Geometric insight into linear augmented observers for uncertain systems

Linear augmented observers (LAOs) are widely used for state and uncertainty estimation in practical control systems, yet their structural limitations remain unclear. Existing analyses typically assume that the augmented system is observable a priori and do not examine how the nominal model structure limits estimation performance. This paper develops a geometric framework to characterize the structural limitations of LAOs for uncertain systems with linear nominal models. A necessary and sufficient condition for the observability of the augmented system is established in terms of the invariant zeros of the nominal model, which removes the need for prior observability assumptions. The estimation error is shown to be confined to a subspace structurally determined by the weakly unobservable subspace of the nominal model, which explicitly reveals how the nominal system structure limits attainable estimation accuracy. In particular, arbitrarily small estimation error is achievable if and only if the nominal model has no invariant zeros. The complete set of state-uncertainty combinations that are estimable via high-bandwidth LAOs is explicitly characterized. In addition, the exponential amplification of measurement noise as observer bandwidth increases is analytically quantified. These results provide explicit and verifiable structural criteria for assessing the feasibility and attainable accuracy of state and uncertainty estimation using LAOs. Numerical simulations validate the theoretical findings.