Aiming at the trajectory tracking problem with unknown uncertainties, a novel controller composed of proportional-integral-differential sliding mode surface (PIDSM) and variable gain hyperbolic reaching law is proposed. A PID-type sliding mode surface with an inverse hyperbolic integral terminal sliding mode term is proposed, which has the advantages of global convergence of integral sliding mode (ISM) and finite time convergence of terminal sliding mode (TSM), and the control effect is significantly improved. Then, a variable gain hyperbolic approach law is proposed to solve the sliding mode approaching velocity problem. The variable gain term can guarantee different approaching velocities at different distances from the sliding mode surface, and the chattering problem is eliminated by using a hyperbolic function instead of the switching function. The redesign of the sliding mode surface and the reaching law ensures the robustness and tracking accuracy of the uncertain system. Adaptive estimation can compensate for uncertain disturbance terms in nonlinear systems, and the combination with sliding mode control further improves the tracking accuracy and robustness of the system. Finally, the Lyapunov stability principle is used for stability analysis, and the simulation study verifies that the proposed control scheme has the advantages of fast response, strong robustness, and high tracking accuracy.

This article presents a modified active-disturbance-rejection control (ADRC) combined with a sliding mode control (SMC) regarding the tracking control problems for plants with unmatched uncertainty. The proposed modified active-disturbance-rejection control with sliding mode (ADRC-SM) employs a reduced-order extended state observer (ESO) for estimating various uncertainties of system in time, including unmatched and matched uncertainties. Meanwhile, a novel reaching law of SMC was designed by using the cycloid function as the main controller of ADRC, which ensures the robustness of the uncertain system. Due to the reduced-order ESO tracking and compensating for various uncertainties in the system as a total disturbance, the upper bound of the disturbance in the SMC is relaxed. The gain coefficient of the reaching law only needs to be designed to be larger than the limit of the lumped disturbance; thus, the chattering problem is greatly reduced. The designed new reaching law of the cycloid function shortens the time for the system state’s convergence to the sliding mode’s surface. The cycloid function replaces the switching function in the traditional reaching law, making the actual control input continuous and shortening the approach time. Compared with traditional ADRC-SM, the use of multiple ESOs or intelligent algorithms to approximate plant parameters can be avoided, the design is simplified, its robustness is enhanced, computational costs are reduced, and the convergence time is reduced. The controlled object with unmatched uncertainty is transformed into a system with matched uncertainty using state-space transformation, which reduces the complexity of the controller’s design. In addition, the stability analysis of the closed-loop system is carried out based on the Lyapunov method. Simulations and experiments verify that the modified ADRC-SM has the merits of fast response, small overshoot, small steady-state error, strong anti-interference competence, and high control accuracy.

The paper studies the control problem for nonlinear uncertain systems with multiple channel uncertainties, and presents the manifold of a class of second-order systems including unmatched and matched disturbances, which is mainly found in flexible components. The system model is first transformed into a Brunovsky form, where the multiple uncertainties in the system are lumped as one term-equivalent total effects. A backstepping united with a reduced-order ESO control design is proposed, where the intermediate variables need not be measurable, and only the system output is required for the control implementation. Finally, the proposed control structure is equivalent to PID, and its working mechanism is revealed. The use of multiple ESO or iterative learners can be avoided to match plant parameters, leading to reduced computational costs, simpler parameter tuning, and improved convergence as compared to traditional control methods. Finally, the simulation and experimental results illustrate the effectiveness of the proposed method.