Robust Adaptive Control for a Class of Nonlinear Systems with Uncertainties of Unknown Bounds

Abstract

Although linear control theory has been applied in industry successfully over a long period, it has been found to be inadequate in many physical systems having nonlinear properties such as friction, hysteresis, backlash, and saturation. To deal with the nonlinear systems with uncertainties of unknown bounds, adaptive sliding mode control (ASMC) techniques with integral and modified integral adaptation law have been introduced during the past two decades. However, the integral type adaptation laws have a weakness that they can not achieve fast response to the uncertainties and lower chattering simultaneously.

The dissertation first reviews the general idea of ASMC techniques for nonlinear systems with uncertainties of unknown bounds. The necessary and sufficient conditions are discussed. It shows that any positive definite monotonic function of the sliding variable can be used in the integral adaptation law. Moreover, a special type of function is proposed to smooth the chattering.

The dissertation then investigates the convergence and boundedness in the existing classic ASMC techniques. By applying a new Lyapunov method and a new majorant curve approach, it successfully proves the finite-time convergence (FTC) of the sliding variable and uniformly ultimately bounded (UUB) convergence of the switching gain. Moreover, it deduces a new formula for reaching time estimation (RTE). The new RTE shows the relationship that the reaching time is inversely proportional to the square root of the designed integral parameter. The explicit relationship indicates that the classic ASMC techniques cannot achieve fast response and lower chattering simultaneously. Thus, it reveals the inherent reason of the slow response existing in the classic ASMC techniques with integral adaptation laws.

In order to thoroughly resolve the trade-offs involved in system response to the uncertainties and chattering attenuation, a new adaptation law, integral-exponential reaching law, of first order ASMC for nonlinear uncertain systems of uncertainties is proposed. The new algorithm combines the classic integral reaching law with an exponential term. With the newly added exponential term, the system responds to the uncertainties quickly. Moreover, it reduces the final switching gain. Thus, the lower chattering level is achieved simultaneously. Moreover, the new design can deal with the uncertainties bounded with not only unknown constant bounds but also polynomial bounds in the norm of the states.

Illustrative simulations are provided to help understand the existing and the proposed ASMC adaptation processes. The proposed designs are then numerically verified upon different real nonlinear dynamic systems such as a variable-length pendulum, a two-degree-of-freedom (2-DOF) experimental helicopter and a 5-DOF robotic manipulator. Moreover, the experiments are conducted on the 2-DOF experimental helicopter-model-based setup to compare the proposed designs with the existing ASMC designs and the common proportional-integral-derivative (PID) designs. Both numerical and experimental results show that the new proposed ASMC designs for nonlinear dynamic systems with uncertainties of unknown bounds can significantly improve the robustness of the systems and reduce the chattering level compared to currently existing ASMC designs.