Adaptive backstepping for uncertain systems with time-delay on-line update laws

This paper addresses the general problem of the equilibrium regulation of potentially unstable linear systems with an unknown input time-delay and unknown parameters in the plant. We extend recent results from the literature where such systems are treated using a backstepping approach applied to a distributed parameters system representation of the delay. Furthering previous approaches, our contribution concerns on-line adaptation of the delay. We develop a local asymptotic convergence result for single-input systems. It is illustrated in simulations on the control of the Air-Fuel Ratio in Spark Ignition engines. The results on this particular example stress the merits of the proposed control algorithm which, with reduced implementation difficulties, reveals sympathetic to on-line applications.