# On the Delayed Scaled Consensus Problems
## 5 Scaled Consensus with Signal Processing Delays  
In this section, we examine the signal processing delay, which is also known as internal delay. Signal processing delay has been much studied in the literature of consensus problems; see, e.g., [1; 2; 3]. In parallel with (6), the scaled consensus control scheme can be designed as:  
\[u_{i}(t)=\text{sgn}(\alpha_{i})\sum_{j=1}^{n}a_{ij}(\alpha_{j}x_{j}(t-\tau)- \alpha_{i}x_{i}(t-\tau)), \tag{16}\]  
where \(\tau\geq 0\) indicates the time delay. The distributed-delay version is:  
\[u_{i}(t)=\text{sgn}(\alpha_{i})\sum_{j=1}^{n}a_{ij}\int_{-\tau}^{0}\big{(} \alpha_{j}x_{j}(t+s)-\alpha_{i}x_{i}(t+s)\big{)}\,d\eta(s), \tag{17}\]  
where \(\eta:[-\tau,0]\rightarrow\mathbb{R}\) is a function of bounded variation describing the distributed delays. In analogy to (8), the switched multi-agent system (1) with protocol (17) can be written as:  
\[\dot{x}(t)=-\text{sgn}(\alpha)\mathcal{L}\alpha\int_{-\tau}^{0}x(t+s)d\eta(s). \tag{18}\]  
The characteristic equation is given by:  
\[\chi(\omega):=\det\left(\omega I_{n}+G(\omega)\text{sgn}(\alpha)\mathcal{L} \alpha\right)=0, \tag{19}\]  
where \(\omega\in\mathbb{C}\) and \(G(\omega)=\int_{-\tau}^{0}e^{\omega s}d\eta(s)\). Similarly as the transmission delay case, we have \(G(0)=1\), \(\chi(0)=0\) and \(G^{\prime}(0)=-\tau:=\int_{-\tau}^{0}sd\eta(s)\). Here, we will only study the system (1) with discrete delay (16), which corresponds to a special form \(G(\omega)=e^{-\omega\tau}\). We assume the following detailed balanced condition. **Assumption 1**. : \(|\alpha_{i}|a_{ij}=|\alpha_{j}|a_{ji}\) _holds for all \(i,j=1,\cdots,n\)._  
The detailed balanced condition has proven to be instrumental in studying coupled dynamics; see, e.g., [19; 20] for details. Our main result in this section reads as follows. **Theorem 3**. : _Suppose that \(\mathcal{G}\) has a spanning tree and that Assumption 1 holds. The multi-agent system (1) with Protocol (16) achieves scaled consensus to \((\alpha_{1},\cdots,\alpha_{n})\) if and only if \(0\leq\tau<\frac{\pi}{2\max_{1\leq j\leq n}\lambda_{i}}\), where \(\{\lambda_{i}\}_{i=1}^{n}\) are the eigenvalues of \(|\alpha|\mathcal{L}\)._  
Proof. : Note that \(|\alpha|\mathcal{L}\) is symmetric and positive semidefinite under Assumption 1. Let \(\lambda_{1}=0\). The spanning tree condition implies \(\lambda_{i}>0\) for \(i=2,\cdots,n\). We first show that zero is a simple characteristic value of (19). Indeed, we already see that \(\chi(0)=0\). Arguing similarly as in Theorem 2, we have:  
\[\chi^{\prime}(0)= \text{tr}\left(\text{adj}(\text{sgn}(\alpha)\mathcal{L}\alpha) \cdot(I_{n}-\text{rsgn}(\alpha)\mathcal{L}\alpha)\right)\] \[= \text{tr}\left(\text{adj}(\text{sgn}(\alpha)\mathcal{L}\alpha) \right)=k\sum_{i=1}^{n}\frac{w_{1i}}{\alpha_{i}}\
eq 0,\]  
where \(k\
eq 0\) and \(w_{1}=(w_{11},\cdots,w_{1n})^{\text{T}}\) are interpreted in the same way as in (12). This means zero is a simple root of (19). Next, we show that \(0<\lambda_{i}\tau<\frac{\pi}{2}\) for all \(i\geq 2\) if and only if all nonzero characteristic values of (19) have negative real parts. Suppose that \(\chi(\omega)=0\) for some \(\omega\in\mathbb{C}\). By (19), we know that \(\frac{-\omega}{G(\omega)}=-\omega e^{\omega\tau}\) is an eigenvalue of \(|\alpha|\mathcal{L}\). That is, \(\omega e^{\omega\tau}=\lambda_{i}\)\((i=2,\cdots,n)\). Under Assumption 1, Theorem 1 in [21] implies that \(\mathrm{Re}(\omega)<0\) for all solutions \(\omega\) of the above system if and only if \(0<\lambda_{i}\tau<\frac{\pi}{2}\) for \(i=2,\cdots,n\). It is easy to see that the sufficiency of Theorem 3 holds by the same argument at the end of Theorem 2. The necessity of Theorem 3 follows similarly from the necessity proof of Theorem 1. **Remark 6**. : _It follows from the Gerslgorin disk theorem that \(0\leq\lambda_{i}\leq 2\max_{1\leq i\leq n}|\alpha_{i}|d_{i}\). Therefore, the eigenvalue condition in Theorem 3 can be replaced by a more geometric (sufficient) condition \(0\leq\tau<\frac{\pi}{4\max_{1\leq i\leq n}|\alpha_{i}|d_{i}}\).