Design and analysis of a novel ℒ₁ adaptive controller, part I: Control signal and asymptotic stability

In this paper, we develop a novel adaptive control architecture that ensures that the input and output of an uncertain linear system track the input and output of a desired linear system during the transient phase, in addition to the asymptotic tracking. These features are established by first performing an equivalent reparametrization of MRAC, the main difference of which from MRAC is in definition of the error signal for adaptive laws. This new architecture, called companion model adaptive controller (CMAC), allows for incorporation of a low-pass filter into the feedback-loop that enables to enforce the desired transient performance by increasing the adaptation gain. For the proof of asymptotic stability, the L₁ gain of a cascaded system, comprised of this filter and the closed-loop desired reference model, is required to be less than the inverse of the upper bound of the norm of unknown parameters used in projection based adaptation laws. Moreover, the new L₁ adaptive controller is guaranteed to stay in the low-frequency range. Simulation results illustrate the theoretical findings.