Adaptive Control for Weakly Minimum Phase Linear Infinite-Dimensional Systems in Hilbert Space Using a Zero Filter

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 persistent disturbance rejection and robustness properties. The plant is described by a closed, densely defined linear operator that generates a continuous semigroup of bounded operators on the Hilbert space of states. For this paper, the plant will be weakly minimum phase, i.e., there will be a finite number of unstable zeros with real part equal to zero. All other zeros will be exponentially stable. The central result will show that all errors will converge to a prescribed neighborhood of zero in an infinite-dimensional Hilbert space even though the plant is not truly minimum phase. The result will not require the use of the standard Barbalat’s lemma which requires certain signals to be uniformly continuous. This result is used to determine conditions under which a linear infinite-dimensional system can be directly adaptively controlled to follow a reference model. In particular we examine conditions for a set of ideal trajectories to exist for the tracking problem. Our principal result will be that the direct adaptive controller can be compensated with a zero filter for the unstable zeros which will produce the desired robust adaptive control results even though the plant is only weakly minimum phase. Our results are applied to adaptive control of general linear infinite-dimensional systems described by self-adjoint operators with compact resolvent.