Angle tracking consensus and vibration control of multiple manipulator-string systems with actuator fault over directed graph

In this paper, we address the angle tracking consensus and vibration suppression problems of the leader-following multiple marine detecting systems subject to actuator fault over directed graph, where each follower is composed of a rigid manipulator, a flexible string and a detecting probe, with the desired angle treated as the virtual leader. First, the dynamics of the systems are characterized by a set of partial differential equations (PDEs). Second, in order to tackle the difficulty in selecting Lyapunov function caused by directed graph, we design a distributed observer for each follower to estimate the leader’s angle, facilitating the control design. Moreover, we develop an adaptive fault-tolerant control law to ensure the robustness of the systems. Based on the above preparations, we propose the distributed boundary control strategy to achieve the angle tracking consensus of the rigid manipulators and vibration elimination of the flexible strings. Finally, some simulations illustrate the effectiveness of the suggested theoretical findings.