ADAPTIVE ROBUST CONTROL OF SECOND-ORDER NONLINEAR SYSTEMS WITH UNDETERMINED PARAMETERS BASED ON THE BACKSTEPPING METHOD
The article investigates the control problem for a class of nonlinear strict-feedback systems with uncertain parameters and external disturbances. The main objective of the study is to develop an adaptive control algorithm that ensures system robustness and guarantees semi-global uniform boundedness of all signals. To compensate for unknown dynamic functions, the recursive synthesis method, namely backstepping, is combined with adaptive approximation laws. The semi-global stability of the closed-loop system is analytically proven using the Lyapunov function method, and it is shown that the tracking error converges to a bounded neighborhood of zero. As a result of the study, the control problem is formulated for second-order nonlinear strict-feedback systems with uncertain parameters and bounded external disturbances, and it is constructively demonstrated that this problem can be solved by means of adaptive robust control. In particular, the existence of a control law ensuring tracking of a given reference trajectory is established on the basis of the backstepping method, and a step-by-step synthesis procedure for its construction is proposed: first, a virtual control is designed, and then adaptive control laws are defined. Using the Lyapunov function method, the stability of the closed-loop system, the uniform boundedness of all signals, and the convergence of the tracking error to a bounded neighborhood of zero are proven. Thus, the work proposes not only a specific control algorithm, but also a theoretical and constructive approach that substantiates the solvability of the control problem for a class of uncertain nonlinear systems. To verify the proposed method in practice, numerical simulation was carried out for the dynamics of a single-link robotic manipulator. The results showed that the proposed adaptive backstepping algorithm preserves the bounded motion mode of the system under sudden changes in the load parameter and in the presence of external disturbances. A numerical comparison was performed using the MSE, maximum error, and settling time criteria, and it was found that the control performance depends on the choice of algorithm parameters. The proposed method can be applied to the control of mechatronic and robotic systems with parametric uncertainty; however, additional tuning of the control gains is required before practical implementation.