Discussion on 'Infinite Dimensional Backstepping-Style Feedback Transformations for a Heat Equation with Arbitrary Level of Instability' by A. Balogh and M. Krstic

In this paper, the authors derive a Dirichlet boundary control law that stabilizes a reaction-diffusion equation (a more general one than what is usually understood by``heat equation'') with arbitrary number of unstable modes in open-loop. Although interesting in its own right, this result is not as important as the technique used, which, the authors claim, generalizes backstepping/feedback linearization to infinite-dimensional systems. In our opinion two important points still need to be elucidated for this method to be generalized: From a practical standpoint, the fact that the stabilizing control law requires full state feedback (the whole ®eld u…:; t† has to be measured) is unrealistic. It is all the more so as thè`kernel function'' k is not rapidly decreasing outside a sub-interval of ‰0; 1Š, which would have indicated a possibility of restricting the number of and/or localizing the sensors. The numerical simulations of Part 5 give interesting information regarding this issue: it appears that the number of sensors can be dramatically lowered without losing stability of the closed-loop system. This is good news on the implementation side but it also means that the backstepping control scheme used in practice does not involve an in®nite number of steps. Thus the idea of a really in®nite-dimensional backstepping should be seen more as a theoretical tool to derive what the most ef®cient stabilizing control law is, than as a method to determine directly implementable controllers. Given the complicated recursive equations that had to be solved for the change of variables , it seems natural to ask the following: ``For a given general problem, what should a rea-sonablètarget system' and coordinate transformation be, so that it is possible to show that the former is, say, L 2-stable and the latter can be determined analytically or, at least, proven to be invertible?'' We would agree that aiming for a closed-loop system that has the same spatial-interconnection structure as the plant, after changing variables, is the right thing to do. However it is not clear why the classical chained changes of variables (i.e. the new variable w i depends on the old variables u 0 ; :::; u i only) of backstepping should be used. We suspect that if the change of coordinates for the semi-discretized system was also chosen to reflect the plant's spatial structure, the divergence of the gain as n 3 I would disappear, because the state transformations would then be purely local …