A Study on the L-1 Optimal PD Controller with Application to Joint Motion Control of a Robot Manipulator

Abstract

In this paper, we consider the L-1 optimal proportional-derivative (PD) controller synthesis by which the L-1 gain of trajectory tracking systems can be reduced. The notion of input-to-state stability (ISS), which has been equipped with the L-2 norm of a vector-valued signal, plays important roles in evaluating the effect of disturbances on the system states. In connection with this, we first redefine the conventional ISS with the L-1 norm to deal with bounded persistent disturbances because the disturbances should be mathematically regarded as elements of the L-1 space. A tractable model for trajectory tracking control of robot systems is then given by using ideas of extended disturbance and composite error. We next introduce a design method of the L-1 optimal PD controller for such a model, by which the trajectory tracking system satisfies the redefined ISS and the L-1 gain is less than a performance level gamma. Finally, we examine the effectiveness of the design method through experimental results for a typical robot manipulator.