## 4 Hybrid Systems
### Stabilization of Non-Autonomous Hybrid Systems  
Next, consider the case where impulsive positive hybrid systems (3) based on FSM are non-autonomous. Designing the output feedback controller \(u(t)=K_{q(t)}(t)y(t)\) and taking it into systems (3), then the closed-loop systems (4) can be obtained.  
**Theorem 7**.: _The constants are \(\mu\geq 1\), \(T,\alpha,\lambda>0\) with \(\lambda\alpha<1\), and suppose \(E\) is the desired set. If there exists a vector-valued function \(v_{q(t)}(t)\succ 0\) such that_  
_(1)_ \(\|x_{d}(t_{k})\|_{1}\) _is strictly monotonically decreasing w.r.t._ \(k\)_,_  
_(2)_ \(\tilde{A}_{q(t)}(t)+\delta I_{n}\succeq 0\)_,_  
_(3)_ \(v^{\mathsf{T}}_{q(t)}(t)+v^{\mathsf{T}}_{q(t)}(t)\tilde{A}_{q(t)}(t)\prec 0\)_,_ \(q(t^{-})=q(t)\)__  
_(4)_ \(v^{\mathsf{T}}_{q(t)}(t)H_{q(t)}(t)-\lambda\delta^{\mathsf{T}}(t)\preceq 0\)_,_ \(q(t^{-})=q(t)\)__  
_(5)_ \(v^{\mathsf{T}}_{q(t)}(t)D_{q(t^{-})}(t)\preceq\mu_{q(t)}v^{\mathsf{T}}_{q(t^{-})} (t)\)_,_ \(q(t^{-})\
eq q(t)\) _and_  
_(6)_ \(\eta^{\mathsf{T}}(t)C_{q(t)}(t)\preceq\alpha v^{\mathsf{T}}_{q(t)}(t)\)_,_  
_hold, then system (4) is positive and there is IO-FTS w.r.t. \((E,\omega_{1},\eta(\cdot),T)\) under MDADT (13); that is to say, system (3) is positive and stabilizable, where \(\tilde{A}_{q(t)}(t)=A_{q(t)}(t)+B_{q(t)}(t)K_{q(t)}(t)C_{q(t)}(t)\), \(H_{q(t)}(t)\succeq 0\), \(D_{q(t)}(t)\succeq 0\), \(C_{q(t)}(t)\succeq 0\), \(\delta>0\), \(x(t_{0})=0\), \(v(\cdot)\in\omega_{1}\), \(q(t)\in\Xi\), and \(t\in[t_{0},T]\)._Proof.: From Lemma 2 and condition (2) of Theorem 7, we know \(\tilde{A}_{q(t)}(t)\) is a Metzler matrix, \(H_{q(t)}(t)\succeq 0\), \(D_{q(t)}(t)\succeq 0\), \(C_{q(t)}(t)\succeq 0\), and \(v_{q(t)}(t)\succeq 0\) for each \(q(t)\in\Xi\). According to Lemma 1, system (4) is positive. Then, under the output feedback controller \(u(t)=K_{q(t)}(t)y(t)\), replacing \(A_{q(t)}(t)\) in Theorem 5 with \(\tilde{A}_{q(t)}(t)\), condition (3) of Theorem 7 can be obtained. By Theorem 5, we can easily find that system (4) is positive and there is IO-FTS w.r.t. \((E,\omega_{1},\eta(\cdot),T)\) under MDADT (13). Therefore, system (3) is positive and stabilizable.  
**Theorem 8**.: _The constants are \(\mu\geq 1\), \(T,\alpha,\lambda>0\) with \(\lambda\alpha(T-t_{0})<1\), and suppose \(E\) is the desired set. If there exists a vector-valued function \(v_{q(t)}(t)\succ 0\) such that_  
\((1)\quad\|x_{d}(t_{k})\|_{1}\) _is strictly monotonically decreasing w.r.t._ \(k\)_,_  
\((2)\quad\tilde{A}_{q(t)}(t)+\delta I_{k}\succeq 0\)_,_  
\((3)\quad\tilde{v}_{q(t)}^{\mathbb{T}}(t)+v_{q(t)}^{\mathbb{T}}(t)\tilde{A}_{q (t)}(t)\prec 0\)_,_ \(\qquad\qquad q(t^{-})=q(t)\)__  
\((4)\quad v_{q(t)}^{\mathbb{T}}(t)H_{q(t)}(t)-\lambda\xi^{\mathbb{T}}(t) \preceq 0\)_,_ \(\qquad\qquad q(t^{-})=q(t)\)__  
\((5)\quad v_{q(t)}^{\mathbb{T}}(t)D_{q(t^{-})}(t)\preceq\mu_{q(t)}v_{q(t^{-})} ^{\mathbb{T}}(t)\)_,_ \(\qquad\qquad q(t^{-})\
eq q(t)\) _and_  
\((6)\quad\eta^{\mathbb{T}}(t)C_{q(t)}(t)\preceq\alpha v_{q(t)}^{\mathbb{T}}(t)\)_,_  
_hold, then system (4) is positive and there is IO-FTS w.r.t. \((E,\omega_{\infty},\eta(\cdot),T)\) under MDADT (17). That is to say, system (3) is positive and stabilizable, where \(\tilde{A}_{q(t)}(t)=A_{q(t)}(t)+B_{q(t)}(t)K_{q(t)}(t)C_{q(t)}(t)\), \(H_{q(t)}(t)\succeq 0\), \(D_{q(t)}(t)\succeq 0\), \(C_{q(t)}(t)\succeq 0\), \(\delta>0\), \(x(t_{0})=0\), \(v(\cdot)\in\omega_{\infty}\), \(q(t)\in\Xi\), and \(t\in[t_{0},T]\)._  
By similar analysis of Theorem 7, Theorem 8 can be easily proved.  
**Remark 6**.: _Compared with the literature [28], the MDADT technique is used for Theorems 5-8 in this paper. It allows every node of FSM to have its own ADT, therefore, the sufficient conditions of IO-FTS have good flexibility and weak conservatism._