Stability analysis of discrete-time adaptive controllers for systems with unknown delay

The stability of discrete-time adaptive controllers when the process delay is unknown is studied. Conditions for stability and instability are derived based on the analysis of the homogeneous part of the linear time-varying difference equation describing the behavior of the parameter estimates. First, using a Lyapunov approach, it is shown that for slow adaptation the inclusion of a leakage [1] and a new normalization factor in the estimation law (2), (8) allows us to preserve stability when the delay is overestimated. Second, we prove that, when the delay is underestimated and no leakage is used, the parameters obtained with a projection estimator grow unbounded for sufficiently high adaptation gains.