On the Lack of Robustness of Observers for Systems with Uncertain, Unstable Dynamics

We consider the robustness of state estimation for linear, time-invariant systems. Since state estimation is dual to full-state feedback, it may be expected that stability of the error dynamics depends continuously on perturbations of the dynamics matrix. This paper shows, however, that, if the system dynamics are unstable, then—regardless of how the filter gain is chosen—there always exist arbitrarily small perturbations of the system dynamics that give rise to unbounded state-estimation error. Since this phenomenon cannot occur in full-state feedback control, this result reveals a surprising break-down in the duality between estimation and control.

I. Introduction

In the classical full-state-feedback control problem for linear, time-invariant plants, the goal is to determine a feedback gain K such that the closed-loop dynamics A+BK are asymptotically stable. Since the eigenvalues of a matrix are continuous functions of its entries, it follows that, if A+BK is asymptotically stable, then, for every perturbation ∆A of sufficiently small norm, the perturbed dynamics A + ∆A + BK are also asymptotically stable. In other words, by virtue of feedback control, A + BK is inherently robust to plant uncertainty, at least to some extent. For the dual problem of state estimation, A + BK is replaced by A − F C, where F is the filter gain. Consequently, one might expect an analogous result to hold, namely, that, if A−F C is asymptotically stable, then, for every perturbation ∆A of sufficiently small norm, the perturbed dynamics A + ∆A − F C are also asymptotically stable. The present paper shows that this expectation is false.