## 2 Problem Description and Preliminaries  
Consider the continuous-time Markov jump system. The model of the system is defined by the following differential equations:  
\[\left\{\begin{array}{l}\dot{x}(t)=A(\gamma_{t})x(t)+B(\gamma_{t})u(t),\\ y(t)=C(\gamma_{t})x(t),\end{array}\right. \tag{1}\]where \(A(\gamma_{t})\), \(B(\gamma_{t})\) and \(C(\gamma_{t})\) are the parameter matrices of the system. \(x(t)\in\mathbb{R}^{n}\), \(u(t)\in\mathbb{R}^{m}\), \(y(t)\in\mathbb{R}^{d}\) are the state vector, input vector, and output vector, respectively. \(\{\gamma_{t},t>0\}\) is the Markov process that represents the jump mode of system (1), which takes values in a finite set \(U=\{1,2,\cdots,N\}\). Moreover, the transition probability matrix \(\Pi\) is given by:  
\[\Pr\{\gamma_{t+\Delta}=j\mid\gamma_{t}=i\}=\left\{\begin{array}{cc}\pi_{ij} \Delta+o(\Delta),&i\
eq j,\\ 1+\pi_{ij}\Delta+o(\Delta),&i=j,\end{array}\right. \tag{2}\]  
where \(\Delta>0\). \(\lim\limits_{\Delta\to 0}\dfrac{o(\Delta)}{\Delta}=0\), \(\pi_{ij}>0,\forall i\
eq j\). Denote the transition rate from i to j and \(\pi_{ii}=-\sum\limits_{j\
eq i}\pi_{ij}\) for i, j. When transition rates are partly unknown, the transition matrix can be described as follows:  
\[\Pi=\left[\begin{array}{cccc}\pi_{11}&?&\cdots&\pi_{1N}\\?&\pi_{22}&\cdots&\pi_{2N}\\ \vdots&\vdots&\ddots&\vdots\\ \pi_{N1}&?&\cdots&\pi_{NN}\end{array}\right], \tag{3}\]  
where the \"?\" denotes the unknown transition rates. To facilitate the future discussion, for all \(i\in N\), the set \(U^{(i)}\) denotes \(U^{(i)}=U^{(i)}_{k}\cup U^{(i)}_{uk}\)  
\[U^{(i)}_{k}=\{j|\pi_{ij}\text{ is known for every }\quad j\in U\}, \tag{4}\]  
\[U^{(i)}_{uk}=\{j|\pi_{ij}\text{ is unknown for every }\quad j\in U\}. \tag{5}\]  
Moreover, if \(U^{(i)}_{uk}\
eq\emptyset\). It is further described as \(U^{(i)}_{k}=\{k^{i}_{1},k^{i}_{2},\cdots k^{i}_{m}\}\). Where \(k^{i}_{j}\) represent the \(j\)th bounded-known element with the index \(k^{i}_{j}\) in the \(i\)th row of \(\Pi\). Denote \(A_{i}\), \(B_{i}\), and \(C_{i}\) to replace \(A(\gamma_{t})\), \(B(\gamma_{t})\), and \(C(\gamma_{t})\), when \(\gamma_{t}=i\).  
**Definition 1** ([44]).: _The closed loop system \(\dot{x}(t)=A_{i}x(t)+B_{i}u(t)\) with \(u(t)=0\) is stochastically stable (SS) if for every initial condition \(x(0)=x_{0}\) and \(\gamma(0)=\gamma_{0}\) the following holds_  
\[\mathbb{E}\{\int_{t_{0}}^{\infty}\|x(t)\|^{2}dt|x_{0},\gamma_{0}\}<\infty. \tag{6}\]  
\(\mathbb{E}\) represents the mathematical expectation operator. This paper researches the model reference tracking control problem for continuous-time systems with a circular-like orbit. Without losing generality, the reference signal can be generated by the following reference model:  
\[\left\{\begin{array}{ll}\dot{x}_{m}(t)=A_{m}(\gamma_{t})x_{m}(t),\\ y_{m}(t)=C_{m}(\gamma_{t})x_{m}(t).\end{array}\right. \tag{7}\]  
where \(x_{m}(t)\in\mathbb{R}^{n_{1}}\), \(y_{m}(t)\in\mathbb{R}^{n_{2}}\) are the state vector and output vector of the reference mode. \(A_{m}\) and \(C_{m}\) are the known matrices with appropriate dimensions. The purpose is to design a controller that can make the output \(y(t)\) track the reference model's output \(y_{m}(t)\) in the meaning square:  
\[\mathbb{E}\{\int_{t_{0}}^{\infty}\|y(t)-y_{m}(t)\|^{2}dt|x_{0},\gamma_{0}\}<\infty. \tag{8}\]  
For arbitrary initial values \(x_{0}\), \(\gamma_{0}\), and \(x_{m_{0}}\). The following problem needs to be done to realize the purpose of the paper. For systems (1) and (7), a controller is designed in the form of  
\[u(t)=K_{i}x(t)+K_{mi}x_{m}(t). \tag{9}\]Using this algorithm, the closed-loop system can be stochastic stable and \(y(t)\) can track \(y_{m}(t)\) in the meaning square.