Analysis and design of polynomial control systems using dissipation inequalities and sum of squares

Polynomial control systems are control systems where the control system description is given in terms of polynomial nonlinearities. One encounters this class of control systems in a wide range of applications. In particular, in process control and systems biology, many control problems can be modeled as, transformed into, or approximated by polynomial control systems. The purpose of this paper is to give an introduction to and an overview over analysis and design of polynomial control systems using dissipation inequalities. Recent results for rather classical analysis problems, like in stability analysis, are presented as well as new results in emerging design problems, like synchronization. In particular, the analysis of the minimum phase behavior, the stability analysis of differential-algebraic systems with higher index, and the design of synchronizing feedbacks is carried out using dissipation inequalities. To ensure practical applicability, semidefinite programming and the sum of squares decomposition is used, to solve the obtained analysis and design dissipation inequalities in a numerically reliable and efficient way.