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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 ̇ () L^2 in the feedback path.
Narendra et al. (Sun,) studied this question.