Global Robust Output Regulation

Abstract

We already investigated in Chap. 7 that, under some suitable assumptions, the GRORP for a given nonlinear system can be converted into a GRSP/GASP for an augmented system. However, the GRSP/GASP itself is a challenging task. So far, the results are mainly limited to nonlinear systems with special structures such as those studied from Chaps. 2–6. In this chapter, using the framework established in Chap. 7, we will first study the GRORP for both output feedback systems and lower triangular systems for the case where $v(t)\in {\mathbb V}$ and $w\in {\mathbb W}$ for two compact sets ${\mathbb V}$ and ${\mathbb W}$ whose boundary is known, and the exosystem is known exactly. It turns out that, for this case, the GRORP of the output feedback systems and lower triangular systems can be converted to the GRSP of the systems studied in Chap. 4. Then, we will further consider the GRORP for the case where the boundary of ${\mathbb V}$ and ${\mathbb W}$ is unknown. The chapter is organized as follows. In Sect. 8.1–8.3, we study systems with relative degree one, output feedback systems, and lower triangular systems, respectively, under the assumption that the solution of the regulator equations satisfies the nonlinear immersion condition. In Sect. 8.4, we further consider the GRORP under the assumption that the solution of the regulator equations satisfies the generalized linear immersion condition which allows the exosystem to be nonlinear. In Sect. 8.5, we consider the GRORP with the boundary of ${\mathbb V}$ and ${\mathbb W}$ unknown, or what is the same, both the unknown parameter $w$ and the initial state of the exosystem can be arbitrary large. Our approach will be illustrated via the Lorenz system in Sect. 8.6. Finally, the notes and references are given in Sect. 8.7.