# Multivariable Adaptive Super-Twisting Guidance Law Based on Barrier Function
## 4 Simulation
### Simulation Compared with Other Super-Twisting Guidance Laws  
Compared with a classical super-twisting controller, the barrier function-based controller does not need the information of the upper bound of disturbance and the super-twisting state variable \(\mathbf{y}(t)\) of it can converge fast. To further verify the distinguished property of the proposed barrier function-based guidance law, two other kinds of super-twisting guidance laws are considered. The first is a fixed-gain super-twisting (FGST) guidance law, which is given by [31]:  
\[\left\{ \begin{array}{l}\mathbf{a}_{M}=-\mathbf{B}^{-1}(\overset{\ast}{F}(\mathbf{x})+c_ {1}\frac{\mathbf{s}}{\|\mathbf{s}\|^{1/2}}-c_{2}\mathbf{s}-\mathbf{y}(t))\\ \dot{\mathbf{y}}(t)=-c_{3}\frac{\mathbf{s}}{\|\mathbf{s}\|}-c_{4}\mathbf{s}\end{array} \right. \tag{46}\]  
where \(c_{i},\ i=1,2,3,4\) is positive constant, it is selected as \(c_{1}=1.6\), \(c_{2}=0.1\), \(c_{3}=0.4\) and \(c_{4}=0.1\) in this paper. The second is an equivalent control based super twisting (ECST) guidance law, which has the following expression [32]:  
\[\left\{ \begin{array}{l}\mathbf{a}_{M}=-\mathbf{B}^{-1}(\overset{\ast}{F}(\mathbf{x})+d_ {1}\sqrt{\overset{\ast}{L}}\circ\frac{\mathbf{s}}{\|\mathbf{s}\|^{1/2}}-d_{2}(t) \overset{\ast}{L}\circ\mathbf{s}-\mathbf{y}(t))\\ \dot{\mathbf{y}}(t)=-d_{3}(t)\overset{\ast}{L}\circ\frac{\mathbf{s}}{\|\mathbf{s}\|}-d_{4} (t)\overset{\ast}{L}\circ\mathbf{s}\end{array} \right. \tag{47}\]  
where \(\overset{\ast}{L}\) is a dual-layer gain, its detailed expression is given in Appendix A; \(d_{i},\ i=1,2,3,4\) is positive constant, it is selected as \(d_{1}=5\), \(d_{2}=0.1\), \(d_{3}=1.2\) and \(d_{4}=1\) in this paper. In order to further demonstrate the effectiveness and applicability of the proposed guidance law, the comparison simulation is designed to contain two cases with various target maneuvers:  
1. The target accelerations in LOS coordinate \(\mathbf{a}_{T\theta}=\mathbf{a}_{T\phi}=-30(2\pi t)\) m/s\({}^{2}\);
2. The target accelerations in LOS coordinate \(\mathbf{a}_{T\theta}=\mathbf{a}_{T\phi}=-30\) m/s\({}^{2}\). The other initial parameters of these two scenarios are the same as those in Section 4.1. Figures 6 and 7 demonstrate the comparison results of case 1 and case 2, respectively. Figures 6a and 7a show the relative distance trajectories between missile and target. It can be clearly seen that the trajectories of the three guidance laws can all converge to zero. Moreover, the curves of the proposed are shorter than others. Figures 6b and 7b depict the acceleration commands. It can be noticed that the acceleration commands of FGST and ECST reach the ultimate maximum lateral acceleration during the flight time. Moreover the command curves of FGST have distinct chattering. Contrarily, the curve of the proposed has smaller maximum values and smooth properties. The LOS angles are illustrated in Figures 6c and 7c, and the LOS angular rates are illustrated in Figures 6d and 7d, respectively. The LOS angles of the proposed reach the desired angles faster than the others. Moreover, its angular rates have smaller sizes in the vicinity of zero. The sliding variables of the two cases are shown in Figures 6e and 7e, respectively, which all converge to zero in finite time. Figures 6f and 7f show the convergence of super-twisting state variable \(\mathbf{y}(t)\). It is obvious that the range values to which the proposed guidance law converges are much smaller than the others in these cases. Furthermore, the detailed information of the two cases is listed in Tables 2 and 3, respectively. It can be obviously seen that the flight time of missiles utilizing the proposed guidance law is the shortest compared with the ones using FGST and ECST. In addition, the acceleration commands \(\left|a_{M\phi}\right|\) of FGST and ECST reach the ultimate maximum lateral acceleration \(a_{M\phi,\max}\) during the flight time, which sustain 8.01 s and 3.57 s (case 1), and 7.16 s and 3.38 s (case 2), respectively.