Adaptive tracking of infinite-dimensional reference models for linear infinite-dimensional systems in Hilbert space

This paper is focused on adaptively controlling a linear infinite-dimensional system to cause it to track an infinite-dimensional reference model. Given a linear continuous-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 direct model reference adaptive control law with certain disturbance rejection and robustness properties. Both the plant and the reference model are described by closed, densely defined linear operators that generate continuous semigroups of bounded operators on the Hilbert space of states. An extension of the Barbalat-Lyapunov result for infinite dimensional Hilbert spaces is used to determine conditions under which a linear Infinite-dimensional system can be directly adaptively controlled to follow such a reference model. In particular we examine conditions for a set of ideal trajectories to exist for the tracking problem and show the solvability of the infinite dimensional matching conditions under the simple condition that the high frequency gain CB is nonsingular and the reference system is a normal operator with compact resolvent.