# Compensation Observer-Based Adaptive Output Feedback Control for Multi-Agent Systems
## 2 Establishing the Problem Model
### Basic Model and Problem Hypothesis  
We consider a set of nonlinear systems defined as follows. First, Equation (6) delineates the state equation:  
\[ \begin{array}{l}\dot{x}_{ij}=A_{i}x_{ij}+\varphi_{0,i,j}(y_{i})+\sum_{k=1}^ {r_{i}}\varphi_{k,i,j}(y_{i})a_{k,i,j}+F_{i,j}(y_{i})x_{ij}+b_{i}H_{i,j}(y_{i} )w_{i}\\ y_{i}=c_{i}^{T}x_{i}\end{array} . \tag{6}\]  
Let \(x  
\[\varphi_{0,i,j}(y_{i})=\big{[}\varphi_{0,1,j}(y_{i})\quad\varphi_{0,2,j}(y_{i}) \quad\cdots\quad\varphi_{0,n,j}(y_{i})\big{]}^{T} \tag{11}\]  
\[\varphi_{k,i,j}(y_{i})=\big{[}\varphi_{k,1,j}(y_{i})\quad\varphi_{k,2,j}(y_{i}) \quad\cdots\quad\varphi_{k,n,j}(y_{i})\big{]}^{T} \tag{12}\]  
where \(w_{i}\) is an input for hysteresis. The Prandtl-Ishlinskii model was selected as the hysteresis model in this study. The relevant formula and mathematical model are as follows:  
\[w_{i}(t)=\sigma_{i}u_{i}(t)+\mathcal{F}[u_{i}](t). \tag{13}\]  
In the expression \(\mathcal{F}[u_{i}](t)=\)\(\int_{0}^{R}k(r)\mathcal{F}_{r}^{\Psi}[u_{i}](t)dr\), the function \(k(r)\) is a continuous, non-negative known function, while \(\sigma_{i}\) is an unknown, non-zero constant. This equation represents an area-based hysteresis model. The design parameter \(R\) serves as the upper limit of integration. In practical analysis, the influence of \(k(r)\) can be neglected when \(R\) approaches infinity (\(R=\infty\)). Consequently, both \(k(r)\) and \(\mathcal{F}_{r}^{\Psi}[u_{i}](t)dr\) are bounded, and \(k(r)\) is determined by experimental data. As a result, the functions \(\mathcal{F}[u_{i}](t)\) are also bounded. Thus, the range of \(\mathcal{F}[u_{i}](t)\) is constrained to \(\mathcal{F}\), or \(\mathcal{F}_{r}^{\Psi}[u_{i}](t)dr\leq\mathcal{F}\). This formulation describes the saturation play operator \(\mathcal{F}_{r}^{\Psi}[u_{i}]\). The expression uses \(u_{i}(t)\) and \(w_{i}\), respectively, as the input and output of symmetrical hysteresis, which is constrained by the curve \(\Psi\) and represented by a well-known, continuous, non-decreasing function. Here, \(R\) indicates the hysteresis threshold:  
\[\mathcal{F}_{r}^{\Psi}[u_{i}](0)=f_{r}^{\Psi}(u_{i}(0)) \tag{14}\]  
\[\mathcal{F}_{r}^{\Psi}[u_{i}](t)=f_{r}^{\Psi}(u_{i}(t),\mathcal{F}_{r}^{\Psi} [u_{i}](t_{k})) \tag{15}\]  
\[f_{r}^{\Psi}(u_{i}(t),w_{i})=\max(\Psi(u_{i})-r,\min(\Psi(u_{i})+r,w_{i})) \tag{16}\]  
\[t_{k}<t<t_{k+1}\,,0\leq k\leq M-1. \tag{17}\]  
Next, we provide a monotone input signal, which is monotone in the interval \([t_{k},t_{k+1}]\), and select the parameter number using the formula above, as follows:  
\[ \begin{split}\sigma_{i}=0.6,\Psi(u_{i})=6\text{tanh}(u_{i}),R=70,k(r)=0.9e^{-0.04(r-0.5)^{2}}\\ u_{i}=4\sin(10\pi t)+\cos(0.5\pi t)\end{split}  \tag{18}\]  
In this way, the hysteresis model diagram shown in Figure 1 can be obtained. Figure 1: PI hysteresis input relation model diagram. The hysteresis effect set in this study has an upper and lower limit, as shown in Figure 1. Here, we use the parameter \(T\) to represent the range of the hysteresis effect, which is \(T=H_{ij}(y_{i})\mathcal{F}[u_{i}](t)\). The parameter \(H\) represents the last term of the hysteresis effect, and the parameter \(h\) represents the minimum value of the hysteresis effect. Here, \(H=20\), \(h=-20\). **Remark 1**. : _The hysteresis phenomenon  
The definition error is \(\dot{\epsilon}_{ij}=x_{ij}-\hat{x}_{ij}\), and its derivative is expressed as  
\[\dot{\epsilon}_{ij}=\left(A_{i}-l_{i}L_{i}q_{i}c_{i}^{T}\right)\epsilon_{ij}+F_{ ij}(y_{i})\epsilon_{ij}+b_{i}T-b_{i}\frac{TH}{h^{2}+\sigma_{\infty}^{2}}. \tag{23}\]  
Transformation of this error is achieved as follows:  
\[\overline{\epsilon}_{ij}=l_{i}^{-\Delta}L_{i}^{-1}\epsilon_{ij}. \tag{24}\]  
We next define the Lyapunov equation as an expression of \(V_{\dot{\epsilon}_{ij}}=\big{(}\overline{\epsilon}_{ij}\big{)}^{T}P_{i} \overline{\epsilon}_{ij}\) and \(A_{i}-l_{i}L_{i}q_{i}c_{i}^{T}=l_{i}L_{i}\big{(}A_{i}-q_{i}c_{i}^{T}\big{)}L_ {i}^{-1}\) and combine Formulas (14)-(18):  
\[