Minimax state estimation for linear descriptor systems

Author’s Summary of the dissertation for the degree of the Candidate of Science (physics and mathematics). The aim of the dissertation is to develop a generalized Kalman Duality concept applicable for linear unbounded non-invertible operators and introduce the minimax state estimation theory and algorithms for linear differential-algebraic equations. In particular, the dissertation pursues the following goals: - develop generalized duality concept for the minimax state estimation theory for DAEs with unknown but bounded model error and random observation noise with unknown but bounded correlation operator; - derive the minimax state estimation theory for linear DAEs with unknown but bounded model error and random observation noise with unknown but bounded correlation operator; - describe how the DAE model propagates uncertain parameters; - estimate the worst-case error; - construct fast estimation algorithms in the form of filters; - develop a tool for model validation, that is to assess how good the model describes observed phenomena. The dissertation contains the following new results: - generalized version of the Kalman duality principle is proposed allowing to handle unbounded linear model operators with non-trivial null-space; - new definitions of the minimax estimates for DAEs based on the generalized Kalman duality principle are proposed; - theorems of existence for minimax estimates are proved; - new minimax state estimation algorithms (in the form of filter and in the variational form) for DAE are proposed.

💡 Research Summary

The dissertation “Minimax State Estimation for Linear Descriptor Systems” develops a comprehensive theory and practical algorithms for state estimation in linear differential‑algebraic equations (DAEs), a class of models that frequently appear in engineering domains such as vehicle dynamics, robotics, and biomedical imaging. Classical minimax state estimation (MSE) relies on the Kalman duality principle, which assumes that the system operator is bounded and invertible. This assumption fails for most DAEs because the underlying linear operator can be unbounded, non‑invertible, and possess a non‑trivial null‑space.

To overcome this limitation the author introduces a Generalized Kalman Duality (GKD) framework. GKD extends the duality concept to unbounded, non‑invertible operators by formulating the original minimax problem as a pair of dual optimization problems that remain well‑posed even when the operator has a null‑space. The framework accommodates unknown but bounded model errors and random observation noise with unknown but bounded correlation.

Two types of minimax estimates are defined:

1. A‑priori estimate – based on a priori knowledge of the uncertainty sets for the model error (f) and the noise covariance (R_\eta). The estimate minimizes the worst‑case mean‑square error over all admissible (f) and (R_\eta). For ellipsoidal uncertainty sets (G={f\mid (Q_1 f,f)\le 1}) and (G_2={R_\eta\mid \operatorname{tr}(Q_2 R_\eta)\le 1}) the author derives explicit formulas (equations (4)–(5)) involving solutions of linear equations (F p = B Q_1^{-1} B^\top \hat z) and (F^\top \hat z = \ell - H^\top Q_2 H p). The worst‑case error is given by ((\ell,p)) and becomes infinite when the functional (\ell) does not belong to the range of (F^\top).

2. A‑posteriori estimate – assumes deterministic but bounded disturbances ((f,g)) and defines a feasible state set (X). The minimax a‑posteriori estimate minimizes the maximum deviation of a linear functional (\ell(x)) over (X). Closed‑form expressions are obtained for the same ellipsoidal sets, leading to equations (7)–(9).

The theory is applied to both discrete‑time and continuous‑time DAEs.

Discrete‑time case: By stacking the DAE equations into block matrices (F, H, B) the problem reduces to a standard linear algebraic form (F x = B f), (y = H x + g). The author presents a recursive minimax filter (Theorem 4) that computes the estimate (\hat x_k) and the associated error covariance (P_k) using forward–backward recursions reminiscent of Kalman filtering but adapted to the singular structure of (F). The algorithm requires only that the combined matrix (