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.

Novel Adaptive law modification termed “weighted e-modification” for MRAC is presented in this paper. Speed control of DC motor with mismatched disturbance load torque is considered; totally unknown model parameters and only upper bounded fast load torque are dealt with. The proposed modification approached the advantages of the performance of the non-modified MRAC and the robustness of the e-modified MRAC. The simulations results showed that weighted e-modification technique speed response has better performance than classical e-modification technique and the non-modified MRAC because it gathered the advantage of the two of performance and robustness; control action resulting from weighted e-modification had better performance than e-modification and avoided the robustness failure due to adaptive gains drift of the non-modified MRAC. MRAC adaptive gains drifted up to unbounded limit due to disturbance while uniform boundedness of adaptive gains are achieved by e-modification and weighted e-modification.

This paper develops Block-Backstepping algorithms for balancing and tracking control of ball and arc system. Three design procedures are presented and, as such, three proposed controllers would arise. The design of Block-Backstepping controllers has been developed for a linearized model of the system and comprises two main control objectives; firstly to stabilize the ball on the top position of the arc-shaped and, secondly, to place the cart at the desired position. The block-backstepping control law is further enhanced by including an integral action to improve both the steady state characteristics and the robustness of the overall system. Moreover, the internal stability of the system has been analyzed using zero dynamic criteria to guarantee the global asymptotic stable at desired equilibrium point. The results based on simulation are used to assess the performance of the proposed controllers for the suggested underactuated mechanical system.