## 2 Materials and Methods
### Simulation  
The ARISE control framework is presented in Figure 1 to provide a better understanding of the simulation flow required by the control architecture. This flowchart outlines the error computation, update law, auxiliary filter, ARISE controller, and plant, as implemented in the subsequent simulations. A series of simulations were performed to demonstrate the effectiveness of the proposed ARISE controller. To demonstrate ARISE's performance compared to SM control, simulations were performed using the developed ARISE controller and a traditional SM controller. The dynamics considered during the simulations were similar to a two-state Van der Pol oscillator; however, the dynamics were modified to be switched, to include disturbances, and to have a control-affine control input. Specifically, the unknown drift dynamics used in the simulations were  
\[f_{\sigma}(x)=\left[ \begin{array}{c}\mu\sigma\Big{(}x_{1}-\frac{1}{3}x_{1}^{ 3}-x_{2}\Big{)}\\ \frac{1}{\mu\sigma}x_{1}\end{array} \right],\]  
where \(\mu\in\mathbb{R}_{>0}\) is an unknown system parameter, \(x=\left[x_{1},x_{2}\right]^{T}\), and  
\[\sigma=\left\{ \begin{array}{ll}1,&x_{1}\geq 0\\ 2,&\text{otherwise}\end{array} \right. \tag{30}\]  
is an unknown state-dependent switching signal that indicates the index of the drift dynamics. The disturbances used during the simulation were designed as  
\[d(t)=\left[ \begin{array}{c}d_{1}\\ d_{2}\end{array} \right],\]  
where \(d_{1},d_{2}\in\mathbb{R}_{>0}\) are uncertain parameters. The control effective matrix used in the simulations was  
\[g_{\phi}(x,t)=\left[ \begin{array}{cc}c_{1}\phi_{1}(x)+c_{2}\phi_{2}(x)&0\\ 0&c_{3}\phi_{1}(x)+c_{4}\phi_{2}(x)\end{array} \right],\]  
where \(c_{1},c_{2},c_{3},c_{4}\in\mathbb{R}\) are unknown, constant, and non-zero parameters and  
\[\phi_{1}(x)\triangleq\left\{ \begin{array}{ll}1,&x_{2}\geq 0\\ 0,&\text{otherwise}\end{array} \right.\text{ and } \tag{31}\]  
\[\phi_{2}(x)\triangleq(1-\phi_{1}(x)) \tag{32}\]  
are known state-dependent switching signals. The index of the control effectiveness matrix is  
\[\phi=\left\{ \begin{array}{ll}1,&\phi_{1}=1\\ 2,&\phi_{1}=0\end{array} \right.,\]  
based on the switching signals defined in (31) and (32). Additionally, satisfying Assumption 4, the product \(g_{\phi}u\) can be linearly parameterized as  
\[g_{\phi}u=\underbrace{\left[ \begin{array}{cccc}\phi_{1}u_{1}&(1-\phi_{1})u _{1}&0&0\\ 0&0&\phi_{1}u_{2}&(1-\phi_{1})u_{2}\end{array} \right]}_{\mathcal{Y}}\underbrace {\left[ \begin{array}{c}c_{1}\\ c_{2}\\ c_{3}\\ c_{4}\end{array} \right]}_{\theta},\]  
where \(u=[u_{1},u_{2}]^{T}\). The initial conditions of the states and the desired trajectory for the simulation were \(x(0)=\left[ \begin{array}{cc}3&2\end{array} \right]^{T}\) and \(x_{d}=\left[5\cos(t),5\sin(t)\right]^{T}\), respectively. The control gains for the ARISE controller were selected as \(\alpha=0.1\), \(k_{1}=3.5\), \(k_{2}=100\), \(k_{3}=20\), and \(\beta=10\). The parameter estimates were initialized as \(\hat{\theta}(0)=\left[ \begin{array}{cc}5&5&5&5\end{array} \right]^{T}\). Two sets of unknown constants were selected for the control effectiveness matrix to analyze the change in performance of the control law across varying parameter selections. For the first set, the values of \(c_{1}\), \(c_{2}\), \(c_{3}\), and \(c_{4}\) were selected as 5, 10, 3, and 6, respectively, and the set was named GA. For the other set of unknown constants, the parameters were, respectively, set as 7, 10, 3, and 4, and the set was named GB. Similarly, the performances of the control laws were analyzed for 8 different sets of disturbances: 4 continuous disturbances and 4 constant disturbances, as presented in Table 1. The control performance was also analyzed for 7 different values of \(\mu\) evenly spaced from 0.5 to 3.5. Analyzing the performance at different \(\mu\) values helps to ensure the robustness of the controllers across varying dynamic parameters.