# Finite-Time Control of Singular Linear Semi-Markov Jump Systems
## 2 Problem Formulation  
Consider a class of SSMJSs with an external disturbance:  
\[E\hat{x}(t)=A(r(t))x(t)+B(r(t))u(t)+C(r(t))\omega(t), \tag{1}\]  
where \(x(t)\in\mathbb{R}^{n}\) is the state vector, \(u(t)\in\mathbb{R}^{m}\) is the control input vector and \(\omega(t)\in\mathbb{R}^{p}\) is the disturbance input satisfying \(\int_{0}^{T}\omega^{T}(t)\omega(t)\mathrm{d}t\leq d\) for given \(T>0\) and \(d>0\). \(r(t)\) is a semi-Markov chain taking values in the finite space \(\mathcal{S}=\{1,2,\ldots,N\}\), which governs the jump parameters in the probabilistic space \((\Omega,\mathcal{F},\mathcal{P})\). \(A(r(t))\), \(B(r(t))\), \(C(r(t))\) and \(r(t)\in\mathcal{S}\) are known matrices with compatible dimensions, which depend on \(r(t)\). The matrix \(E\) is singular, and we assume \(\mathrm{rank}(E)=r\leq n\). The evolution of the semi-Markov process \(r(t),t\geq 0\) in system (1) is governed by the following probability transitions. \[P\{r(t+\Delta)=j\mid r(t)=i\}= \begin{cases}\mu_{ij}(h)\Delta+o(\Delta),&r(t) \text{ jumps from }i\text{ to }j,\\ 1+\mu_{ii}(h)\Delta+o(\Delta),&r(t)\text{ stays at mode }i,\end{cases}  \tag{2}\]  
where \(h\geq 0\) is the sojourn time; \(\Delta>0\) and \(\lim_{\Delta\to 0}\frac{o(\Delta)}{\Delta}=0\); \(\mu_{ij}(h)\) is the transition rate when \(r(t)\) jumps from the mode \(i\) to mode \(j\) where \(i\
eq j\) and \(\mu_{ii}(h)=-\sum_{j=1,j\
eq i}^{N}\mu_{ij}(h)\). Here, it is also assumed that the transition rate \(\mu_{ij}(h)\) is not exactly known but bounded by \(\underline{\mu}_{ij}\leq\mu_{ij}(h)\leq\bar{\mu}_{ij}\) for \(\underline{\mu}_{ij}\geq 0\) and \(\bar{\mu}_{ij}\geq 0\). The objective of this paper is to walk through the design pf a state feedback controller for a SSMJS (1), such that the closed-loop system is regular, impulse-free and stochastically finite-time bounded. To this end, some definitions and lemmas are introduced first. **Definition 1** ([PERSON] et al. [34]). : _1. The pair \((E,A_{i})\) is said to be regular, if for each \(i\in\mathcal{S}\), \(\det(sE-A_{i})\
ot\equiv 0\), that is, \(\exists s\in\mathbb{C}\), \(\det(sE-A_{i})\
ot\equiv 0\)._
2. _The pair_ \((E,A_{i})\) _is said to be impulse-free, if for each_ \(i\in\mathcal{S}\)_,_ \(\deg(\det(sE-A_{i}))=\text{rank}(E)\)_._
3. _The SSMJS in_ (1) _with_ \(u(t)=0\) _and_ \(\omega(t)=0\) _is said to be regular and impulse-free, if for each_ \(i\in\mathcal{S}\)_, the pair_ \((E,A_{i})\) _is regular and impulse-free._  
**Lemma 1** ([PERSON] et al. [20]). : _The solution of the SSMJS in (1) with \(u(t)=0\) and \(\omega(t)=0\) exists and is unique and impulse-free, if this system is regular and impulse-free._  
**Definition 2** ([PERSON] et al. [35]). : _Consider the SSMJS in (1) with \(u(t)=0\) and \(\omega(t)=0\). For given positive scalars \(c_{1}\), \(c_{2}\) and \(T\) with \(c_{1}<c_{2}\), and symmetric positive-definite matrix \(R\), this system is said to be stochastically finite-time-stable with respect to \((c_{1},c_{2},T,R)\), if for \(\forall t\in[0,T]\), this system is regular, impulse-free and_  
\[\mathbb{E}\{x^{T}(0)E^{T}REx(0)\}\leq c_{1}\Longrightarrow\mathbb{E}\{x^{T}(t)E ^{T}REx(t)\}<c_{2}. \tag{3}\]  
**Definition 3**. : _Consider the SSMJS in (1) with \(\omega(t)=0\). For given positive scalars \(c_{1}\), \(c_{2}\) and \(T\) with \(c_{1}<c_{2}\), and symmetric positive-definite matrix \(R\), this system is said to be stochastically finite-time stabilizable with respect to \((c_{1},c_{2},T,R)\), if there exists a state feedback control law such that the closed-loop system is stochastically finite-time-stable with respect to \((c_{1},c_{2},T,R)\)._  
**Definition 4** ([PERSON] et al. [35]). : _Consider the SSMJS in (1).