Duality of Stochastic Observability and Constructability and Links to Fisher Information

TL;DR

This work establishes a formal duality between stochastic observability and constructability for discrete-time linear systems under process and measurement noise, linking these Gramians to the Fisher information matrix and the posterior Cramér-Rao bound. By introducing a dual system, the authors derive a numerically robust recursive formula for the stochastic observability Gramian from the existing stochastic constructability recursion, and show equivalences between the dual and original problems for both time-varying and time-invariant systems. The key contributions include a complete duality framework, recursive computation that avoids large-scale inversions, and a demonstration of convergence in the LTI case via a Riccati-type equation, supported by a numerical example highlighting robustness and memory efficiency. The results enable method interchange between observability and constructability, with potential extensions to nonlinear and data-driven contexts and connections to smoother posterior bounds. Overall, the paper provides a principled bridge between observability theory and information-theoretic bounds, with practical algorithms for robust state-estimation analysis under uncertainty.

Abstract

Given a set of measurements, observability characterizes the distinguishability of a system's initial state, whereas constructability focuses on the final state in a trajectory. In the presence of process and/or measurement noise, the Fisher information matrices with respect to the initial and final states$\unicode{x2013}$equivalent to the stochastic observability and constructability Gramians$\unicode{x2013}$bound the performance of corresponding estimators through the Cramér-Rao inequality. This letter establishes a connection between stochastic observability and constructability of discrete-time linear systems and provides a more numerically robust way for calculating the stochastic observability Gramian. We define a dual system and show that the dual system's stochastic constructability is equivalent to the original system's stochastic observability, and vice versa. This duality enables the interchange of theorems and tools for observability and constructability. For example, we use this result to translate an existing recursive formula for the stochastic constructability Gramian into a formula for recursively calculating the stochastic observability Gramian for both time-varying and time-invariant systems, where this sequence converges for the latter. Finally, we illustrate the robustness of our formula compared to existing (non-recursive) formulas through a numerical example.

Figures (2)

• Figure 1: Illustration of key definitions and organization. Middle panel shows the total information for the first state variable of a discrete-time linear time-invariant (LTI) system, i.e., the first diagonal entry of the FIM of all measurements with respect to $\mathbf{x}_k$. The discrete-time LTI system matrices are $\Phi= \hbox{$\begin{bmatrix} 1 & -1 \\ 0 & 1 \end{bmatrix}$}$, $C=10$, $Q= \hbox{$\begin{bmatrix} 1\times 10^{-11} & -5\times 10^{-18} \\ -5\times 10^{-18} & 1\times 10^{-17} \end{bmatrix}$}$, and $R=2.89\times 10^{-10}$, adapted from crassidis2004 to better illustrate the concepts. Present measurement information is constant as $C$ and $R$ are time-invariant, contributing to both observability and constructability: observability reflects the combined information from the future and present, while constructability accounts for past and present measurements. In Sec. \ref{['sec:duality']}, we construct a dual system in which the stochastic observability Gramian is equivalent to the original system's stochastic constructability Gramian, and vice versa. Due to process noise, distant measurements from the state of interest (whether in the future or past) become irrelevant.
• Figure 2: Comparison of three approaches for calculating the stochastic observability Gramian highlighting the numerical stability of our recursive formulation. Each panel shows one of the four entries of the stochastic observability Gramian over time, sharing the same x-axis label. The 11- and 31-step dual system's constructability Gramians entries are also shown to illustrate that they reach the same final value as $F^{\mathbf{x}_0}_{\downarrow_w}$, despite differences in intermediate values. Although the observability calculations for each $w$ require unique dual systems, these calculations can be run in parallel. Negative values of the off-diagonal entries of $F^{\mathbf{x}_0}_{\downarrow_w}$ are acceptable as long as $F^{\mathbf{x}_0}_{\downarrow_w}$ remains positive definite.

Theorems & Definitions (7)

• Definition 1
• Theorem 1
• Remark 1
• Lemma 1
• Remark 2
• Definition 2
• Theorem 2