  Absolute stability is a technique for analyzing the stability of Lur'e
systems, which arise in diverse applications, such as oscillators with
nonlinear damping or nonlinear stiffness. A special class of Lur'e systems
consists of self-excited systems (SES), in which bounded oscillations arise
from constant inputs. In many cases, SES can be stabilized by linear
controllers, which motivates the present work, where the goal is to evaluate
the effectiveness of adaptive model predictive control for Lur'e systems. In
particular, the present paper considers predictive cost adaptive control
(PCAC), which is equivalent to a linear, time-variant (LTV) controller. A
closed-loop Lur'e system comprised of the positive feedback interconnection of
the Lur'e system and the PCAC-based controller can thus be derived at each
step. In this work, the circle and Tsypkin criteria are used to evaluate the
absolute stability of the closed-loop Lur'e system, where the adaptive
controller is viewed as instantaneously linear time-invariant. When the
controller converges, the absolute stability criteria guarantee global
asymptotic stability of the asymptotic closed-loop dynamics.
