# Two-Phase Non-Singular Terminal Sliding Mode Control of Nonlinear Systems
## 3 Problem Formulation
### Extensions to MIMO Systems  
Consider a class of multi-input multi-output (MIMO) nonlinear systems,  
\[\left\{ \begin{array}{l}\dot{X}_{1}=F_{1}(X_{1},X_{2})\\ \dot{X}_{2}=F_{2}(X_{1},X_{2})+B(X_{1},X_{2})U+D(X_{1},X_{2})\end{array} \right. \tag{24}\]  
where \(X_{1}=[x_{11},x_{12},\ldots,x_{1,n}]\in\mathbb{R}^{n}\), \(X_{2}=[x_{21},x_{22},\ldots,x_{2,n}]\in\mathbb{R}^{n}\) are the states, \(F_{1}(\cdot),F_{2}(\cdot)\) are nonlinear vector functions, and \(B(\cdot)\) is a non-singular matrix, and \(U\in\mathbb{R}^{n}\) is the control input, and \(D(\cdot)\) stands for the external disturbances. There are many practical systems that can be expressed by referring to (24). Therefore, the presented scheme can be applied to these plants. Following assumptions are made:  
**Assumption 1**. : _The disturbance \(D(\cdot)\) is bounded and satisfies \(\|D(X_{1},X_{2})\|=\bar{d}(X_{1},X_{2})\)._  
**Assumption 2**. : \(\frac{\partial F_{1}}{\partial X_{2}}B(X_{1},X_{2})\) _is a non-singular matrix._  
The two-phase NTSM is constructed for system (24),  
\[\Xi=\left\{ \begin{array}{ll}\dot{X}_{1}+\Lambda_{1}X_{1}^{\Phi_{1}}+\Lambda _{2}X_{1}^{2-\Phi_{1}},&\|X_{1}\|\geq X_{t}\\ \dot{X}_{1}+\Lambda_{1}X_{1}+\Lambda_{1}X_{1}^{\Phi_{1}},&\|X_{1}\|<X_{t} \end{array} \right. \tag{25}\]  
where \(\Xi\) represents the n-dimensional sliding mode manifold, and the matrices are defined as \(\Lambda_{1}=\text{diag}(\lambda_{11},\lambda_{12},\ldots,\lambda_{1n})\), \(\Lambda_{2}=\text{diag}(\lambda_{21},\lambda_{22},\ldots,\lambda_{2n})\), \(\Phi_{1}=\text{diag}(\phi_{11},\phi_{12},\ldots,\phi_{1n})\), in which \(\lambda_{1i}>0,\lambda_{2i}>0,i=1,\ldots,n\) and \(0<\phi_{1i}<1,i=1,\ldots,n\) are fractions in which the numerator and denominator are positive odd integers. \(X_{t}\) is the transition state matrix given by the designer. The derivative of \(X_{1}^{\Phi_{1}}\) is denoted as \(\frac{d}{dt}(X_{1}^{\Phi_{1}})=\Phi_{1}\text{diag}(|X_{1}|^{\Phi_{1}-I_{n}}) \dot{X}_{1}\), where \(I_{n}\) is the \(n\)-dimensional unit matrix. In the presence of Assumptions 1-3 and the two-phase manifold (25), the NTSM control input is developed as  
\[U = -(\frac{\partial F_{1}}{\partial X_{2}}B)^{-1}\bigg{(}\frac{ \partial F_{1}}{X_{1}}F_{1}+\frac{\partial F_{1}}{\partial X_{2}}F_{2}+\|\frac{ \partial F_{1}}{\partial X_{2}}\|\tilde{d}+ \tag{26}\] \[M\frac{\Xi}{\|\Xi\|}+\Lambda_{1}\Phi_{1}\text{sat}(X_{1}^{\Phi_ {1}-I_{n}}\dot{X}_{1},h)+\] \[((\frac{\Lambda_{1}}{2}+\frac{\Lambda_{2}}{2})+(\frac{\Lambda_{2 }}{2}-\frac{\Lambda_{1}}{2})\text{sign}(|X_{1}|-X_{t}))\times\] \[(\frac{3I_{n}-\Phi_{1}}{2}+\frac{I_{n}-\Phi_{1}}{2}\text{sign}(|X _{1}|-X_{t}))\times\] \[X_{1}^{\frac{I_{n}-\Phi_{1}}{2}+\frac{I_{n}-\Phi_{1}}{2}\text{ sign}(\|X_{1}\|-X_{t})}\dot{X}_{1}+\] \[\Gamma_{1}\[PERSON]^{\[PERSON]_{2}}+((\frac{\Gamma_{1}}{2}+\frac{\Gamma_{2 }}{2})+(\frac{\Gamma_{2}}{2}-\frac{\Gamma_{1}}{2})\times\] \[\text{sign}(\|\[PERSON]\|-\[PERSON]_{t}))\[PERSON]^{\frac{3I_{n}-\[PERSON]_{2}}{2}+ \frac{I_{n}-\[PERSON]_{2}}{2}\text{sign}(\|\[PERSON]\|-\[PERSON]_{t})}\]  
where \(\Gamma_{1}=\text{diag}(\gamma_{11},\gamma_{12},\ldots,\gamma_{1n}),\Gamma_{2}= \text{diag}(\gamma_{21},\gamma_{22},\ldots,\gamma_{2n}),\Phi_{2}=\text{diag}( \phi_{21},\phi_{22},\ldots,\phi_{2n})\), in which \(\lambda_{2i}>0,i=1,\ldots,n\) and \(0<\phi_{2i}<1,i=1,\ldots,n\) are fractions in which numerator and denominator are positive odd integers, and \(h\) is the bound of \(\|\text{diag}(|X_{1}|^{\Phi_{1}-I_{n}})\dot{X}_{1}\|\). The analysis is conducted in line with the proof of Theorem 1, and is given as follows. For \(\|\text{diag}(|X_{1}|^{\Phi_{1}-I_{n}})\dot{X}_{1}\|<h\), the Lyapunov candidate function is chosen as \(\tilde{V}=\Xi^{T}\Xi\). Along with the manifold (25) and the control input (26), the derivative of \(\tilde{V}\) can be expressed as  
\[\hat{V} = \bigg{\{}