## 2 Problem Statement  
Consider an uncertain system with uncertainty and noise described by:  
\[\dot{x}(t) =(A+\Delta A(x(t),\alpha(t),t))x(t)+Bu(t)+B_{1}w(t),\] \[y(t) =Cx(t)+Du(t)+D_{1}w(t), \tag{1}\] \[x(0) =x_{0}.\]  
Here, \(t\)\(\in\)\(\mathbb{R}\) is time, \(x(t)\)\(\in\)\(\mathbb{R}^{n}\) is the state, \(u(t)\)\(\in\)\(\mathbb{R}^{m}\) is the control input, \(w(t)\)\(\in\)\(\mathbb{R}^{k}\) is the noise and \(y(t)\)\(\in\)\(\mathbb{R}^{l}\) is the output. \(A\)\(\in\)\(\mathbb{R}^{n\times n}\), \(B\)\(\in\)\(\mathbb{R}^{n\times m}\), \(B_{1}\)\(\in\)\(\mathbb{R}^{n\times k}\), \(C\)\(\in\)\(\mathbb{R}^{l\times n}\), \(D\)\(\in\)\(\mathbb{R}^{l\times m}\), \(D_{1}\)\(\in\)\(\mathbb{R}^{l\times k}\) are known constant real matrices; \(\Delta A(x(t),\alpha(t),t))\) is a matrix function representing time-varying parameter uncertainties in the system model and proposed by the following assumption. **Assumption 1**. : _There exist real matrices \(E(t)\)\(\in\)\(\mathbb{R}^{m\times n}\) and \(G\)\(\in\)\(\mathbb{R}^{m\times k}\), such that:_  
\[ \begin{split}&\Delta A(x(t),\alpha(t),t)=BE(x(t),\alpha(t),t),\\ & B_{1}=BG,\end{split}  \tag{2}\]  
_for all \((x,w,t)\)\(\in\)\(\mathbb{R}^{n}\times\mathbb{R}^{k}\times\mathbb{R}\)._  
Define:  
\[v(t):=E(x(t),\alpha(t),t)x(t)+Gw(t). \tag{3}\]  
From (1), \(v(t)\) can be considered as the whole uncertain element. **Assumption 2**. : _Consider that the uncertain system (1) subjected to Assumption 1, for all \((x,w,t)\in\mathbb{R}^{n}\times\mathbb{R}^{k}\times\mathbb{R}\), \(v(t)\) satisfies:_  
\[ \begin{split}\|v(t)\|&\leq\|E\|\|x(t)\|+\|G\|\|w(t)\|\\ &\leq\|E\|_{max}\|x(t)\|+\|G\|w_{max}=:\rho(x,w,t),\end{split}  \tag{4}\]  
_where \(\|E\|_{max}\) is the maximum value of \(\|E\|\), \(w_{max}\) is the maximum value of \(\|w(t)\|\) and \(\rho(x,w,t):\mathbb{R}^{n}\times\ \mathbb{R}^{k}\times\ \mathbb{R}\to\mathbb{R}_{+}\) is a known function._  
Consider a reference model described as:  
\[ \begin{split}\dot{x}_{ss}(t)&=Ax_{ss}(t)+Bu_{ss}(t )\\ y_{ss}(t)&=Cx_{ss}(t)+Du_{ss}(t),\end{split}  \tag{5}\]  
where \(x_{ss}(t)\in\mathbb{R}^{n}\), \(u_{ss}(t)\in\mathbb{R}^{m}\) and \(y_{ss}(t)\in\mathbb{R}^{l}\) are, respectively, the state, input and the output of the reference model. By (1) and (3), we have a deviation system as:  
\[ \begin{split}\dot{\bar{x}}(t)&=A\bar{x}(t)+\Delta A (x(t),\alpha(t),t)x(t)+B\bar{u}(t)+B_{1}w(t)\\ \tilde{y}(t)&=C\bar{x}(t)+D\bar{u}(t)+D_{1}w(t)\\ \tilde{x}(0)&=x_{0}-x_{ss0},\end{split}  \tag{6}\]  
where \(\bar{x}(t)=x(t)-x_{ss}(t)\), \(\bar{u}(t)=u(t)-u_{ss}(t)\), \(\tilde{y}(t)=y(t)-y_{ss}(t)\) are the state, control input and output of the uncertain system. For the robust control design, we divide the system (4) into two parts: the nominal system (the system without the uncertainty) and the uncertain portion. Consider System (4); we propose our robust control as:  
\[\bar{u}(t)=\tilde{p}(t)+\tilde{q}(t), \tag{7}\]  
where \(\tilde{q}(t)\in\mathbb{R}^{m}\) is the control for the nominal system and \(\tilde{p}(t)\in\mathbb{R}^{m}\) is the control for the uncertain portion. We describe the nominal system as:  
\[ \begin{split}\dot{\bar{x}}_{n}(t)&=A\bar{x}_{n}(t )+B\tilde{q}(t)+B_{1}w_{n}(t)\\ \tilde{y}_{n}(t)&=C\bar{x}_{n}(t)+D\tilde{q}(t)+D_{1 }w_{n}(t)\\ \tilde{x}_{n}(0)&=x_{n0}-x_{ns0},\end{split}  \tag{8}\]  
where \(\bar{x}_{n}(t)=x_{n}(t)-x_{nss}(t)\), \(\tilde{y}_{n}(t)=y_{n}(t)-y_{nss}(t)\) are the state and output of the nominal system (1). **Assumption 3**.