# Observer-Based Consensus Control for Heterogeneous Multi-Agent Systems with Output Saturations
## 3 Main Results
### Leaderless Case  
In this subsection, we consider the output consensus without the leader agent. We propose the observer-based consensus algorithm as follows:  
\[\dot{\mathbf{\dot{x}}}_{i}= A_{i}\dot{\mathbf{\dot{x}}}_{i}+B_{i}u_{i}+H_{i}\dot{\mathbf{\dot{z}}}_{i} \tag{8a}\] \[\dot{w}_{i}= Sw_{i}+F\left(\sum_{j=1}^{N}\alpha_{ij}(w_{j}-w_{i})\right)\] (8b) \[u_{i}= K_{i}(\dot{\mathbf{\dot{x}}}_{i}-\Pi_{i}w_{i})+\Gamma_{i}w_{i}\] (8c) \[\dot{\mathbf{\dot{z}}}_{i}= \sigma(\mathrm{C}_{i}\dot{\mathbf{x}}_{i})-\sigma(\mathrm{C}_{i}\dot{ \mathbf{\dot{x}}}_{i}),\quad i\in\mathcal{V}, \tag{8d}\]  
where \(\dot{\mathbf{x}}_{i}\in\mathbf{R}^{n_{i}}\) and \(w_{i}\in\mathbf{R}^{n_{0}}\) are the states of the observer and the reference generator of the \(i\)th agent, respectively; \(H_{i}\), \(F\) and \(K_{i}\) are the gain matrices with compatible dimensions that will be determined later; and \(S\), \(\Pi_{i}\) and \(\Gamma_{i}\) are the constant matrices satisfying Assumption 1. Before we present our main result, we consider the following lemma [11]. **Lemma 4**. : _Suppose that the pair \((S,Q)\) is stabilizable. Then, there exists a symmetric positive-definite matrix \(W\) such that_  
\[WS+S^{T}W-2 WQQ^{T}W<0. \tag{9}\]  
Then, we can solve the leaderless output consensus problem from the following theorem. **Theorem 1**. : _Consider a group of \(N\) heterogeneous agents (1), and suppose that Assumptions 1-3 hold. Then, we can achieve the output consensus using the observer-based consensus algorithm (8) if the gain matrices satisfy the following conditions:_  
1. \(H_{i}=\beta_{i}\bar{T}_{i}^{-1}P_{i}^{-1}\mathrm{C}_{i}^{T}\)_, where_ \(\beta_{i}\) _is any positive constant,_ \(T_{i}\) _and_ \(\bar{C}_{i}\) _are given in (_5_), and_ \(P_{i}\) _is given in Lemma_ 1_._
2. \(F=\tau QQ^{T}W\)_, where_ \(\tau\) _is a positive constant such that_ \(\tau\lambda_{2}(L)>1\)_, and_ \(Q\) _and_ \(W\) _are the solution of (_9_) in Lemma_ 4_._
3. \(K_{i}\) _is a constant matrix such that_ \(A_{i}+B_{i}K_{i}\) _is Hurwitz._
4. \(w_{i}(0)\) _is bounded such that_ \(\|R\varpi(t)\|_{\infty}\leq 1-\delta\)_, for_ \(0<\delta<1\)_,_ \(\forall t\geq 0\)_, where_ \(\varpi\) _is the solution of the following dynamics:_ \[\hat{\mathbf{w}}=S\bar{w},\quad\bar{w}(0)=\frac{1}{N}\sum_{i=1}^{N}w_{i}(0).\] (10)  
**Proof of Theorem 1. ** To prove the consensus, we investigate the asymptotic stability of the error dynamics. Then, we first consider the reference generator (8b) and define the state vector \(w=[w_{1}^{T},...,w_{N}^{T}]^{T}\). Then, from (8b), we have  
\[\hat{w}= (I_{N}\otimes S-L\otimes F)w. \tag{11}\]  
Since the graph is undirected and connected, there exists an orthogonal matrix \(U=\left[\frac{1}{\sqrt{N}}1_{N}U_{2}\right]\), such that \(U^{T}LU=\Lambda=\mathrm{diag}(0,\lambda_{2}(L),...,\lambda_{N}(L))\) with \(U_{2}\in\mathbf{R}^{N\times(N-1)}\) and \(U_{2}^{T}U_{2}=I_{N-1}\)[14]. Let \(\zeta=(U^{T}\otimes I_{n_{0}})w\) with \(\zeta=[\zeta_{1}^{T},\zeta_{2}^{T},...,\zeta_{N}^{T}]^{T}\) and \(\bar{\zeta}=(U_{2}^{T}\otimes I_{n_{0}})w\) with \(\zeta=[\zeta_{2}^{T},\zeta_{3}^{T},...,\zeta_{N}^{T}]^{T}\). In [12], it was shown that \((w_{i}-w_{j})\to 0\), \(\forall i,j=1,...,N\), if and only if \(\xi\to 0\). Moreover, if \((w_{i}-w_{j})\to 0\), \(\forall i,j=1,...,N,w_{i}(t)\to\bar{w}(t)\), where \(\bar{w}(t)\) is the solution of (10) [10]. Therefore, in what follows, we derive the asymptotic convergence of \(\bar{\zeta}\) to the origin. After the coordinate transformation, we have  
\[\bar{\zeta}=(I_{N-1}\otimes S-\bar{\Lambda}\otimes F)\bar{\zeta}, \tag{12}\]  
where \(\bar{\Lambda}=\mathrm{diag}(\lambda_{2}(L),\lambda_{3}(L),...,\lambda_{N}(L))\). We next consider the observer (8a) and the regulator (8c) and define the observer error and the regulation error as \(e_{i}=T_{i}(x_{i}-\hat{x}_{i})\) and \(e_{i}=\hat{x}_{i}-\Pi_{i}w_{i}\), respectively. Then, from (5), the observer error dynamics can be written as  
\[