# Observer-Based Consensus Control for Heterogeneous Multi-Agent Systems with Output Saturations
## 3 Main Results
### Leader-Following Case  
In this subsection, we consider the leader agent, which generates the reference trajectory. The dynamics of the leader is given by  
\[ \begin{split}\dot{x}_{0}=& Sx_{0}\\ y_{0}=& Rx_{0},\end{split}  \tag{22}\]  
where \(x_{0}\in\mathbf{R}^{n_{0}}\) and \(y_{0}\in\mathbf{R}^{q}\) are the state and output, respectively, of the leader. \(S\) and \(R\) are the constant matrices satisfying Assumption 1. Then, to achieve the consensus, we consider the following assumption to control the follower agents into the leader's trajectory. **Assumption 4**. : _The leader agent (22) satisfies the following conditions:_  
1. \((R,S)\) _is detectable._
2. _For the leader agent, there exists_ \(0<\delta<1\)_, satisfying_ \[x_{0}(0)\in\mathcal{X}_{0}\Rightarrow\lVert Rx_{0}(t)\rVert_{\infty}\leq 1- \delta,\ \ \forall t\geq 0.\] (23)Then, to achieve the consensus, we propose the following consensus algorithm:  
\[\dot{\hat{x}}_{i}= A_{i}\hat{x}_{i}+B_{i}u_{i}+H_{i}\hat{z}_{i} \tag{24a}\] \[\dot{w}_{i}= Sw_{i}+F\left(\sum_{j=1}^{N}\alpha_{ij}(w_{j}-w_{i})+\alpha_{i0}(x_ {0}-w_{i})\right)\] (24b) \[u_{i}= K_{i}(\hat{x}_{i}-\Pi_{i}w_{i})+\Gamma_{i}w_{i}\] (24c) \[\hat{z}_{i}= \sigma(C_{i}x_{i})-\sigma(C_{i}\hat{x}_{i}),\quad i\in\mathcal{V}, \tag{24d}\]  
where \(\hat{x}_{i}\in\mathbf{R}^{n_{i}}\) and \(w_{i}\in\mathbf{R}^{n_{0}}\) are the states of the observer and the reference generator of the \(i\)th follower, respectively; \(H_{i}\), \(F\) and \(K_{i}\) are the gain matrices with compatible dimensions that will be determined later; and \(\Pi_{i}\) and \(\Gamma_{i}\) are the solution of linear matrix equations in Assumption 1. We next consider the following lemma [10], which is the dual problem of Lemma 4:  
Lemma 5: _Suppose that the pair \((R,S)\) is detectable. Then, there exists a symmetric positive-definite matrix \(W\) such that_  
\[WS+S^{T}W-2R^{T}R<0. \tag{25}\]  
Then, we can solve the leader-following output consensus from the following theorem. Theorem 2: _Consider a group of \(N\) follower agents (1) with the leader (22). Suppose that Assumptions 1-4 hold, and there exists at least one follower that can receive the information from the leader. Then, we can achieve the leader-following output consensus using the observer-based consensus algorithm (24) if the gain matrices satisfy the following conditions:_  
1. \(H_{i}=\beta_{1}T_{i}^{-1}P_{i}^{-1}\mathbb{C}_{i}^{T}\)_, where_ \(\beta_{i}\) _is any positive constant,_ \(T_{i}\) _and_ \(\mathbb{C}_{i}\) _are given in (_5_), and_ \(P_{i}\) _is given in Lemma_ 1_._
2. \(F=\tau W^{-1}R^{T}R\)_, where_ \(\tau\) _is a positive constant such that_ \(\tau\lambda_{1}(\bar{L})>1\)_, where_ \(\bar{L}=L+\operatorname{diag}(\alpha_{10},...,\alpha_{N0})\)_, and the positive definite matrix_ \(W\) _is the solution of (_25_) in Lemma_ 5_._
3. \(K_{i}\) _is a constant matrix such that_ \(A_{i}+B_{i}K_{i}\) _is Hurwitz._  
Proof of Theorem 2.: We apply the same procedure as in the proof of Theorem 1. We define the tracking error of the reference generator, the observer error and the regulation error as \(\zeta_{i}=w_{i}-x_{0}\), \(e_{i}=T_{i}(x_{i}-\hat{x}_{i})\), and \(\epsilon_{i}=\hat{x}_{i}-\Pi_{i}w_{i}\), respectively. Then, the tracking error dynamics is given by  
\[\dot{\zeta}_{i}= Sw_{i}+F\left(\sum_{j=1}^{N}\alpha_{ij}(w_{j}-w_{i})+\alpha_{i0}( x_{0}-w_{i})\right)-Sx_{0} \tag{26}\] \[= S\zeta_{i}+F\left(\sum_{j=1}^{N}\alpha_{ij}(\zeta_{j}-\zeta_{i})- \alpha_{i0}\zeta_{i}\right),\]  
and, from (13) and (14), the observer error dynamics and the regulation error dynamics are given by  
\[\dot{\epsilon}_{i}= \bar{A}_{i}e_{i}-T_{i}H_{i}\hat{z}_{i} \tag{27a}\] \[\dot{\epsilon}_{i}= (A_{i}+B_{i}K_{i})\epsilon_{i}+H_{i}\hat{z}_{i}-\Pi_{i}F\left(\sum _{j=1}^{N}\alpha_{ij}(\zeta_{j}-\zeta_{i})-\alpha_{i0}\zeta_{i}\right), \tag{27b}\]  
where we have used the fact that \((w_{j}-w_{i})=(\zeta_{j}-\zeta_{i})\).