## Problem Formulation and Preliminaries
### Io-Fts  
Consider the following impulsive hybrid systems:  
\[\left\{ \begin{array}{ll}\dot{x}(t)=A_{q(t)}(t)x(t)+H_{q(t)}(t)v(t),&q(t)=q( t^{+})\\ x(t^{+})=D_{q(t)}(t)x(t),&q(t)\
eq q(t^{+})\\ y(t)=C_{q(t)}(t)x(t),&\\ G=(\Xi,\Sigma,\zeta),\end{array} \right. \tag{2}\]  
where \(x(t)\in\mathbb{R}^{n}\) is the system state, \(v(t)\in\mathbb{R}^{r}\) is the particular exogenous input, and \(y(t)\in\mathbb{R}^{h}\) is the system output. \(A_{q(t)}(\cdot):\mathbb{R}^{+}\rightarrow\mathbb{R}^{n\times n}\), \(H_{q(t)}(\cdot):\mathbb{R}^{+}\rightarrow\mathbb{R}^{n\times r}\), \(D_{q(t)}(\cdot):\mathbb{R}^{+}\rightarrow\mathbb{R}^{n\times n}\), and \(C_{q(t)}(\cdot):\mathbb{R}^{+}\rightarrow\mathbb{R}^{n\times n}\). Figure 1: A simple example of the finite state machine (FSM). \(\mathbb{R}^{+}\to\mathbb{R}^{h\times n}\) are continuous matrix-valued functions. \(G=(\Xi,\Sigma,\zeta)\) is an FSM which is introduced in Section 2.1, and \(q(t)\in\Xi\) is the current node. Definition 4 ([12]): The hybrid systems (2) are said to be positive if, for any initial state \(x(t_{0})\succeq 0\), all \(v(t)\succeq 0\) and any jumping signal \(q(t)\in\Xi\), the corresponding trajectory \(x(t)\succeq 0\) and \(y(t)\succeq 0\) hold for all \(t\geq 0\). Lemma 1 ([12]): The hybrid systems (2) are positive if and only if \(A_{q(t)}(t)\) is a Metzler matrix, \(H_{q(t)}(t)\succeq 0\), \(D_{q(t)}(t)\succeq 0\), and \(C_{q(t)}(t)\succeq 0\) for each \(q(t)\in\Xi\). Lemma 2 ([23]): A matrix \(A\in\mathbb{R}^{n\times n}\) is a Metzler matrix if and only if there exists a positive constant \(\delta\) such that \(A+\delta I_{n}\succeq 0\). Let us extend the definition of IO-FTS given in [37] to the impulsive hybrid systems previously introduced. Definition 5: Given a positive scalar \(T\), a positive integer \(\mathcal{K}\), the initial node \(q(t_{0})\in\Xi\), the desired set \(E\), and a vector-valued function \(\eta(\cdot)\succ 0\), the positive hybrid systems (2) are said to be IO-FTS w.r.t. \((E,\omega,\eta(\cdot),T)\) if  
(1) the FSM is stable w.r.t. \((q(t_{0}),E,\mathcal{K})\), \(t_{\mathcal{K}}\leq T\), and  
(2) \(v(\cdot)\in\omega\Rightarrow\eta^{\mathrm{T}}(t)y(t)<1\),  
where \(\omega\) are a class of particular exogenous input signals defined over \([t_{0},T]\), and \(t\in[t_{0},T]\). In this paper, two different classes of exogenous input signals are considered (as has been done in [37]), and the vector-valued function \(\xi(\cdot)\) always satisfies \(\xi(t)\succeq 0\) for \(t\in[t_{0},T]\). (1) The set \(\omega\) coincides with the set of norm-bounded integrable signals over \([t_{0},T]\), defined as follows:  
\(\omega_{1}(t_{0},T,\xi(\cdot)):=\{v(\cdot)\in\mathcal{L}_{1,[t_{0},T]}:|v|_{[ t_{0},T],\xi(\cdot)}\leq 1\}\). (2) The set \(\omega\) coincides with the set of uniformly bounded signals over \([t_{0},T]\), defined as follows:  
\(\omega_{\infty}(t_{0},T,\xi(\cdot)):=\{v(\cdot)\in\mathcal{L}_{\infty,[t_{0}, T]}:\xi^{\mathrm{T}}(t)v(t)\leq 1,t\in[t_{0},T]\}\). The definitions of \(\omega_{1}(t_{0},T,\xi(\cdot))\) and \(\omega_{\infty}(t_{0},T,\xi(\cdot))\) depend on the choice of \(t_{0}\), \(T\), and \(\xi(\cdot)\), but these arguments will be omitted for brevity in the rest of the paper. Consider a class of impulsive hybrid systems:  
\[\left\{ \begin{array}{ll}\dot{x}(t)=A_{q(t)}(t)x(t)+B_{q(t)}(t)u(t)+H_{q(t)} (t)v(t),&q(t)=q(t^{+})\\ x(t^{+})=D_{q(t)}(t)x(t),&q(t)\
eq q(t^{+})\\ y(t)=C_{q(t)}(t)x(t)&\\ G=(\Xi,\Sigma,\zeta),\end{array} \right. \tag{3}\]  
where \(u(\cdot)\in\mathbb{R}^{x}\) is the control input, and \(B_{q(\cdot)}(\cdot):\mathbb{R}^{+}\to\mathbb{R}^{n\times x}\) is continuous matrix-valued function. Given a positive scalar \(T\) and a class of particular exogenous input signals \(\omega\) defined over \([t_{0},T]\), the objective of the paper is to find a output feedback control law \(u(t)=K_{q(t)}(t)y(t)\), where \(K_{q(t)}(t):\mathbb{R}^{+}\to\mathbb{R}^{n\times h}\) is the control gain, such that the closed-loop system  
\[\left\{