The control of a double-pendulum system

In this paper, the control of a double mass and spring system as a physical model of a flexible arm robot is considered. The main goal of design is to use an appropriate control method that produces suitable external torques such that the whole system can be completely stabilized and controlled. For this purpose, mathematical model of the system is required which is achieved by Lagrange´s equations in the first part of the paper. Since the controller design in this paper is based on state space description, these dynamical equations are re-written in the form of state-space equations and are linearized about an operating state. After examining the controllability of the system, a state feedback controller and also a Linear Quadratic Regulator (LQR) are designed. Moreover, since in these controllers the states of the system are required, a suitable full-order state estimator is developed as well. Finally, simulation results are provided to show the effectiveness of the proposed controllers.