A study on the L1 optimal PD controller with application to joint motion control of a robot manipulator

Abstract

In this paper, we consider the L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> optimal proportional-derivative (PD) controller synthesis by which the L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> gain of trajectory tracking systems can be reduced. The notion of input-to-state stability (ISS), which has been equipped with the L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> 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 <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> norm to deal with bounded persistent disturbances because the disturbances should be mathematically regarded as elements of the L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> 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 <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> optimal PD controller for such a model, by which the trajectory tracking system satisfies the redefined ISS and the L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> gain is less than a performance level γ. Finally, we examine the effectiveness of the design method through experimental results for a typical robot manipulator.