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

Abstract

In this paper, we study the problems of robustness and disturbance rejection in a general setting, whereby the signal space in which the inputs and outputs reside is not necessarily L₂. It is well known that, if the signal space is taken as L₂, then both optimal robustness design and optimal disturbance rejection can be formulated as H_∞-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 L₂. It is shown that, 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 time-varying feedback elements belonging to a specified sector, then the feedback system remains stable for all nonlinear, possibly time-varying feedback elements belonging to the same sector. It is shown that the remaining two conjectures are equivalent to each other, and necessary and sufficient conditions for each of the two conjectures to hold are derived. Next, it is shown that, in general, the problems of optimal disturbance rejection and optimal robustness design are quite distinct. In the special case where the signal space is L₂ and the corresponding Banach algebra of causal stable LTI systems is H_∞, both problems coincide. But in general, the problem of optimal disturbance rejection is that of minimizing the norm of the weighted sensitivity matrix, whereas the problem of optimal robustness design is that of minimizing something like the spectral radius of the weighted complementary sensitivity matrix.