###### Abstract
Though SM controllers are known for their robustness to uncertainties, this rapid switching between differing control actions requires infinite control frequency and introduces chattering, all of which hinder the practical application of SM control [13; 20; 21]. Robust integral of the sign of the error (RISE) control, which is a high-gain feedback control approach, was developed to address the shortcomings of SM control while still exploiting its advantages and robustness. Unlike conventional SM control development methods where a signum function is included directly in the control input, the signum function is instead included in the time derivative of an auxiliary error system in the RISE-based approach. Incorporating the auxiliary error in the RISE control input introduces an integral SM term in the control law, rather than an SM term itself, making the control input continuous. The continuous RISE controller eliminates the SM controller's requirement for infinite control frequency and reduces chattering, bolstering the RISE controller's utility in real-world systems. In fact, RISE controllers have been implemented in numerous systems; for example, autonomous flight, delayed systems, underwater vehicles, and many more [13; 22; 23; 24; 25; 26]. Additionally, many researchers have demonstrated that RISE-based methods have produced excellent results in a variety of applications, including estimation, control design, and optimization [23; 27; 28; 29]. Across a broad class of smooth systems, RISE-based control laws typically produce an asymptotic tracking outcome [22; 30; 31; 32; 33; 34; 35; 36; 37; 38], and after some modification, researchers have also been successful in achieving an exponential error convergence [39]. Though RISE controllers are highly robust and leverage the benefits of SM all while mitigating its downsides, the RISE-based method requires the dynamic model including the control input to be continuously differentiable, which precludes its application in switched/discontinuous systems. To counter the limitations of RISE control, the authors developed a novel auxiliary RISE (ARISE) control approach, upon which this work is built, for a general, nonsmooth, and uncertain control-affine nonlinear dynamic system with the goal of developing a chatter-mitigating controller that can be applied to a wide range of uncertain nonlinear systems [40]. Similar to [40], this work developed a filtered error signal to inject an SM term into the closed-loop error system, whereas the controller itself includes an integral SM term to mitigate chattering and to reduce the need for an infinite control frequency. The control effectiveness used in the dynamics in this work is considered to be unknown and potentially switched or nonsmooth. To account for the uncertain control effectiveness matrix, an adaptive estimate is proposed. A nonsmooth Lyapunov-like stability analysis is performed to confirm the efficacy of the proposed adaptive ARISE control system. The stability result shows that, if the gain and initial state requirements are satisfied, the designed control system produces semi-global exponential trajectory tracking towards an ultimate bound. In contrast to our work in [40], this work includes a series of numerical simulations using a modified two-state Van der Pol oscillator as the dynamic model to validate the performance of the proposed control system. The oscillator was specifically modified by including control-affine control inputs and by enabling the model parameters to be switched. The performance of the proposed ARISE controller was compared with a traditional SM controller to determine their relative performances when implemented on a switched system. The root mean square (RMS) error was calculated for the trajectory tracking errors for both the ARISE and SM controllers across different disturbances and dynamic parameters to compare both approaches robustness.