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Geometric insight into linear augmented observers for uncertain systems.

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Title: Geometric insight into linear augmented observers for uncertain systems.
Authors: Li, Junhui1 (AUTHOR) jh.li@gxu.edu.cn, Gong, Beili1 (AUTHOR) aublgong@gxu.edu.cn
Source: ISA Transactions. May2026, Vol. 172, p211-220. 10p.
Subjects: Observability (Control theory), Invariant subspaces, Noise measurement, Uncertain systems, Estimation bias
Abstract: 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. • A necessary and sufficient structural condition for the observability of the augmented system is established, providing a rigorous geometric characterization of LAO feasibility in terms of invariant zeros of the nominal system, without relying on ad hoc observability assumptions. • The estimation error of high-bandwidth LAOs is shown to be confined to a subspace structurally determined by the weakly unobservable subspace of the nominal system, highlighting that the nominal system structure imposes intrinsic limitations on the estimation accuracy and providing guidance for observer bandwidth selection, independent of specific disturbance realizations. • The estimable combinations of system states and uncertainties for high-bandwidth LAOs are geometrically characterized and shown to correspond to those whose combination matrices project out the estimation-error component associated with the weakly unobservable subspace, thereby identifying the quantities accessible to LAO-based estimation and admissible for observer-based control. • A quantitative analysis of measurement noise sensitivity in LAO-based estimation is developed, showing how increasing observer bandwidth can exponentially amplify noise and revealing the fundamental trade-off between bandwidth and achievable estimation performance. [ABSTRACT FROM AUTHOR]
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Abstract: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. • A necessary and sufficient structural condition for the observability of the augmented system is established, providing a rigorous geometric characterization of LAO feasibility in terms of invariant zeros of the nominal system, without relying on ad hoc observability assumptions. • The estimation error of high-bandwidth LAOs is shown to be confined to a subspace structurally determined by the weakly unobservable subspace of the nominal system, highlighting that the nominal system structure imposes intrinsic limitations on the estimation accuracy and providing guidance for observer bandwidth selection, independent of specific disturbance realizations. • The estimable combinations of system states and uncertainties for high-bandwidth LAOs are geometrically characterized and shown to correspond to those whose combination matrices project out the estimation-error component associated with the weakly unobservable subspace, thereby identifying the quantities accessible to LAO-based estimation and admissible for observer-based control. • A quantitative analysis of measurement noise sensitivity in LAO-based estimation is developed, showing how increasing observer bandwidth can exponentially amplify noise and revealing the fundamental trade-off between bandwidth and achievable estimation performance. [ABSTRACT FROM AUTHOR]
ISSN:00190578
DOI:10.1016/j.isatra.2026.03.015