# On Consensus-Based Distributed Blind Calibration of Sensor Networks

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## Abstract

**:**

## 1. Introduction

## 2. Related Work

## 3. Problem Definition and the Basic Algorithm

## 4. Convergence Analysis

**Lemma**

**1**

**Lemma**

**2**

**.**Let $T=\left[\begin{array}{ccc}{i}_{1}& {i}_{2}& {T}_{2n\times (2n-2)}\end{array}\right]$, where ${T}_{2n\times (2n-2)}$ is an $2n\times (2n-2)$ matrix, such that span$\left\{{T}_{2n\times (2n-2)}\right\}=\mathrm{span}\left\{\overline{B}\right\}$ ($\mathrm{span}\left\{A\right\}$ denotes a linear space spanned by the columns of matrix A). Then, T is nonsingular and

**Lemma**

**3**

**.**For the matrix $B\left(t\right)$ in (10) it holds that, for all t,

**Theorem**

**1**

**.**Assume that Assumptions (A1)–(A4) hold. Then there exists ${\delta}^{\prime}>0$ such that for all $\delta \le {\delta}^{\prime}$ in (10)

**Theorem**

**2**

**.**Assume that the assumptions (A1), (A2’), (A3) and (A4) hold. Then there exists ${\delta}^{\u2033}>0$ such that for all $\delta \le {\delta}^{\u2033}$ in (10) ${lim}_{t\to \infty}\widehat{\varphi}\left(t\right)=({i}_{1}{\rho}_{1}+{i}_{2}{\rho}_{2})\widehat{\varphi}\left(0\right)$ in the mean square sense and w.p.1.

## 5. Extensions of the Basic Algorithm

#### 5.1. Communication Errors

**Theorem**

**3**

**.**Let Assumptions (A1’), (A2)–(A7) be satisfied. Then, $\widehat{\varphi}\left(t\right)$ generated by (21) converges to ${i}_{1}{w}_{1}+{i}_{2}{w}_{2}$ in the mean square sense and w.p.1, where ${w}_{1}$ and ${w}_{2}$ are scalar random variables satisfying $E\left\{{w}_{1}\right\}={\rho}_{1}^{\prime}\widehat{\varphi}\left(0\right)$ and $E\left\{{w}_{2}\right\}={\rho}_{2}^{\prime}\widehat{\varphi}\left(0\right)$.

#### 5.2. Measurement Noise

**Theorem**

**4**

**.**Assume that the assumptions (A1’), (A2)–(A8) hold. Then, $\widehat{\varphi}\left(t\right)$, given by (25), converges to ${i}_{1}{w}_{1}+{i}_{2}{w}_{2}$ in the mean square sense and w.p.1, where ${w}_{1}$ and ${w}_{2}$ are scalar random variables satisfying $E\left\{{w}_{1}\right\}={\rho}_{1}^{\prime}\widehat{\varphi}\left(0\right)$ and $E\left\{{w}_{2}\right\}={\rho}_{2}^{\prime}\widehat{\varphi}\left(0\right)$.

**Theorem**

**5**

**.**Assume that the assumptions (A1’), (A2’), (A3), (A4’), (A5)–(A8) hold. Then $\widehat{\varphi}\left(t\right)$, given by (28) with $d={d}_{0}$, converges to ${i}_{1}{w}_{1}+{i}_{2}{w}_{2}$ in the mean square sense and w.p.1, where ${w}_{1}$ and ${w}_{2}$ are scalar random variables satisfying $E\left\{{w}_{1}\right\}={\rho}_{1}^{\u2033}\widehat{\varphi}\left(0\right)$ and $E\left\{{w}_{2}\right\}={\rho}_{2}^{\u2033}\widehat{\varphi}\left(0\right)$.

#### 5.3. Asynchronous Broadcast Gossip Communication

- ${\widehat{\theta}}_{i}\left(k\right)={\left[{\widehat{a}}_{i}\left(k\right)\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{\widehat{b}}_{i}\left(k\right)\right]}^{T}$,
- ${\delta}_{i}\left(k\right)$ is the step size given by ${\delta}_{i}\left(k\right)={\nu}_{i}{\left(k\right)}^{-c}$, where ${\nu}_{i}\left(k\right)={\sum}_{m=1}^{k}I\{i\in J\left(m\right)\}$ is the number of parameter updates of node i up to the iteration k, with $1/2<c\le 1$ ($I\{\xb7\}$ denotes the indicator function),
- ${\u03f5}_{i,j\left(k\right)}\left(k\right)={\widehat{z}}_{j\left(k\right)}\left(k\right)-{\widehat{z}}_{i}\left(k\right)$, where$$\begin{array}{cc}\hfill {\widehat{z}}_{j\left(k\right)}\left(k\right)=& {\widehat{a}}_{j\left(k\right)}(k-1){y}_{j\left(k\right)}\left(k\right)+{\widehat{b}}_{j\left(k\right)}(k-1),\hfill \end{array}$$$$\begin{array}{cc}\hfill {\widehat{z}}_{i}\left(k\right)=& {\widehat{a}}_{i}(k-1){y}_{i}\left(k\right)+{\widehat{b}}_{i}(k-1)\hfill \end{array}$$

- $\widehat{\varphi}\left(k\right)={[{\widehat{\varphi}}_{1}{\left(k\right)}^{T}\dots {\widehat{\varphi}}_{n}{\left(k\right)}^{T}]}^{T},$
- $\Delta \left(k\right)=\mathrm{diag}\{{\delta}_{1}\left(k\right),\dots ,{\delta}_{n}\left(k\right)\}$,
- $\Omega \left(k\right)=\mathrm{diag}\{{\Omega}_{1}\left(k\right),\dots ,{\Omega}_{n}\left(k\right)\}$,
- $\Gamma \left(k\right)=[\Gamma {\left(k\right)}_{lm}]$, with $\Gamma {\left(k\right)}_{ll}=-{\gamma}_{l,j\left(k\right)}$ and $\Gamma {\left(k\right)}_{l,j\left(k\right)}={\gamma}_{l,j\left(k\right)}$ for all $l\in J\left(k\right)$, $\Gamma {\left(k\right)}_{lm}=0$, otherwise,
- $\Psi \left(k\right)=\mathrm{diag}\{{\Psi}_{1}\left(k\right),\dots ,{\Psi}_{n}\left(k\right)\}$,
- $\tilde{N}\left(k\right)=\left[{\tilde{N}}_{lm}\left(k\right)\right]$, where ${\tilde{N}}_{ll}\left(k\right)=-{\gamma}_{l,j\left(k\right)}{N}_{ll}\left(k\right)$ and ${\tilde{N}}_{l,j\left(k\right)}\left(k\right)={\gamma}_{l,j\left(k\right)}{N}_{l,j\left(k\right)}\left(k\right)$, for all $l\in J\left(k\right)$, $\tilde{N}{\left(k\right)}_{lm}=0$, otherwise.

**Theorem**

**6**

**.**Let Assumptions (B1)–(B4) be satisfied. Then $\widehat{\varphi}\left(k\right)$ given by (37) converges to ${\widehat{\varphi}}_{\infty}={\chi}_{1}{i}_{1}+{\chi}_{2}{i}_{2}$ in the mean square sense and w.p.1, where ${\chi}_{1}$ and ${\chi}_{2}$ are random variables with bounded second moments.

## 6. Discussion

#### 6.1. Rate of Convergence

**Theorem**

**7**

**.**Under the assumptions of any of the Theorems 3, 4 or 5, together with ${lim}_{t\to \infty}(\delta {(t+1)}^{-1}-\delta {\left(t\right)}^{-1})=d\ge 0$, there exists such a positive number ${\sigma}^{\prime}<1$ that for all $0<\sigma <{\sigma}^{\prime}$ the asymptotic consensus is achieved by the presented algorithms with the convergence rate $o\left(\delta {\left(t\right)}^{\sigma}\right)$.

#### 6.2. Stationarity of the Measured Signal

#### 6.3. Network Weights Design

- By reducing the values of all the elements in the i-th row of $\overline{\Gamma}$, or
- By increasing the values ${\gamma}_{ji}$, $j\ne i$, from the i-th column (keeping in mind that $\overline{\Gamma}$ must be row stochastic).

#### 6.4. Macro Calibration for Networks with Reference Nodes

**Theorem**

**8**

**.**Let Assumptions (B1)–(B4) be satisfied and let all the nodes from $\mathcal{N}-{\mathcal{N}}^{f}$ be reachable from all the nodes in ${\mathcal{N}}^{f}$. Then the algorithm (31), in which ${\gamma}_{ij}=0$ for all $i\in {\mathcal{N}}^{f}$, provides convergence of ${\widehat{\varphi}}^{v}\left(k\right)$ in the mean square sense and w.p.1 to the limit defined by

#### 6.5. Autonomous Gain Correction and Relationship with Time Synchronization

## 7. Simulation Results

## 8. Conclusions

#### Future Work

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**An example sensor network used in smart-city applications with decentralized communication topology. The inter-node communication is performed according to the depicted directed graph. The introduced distributed calibration algorithm achieves asymptotic calibration of all the sensor nodes in the network without using any type of fusion center.

**Figure 2.**An example sensor network used in smart-city applications with multiple (four) reference nodes. The reference nodes (RNs) have fixed calibration parameters: only the rest of the nodes implement the given distributed sensor calibration recursions.

**Figure 3.**Noiseless synchronous algorithm without references: convergence to consensus is achieved for corrected gains and corrected offsets.

**Figure 4.**Noiseless synchronous algorithm with one reference sensor: convergence to the reference is chieved.

**Figure 5.**The modified algorithm (25): convergence to consensus is achieved for corrected gains and corrected offsets despite measurement noise presence.

**Figure 6.**The asynchronous algorithm based on instrumental variables without reference sensors: convergence to consensus is achieved for corrected gains and corrected offsets.

**Figure 8.**The asynchronous algorithm with two reference sensors with different characteristics: both the corrected gains and the corrected offsets converge to different values determined by (38).

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**MDPI and ACS Style**

Stanković, M.S.; Stanković, S.S.; Johansson, K.H.; Beko, M.; Camarinha-Matos, L.M. On Consensus-Based Distributed Blind Calibration of Sensor Networks. *Sensors* **2018**, *18*, 4027.
https://doi.org/10.3390/s18114027

**AMA Style**

Stanković MS, Stanković SS, Johansson KH, Beko M, Camarinha-Matos LM. On Consensus-Based Distributed Blind Calibration of Sensor Networks. *Sensors*. 2018; 18(11):4027.
https://doi.org/10.3390/s18114027

**Chicago/Turabian Style**

Stanković, Miloš S., Srdjan S. Stanković, Karl Henrik Johansson, Marko Beko, and Luis M. Camarinha-Matos. 2018. "On Consensus-Based Distributed Blind Calibration of Sensor Networks" *Sensors* 18, no. 11: 4027.
https://doi.org/10.3390/s18114027