## 3 Offset Correction
### Full-Consensus Algorithm  
Consider a special case of the model (1) and the calibration function (2) in which \(\alpha_{i}=1\), \(\xi^{i}_{t}=0\), and \(x_{t}\) is an unknown deterministic function of \(t\), bounded for all \(t\). According to the general consensus-based calibration methodology presented in [34; 35; 36; 44; 45; 46], we define the following criterion function:  
\[f^{i}_{b}(b^{i})=\sum_{j\in\mathcal{N}^{in}_{t}}\gamma_{ij}(z^{j}_{t}-z^{i}_{t })^{2}, \tag{3}\]  
\(i=1,\ldots,n\), where \(\gamma_{ij}\) are _non-negative weights_ reflecting the relative importance of neighboring nodes from \(\mathcal{N}_{i}\) for all \(i\). After computing the derivatives of \(z^{i}\) with regard to \(b^{i}\), we can directly formulate the following distributed _gradient-type algorithm_ for estimating \(b^{i}\) in real time [49; 50; 51], using the current measurements of the sensor outputs:  
\[\hat{b}^{i}_{t+1}=\hat{b}^{i}_{t}+\delta^{i}_{t}\sum_{j\in\mathcal{N}^{in}_{t }}\gamma_{ij}\varepsilon^{ij}_{t}, \tag{4}\]  
where \(\hat{b}^{i}_{t}\) is an estimate of \(b^{i}\) at time \(t\), \(\delta^{i}_{t}\) is a positive step-size sequence influencing the convergence properties of the algorithm, and \(\varepsilon^{ij}_{t}=\hat{z}^{j}_{t}-\hat{z}^{i}_{t}\), with \(\hat{z}^{i}_{t}=y^{i}_{t}+\hat{b}^{i}_{t}\), where \(y^{i}_{t}\) is defined by (1). The initial conditions are determined by the available a priori information; without any information, it is natural to adopt \(\hat{b}^{i}_{0}=0\), \(i=1,\ldots,n\). Notice that, to the authors' knowledge, the nonlinear term \(v^{i}_{t}=v^{i}(x_{t})\) has not been treated in this context in the literature. It is convenient to reformulate (4) in the following way:  
\[\hat{f}_{t+1}^{i}=\hat{f}_{t}^{i}+\delta_{t}^{i}\sum_{j\in\mathcal{N}_{i}^{in}} \gamma_{ij}\{(\hat{f}_{t}^{j}-\hat{f}_{t}^{i})+(v_{t}^{j}-v_{t}^{i})\}, \tag{5}\]  
where \(\hat{f}_{t}^{i}\) is an estimate of the corrected offset \(f^{i}\). All the recursions (5) can be represented by the following global vector-matrix model at the network level:  
\[\hat{f}_{t+1}=(I+\Delta_{t}\Gamma)\hat{f}_{t}+\Delta_{t}\Gamma v_{t} \tag{6}\]  
where \(\hat{f}_{t}=[\hat{f}_{t}^{1}\cdots\hat{f}_{t}^{n}]^{T}\), \(\Delta_{t}=\text{diag}\{\delta_{1}^{1},\ldots,\delta_{t}^{n}\}\),  
\[\Gamma=\left[ \begin{array}{cccc}-\sum_{j=2}^{n}\gamma_{1j}&\gamma_{12}&\cdots &\gamma_{1n}\\ \gamma_{21}&-\sum_{j=1,j\
eq 2}^{n}\gamma_{2j}&\cdots&\gamma_{2n}\\ \vdots&\vdots&\vdots&\vdots\\ \gamma_{n1}&\gamma_{n2}&\cdots&-\sum_{j=1,j\
eq n}^{n}\gamma_{nj}\end{array} \right]\]  
and \(v_{t}=[v_{t}^{1}\cdots v_{t}^{n}]^{T}\). The recursion (6) represents a linear dynamic system with time-varying parameters and unknown inputs, which can be analyzed using the general theory of linear time-varying systems, together with the specific methodology developed for dynamic consensus schemes, e.g., [52; 53; 54]. We adopt the following basic assumptions:  
1. \(\delta_{t}^{i}=\delta=\text{const}\). 2. \(\|v_{t}^{i}\|\leq k_{v}^{i}<\infty\), \(i=1,\ldots,n\). 3. Graph \(\mathcal{G}\) possesses a central node. Assumption A1 is adequate for the adopted deterministic framework. In practice, it is advisable to adopt A1 when the tracking of time-varying parameters may be desirable. Assumption A3 is frequently encountered in consensus-based schemes (the central node is a node which can be reached from all the remaining nodes in the graph, \(\mathcal{G}\)) [48]. The following two lemmas are fundamental for all the formal results presented in this paper. Matrix \(\Gamma\), containing the algorithm meta-parameters, plays a fundamental role in defining the properties of the algorithm. A more detailed insight into the influence of \(\Gamma\) can be found in [44; 45]. **Lemma 1** ([55]). : _Let A3 be satisfied. Then, matrix \(\Gamma\) has one eigenvalue at the origin and the remaining ones in the left-half complex plane._  
Define \(\mathbf{1}=[1\cdots 1]^{T}\); vector \(\mathbf{1}\) is, by definition, the right eigenvector of \(\Gamma\), corresponding to the zero eigenvalue.