Conjectures and counterexamples on optimal L2 disturbance attenuation in nonlinear systems

This paper considers the problem of optimal L₂ disturbance attenuation with global asymptotic stability for strict feedback nonlinear systems. It is known from previous results that this problem cannot be solved with an arbitrary level of disturbance attenuation (almost disturbance decoupling) if the disturbance input drives unstable zero dynamics of the system. In this case, the problem can only be solved to achieve a level of disturbance attenuation above a nonzero optimal bound. An explicit expression of this lowest optimal bound is known for linear systems, and an approximate bound exists for a special subclass of nonlinear systems with second order zero dynamics. A more general expression for the lowest bound remains unknown. In this paper we provide background to the problem, and discuss the feasibility of obtaining such a general expression by presenting a series of conjectures, examples and counterexamples. We first present a conjecture that might appear as a natural generalisation of the linear expression but that, as we show by means of a counterexample, is generally false. Finally, we present a second conjecture, which holds generally for the linear case, and also for a class of scalar nonlinear systems. A general proof, or a counterexample, to this conjecture are still questions open to further research.