Some results on adaptive tracking for a class of nonlinear time-delay systems

The tracking control problem is addressed for a class of nonlinear delay systems with unknown parameters. Such a problem is solved in the case that the nonlinear delay system has some geometrical properties, that is full delay relative degree and no unstable internal dynamics when the output and its derivatives are taken bounded by the control law. It is supposed moreover that the unknown parameters do not affect the output and its derivatives until n-1, where n is the dimension of the state Euclidean vector. As usual, the control law here found depends on the state variables in present and past times, and on the control law itself in past times too. Standard Lyapunov methodology is here used to-find out the adaptive control law and the dynamics of estimated parameters. It is proved that when the found out control law is applied to the system, the tracking error asymptotically goes to zero. The case of unknown delay is considered too. An upper bound for the error in delay knowledge is found which can be tolerated when controlling the system by recent methodologies based on standard nonlinear analysis. Simulation results are here shown for a prey-predator Lotka Volterra system