# Multivariable Adaptive Super-Twisting Guidance Law Based on Barrier Function
## 3 Barrier Function-Based Multivariable Super-Twisting Guidance Law Design
### Design of Sliding Mode Guidance Law Based on Adaptive Super-Twisting Control  
In order to improve the convergence speed of the sliding variable and avoid the singularity in the meanwhile, the following sliding manifold for system (5) is taken into account, which is formulated as:  
\[\mathbf{s}=\mathbf{x}_{2}+k_{1}\mathbf{x}_{1}+k_{2}\mathbf{w}(\mathbf{x}_{1}) \tag{16}\]  
where \(\mathbf{s}=\left[ \begin{array}{cc}s_{1}&s_{2}\end{array} \right]^{T}\) is the sliding manifold vector; \(k_{1}\) and \(k_{2}\) are designed positive constants; and term \(\mathbf{w}(\mathbf{x}_{1})=\left[ \begin{array}{cc}w(x_{11})&w(x_{12})\end{array}  \right]^{T}\) is given by:  
\[w(x_{1i})=\left\{ \begin{array}{ll}|x_{1i}|^{k_{3}}sign(x_{1i}),&s_{i}=0\; \text{or}\;\overline{s}_{i}\
eq 0,|x_{1i}|\geq\xi\\ k_{4}\mathbf{x}_{1i}+k_{5}\mathbf{x}_{1i}^{2}sign(x_{1i}),&\overline{s}_{i}\
eq 0,|x_{ 1i}|<\xi\end{array} \right.\quad(i=1,2) \tag{17}\]  
where \(\overline{s}_{i}=x_{2i}+k_{1}\mathbf{x}_{1i}+k_{2}|x_{1i}|^{k_{3}}sign(x_{1i})\); \(\xi\) and \(k_{3}\) are designed constants, which satisfy \(0<k_{3}<1\), \(\xi>0\); \(k_{4}\) and \(k_{5}\) are designed to keep \(w(x_{1i})\) continuous, which satisfy:  
\[k_{4}=(2-k_{3})\xi^{k_{3}-1},\;k_{5}=(2-k_{3})\xi^{k_{3}-2} \tag{18}\]  
Then the time derivative of \(\mathbf{s}\) can be formulated as:  
\[\dot{\mathbf{s}}=\dot{\mathbf{x}}_{2}+k_{1}\mathbf{x}_{2}+k_{2}\dot{w}(\mathbf{x}_{1}) \tag{19}\]  
where the term \(\dot{w}(x_{1})\) is given by:  
\[\dot{w}(x_{1i})=\left\{ \begin{array}{ll}k_{3}x_{1i}^{k_{3}-1}x_{2i},&s_{i}=0 \;\text{or}\;\overline{s}_{i}\
eq 0,|x_{1i}|\geq\xi\\ k_{4}x_{2i}+2k_{5}x_{1i}x_{2i},&\overline{s}_{i}\
eq 0,|x_{1i}|<\xi\end{array}  \right.\quad(i=1,2) \tag{20}\]  
Substituting (5) into (19) yields:  
\[\dot{\mathbf{s}}=\left[ \begin{array}{cc}\dot{s}_{1}&\dot{s}_{2}\end{array}  \right]^{T}=\mathbf{F}(\mathbf{x})+\mathbf{B}\mathbf{a}_{M}+k_{1}\mathbf{x}_{2}+k_{2}\dot{w}(\mathbf{x} _{1})+\Delta \tag{21}\]  
Introducing a super-twisting controller as \(\mathbf{u}_{e}(t)=\mathbf{F}(\mathbf{x})+\mathbf{B}\mathbf{a}_{M}+k_{1}\mathbf{x}_{2}+k_{2}\dot{w}(\mathbf{ x}_{1})\), Equation (21) can be rewritten as:  
\[\dot{\mathbf{s}}=\mathbf{u}_{e}(t)+\Delta \tag{22}\]  
And the missile acceleration command \(\mathbf{a}_{M}\) can be formulated as:  
\[\mathbf{a}_{M}=-\mathbf{B}^{-1}(\mathbf{F}(\mathbf{x})+k_{1}\mathbf{x}_{2}+k_{2}\dot{w}(\mathbf{x}_{1}) -\mathbf{u}_{e}(t)) \tag{23}\]  
Then, according to (15), regarding \(\Delta\) as the system disturbance, the equivalent multivariable super-twisting controller based on barrier function is given by:  
\[\left\{ \begin{array}{l}\mathbf{u}_{e}(t)=-\mathbf{L}(t)\circ\frac{\mathbf{s}}{\|\mathbf{s}\| ^{1/2}}+\mathbf{y}(t)\\ \dot{\mathbf{y}}(t)=-\mathbf{L}^{2}(t)\circ\frac{\mathbf{s}}{\|\mathbf{s}\|}\end{array} \right. \tag{24}\]where the barrier function gain is defined by:  
\[\mathbf{L}(t,s)=\left\{ \begin{array}{cc}\tilde{\mathbf{L}}(t),&0\leq t<\tilde{t}(s_{0}) \\ \mathbf{L}_{b}(s),&t\geq\tilde{t}(s_{0})\end{array} \right. \tag{25}\]  
\[\tilde{\mathbf{L}}(t)=\left[ \begin{array}{cc}\overline{L}_{1}&\overline{L}_{2} \end{array} \right]^{T}=\left[ \begin{array}{c}l_{10}+l_{11}t\\ l_{20}+l_{21}t\end{array} \right] \tag{26}\]  
\[\mathbf{L}_{b}(s)=\left[ \begin{array}{cc}L_{b1}&L_{b2}\end{array} \right]^{T}= \left[ \begin{array}{c}\sqrt{\varepsilon_{1}}b_{1}(\varepsilon_{1}-|s_{1}|)^{ -1/2}\\ \sqrt{\varepsilon_{2}}b_{2}(\varepsilon_{2}-|s_{2}|)^{-1/2}\end{array}