Discrete-time infinite-dimensional adaptive control and rejection of persistent disturbances: To D or not to D?

In many cases an adaptive control must be implemented in discrete-time rather than continuous-time, aerospace applications. In the case of infinite-dimensional systems, the adaptive control theoretic problem becomes substantially different; we will emphasize those anomalies here. Given a linear discrete-time infinite dimensional plant on a Hilbert space and disturbances of known waveform but unknown amplitude and phase, we show that there exists a stabilizing discerete-time direct model reference adaptive control law with certain disturbance rejection and robustness properties. Our central result is a discrete-time version of Barbalat-Lyapunov result for infinite-dimensional Hilbert spaces. This is used to determine conditions under which a linear infinite-dimensional system can be directly adaptively regulated. Our results are illustrated on a system described by a compact self-adjoint operator; such a description fits many discrete-time applications.