Adaptive regulation: Lyapunov design with a growth condition

We propose a new Lyapunov design of an adaptive regulator under some restriction on the dependence of a Lyapunov function on the parameters. This restriction has been introduced by Praly et al. Its interest is to involve only a Lyapunov function and not explicitly the system non‐linearities. We show it is satisfied by strict pure feedback systems with polynomial growth non‐linearities and some other non‐feedback linearizable systems. Our new Lyapunov design leads to an adaptive regulator where the adapted parameter vector is transformed before being used in the control law; namely, the so‐called certainty equivalence principle is not applied. Unfortunately, the implementation of this regulator needs the explicit solution of a fixed point problem, so in a second stage we propose a more practical solution obtained by replacing the fixed point static equation by a dynamical system with this fixed point as equilibrium.