Adaptive regulation in the presence of persistent disturbances for linear infinite-dimensional systems in Hilbert space: Conditions for almost strict dissipativity

This paper is focused on adaptively controlling a linear infinite-dimensional system to cause it to regulate the output to zero in the presence of persistent disturbances. The plant (A, B, C) is described by a closed, densely defined linear operator A that generates a continuous semigroup of bounded operators on a Hilbert space of states; the input-output operators B & C are finite rank linear operators. We show that there exists a direct model reference adaptive control law that regulates the output in the presence of disturbances of known waveform but unknown amplitude and phase. The conditions needed for the success of the direct adaptive controller include the need for (A, B, C) to be almost strictly dissipative (ASD). In finite dimensional space, ASD is equivalent to two simple open-loop requirements: the high frequency gain CB is sign-definite and the open-loop transfer function P(s) is minimum phase. Our main result will prove infinite-dimensional versions of these conditions for a large class of infinite-dimensional systems.