Stable adaptive controller design, part II: Proof of stability

The paper presents a proof of stability of model reference adaptive control systems using direct control. The structure of the adaptive system is similar to that considered by Monopoli [1] and Narendra and Valavani [2] but contains an additional feedback term which ensures that the time derivative of the parameter error vector belongs to the L
²
space. The output of the plant is shown to be bounded by expressing the plant feedback loop as an exponentially stable system with a time-varying gain
\dot{\phi}(\cdot)\in L^{2}
in the feedback path.