Control design for nonlinear systems via optimization and linearization: Application to single flexible link control

Most nonlinear dynamic systems do not lend themselves to simple closed-loop control strategies. However, computational methodologies are now available to analyze such nonlinear systems and design/optimize their open-loop control strategies. In this work, we augment the optimal open-loop scheme with a closed-loop scheme. We linearize the equations of motion about the optimal open-loop trajectory. The resultant linear time-varying system is controlled by a simple feedback law. The proposed control scheme can be used to design controllers for a wide array of nonlinear dynamic systems, e.g. manufacturing systems. Here, we design and implement a controller for a single flexible link. It is well known that noncollocated control of a flexible link results in unstable closed-loop behavior and that stable collocated feedback fails to adequately suppress the residual tip vibration. The proposed control methodology combines the optimal open-loop scheme that is designed to suppress residual tip vibration with stable collocated feedback. A nonlinear dynamic model, based on finite strain rod theory, is used with sensitivity analysis and numerical optimization to obtain the optimal open-loop scheme. Simulation and experiment are used to evaluate and compare the performances of the proposed open-loop controller, the proposed closed-loop controller, and a traditional proportional-derivative collocated feedback controller.