Low gain control of uncertain distributed parameter systems: I and II

Considers a low gain tracking problem for a general class of open loop stable systems or plants. It is well known that closing the loop around a stable, finite dimensional, single input, single output plant, with transfer function G(s), compensated by a pure integral controller will result in a stable closed loop system which achieves asymptotic tracking of arbitrary constant reference signals, provided that the gain is small enough. So, if a plant is known to be stable and if the sign of G(O) is known, the solution of the low-gain integral tracking problem reduces to the tuning of the gain parameter. Similar results exist for multi input, multi output systems, under suitable assumptions on G(O). A number of natural and interesting questions arise in low-gain integral control: (a) How large can the integral gain be? Or, what is the smallest gain for which the closed-loop system becomes unstable? (b) Can stabilizing integral gains be found adaptively? (c) Can the finite dimensionality assumptions be relaxed to include neglected distributed parameter effects? The authors deal with the questions outlined in (b) and (c) in a natural and most general setting, that of abstract linear regular systems (with infinite dimensional state space).