Analyse und Synthese nichtlinearer dissipativer Systeme: Ein Überblick (Teil 2)

The second part of this contribution consisting of the sections 5 to 9 continues the first part published in the last issue. In Section 5 a passive canonical form which allows a pleasing interpretation of the energy flows in the system and with the system environment is introduced. This class of systems, als known as PCHD-systems (port-controlled Hamiltonian systems with dissipation), is the starting point for the passivity-based controller design with damping injection being presentd in Section 6. From their definitions the concepts of dissipativity and passivity are related with the state of the system. In contrast to this, the notion of positive realness introduced in Section 7 is based on the input-output behavior of the system and hence is independent of the state. Furthermore, the relation between positive realness, the L2-stability and the passivity together with the celebrated Kalman-Yakubovich-Popov-Lemma formulated for non-linear systems are discussed in detail in Section 7. Section 8 is devoted to the classical concept of absolute stability. It will be shwon that the well known circle and Popov-criterion are just the application of a far more general theorem, which is based on the feedback interconnection of two passive systems. The results are well known, but the considerations being presented allow not only a deep insight into the ideas behind these concept but can also be understood as a starting point for new modified stability criteria. Finally, in the last section, Section 9, we will give a short conclusion.