Novel Auxiliary RISE (ARISE) Robust Control Approaches for Nonsmooth or Switched Nonlinear Systems

Abstract

To deal with uncertainties in a dynamic system, many nonlinear control approaches have been considered. Unique challenges arise from uncertainties that are bounded by constants, which led to the development of both continuous and discontinuous control methods. One such approach that yields a favorable transient performance is sliding mode (SM) control. However, SM control requires rapid switching within the control input due to a discontinuous signum function in the control architecture, which introduces chattering during real-world application. To address the limitations of SM control while still exploiting its advantages and robustness, a class of continuous robust controllers named Robust Integral of the Sign of the Error (RISE) was developed. RISE controllers include an integral of the signum function, not a signum function, in the control law. Filtering the signum function with an integral makes the signum function continuous and thereby mitigates chatter that would result from a discontinuous SM term. Even though RISE controllers exhibit a significant level of robustness, it is limited to classes of smooth nonlinear dynamic models. In the first part of this thesis, a novel auxiliary RISE (ARISE) controller is proposed to prevent chattering and deal with uncertainties (even those bounded by constants) for general, switched, and nonsmooth control-affine nonlinear systems. The ARISE control system includes a unique auxiliary error that is designed to inject a discontinuous SM term directly into the error system without including a SM term in the controller itself akin to the RISE control architecture. Consequently, the ARISE control law minimizes the chattering effect that results from discontinuous SM terms. The proposed ARISE control system is augmented with an adaptive update law to deal with the unknown control effectiveness matrix in the dynamic model. To prove the effectiveness of the ARISE controller, a nonlinear stability analysis was conducted and resulted in semi-global exponential tracking towards an ultimate bound. Furthermore, the performance of the proposed controller was evaluated and compared against a traditional SM controller through simulations using a switched Van der Pol oscillator model. The simulations were conducted in Matlab/Simulink, and the results of the comparisons are shown in various graphs. It is concluded that the proposed ARISE controller performs better for a switched system than the SM controller. The improved performance of the ARISE controller was consistent across different dynamic parameters and disturbances. In the second part of this thesis, the ARISE controller is modified to account for the physical limitations of actuators. Specifically, to enhance the applicability of the ARISE control approach, a saturated hyperbolic tangent function is incorporated into the control input to allow the controller to be bounded a priori, which will account for the limitations of physical actuators during real-world applications. A nonsmooth Lyapunov-like stability analysis is performed for the proposed saturated ARISE control system to prove that the closed-loop error system is uniformly ultimately bounded for a switched dynamic model. A series of numerical simulations were performed to compare the performance of the saturated ARISE controller with a standard saturated SM controller across different dynamic parameters and disturbances.