Beyond H-infinity-design: robustness, disturbance rejection and Aizerman-Kalman type conjectures in general signal spaces

In this paper, we study the problems of robustness and disturbance rejectio n in a general setting, whereby the signal space in which the inputs and ou tputs reside is not necessarily L-2. It is well known that, if the signal s pace is taken as L-2, then both optimal robustness design and optimal distu rbance rejection can be formulated as H-infinity-norm minimization problems . Three distinct 'Aizerman' type of conjectures regarding the stability of nonlinear feedback systems are formulated, each of which happens to be true in the special case when the underlying signal space is L-2. It is shown t hat, in a general setting, only one of the three Aizerman type conjectures is true, namely: If a feedback system is stable for all linear, possibly ti me-varying feedback elements belonging to a specified sector, then the feed back system remains stable for all nonlinear, possibly lime-varying feedbac k elements belonging to the same sector. It is shown that the remaining two conjectures are equivalent to each other, and necessary and sufficient con ditions for each of the two conjectures to hold are derived. Next, it is sh own that, in general, the problems of optimal disturbance rejection and opt imal robustness design are quite distinct. In the special case where the si gnal space is L-2, and the corresponding Banach algebra of causal stable LT I systems is H-infinity, both problems coincide. But in general, the proble m of optimal disturbance rejection is that of minimizing the norm of the we ighted sensitivity matrix, whereas the problem of optimal robustness design is that of minimizing something like the spectral radius of the weighted c omplementary sensitivity matrix. Copyright (C) 2000 John Wiley & Sons, Ltd.