LMI-based Optimization for the Position Feedback Control of Underactuated Robotic Systems via an Affine PD Controller: Case of the Pendubot

The set-point control problem of underactuated mechanical systems is a challenging issue and needs to be further investigated. This research work focuses on such problem for underactuated Lagrangian-type robotic systems, and then the objective lies in designing a suitable controller to change the robot’s current position to a desired state. Thus, our main idea is to control the robotic systems through an affine-type PD controller and by virtue of the LMI-based optimization techniques. Thus, we will develop LMI conditions on the controller gains to ensure the stability of the underactuated robot at the desired state. Then, to achieve this goal, we will consider the rescaled dynamical model describing the difference between the nonlinear model and its approximate linear dynamics, where we assume that this rescaled dynamics satisfies such some Lipschitz condition. Finally, the pendubot manipulator is adopted in order to verify the efficiency of the proposed controller.