A Distributed Consensus-Based Clock Synchronization Protocol for Wireless Sensor Networks

Abstract

The birth of computer networks and distributed systems has led to the appearance of the clock synchronization problem. This issue has gained increasing importance with the emergence of new resource constrained networks such as wireless sensor networks. In this paper, we propose a new distributed clock synchronization algorithm, referred to as Weighted Consensus Clock Synchronization (WCCS), whose objective is to achieve a consensus clock among network nodes. In this distributed approach and in contrast to centralized schemes, each node periodically exchanges the local clock reading with its immediate neighbor nodes. Then, each node employs these time informations to calculate its relative offset and skew with respect to its neighbor nodes using a weighted average consensus based technique. The effectiveness of WCCS is proved through both simulations and an experimental study on TelosB mote using TinyOS.

Appendices

Appendix 1: Proof of Lemma 1

Proof

Once node i receives a synchronization packet at time $t_1$, it tries to synchronize itself with the received virtual compensated clock ${\mathcal {C}}_j(t)$. The relative clock rate is the ratio of the virtual compensated clock frequency of node j to the local clock frequency of node i. Thus, from Eqs. (4) and (7) we have:

$${\mathcal {C}}_j(t_1)=\hat{\alpha }_{ij}\tau _i(t_1)+\hat{\beta }_{ij}$$

Subtracting $\tau _i(t_1)$ from each side of this equation leads to:

$${\mathcal {C}}_j(t_1)-\tau _i(t_1)=\hat{\alpha }_{ij}\tau _i(t_1)-\tau _i(t_1)+\hat{\beta }_{ij}$$

As a result:

$$offset ^{t_1}_{ij}=(\hat{\alpha }_{ij}-1)\tau _i(t_1)+\hat{\beta }_{ij}$$

(17)

Similarly, at time $t_2$ we have:

$$offset ^{t_2}_{ij}=(\hat{\alpha }_{ij}-1)\tau _i(t_2)+\hat{\beta }_{ij}$$

(18)

and subtracting (17) from (18) gives:

$$offset ^{t_2}_{ij}- offset ^{t_1}_{ij}=(\hat{\alpha }_{ij}-1)(\tau _i(t_2)-\tau _i(t_1))$$

(19)

Consequently,

$$\alpha _{ij}=\frac{ offset ^{t_2}_{ij}- offset ^{t_1}_{ij}}{\tau _i(t_2)-\tau _i(t_1)}$$

$\square$

Appendix 2: Proof of Theorem 1

Proof

According to Lemma 1 and Eq. (11), we have:

$$\begin{aligned} \alpha _{i}(k+1)&=\sum \limits _{j\in {\mathcal {N}}_i} w_{ij}(k)\left( \frac{ offset ^{t_2}_{ij}- offset ^{t_1}_{ij}}{\tau _i(t_2)-\tau _i(t_1)}\right) (k)\\&=\sum \limits _{j\in {\mathcal {N}}_i} w_{ij}(k)\left( \frac{({\mathcal {C}}_j(t_2)-\tau _i(t_2))-({\mathcal {C}}_j(t_1)-\tau _i(t_1))}{\tau _i(t_2)-\tau _i(t_1)}\right) (k)\\&=\sum \limits _{j\in {\mathcal {N}}_i} w_{ij}(k)\left( \frac{{\mathcal {C}}_j(t_2)-{\mathcal {C}}_j(t_1)}{\tau _i(t_2)-\tau _i(t_1)}-1\right) (k) \end{aligned}$$

By adding 1 to each side of the equation, we obtain:

$${\alpha }_{i}(k+1)+1=1+\sum \limits _{j\in {\mathcal {N}}_i} w_{ij}(k)\left( \frac{{\mathcal {C}}_j(t_2)-{\mathcal {C}}_j(t_1)}{\tau _i(t_2)-\tau _i(t_1)}-1\right) (k)$$

As $\hat{\alpha }_{i}=\alpha _{i}+1$, the above equation can be written as:

$$\hat{\alpha }_{i}(k+1)=1+\sum \limits _{j\in {\mathcal {N}}_i} w_{ij}(k)\left( \frac{{\mathcal {C}}_j(t_2)-{\mathcal {C}}_j(t_1)}{\tau _i(t_2)-\tau _i(t_1)}-1\right) (k)$$

From Eq. (12), we can easily prove that $\sum \nolimits _{j\in {\mathcal {N}}_i} w_{ij}=1$, implying:

$$\begin{aligned} \hat{\alpha }_{i}(k+1)&=\sum \limits _{j\in {\mathcal {N}}_i} w_{ij}(k)+\sum \limits _{j\in {\mathcal {N}}_i} w_{ij}(k)\left( \frac{{\mathcal {C}}_j(t_2)-{\mathcal {C}}_j(t_1)}{\tau _i(t_2)-\tau _i(t_1)}-1\right) (k)\\&=\sum \limits _{j\in {\mathcal {N}}_i} w_{ij}(k)\left( \frac{{\mathcal {C}}_j(t_2)-{\mathcal {C}}_j(t_1)}{\tau _i(t_2)-\tau _i(t_1)}\right) (k) \end{aligned}$$

According to (3) and (8), we can substitute the values of both ${\mathcal {C}}_j(t)$ and $\tau _i(t)$ into the above equation to obtain:

$$\begin{aligned} \hat{\alpha }_{i}(k+1)&=\sum \limits _{j\in {\mathcal {N}}_i} w_{ij}(k)\left( \frac{(\hat{\alpha }_{j}a_jt_2+\hat{\alpha }_{j}b_j+\hat{\beta }_{j})-(\hat{\alpha }_{j}a_jt_1+\hat{\alpha }_{j}b_j+\hat{\beta }_{j})}{(a_it_2+b_i)-(a_it_1+b_i)}\right) (k)\\&=\sum \limits _{j\in {\mathcal {N}}_i} w_{ij}(k)\frac{\hat{\alpha }_{j}a_j}{a_i}(k) \end{aligned}$$

If we multiply each side of the equation by $a_i$ we obtain:

$$\hat{\alpha }_{i}a_i(k+1)=\sum \limits _{j\in {\mathcal {N}}_i} w_{ij}(k)\hat{\alpha }_{j}a_j(k)$$

If we denote $\hat{\alpha }_{i}a_i$ by $x_i$, we can rewrite the above equation as follows:

$$x_i(k+1)=\sum \limits _{j\in {\mathcal {N}}_i} w_{ij}(k)x_j(k)$$

which can be rewritten more compactly as:

$${\mathbf {x}}(k+1)={\mathcal {W}}(k){\mathbf {x}}(k)$$

where ${\mathcal {W}}$ denotes the $n\times n$ matrix whose elements are the weights $w_{ij}$ defined by Eq. (12), and where ${\mathbf {x}}(k)= [x_1(k),\ldots , x_n(k)]^T$ defines a column vector whose elements represent the clock rates of all compensated clocks of the network at time slot k. According to Eq. (12), the elements of ${\mathcal {W}}$ are nonnegative. In addition, ${\mathcal {W}}$ is row-stochastic since it is nonnegative and the sum of the elements of each row is equals 1, i.e., ${\mathcal {W}}{\mathbf {1}}={\mathbf {1}}$ where ${\mathbf {1}} =(1, 1, 1,\ldots , 1)^T$. Also, the state of each node $i\in {\mathcal {V}}$ will be influenced directly or indirectly by every node $j\in {\mathcal {V}}$ through a sequence of communications as ${\mathcal {G}}$ is connected according to Assumption 1. Following [28, 29], all this provides a sufficient condition to guarantee that all $x_i$ will converge to a steady-state value c:

$$\lim _{k \rightarrow \infty }{\mathbf {x}}(k)={\mathbf {c}}{\mathbf {1}}$$

Consequently, as $x_i=\hat{\alpha }_{i}a_i$,

$$\lim _{k \rightarrow \infty }\hat{\alpha }_{i}(k)a_i=\alpha _v{\mathbf {1}}$$

$\square$

Appendix 3: Proof of Lemma 2

Proof

According to Eq. (19), we can write:

$$\begin{aligned} offset ^{t}_{ij}- offset ^{old}_{ij}&=(\hat{\alpha }_{ij}-1)(\tau _i(t)-\tau _i(t_{old}))\\ offset ^{t}_{ij}&= offset ^{old}_{ij}+(\hat{\alpha }_{ij}-1)(\tau _i(t)-\tau _i(t_{old}))\\ {\mathcal {C}}_j(t)-\tau _i(t)&= offset ^{old}_{ij}+(\hat{\alpha }_{ij}-1)(\tau _i(t)-\tau _i(t_{old})) \end{aligned}$$

Consequently,

$${\mathcal {C}}_j(t)=\hat{\alpha }_{ij}\tau _i(t)+ offset ^{old}_{ij}-(\hat{\alpha }_{ij}-1)\tau _i(t_{old})$$

(20)

$\square$

Appendix 4: Proof of Theorem 2

Proof

The Eq. (14) can be written as:

$${\mathcal {C}}_i(t)=\hat{\alpha }_{i}\overline{a}_it+\hat{\alpha }_{i}\overline{b}_i+ offset ^{old}_{i}-(\hat{\alpha }_{i}-1)\overline{\tau }_i(t_{old})$$

(21)

where $\overline{a}_it= \sum \nolimits _{j\in {\mathcal {N}}_i} w_{ij}a_it$ and $\overline{b}_i= \sum \nolimits _{j\in {\mathcal {N}}_i} w_{ij}b_i$ as stated by Eq. (16). According to Eqs. (8) and (21), we have:

$$\hat{\alpha }_{i}(k+1)b_i+\hat{\beta }_{i}(k+1)=\hat{\alpha }_{i}(k)\overline{b}_i+ offset ^{old}_{i}(k)-(\hat{\alpha }_{i}-1)\overline{\tau }_i(t_{old})(k)$$

We denote $\hat{\alpha }_{i}(k+1)b_i+\hat{\beta }_{i}(k+1)$ by $x_i(k+1)$ to obtain:

$$x_i(k+1)=\hat{\alpha }_{i}(k)\overline{b}_i+ offset ^{old}_{i}(k)-(\hat{\alpha }_{i}-1)\overline{\tau }_i(t_{old})(k)$$

Using Eqs. (15) and (16), we can substitute the values of both $offset ^{old}_{i}$ and $\overline{\tau }_i(t_{old})$ into the above equation to obtain:

$$\begin{aligned} x_i(k+1)&=\hat{\alpha }_{i}(k)\overline{b}_i+\sum \limits _{j\in {\mathcal {N}}_i} w_{ij} offset _{ij}(k)-(\hat{\alpha }_{i}-1)\sum \limits _{j\in {\mathcal {N}}_i} w_{ij}\tau _i(t)(k)\\&=\hat{\alpha }_{i}(k)\overline{b}_i+\sum \limits _{j\in {\mathcal {N}}_i} w_{ij}({\mathcal {C}}_j(t)-\tau _i(t)-(\hat{\alpha }_{i}-1)\tau _i(t))(k) \end{aligned}$$

According to Eq. (7), we can substitute the value of ${\mathcal {C}}_j(t)$ into the above equation to obtain:

$$\begin{aligned} x_i(k+1)&=\hat{\alpha }_{i}(k)\overline{b}_i+\sum \limits _{j\in {\mathcal {N}}_i} w_{ij}(\hat{\alpha }_{j}\tau _j(t)+\hat{\beta }_{j}-\hat{\alpha }_{i}\tau _i(t))(k)\\&=\hat{\alpha }_{i}(k)\overline{b}_i+\sum \limits _{j\in {\mathcal {N}}_i} w_{ij}(\hat{\alpha }_{j}a_jt+\hat{\alpha }_{j}b_j+\hat{\beta }_{j}-\hat{\alpha }_{i}a_it-\hat{\alpha }_{i}b_i)(k)\\&=\hat{\alpha }_{i}(k)\overline{b}_i-\hat{\alpha }_{i}(k)\sum \limits _{j\in {\mathcal {N}}_i} w_{ij}b_i+\sum \limits _{j\in {\mathcal {N}}_i} w_{ij}(\hat{\alpha }_{j}a_jt-\hat{\alpha }_{i}a_it)(k)+\sum \limits _{j\in {\mathcal {N}}_i} w_{ij}(\hat{\alpha }_{j}b_j+\hat{\beta }_{j})(k)\\&=\hat{\alpha }_{i}(k)\overline{b}_i-\hat{\alpha }_{i}(k)\overline{b}_i+\sum \limits _{j\in {\mathcal {N}}_i} w_{ij}(\hat{\alpha }_{j}a_jt-\hat{\alpha }_{i}a_it)(k)+\sum \limits _{j\in {\mathcal {N}}_i} w_{ij}x_j(k)\\&=\sum \limits _{j\in {\mathcal {N}}_i} w_{ij}x_j(k)+\sum \limits _{j\in {\mathcal {N}}_i} w_{ij}(\hat{\alpha }_{j}a_jt-\hat{\alpha }_{i}a_it)(k) \end{aligned}$$

which can be rewritten more compactly as:

$${\mathbf {x}}(k+1)={\mathcal {W}}(k){\mathbf {x}}(k)+\varvec{\phi }(k)$$

According to Theorem 1, $\lim \nolimits _{k \rightarrow \infty }a_i\hat{\alpha }_{i}(k)=\alpha _v, \forall i\in {\mathcal {V}}$ , and thus $\lim \nolimits _{k \rightarrow \infty }\varvec{\phi }(k)=0$. Likewise, ${\mathcal {W}}$ is row-stochastic since it is nonnegative and the sum of the entries of each row equals 1. Also, according to Assumption 1, each node $i\in {\mathcal {V}}$ can communicate its values to every other node $j\in {\mathcal {V}}$. As a result [28, 29], $x_i$ will converge to a steady-state value c:

$$\lim _{k \rightarrow \infty }{\mathbf {x}}(k)={\mathbf {c}}{\mathbf {1}}$$

Consequently, as $x_i=\hat{\alpha }_{i}b_i+\hat{\beta }_{i}$, we obtain:

$$\lim \limits _{k \rightarrow \infty } \hat{\alpha }_{i}(k)b_i+\hat{\beta }_{i}(k)=\beta _v{\mathbf {1}}$$

$\square$