In this work, a new adaptive block-backstepping control design algorithm was developed for an under-actuated model (represented by a ball–arc system) to enhance the transient and steady-state behaviors and to improve the robustness characteristics of the controlled system against parameter variation (load change and model uncertainty). For this system, the main mission of the proposed controller is to simultaneously hold the ball at the top of the arc and retain the cart at the required position. The stability of a controlled system based on the proposed adaptive controller was analyzed, and its globally asymptotic stability was proven based on the Lyapunov theorem. A comparative study of adaptive and non-adaptive block-backstepping controllers was conducted in relation to the transient, steady-state, and robustness characteristics. The effectiveness of the controller was verified via simulation within a MATLAB/SIMULINK environment. The simulated results show that the proposed adaptive control strategy could successfully stabilize the under-actuated ball–arc system, regardless of both the regulation problem and the tracking problem. This provides a better dynamic performance and a better load rejection capability, and it performs well in solving the uncertainty problem in the model parameter.

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