# On the Delayed Scaled Consensus Problems
## 4 Scaled Consensus with Signal Transmission Delays
Consequently, zero is a simple characteristic value of (9). Since all solutions of (8) involve a factor \(e^{\omega t}\) with \(\omega\) being a root of the characteristic Equation (9). Note that the root \(\omega=0\) corresponds to the kernel of \(\mathrm{sgn}(\alpha)\mathcal{L}\alpha\), i.e., \(\mathrm{span}\{v_{1}\}\). Moreover, the dynamicson the subspace \(\mathrm{span}\{v_{1}\}\) is constant, to wit, \(\dot{x}(t)\equiv 0\). Combining the above discussions, we are led to the conclusion that:  
\[u_{i}(t)=\mathrm{sgn}(\alpha_{i})\sum_{j=1}^{n}a_{ij}\left(\int_{-\tau}^{0} \alpha_{j}x_{j}(t+s)d\eta(s)-\alpha_{i}x_{i}(t)\right), \tag{7}\]0  
for some \(c\in\mathbb{R}\). The proof is complete. **Remark 4**. : _The quantity \(c\) in (13) can be determined. In fact, noting that \(w_{1}^{\mathrm{T}}\mathrm{sgn}(\alpha)\mathcal{L}\alpha=0\), we obtain from (8) that:_  
\[u_{i}(t)=\mathrm{sgn}(\alpha_{i})\sum_{j=1}^{n}a_{ij}\left(\int_{-\tau}^{0} \alpha_{j}x_{j}(t+s)d\eta(s)-\alpha_{i}x_{i}(t)\right), \tag{7}\]1 \[u_{i}(t)=\mathrm{sgn}(\alpha_{i})\sum_{j=1}^{n}a_{ij}\left(\int_{-\tau}^{0} \alpha_{j}x_{j}(t+s)d\eta(s)-\alpha_{i}x_{i}(t)\right), \tag{7}\]2 \[u_{i}(t)=\mathrm{sgn}(\alpha_{i})\sum_{j=1}^{n}a_{ij}\left(\int_{-\tau}^{0} \alpha_{j}x_{j}(t+s)d\eta(s)-\alpha_{i}x_{i}(t)\right), \tag{7}\]3  
_This means that the quantity \(w_{1}^{\mathrm{T}}\left(x(t)+\mathrm{sgn}(\alpha)\mathcal{D}\alpha\int_{- \tau}^{0}\int_{t+s}^{t}x(t^{\prime})dt^{\prime}d\eta(s)\right)\) remains unchanged with respect to time. By setting \(t\to\infty\), we have:_  
\[u_{i}(t)=\mathrm{sgn}(\alpha_{i})\sum_{j=1}^{n}a_{ij}\left(\int_{-\tau}^{0} \alpha_{j}x_{j}(t+s)d\eta(s)-\alpha_{i}x_{i}(t)\right), \tag{7}\]4 \[u_{i}(t)=\mathrm{sgn}(\alpha_{i})\sum_{j=1}^{n}a_{ij}\left(\int_{-\tau}^{0} \alpha_{j}x_{j}(t+s)d\eta(s)-\alpha_{i}x_{i}(t)\right), \tag{7}\]5 \[u_{i}(t)=\mathrm{sgn}(\alpha_{i})\sum_{j=1}^{n}a_{ij}\left(\int_{-\tau}^{0} \alpha_{j}x_{j}(t+s)d\eta(s)-\alpha_{i}x_{i}(t)\right), \tag{7}\]6 \[u_{i}(t)=\mathrm{sgn}(\alpha_{i})\sum_{j=1}^{n}a_{ij}\left(\int_{-\tau}^{0} \alpha_{j}x_{j}(t+s)d\eta(s)-\alpha_{i}x_{i}(t)\right), \tag{7}\]7  
_Hence, we have:_  
\[u_{i}(t)=\mathrm{sgn}(\alpha_{i})\sum_{j=1}^{n}a_{ij}\left(\int_{-\tau}^{0} \alpha_{j}x_{j}(t+s)d\eta(s)-\alpha_{i}x_{i}(t)\right), \tag{7}\]8  
_In the case of no delay, i.e., \(\bar{\tau}=0\), (14) reduces to \(c=w_{1}^{\mathrm{T}}x(0)\), which agrees with Equation (4) in [6]. In the case of discrete delay delineated by (6), we have:_  
\[u_{i}(t)=\mathrm{sgn}(\alpha_{i})\sum_{j=1}^{n}a_{ij}\left(\int_{-\tau}^{0} \alpha_{j}x_{j}(t+s)d\eta(s)-\alpha_{i}x_{i}(t)\right), \tag{7}\]9  
**Remark 5**. : _From (14), we observe that the scaled consensus values rely on the initial history of the agents' state \(x\) over the interval \([-\tau,0]\) and that the scaled consensus is guaranteed independent of the magnitude of transmission delays. An important implication is that, in the event of additive measurement noise with zero temporal mean, the delayed system may have better performance by choosing a suitable \(\tau\) to reduce the noise effect by averaging over \([-\tau,0]\) via (14). In [18], a similar procedure for delayed standard consensus problems has been discussed._  
Finally, we consider a tracking dynamics on the scaled consensus manifold as:  
\[\dot{x}(t)=-\mathrm{sgn}(\alpha)\mathcal{D}\alpha x(t)+\mathrm{sgn}(\alpha) \mathcal{A}\alpha\int_{-\tau}^{0}x(t+s)d\eta(s). \tag{8}\]0  
where the forcing input \(f\) is defined as in Section 3. We can similarly derive the following useful corollary. **Corollary 2**. : _If \(\mathcal{G}\) has a spanning tree, then the multi-agent system (15) achieves scaled consensus to \((\alpha_{1},\cdots,\alpha_{n})\). In particular, \(x(t)\) converges to the time function \((c+f(t))v_{1}\) as \(t\) grows, where \(c\) is given by (14)._