# Multivariable Adaptive Super-Twisting Guidance Law Based on Barrier Function
## 3 Barrier Function-Based Multivariable Super-Twisting Guidance Law Design
### Stability Analysis of the Proposed Guidance Law  
The main conclusion of the stability property of the proposed guidance law is summarized as the following theorem. **Theorem 1**: _In consideration of the three-dimension missile-target engagement system with terminal angle constraint (5), the proposed adaptive multivariable super-twisting guidance law (29) with barrier function gains (25) can drive the sliding surface to converge to a small neighborhood around zero._  
**Proof. ** Substituting (29) into (21) and employing \(\tilde{\mathbf{y}}(t)=\left[ \begin{array}{cc}\overline{y}_{1}&\overline{y}_{2} \end{array} \right]=\mathbf{y}(t)+\Delta\) instead of \(\mathbf{y}(t)\) yields:  
\[\left\{ \begin{array}{l}\dot{\mathbf{s}}=-\mathbf{L}(t)\circ\frac{\mathbf{s}}{\|\mathbf{s}\|^ {1/2}}+\ddot{\mathbf{y}}(t)\\ \vdots\\ \dot{\mathbf{y}}(t)=-\mathbf{L}^{2}(t)\circ\frac{\mathbf{s}}{\|\mathbf{s}\|}+\dot{\mathbf{\Delta}} \end{array} \right. \tag{30}\]  
Thus, the statement of Theorem 1 can be rewritten as: for system (30), there exists a first time \(\tilde{t}(s_{0})\) so that \(s(t)<\varepsilon/2\); then for all \(t\geq\tilde{t}(s_{0})\), \(|s(t)|<\varepsilon\); furthermore, for the trajectory of (30), one has \(\text{limsup}_{t\to\infty}|\overline{y}_{i}(t)|\leq v\) (\(i=1,2\)), where \(v\) is a positive constant. It has been proven that there exists a finite time \(\tilde{t}(s_{0})\) for system (30) under the adaptive barrier function gains (25) [35]. The derivate of the barrier function gains has the following form:  
\[\dot{L}_{bi}=\frac{b\sqrt{\varepsilon_{i}}sign(s_{i})\dot{s}_{i}}{2(\varepsilon_{ i}-\left|s_{i}\right|)^{3/2}},i=1,2 \tag{31}\]  
To simplify the operations in the proof, consider the barrier function gains as scalar quantity forms and assume \(L_{b}=L_{b1}=L_{b2}\)  
Introduce an auxiliary vector \(\mathbf{z}\) which is formulated as:  
\[\mathbf{z}=\left[ \begin{array}{c}\mathbf{z}_{1}\\ \mathbf{z}_{2}\end{array} \right]=\left[ \begin{array}{c}L_{b}^{2}\mathbf{g}(t)\\ \mathbf{y}(t)\end{array} \right] \tag{32}\]  
The derivative of \(\mathbf{z}\) can be expressed as:  
\[\dot{\mathbf{z}}=\left[ \begin{array}{c}\dot{\mathbf{z}}_{1}\\ \dot{\mathbf{z}}_{2}\end{array} \right]=\left[ \begin{array}{c}2\frac{\dot{L}_{b}} {L_{b}}\mathbf{z}_{1}+L_{b}^{2}\mathbf{g}\\ \vdots\\ \mathbf{y}(t)\end{array} \right] \tag{33}\]  
Denoting \(C_{0}=1/2b^{2}\varepsilon\), expression (33) can be written as:  
\[\dot{\mathbf{z}}=\left[ \begin{array}{c}L_{b}^{2}(-\frac{z_{1}}{\|\mathbf{z}_{1}\|^{ 1/2}}+\mathbf{z}_{2})\circ(1+2C_{0}\|\mathbf{z}_{1}\|)\\ -L_{b}^{2}\frac{z_{1}}{\|\mathbf{z}_{1}\|}+\ \dot{\Delta}\end{array} \right] \tag{34}\]  
Define a new time scale \(\tau\) which satisfy \(\tau(\widetilde{t}(s_{0}))=0\) and \(d\tau=L_{b}^{2}dt\), and denote \(x^{\prime}\) as the derivative of the any given variable \(x\) with respect to \(\tau\). Then, (34) can be rewritten as:  
\[\mathbf{z}^{\prime}=\left[ \begin{array}{c}(-\frac{z_{1}}{\|\mathbf{z}_{1}\|^{1/2}}+ \mathbf{z}_{2})\circ(1+2C_{0}\|\mathbf{z}_{1}\|)\\ -\frac{z_{1}}{\|\mathbf{z}_{1}\|}+\frac{\dot{\Delta}}{L_{b}^{2}}\end{array}