On Multipliers Theory

Multipliers are often used to find conditions for the absolute stability of Lur'e systems. They can be used either in conjunction with passivity theory or within the more recent framework of integral quadratic constraints (IQCs). This seminar presents two equivalence results within multiplier theory. In the first part of the seminar, passivity with multipliers and IQC theory are compared. The passivity theorem is developed from an energetic point of view. In short, the energy of the system cannot increase by more than the energy provided at the inputs. The concept of causality is essential to this argument. The causality in passivity theory requires that any multipliers must have a canonical factorization. On the other hand, the IQC theorem is developed using a homotopy argument and causality is not required. It has been suggested in the literature that this represents an advantage of the IQC theory. However, under some mild conditions the factorization is ensured and equivalence between both theories can be stated. The second part of the seminar focuses on slope-restricted nonlinearities and the Zames—Falb multipliers. Several other classes of multipliers can be found in the recent literature. Some of them are referred to as extensions of the Zames— Falb multipliers. Nevertheless, it can be demonstrated that all the classes of multipliers presented in the literature are “phase-equivalent” to the class of Zames—Falb multipliers. Therefore, the class of Zames—Falb multipliers remains the widest available class of multipliers.