Source-linked AI summary
On asymptotic stability of the time-varying Kalman filter for unstabilizable linear systems: an optimization perspective
James B. Rawlings, Titus Quah, Matthias A. Müller
TL;DR
The paper asks when covariance convergence is insufficient to establish stability of the time-varying Kalman filter’s estimation error. It analyzes the equivalent optimization problem and proves necessary and sufficient conditions for asymptotic stability. The paper also notes that this asymptotic stability is not robust.
Problem
Riccati covariance convergence does not by itself determine whether the time-varying estimator error decays.
Source: §I · p. 1Method
Results
Takeaways & Limitations
The optimization problem provides a framework for studying estimator stability distinct from covariance convergence.
Source: §I · p. 2, §V · p. 6The result establishes asymptotic stability, which is not robust; the paper defers necessary and sufficient conditions for exponential stability to future work.
Source: §V · p. 7
Abstract
from arXiv · showhide
This paper establishes the necessary and sufficient conditions for asymptotic stability of the time-varying Kalman filter applied to a linear time invariant system with semidefinite initial state covariance and positive definite process and measurement noise. Rather than analyze the discrete Riccati equation as in the classic literature, the equivalent state smoothing optimization problem is stated and all results are established using properties of this optimization problem. A Lyapunov-like function, termed a modified $Q$-function is derived and used for this analysis. This optimization approach removes the need for the classic but cumbersome Riccati iteration algebra and provides better generalization and application for nonlinear systems.
I. INTRODUCTION, MOTIVATION, AND LITERATURE REVIEW
The paper asks when the time-varying Kalman filter’s estimation error is asymptotically stable, a question not settled by covariance convergence alone. It gives necessary and sufficient conditions and develops an optimization-based analysis for detectable but unstabilizable systems.
I. INTRODUCTION, MOTIVATION, AND LITERATURE REVIEW
- Earlier work progressively weakened observability and controllability assumptions for Riccati convergence, but covariance convergence and stable estimator error remain distinct questions.The literature cited includes detectability and stabilizability conditions as well as later results relaxing stabilizability for covariance convergence.Source: §I · p. 1
- Covariance convergence does not guarantee estimator-error decay, and the time-varying filter can be stable even when its limiting steady-state filter is not.When the prior misses a unit-circle mode, covariance may converge while estimation error does not decay.Source: §I · pp. 1–2
- The estimator is asymptotically stable if and only if conditions C1 and C2 hold.The paper distinguishes this criterion from conditions for Riccati covariance convergence.Source: §I · p. 2
- The authors establish the equivalence through the filter’s state-estimation optimization problem rather than Riccati recursion.A modified Q-function accommodates unstabilizable systems and semidefinite priors; the original Q-function lacks the required upper bound in this setting.Source: §I · p. 2
- The result proves asymptotic stability, not robustness: a bounded disturbance decaying to zero can still cause error divergence in the stated unit-circle setting.Exponential stability, which the paper identifies as necessary for robustness, is deferred to later work.Source: §I · p. 2
II. DISCUSSION OF ANALYSIS TOOLS
The section motivates analyzing the estimator through optimization rather than Riccati algebra. Existing Lyapunov candidates have shortcomings in the unstabilizable setting, and the Q-function approach also requires modification.
II. DISCUSSION OF ANALYSIS TOOLS
- The candidate V_k(e) = e′Σ(k)^−1e is not a Lyapunov function for the case considered in Proposition 1.The section points to an example demonstrating this failure.Source: §II · p. 2
- Riccati-iteration approaches require complex algebraic manipulations and can impose conservative restrictions on the initial variance.The authors favor optimization-problem analysis partly because it can extend to nonlinear systems.Source: §II · p. 2
- For unstabilizable systems, the Q-function approach lacks the required upper bound, motivating the modified construction developed later.For stabilizable systems, the cited approach establishes exponential stability in the same way as a Lyapunov function.Source: §II · p. 2
A. Optimization problems.
The section formulates the Kalman filter as a finite-horizon optimization problem, adapting it to semidefinite initial covariance. It states that the resulting problem has an existing, unique solution without requiring detectability or stabilizability.
A. Optimization problems
- For semidefinite initial covariance, the standard objective is undefined because Σ0 has no inverse.The construction uses the covariance pseudoinverse and adds an equality constraint to obtain a well-defined optimization problem.Source: A. Optimization problems · pp. 2–3
- The optimization problem corresponds one-to-one with the time-varying Kalman filter when the initial covariance is semidefinite.Its decision variables include the initial state and a finite-horizon process-noise sequence.Source: A. Optimization problems · pp. 2–3
- When Σ0 is positive definite, the added U2 constraint is removed; when Σ0 is zero, the initial state is fixed at x0.These are the two edge cases described for the optimization formulation.Source: A. Optimization problems · p. 3
- The finite-horizon optimization problem has an existing, unique minimizer for all system matrices when Q and R are positive definite and Σ0 is semidefinite.The existence and uniqueness claim does not assume detectability or stabilizability.Source: A. Optimization problems · p. 3
B. Linear systems
For nominal measurements, the section establishes a uniform upper bound on the optimization value under C2 and shows that the terminal residual cost tends to zero. The argument uses properties of linear systems and the estimator’s error dynamics.
B. Linear systems
- The analysis specializes to the linear estimator error system and assumes measurements from the nominal system without disturbances.The section identifies linear-system properties as part of its stability analysis.Source: B. Linear systems · p. 3
- Under C2 and nominal measurements, the optimization value is uniformly bounded above for all horizons.The stated bound applies for all initial states and prior estimates.Source: B. Linear systems · p. 3
- The terminal process- and measurement-noise cost tends to zero as the horizon grows.The convergence argument compares feasible solutions at consecutive horizons and uses bounded, nondecreasing optimal values.Source: B. Linear systems · pp. 3–4
III. UNSTABILIZABLE SYSTEMS AND Q-FUNCTIONS
This section establishes asymptotic stability conditions for the time-varying estimator and shows how a modified Q-function supports global asymptotic stability for unstabilizable systems.
III. UNSTABILIZABLE SYSTEMS AND Q-FUNCTIONS
- When the horizon limits of the modified Q-function exist, its nonincreasing, nonnegative limit and the estimation argument yield convergence of the estimate to the state.The proof also uses convergence of the finite-horizon estimate to its infinite-horizon counterpart.Source: §III · p. 4
- Linearity upgrades global attraction to uniform global attraction, establishing the stronger KL form of GAS.The argument first establishes uniform global stability and then uses linearity to obtain uniformity of attraction.Source: §III · p. 4
- Under C1 and C2, the filter admits a modified Q-function whose K∞ functions can be chosen quadratic.This supplies the modified-Q existence result used in the stability proof.Source: §III · p. 5
- Although the modified Q-function lacks the upper bound available for stabilizable systems, the filter’s linearity allows the argument to establish GAS.The loss of the upper bound would otherwise appear to obstruct even asymptotic stability.Source: §III · pp. 4–5
- The time-varying linear estimator is asymptotically stable if and only if C1 and C2 hold.The proof establishes sufficiency through the modified Q-function argument and necessity of both conditions.Source: §III · p. 5
- If C2 fails, a prior-null witness in the unstable, uncontrollable subspace forces the estimate error not to converge to zero.The proof shows the witness pins a component of the optimizer’s initial estimate and yields nonvanishing average estimate energy.Source: §III · pp. 5–6
IV. EXAMPLES
The examples show that a proposed covariance-based Lyapunov candidate can fail to register error decay, and that asymptotic stability need not ensure robustness to a decaying disturbance.
IV. EXAMPLES
- The candidate Lyapunov function remains constant even as the estimation error converges exponentially to zero.The example’s error evolves as ˆe(k) = α^kˆe(0).Source: §IV · p. 6
Example 2: An asymptotically stable Kalman filter is not robust
- A bounded disturbance that decreases to zero can make the estimate error diverge despite global asymptotic stability, even with perfect measurement.The example uses the unstabilizable case A = 1, C = 1, G = 0, Σ0 = 1, Q = 1, R = 1.Source: Example 2: An asymptotically stable… · p. 6
V. CONCLUSIONS
The paper uses the equivalent optimal-estimation problem to analyze asymptotic stability of the time-varying estimator for detectable but unstabilizable LTI systems. It derives necessary and sufficient conditions, while noting that robustness and exponential stability remain future work.
V. CONCLUSIONS
- The optimization-based analysis establishes necessary and sufficient conditions C1 and C2 for asymptotic stability of the time-varying optimal estimator.The method uses properties of the optimal estimation problem equivalent to the Kalman filter.Source: §V · pp. 6–7
- The alternative analysis is motivated by the difficulty of Riccati-iteration convergence arguments and their lack of an obvious extension to nonlinear systems.The paper presents its linear-setting tools as preparation for future nonlinear applications such as moving horizon estimation.Source: §V · p. 7
- The paper leaves necessary and sufficient conditions for exponential stability to future work; exponential stability provides robustness absent from asymptotic stability.The stated future direction is to derive those conditions.Source: §V · p. 7
VI. PROOFS
The proofs establish optimizer convergence and bounded costs under the stated conditions, then use a modified Q-function to prove global asymptotic stability and the necessity and sufficiency of C1 and C2.
VI. PROOFS
- C2 makes every unstable uncontrollable-state initialization feasible, enabling a uniform upper bound on the finite-horizon optimization cost.The proof uses a stabilizing feedback bound for the stabilizable subsystem and combines it with the initial-stage cost bound.Source: §VI · p. 9
- Under C2, the finite-horizon optimal initial estimates and each fixed process-noise term converge as the horizon grows.The proof establishes these limits using Cauchy sequences; detectability C1 is additionally required for convergence of the terminal estimates and the infinite-horizon cost result.Source: §VI · pp. 9–10
- The modified Q-function decreases by the stage cost, but lacks its usual upper bound when the system is not necessarily stabilizable.The proof pairs this construction with detectability-based IOSS-Lyapunov inequalities.Source: §VI · p. 10