Trajectory tracking control

Abstract

In this chapter we extend the passivity-based method, developed for regulation in the previous chapter, to solve trajectory tracking problems. The first main modification that we have to make is that for tracking, besides reshaping the potential energy of the EL plant, we must also shape the “kinetic energy” function. Whereas modifying the potential energy function means to relocate the equilibria of the system, the modification of the “kinetic energy” function can, roughly speaking, be rationalized as imposing a specific pattern to the transformation of potential into kinetic energy. However, the quotes here are important because the storage function that we assign to the closed loop is not an energy function in the sense that it defines the equations of motion. With an obvious abuse of notation we will still refer to this step as energy shaping, but it is better understood as passivation with a desired storage function (see Appendix A). The damping injection step is added then to make the passivity strict. The passivation objective is achieved invoking the key passive error dynamics Lemma 2.7, which states that we can always factor the workless forces in such a way that, in terms of the error signals s, the EL system behaves like a linear passive system.