Reciprocal relationship between detectability and observability in a non-uniform setting

Building on the recent notion of non-uniform complete observability, and on the fact that this property ensures non-uniform exponential detectability, this paper establishes the converse implication under suitable additional assumptions. Specifically, we investigate conditions under which non-uniform exponential detectability guarantees non-uniform complete observability. Our approach is based on a refined analysis of the associated output feedback systems and on the preservation of non-uniform observability properties under such feedback transformations. These results extend the classical equivalence between observability and detectability beyond the uniform framework and provide new tools for the qualitative analysis of time-varying control systems.

💡 Research Summary

The paper investigates the reciprocal relationship between detectability and observability for linear time‑varying systems in a non‑uniform setting. Classical control theory, rooted in Kalman’s work, establishes that for uniformly time‑varying systems the notions of complete observability and complete controllability are equivalent to the positivity of the observability and controllability Gramians over a fixed interval length σ. Moreover, the uniform Kalman condition—an exponential bound on the state transition matrix—guarantees uniform bounded growth and underpins the equivalence between the uniform versions of these properties.

The authors first review these uniform concepts and then move to the non‑uniform framework, where the transition matrix may grow or decay at rates that depend on time. They introduce two key generalizations: (i) non‑uniform bounded growth, expressed as ‖Φ_A(t,τ)‖ ≤ K₀ e^{η|τ|} e^{a|t−τ|}, and (ii) a non‑uniform Kalman condition, requiring the existence of ν > 0 and a function α belonging to the class B₂ such that ‖Φ_A(t,τ)‖ ≤ e^{ντ} α(|t−τ|). These relax the uniform requirement of a single constant bound and allow the system’s dynamics to be “non‑uniformly stable” or “non‑uniformly unstable” in a controlled manner.

Building on these definitions, the paper defines non‑uniform complete controllability (NUCC) and non‑uniform complete observability (NUCO). Unlike the uniform case, the Gramians are allowed to be scaled by time‑dependent exponential factors e^{±μt} and by positive functions α_i(σ), β_i(σ) that depend on the interval length σ. This scaling captures the fact that the energy needed to steer or observe the system may increase or decrease with time.

The central contribution is a set of “pseudo‑equivalence” results showing that, under suitable additional hypotheses, non‑uniform exponential detectability (NUED) implies NUCO. Detectability means that any unobservable mode of the plant decays exponentially, albeit with a time‑varying rate. The authors consider an output‑feedback law u = K(t) y and analyze the closed‑loop system. By exploiting the structure of the feedback, they prove that the closed‑loop transition matrix inherits the non‑uniform Kalman bound, and the resulting observability Gramian retains a positive lower bound after appropriate scaling. Consequently, the closed‑loop system satisfies the definition of NUCO.

A crucial technical tool is the use of the adjoint system (˙x = −Aᵀ(t)x − Cᵀ(t)u) and the dual system (˙x = Aᵀ(−t)x + Cᵀ(−t)u). The authors establish precise identities linking the controllability Gramians of these dual systems to the observability Gramians of the original system. By showing that the non‑uniform Kalman condition holds simultaneously for the primal and dual systems, they guarantee that both the controllability and observability Gramians possess the required time‑dependent bounds. This duality argument mirrors the classical Kalman duality but is carefully adapted to the non‑uniform context.

In the final section, the paper presents the dual result for controllability: non‑uniform exponential stabilizability, together with the non‑uniform Kalman condition, yields NUCC. The proof follows the same pattern as the observability side, using state‑feedback instead of output‑feedback and invoking the duality between stabilizability and detectability.

Overall, the work extends the classical equivalence between observability and detectability beyond the uniform framework. It provides a rigorous foundation for analyzing and designing observers and controllers for systems whose dynamics exhibit time‑dependent growth or decay rates—situations common in many engineering applications such as adaptive robotics, power‑grid dynamics, and biological networks. By establishing that detectability and observability are essentially interchangeable under the non‑uniform assumptions, the paper opens the door to new synthesis techniques that can exploit whichever property is easier to verify in a given application.