## 2 Materials and Methods  
### Dynamic Equaiton of a Nonlinear MIMO System  
Dynamic equations of a multi-input multi-output (MIMO) nonlinear system are described as follows [38]:  
\[\begin{array}{l}\dot{x}_{i,j_{i}}=f_{i,j_{i}}(\overline{x}_{i,1},\ldots, \overline{x}_{i,n_{i}-1},x_{i,1},\ldots,x_{i,n_{i}-1},u_{1},\ldots,u_{i-1})+g _{i,j_{i}}(\overline{x}_{i,1},\ldots,\overline{x}_{i,n_{i}-1},x_{i,1},\ldots,x _{i,n_{i}-1}u_{1},\ldots,u_{i-1})x_{i,j_{i}+1}+d_{i,j_{i}}\\ =f_{i,j_{i}}(\overline{x}_{i,1},\ldots,\overline{x}_{i,n_{i}-1},x_{i,1},\ldots,x_{i,n_{i}-1},u_{1},\ldots,u_{i-1})+g_{0i,j_{i}}(\overline{x}_{i,1},\ldots, \overline{x}_{i,n_{i}-1},x_{i,1},\ldots,x_{i,n_{i}-1},\\ u_{1},\ldots,u_{i-1})x_{i,j_{i}+1}+g_{0i,j_{i}}(\overline{x}_{i,1},\ldots, \overline{x}_{i,n_{i}-1},x_{i,1},\ldots,x_{i,n_{i}-1},u_{1},\ldots,u_{i-1})x_ {i,j_{i}+1}+l_{i,j_{i}}\,,\\ \vdots\\ \dot{x}_{i,n_{i}}=f_{i,n_{i}}(\overline{x}_{i,1},\ldots,\overline{x}_{i,n_{i}- 1},x_{i,1},\ldots,x_{i,n_{i}},u_{1},\ldots,u_{i-1})+g_{i,n_{i}}(\overline{x}_ {i,1},\ldots,\overline{x}_{i,n_{i}-1},x_{i,1},\ldots,x_{i,n_{i}}u_{1},\ldots,u_ {i-1})u_{i}+d_{i,n_{i}}\\ =f_{i,n_{i}}(\overline{x}_{i,1},\ldots,\overline{x}_{i,n_{i}-1},x_{i,1},\ldots,x_{i,n_{i}},u_{1},\ldots,u_{i-1})+g_{0i,n_{i}}(\overline{x}_{i,1},\ldots, \overline{x}_{i,n_{i}-1},x_{i,1},\ldots,x_{i,n_{i}}u_{1},\ldots,u_{i-1})u_{i} +l_{i,n_{i}}\,,\\ y_{i}=x_{i,1},j_{i}=1,\ldots,n_{i}-1,\ i=1,\ldots,m,\end{array} \tag{1}\]  
where \(x=\left[\overline{x}_{1,n_{1}}^{\mathrm{T}},\ldots,\overline{x}_{m,n_{m}} \right]^{\mathrm{T}}\in R^{n}\) is the state vector, \(\overline{x}_{i,n_{1}}=\left[x_{i,1},\ldots,x_{i,n_{1}}\right]^{\mathrm{T}} \in R^{n_{1}}\), \(\overline{x}_{i,j_{i}}=\left[x_{i,1},\ldots,x_{i,j_{i}}\right]^{\mathrm{T}} \in R^{j_{i}}\), and \(\overline{x}_{i,n_{i}-1}=\left[x_{i,1},\ldots,x_{i,n_{i}-1}\right]^{\mathrm{T} }\in R^{n_{i}-1}\) are the partial state vector; \(u_{i}\in R\) and \(y_{i}\in R\) denote the input and output of the \(i\)th subsystem, respectively; \(f_{i,j_{i}}\) is unknown; \(g_{i,j_{i}}\) is partially known function; \(g_{i,j_{i}}=g_{0i,j_{i}}+\Delta g_{i,j_{i}}\), where \(g_{0i,j_{i}}\) is the known function, \(\Delta g_{i,j_{i}}\) is the perturbation of \(g_{i,j_{i}}\); \(d_{i,j_{i}}\) is the bounded uncertainty, which includes the nonlinear dynamics and external disturbance; and \(l_{i,j_{i}}=\Delta g_{i,j_{i}}x_{i,j_{i}+1}+d_{i,j_{i}}\) and \(l_{i,n_{i}}=\Delta g_{i,n_{i}}u_{i}+d_{i,n_{i}}\) are lumped uncertainties.  
The aim of the design of the virtual controllers \(\alpha_{i,j_{i}}\) and final controllers \(u_{i}\) is to provide controllers for tracking the state variable \(x_{i,j_{i}}\) along with the command trajectory \(x_{f_{i,j_{i}}}\) and the compensating uncertainties \(l_{i,j_{i}}\), while ensuring that all signals are bounded and the errors of estimate and tracking are converged to zero in finite time.  
**Assumption 1**.: _The control gain function \(g_{i,j_{i}},j_{i}=1,\ldots,n_{i},\ i=1,\ldots,m,\) is bounded, and has known sign, and has its inverse matrix._  
**Assumption 2**.: \(\left|l_{i,j_{i}}\right|\) _is bounded and is satisfied with the unknown positive constants \(\left|l_{i,j_{i}}\right|\leq l^{\prime}_{i,j_{i}}\), \(j_{i}=1,\ldots,n_{i},\ i=1,\ldots,m\)._