# On the Delayed Scaled Consensus Problems
## 4 Scaled Consensus with Signal Transmission Delays  
Signal transmission delay, also known as coupling delay, can be introduce to the control scheme (1) by considering:  
\[u_{i}(t)=\mathrm{sgn}(\alpha_{i})\sum_{j=1}^{n}a_{ij}(\alpha_{j}x_{j}(t-\tau) -\alpha_{i}x_{i}(t)), \tag{6}\]  
where \(\tau\geq 0\) represents the time delay. Signal transmission delay often appears in networks of oscillators and has been investigated in, e.g., [14; 18] for standard consensus problems. In fact, (6) can be generalized to accommodate a history dependence over the interval \([t-\tau,t]\) instead of a discrete past time instant \(t-\tau\):  
\[u_{i}(t)=\mathrm{sgn}(\alpha_{i})\sum_{j=1}^{n}a_{ij}\left(\int_{-\tau}^{0} \alpha_{j}x_{j}(t+s)d\eta(s)-\alpha_{i}x_{i}(t)\right), \tag{7}\]  
where \(\eta:[-\tau,0]\to\mathbb{R}\) is a function of bounded variation describing the distributed delays. Assume that \(\eta\) is nondecreasing and appropriately normalized: \(\int_{-\tau}^{0}d\eta(s)=1\). \(\eta\) can be viewed as a probability distribution and admits scaled consensus solutions. In particular, when \(\eta\) is a Heaviside step function, the control input (7) readily reduces to (6). The switched multi-agent system (1) with Protocol (7) can be recast in a compact form as:  
\[\dot{x}(t)=-\mathrm{sgn}(\alpha)\mathcal{D}\alpha x(t)+\mathrm{sgn}(\alpha) \mathcal{A}\alpha\int_{-\tau}^{0}x(t+s)d\eta(s). \tag{8}\]  
The formulation \(x(t)=e^{\omega t}v\) (\(v\in\mathbb{R}^{n}\)) yields the characteristic equation in \(\omega\in\mathbb{C}\) as:  
\[\chi(\omega):=\det\left(\omega I_{n}+\mathrm{sgn}(\alpha)\mathcal{D}\alpha-G( \omega)\mathrm{sgn}(\alpha)\mathcal{A}\alpha\right)=0, \tag{9}\]  
where \(I_{n}\in\mathbb{R}^{n\times n}\) is the \(n\)-dimensional identity matrix and \(G(\omega)=\int_{-\tau}^{0}e^{\omega s}d\eta(s)\). Let \(\bar{\tau}:=-\int_{-\tau}^{0}sd\eta(s)\geq 0\) represent the mean delay. It is clear that \(G(0)=1\) and \(G^{\prime}(0)=-\bar{\tau}\). Moreover, zero is always a characteristic value of (9) since \(\chi(0)=\det(\mathrm{sgn}(\alpha))\det(\mathcal{L})\det(\alpha)=0\). Our main result in this section shows that the stability of scaled consensus is independent of the transmission delays, generalizing the stability result of standard consensus [14] with all \(\alpha_{i}\) being one. **Theorem 2**. : _If \(\mathcal{G}\) has a spanning tree, the multi-agent system (8) achieves scaled consensus to \((\alpha_{1},\cdots,\alpha_{n})\)._  
**Proof. ** We first show that all nonzero characteristic values of (9) have negative real parts. In fact, assume that \(\chi(\omega)=0\) and \(\mathrm{Re}(\omega)\geq 0\). It follows from (9) that \(-\omega\) is an eigenvalue of \(\mathrm{sgn}(\alpha)\mathcal{D}\alpha-G(\omega)\mathrm{sgn}(\alpha)\mathcal{A}\alpha\). Note that the matrix \(\mathrm{sgn}(\alpha)\mathcal{D}\alpha-G(\omega)\mathrm{sgn}(\alpha)\mathcal{A}\alpha\) is similar to \(|\alpha|\mathcal{D}-G(\omega)|\alpha|\mathcal{A}\). In light of the Gershgorin disk theorem, the eigenvalues of \(\mathrm{sgn}(\alpha)\mathcal{D}\alpha-G(\omega)\mathrm{sgn}(\alpha)\mathcal{A}\alpha\) must lie in the union of \(n\) disks centered at \(|\alpha_{i}|d_{i}\) with radius \(|G(\omega)|\cdot|\alpha_{i}|\cdot\sum_{j=1}^{n}a_{ij}\leq|\alpha_{i}|d_{i} \cdot\int_{-\tau}^{0}d\eta(s)\leq|\alpha_{i}|d_{i}\) for \(i=1,\cdots,n\). This implies that the nonzero characteristic values of (9) are contained in the set \(\{\omega\in\mathbb{C}:\mathrm{Re}(\omega)>0\}\cup\{0\}\), and thus, \(\mathrm{Re}(-\omega)\leq 0\). Accordingly, we have \(\omega=0\), which completes the proof of the claim. Next, we show that zero is a simple characteristic value of (9).