this architecture, the implementation of the nodes requires a combination of several PLL chips with the outputs of their phase detectors weighted and the result being the input of a single filter that feeds a single VCO.For the node 1 from Figure 1, the definitions are as follows: i τ 21 : delay from node 2 to node 1; ii τ 31 : delay from node 3 to node 1; iii τ N1 : delay from node N to node 1; iv v 1 t : VCO output of the node 1; ix v d31 t : output of PD 3; x v dN1 t : output of PD N; xi v c1 t : output of F that controls the VCO.
The instantaneous individual VCO phases and the instantaneous phase errors, φ i t and Δφ ji t , are expressed in the forms Δφ ji t φ j t − φ i t .

2.1
The node parameters related to PLL operation can be characterized by the following: ii gains of the VCOs: iii cut-off frequencies of F: The constitutive parameters of the nodes are considered to be the same, in order to simplify the analytical reasoning.If the filters cut-off frequencies are different, but sufficient to avoid double frequency jitter 18 , the results derived here are basically the same and only the acquisition times change 13 .If VCO and PD gains are considered to be different, the problem has to be numerically treated, and the expression of the synchronous state frequency derived here changes 22 .
Under the assumption that all nodes have the same constitutive parameters, the output of each VCO is and signals received by the phase detector of node i from node j, with propagation delays τ ji , can be written as where V is the controlled amplitude of the outputs of VCOs and PDs.Considering that each phase detector j / i, belonging to node i, is a signal multiplier 17 : replacing 2.2 and 2.3 in 2.4 , and neglecting the double frequency components 18-20 as their frequency are much greater than the filter cut-off frequency, the output of each phase detector j / i, belonging to node i, is given by Each resulting signal given by 2.5 is multiplied by 1/ N − 1 and added, in order to compose the filter input as below: The filters are considered to be first-order low-pass implying second-order nodes.This choice is a common practice because second-order PLLs always reach a synchronous state when submitted to phase steps and ramps 17 in spite of the double-frequency jitter 19, 20 .If more accurate transient responses are necessary, second-order filters are used, but complicated behaviors like bifurcation and chaos appear 23 , worsening the operation.
Consequently, equations for the dynamics of the VCO phase are obtained by considering the filter transfer function F i s μ 1 / s μ 1 15 , resulting in the expression Replacing v d i , given by 2.7 , and the VCO control signal v c i θi t /k 0 in 2.8 , the equation for the node phase is Then, considering φ i t w i t θ i t in 2.9 and defining μ 2 k o k d and k μ 1 μ 2 / N − 1 , the dynamics of each VCO phase in a fully connected network is given by Equation 2.10 is similar to the pendulum equation, containing a dissipation component μ 1 φi t , a delayed conservative term k N j 1,j / i sin φ j t − τ ji − φ i t , and a forcing part μ 1 ω i 16 .Consequently, it is reasonable to suppose that the long-term solution of the system is a synchronous state, as shown in 14, 18 , with the phases of all nodes oscillating with the same frequency ω s that can be estimated.Thus, in order to estimate this frequency, the following hypotheses are considered 13 : Therefore, The values of Δφ ji − ω s τ ji are considered to be small because in the majority of the practical situations, the network is operating in the lock-in mode 17 .Consequently, 2.11 can be written as a linear approximation considering sin Δφ ji − ω s τ ji ≈ Δφ ji − ω s τ ji .Hence, for each node i, Using 2.12 for an N-node network, with i, j 1, . . ., N, and j / i, as well as adding the N resulting equations, as the sum of the terms Δφ ji is equal to zero because Δφ ji −Δφ ij , one can write Calculating ω s from 2.13 ,

2.15
Therefore, expression 2.15 is an estimation for the frequency of the synchronous state for a fully connected second-order PLL network, depending on the individual free-running frequencies and propagation delays.Notice that when the delays are zero, ω s is given by the mean value of ω i .As it was considered that all free-running frequencies are different, there are phase shifts between the nodes in the synchronous state.In practical cases, as in communication networks, these phase differences are object of delay compensation techniques 5 .
In previous work 13 , numerical simulations were conducted to investigate the accuracy of expression 2.15 and to analyze how gains and delays change the behavior of the network.Here, the dynamic equations for the phase differences are derived allowing the analytic study of the local stability of the synchronous state.

Phase difference equations
In this section, the equations describing the dynamics of the phase errors, Δφ ji t , are derived.A set of N − 1 second-order ordinary differential equations is obtained, expressing the phase differences between all the nodes and the node 1, taken as reference.
Starting with the individual dynamical equations for each VCO phase and expressing the differences between the nodes 2, 3, . . ., N and node 1, equations for Δφ j1 t , j 2, 3, . . ., N are written as below: . . .

VCOs 1 and N:
These equations give the general expression that describes the dynamical behavior of the phase differences for the network VCOs as below: sin Δφ m1 t − ω s τ m1 0.

3.5
The nonlinear differential equation 3.5 , for small-phase deviations, can be approximately expressed by the linear term of the Taylor series expansion 16 .Then, rewriting 3.5 Expressions 3.5 and 3.6 describe the dynamics of the phase adjustments of a fully connected PLL network depending on the PLL node parameters μ 1 and k, the number of nodes N, as well as the individual free-running frequencies and the delays.These equations allow the research of the synchronous state stability that is conducted in the next section.

Synchronous state stability
In Section 2, an expression for the synchronous state frequency for the fully connected network was derived and, in this section, the stability of the synchronous state is studied, that is, if the reachable synchronous state is robust under small perturbations.The analysis is performed considering that the solutions of the linear equation 3.6 can be topologically equivalent to the solutions of 3.5 in a small neighborhood of the synchronous state if it is a hyperbolic equilibrium point 16 .The procedure will be shown for three-node and four-node networks providing the identification of patterns in the expressions for the eigenvalues of the Jacobian matrix representing the linear equivalent system around the synchronous state.

Three-node network
For a three-node network, the phase differences are Δφ 12 and Δφ 13 .By using 3.5 , the system is described by 4.1 The system composed of two second-order equations 4.1 can be described by the state variables: Considering that Δφ 23 x 3 − x 1 , the following first-order equations result:

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Consequently, the Jacobian matrix J from 4.3 is The eigenvalues of J are the roots of the characteristic polynomial which have multiplicity two, and are given by 4.6

Four-node network
The calculation of the eigenvalues for the four-node network follows the same procedure of the former case.The phase differences are represented by Δφ 12 , Δφ 13 , and Δφ 14 .By using 3.5 , the system is written as

4.7
The second-order expressions 4.7 can be rewritten in terms of state variables: Considering that Δφ 23 x 3 − x 1 , Δφ 24 x 5 − x 1 , and Δφ 34 x 3 − x 5 , the following first-order equations result: 4.9 Consequently, the Jacobian matrix J from 4.9 is The eigenvalues of J are given by the roots of the characteristic polynomial: which have multiplicity 3, and are 4.12

N-node network
As it was shown, for a fully connected PLL network, the Taylor series development around the synchronous state results in Jacobian matrices with a canonical Jordan form 24 .Table 1 shows the characteristic polynomials and the expressions of the eigenvalues and their multiplicities, m, corresponding to fully connected networks dynamic equations, around the synchronous state, for different number, n, of nodes.It can be observed that for any number of nodes, the eigenvalues from Table 1 are with negative real parts, implying that the synchronous state is locally asymptotically stable.

Conclusions
As the fully connected architecture started to be used in large scale in clock distribution systems, 2.15 can be applied by network designers as an estimation for the frequency of the synchronous state when second-order PLLs are used to extract the timing information in the nodes.
Besides, because the synchronous state is a hyperbolic locally asymptotically stable equilibrium point, the network can recover synchronization after perturbation.The eigenvalues presented in Table 1 can be used for determining the transient behavior of such a resynchronization process.

Table 1 :
Eigenvalues for an N-node network.