Nonuniform Complete Observability

• Nonuniform complete observability is defined by time-dependent Gramian bounds that bridge complete observability and uniform complete observability in linear time-varying systems.
• The framework employs exponential weight factors and time-varying window thresholds to capture nonuniform growth, facilitating duality between observability and controllability.
• Its practical implications include robust observer synthesis and state-feedback designs under nonuniform growth conditions, ensuring stability and performance.

Nonuniform complete observability (NUCO) is an observability property for nonautonomous linear control systems in finite dimension; it is more general than uniform complete observability and more restrictive than complete observability. In the recent linear time-varying framework, NUCO is formulated through time-weighted lower and upper bounds on observability Gramians over windows $[t,t+\sigma]$ whose threshold length may depend on the initial time, and it has been developed together with duality results, preservation under output feedback, the implication $\mathrm{NUCO}\Rightarrow\mathrm{NUED}$, and a converse $\mathrm{NUED}\Rightarrow\mathrm{NUCO}$ under additional hypotheses (Huerta, 2024, Huerta et al., 15 Dec 2025).

1. Formal setting and definition

The standard setting is the linear time-varying control system

$$\dot{x}(t)=A(t)x(t)+B(t)u(t), \qquad y(t)=C(t)x(t),$$

where $x(t)\in\mathbb{R}^n$, $u(t)\in\mathbb{R}^p$, $y(t)\in\mathbb{R}^m$, and $A(t),B(t),C(t)$ are measurable matrix-valued functions bounded on finite intervals. Observability is reduced to the output pair $(A,C)$, that is,

$\dot{x}(t)=A(t)x(t), \qquad y(t)=C(t)x(t),$

with transition matrix $\mathrm{NUCO}\Rightarrow\mathrm{NUED}$0 (Huerta et al., 13 Aug 2025).

For the pair $\mathrm{NUCO}\Rightarrow\mathrm{NUED}$1, the observability Gramian over $\mathrm{NUCO}\Rightarrow\mathrm{NUED}$2 is

$\mathrm{NUCO}\Rightarrow\mathrm{NUED}$3

and a second Gramian is

$\mathrm{NUCO}\Rightarrow\mathrm{NUED}$4

with the identities

$\mathrm{NUCO}\Rightarrow\mathrm{NUED}$5

(Huerta et al., 15 Dec 2025).

In the 2024 formulation on $\mathrm{NUCO}\Rightarrow\mathrm{NUED}$6, the pair $\mathrm{NUCO}\Rightarrow\mathrm{NUED}$7 is nonuniformly completely observable if there exist fixed numbers $\mathrm{NUCO}\Rightarrow\mathrm{NUED}$8, $\mathrm{NUCO}\Rightarrow\mathrm{NUED}$9 and functions $\mathrm{NUED}\Rightarrow\mathrm{NUCO}$0 such that for any $\mathrm{NUED}\Rightarrow\mathrm{NUCO}$1, there exists $\mathrm{NUED}\Rightarrow\mathrm{NUCO}$2 with

$\mathrm{NUED}\Rightarrow\mathrm{NUCO}$3

(Huerta, 2024).

In the later whole-line formulation on $\mathrm{NUED}\Rightarrow\mathrm{NUCO}$4, NUCO is defined symmetrically by requiring fixed numbers $\mathrm{NUED}\Rightarrow\mathrm{NUCO}$5 and functions $\mathrm{NUED}\Rightarrow\mathrm{NUCO}$6 such that for any $\mathrm{NUED}\Rightarrow\mathrm{NUCO}$7, there exists $\mathrm{NUED}\Rightarrow\mathrm{NUCO}$8 with, for every $\mathrm{NUED}\Rightarrow\mathrm{NUCO}$9,

$$\dot{x}(t)=A(t)x(t)+B(t)u(t), \qquad y(t)=C(t)x(t),$$0

$$\dot{x}(t)=A(t)x(t)+B(t)u(t), \qquad y(t)=C(t)x(t),$$1

(Huerta et al., 13 Aug 2025).

The terminology is explicit. “Complete” means observability over some finite horizon that may depend on the initial time $$\dot{x}(t)=A(t)x(t)+B(t)u(t), \qquad y(t)=C(t)x(t),$$2 through $$\dot{x}(t)=A(t)x(t)+B(t)u(t), \qquad y(t)=C(t)x(t),$$3. “Non-uniform” means that the Gramian bounds involve factors depending on $$\dot{x}(t)=A(t)x(t)+B(t)u(t), \qquad y(t)=C(t)x(t),$$4, such as $$\dot{x}(t)=A(t)x(t)+B(t)u(t), \qquad y(t)=C(t)x(t),$$5, and that the threshold window length need not be constant (Huerta et al., 15 Dec 2025).

2. Gramians, growth conditions, and duality

NUCO is coupled to nonuniform growth assumptions on the plant. A non-uniform Kalman property is

$$\dot{x}(t)=A(t)x(t)+B(t)u(t), \qquad y(t)=C(t)x(t),$$6

with some $$\dot{x}(t)=A(t)x(t)+B(t)u(t), \qquad y(t)=C(t)x(t),$$7 and $$\dot{x}(t)=A(t)x(t)+B(t)u(t), \qquad y(t)=C(t)x(t),$$8, while a particular case is non-uniform bounded growth (NUBG),

$$\dot{x}(t)=A(t)x(t)+B(t)u(t), \qquad y(t)=C(t)x(t),$$9

with constants $x(t)\in\mathbb{R}^n$0, $x(t)\in\mathbb{R}^n$1, $x(t)\in\mathbb{R}^n$2 (Huerta et al., 13 Aug 2025).

The nonuniform framework contains a three-way equivalence theorem: on $x(t)\in\mathbb{R}^n$3, any two of the following properties imply the third: the nonuniform Kalman property, the NUCO bounds for $x(t)\in\mathbb{R}^n$4, and the NUCO bounds for $x(t)\in\mathbb{R}^n$5. This is the nonuniform analogue of the classical uniform result in which any two of uniform bounded growth, $x(t)\in\mathbb{R}^n$6-bounds, and $x(t)\in\mathbb{R}^n$7-bounds imply the third (Huerta et al., 13 Aug 2025, Huerta et al., 15 Dec 2025).

Duality is a central structural feature. The adjoint and dual systems of $x(t)\in\mathbb{R}^n$8 are

$x(t)\in\mathbb{R}^n$9

with transition matrices

$u(t)\in\mathbb{R}^p$0

If $u(t)\in\mathbb{R}^p$1 and $u(t)\in\mathbb{R}^p$2 denote the controllability Gramians of the adjoint and dual systems, then

$u(t)\in\mathbb{R}^p$3

$u(t)\in\mathbb{R}^p$4

(Huerta et al., 15 Dec 2025).

These identities yield the equivalences

$u(t)\in\mathbb{R}^p$5

$u(t)\in\mathbb{R}^p$6

(Huerta et al., 13 Aug 2025).

The same framework recovers the uniform theory as a special case. When $u(t)\in\mathbb{R}^p$7, $u(t)\in\mathbb{R}^p$8, and $u(t)\in\mathbb{R}^p$9 is constant, the non-uniform results reduce to the uniform case (Huerta et al., 15 Dec 2025).

3. Position between complete and uniform observability

Uniform complete observability (UCO) requires a fixed window length $y(t)\in\mathbb{R}^m$0 and time-independent positive constants such that

$y(t)\in\mathbb{R}^m$1

for all admissible $y(t)\in\mathbb{R}^m$2. NUCO relaxes this by allowing exponential weights in $y(t)\in\mathbb{R}^m$3 and a window threshold $y(t)\in\mathbb{R}^m$4 depending on time. Complete observability (CO), by contrast, only requires that for each initial time there exists some finite horizon on which the Gramian is positive definite. Accordingly, NUCO is strictly between complete observability and uniform complete observability (Huerta et al., 13 Aug 2025, Huerta et al., 15 Dec 2025).

The examples developed in the literature separate these notions sharply. For $y(t)\in\mathbb{R}^m$5,

$y(t)\in\mathbb{R}^m$6

one has $y(t)\in\mathbb{R}^m$7 and

$y(t)\in\mathbb{R}^m$8

For $y(t)\in\mathbb{R}^m$9 and $A(t),B(t),C(t)$0,

$A(t),B(t),C(t)$1

so UCO holds, hence CO holds, and NUCO also holds by suitable non-uniform bounds (Huerta et al., 13 Aug 2025).

A contrasting scalar example yields NUCO while UCO fails. The transition matrix

$A(t),B(t),C(t)$2

satisfies

$A(t),B(t),C(t)$3

which shows NUBG. With $A(t),B(t),C(t)$4, the NUCO bounds follow, so the system is NUCO and completely observable; however, UCO fails because assuming UCO would force a uniform bounded-growth estimate contradicted by a specific choice of $A(t),B(t),C(t)$5 and $A(t),B(t),C(t)$6 with $A(t),B(t),C(t)$7 and $A(t),B(t),C(t)$8 as $A(t),B(t),C(t)$9 (Huerta et al., 15 Dec 2025).

There are also examples showing that CO does not imply NUCO. The adjoint of a scalar control system that is completely controllable but neither uniformly completely controllable nor nonuniformly completely controllable is completely observable but neither UCO nor NUCO. This establishes that NUCO is more restrictive than CO and does not collapse to it (Huerta et al., 13 Aug 2025).

4. Preservation under output feedback

A basic robustness result concerns output feedback of the form

$(A,C)$0

which produces the closed-loop pair

$(A,C)$1

with

$(A,C)$2

(Huerta, 2024).

In the 2024 half-line setting, the preservation theorem assumes NUBG for $(A,C)$3 and the explicit growth bounds

$(A,C)$4

together with the gap conditions

$(A,C)$5

Under these assumptions,

$(A,C)$6

(Huerta, 2024).

In the later whole-line version, the feedback term is required to decay exponentially in $(A,C)$7: $(A,C)$8 Under NUBG, the same equivalence holds: $(A,C)$9 (Huerta et al., 15 Dec 2025).

The proof mechanism is based on a variation-of-constants formula for the closed-loop transition matrix,

$\dot{x}(t)=A(t)x(t), \qquad y(t)=C(t)x(t),$0

and on a “symmetric” comparison between the open-loop and closed-loop observability integrals. In the 2025 proof, Cauchy–Schwarz estimates and NUBG bounds produce lower and upper estimates for

$\dot{x}(t)=A(t)x(t), \qquad y(t)=C(t)x(t),$1

in terms of $\dot{x}(t)=A(t)x(t), \qquad y(t)=C(t)x(t),$2 and exponentially weighted integral terms, which yields the NUCO inequalities for the closed-loop system as well (Huerta et al., 15 Dec 2025).

A related lemma shows that NUBG itself is preserved under exponentially decaying perturbations: if

$\dot{x}(t)=A(t)x(t), \qquad y(t)=C(t)x(t),$3

and the original plant has NUBG parameters $\dot{x}(t)=A(t)x(t), \qquad y(t)=C(t)x(t),$4, then the perturbed system also has NUBG, with parameters $\dot{x}(t)=A(t)x(t), \qquad y(t)=C(t)x(t),$5 (Huerta et al., 15 Dec 2025).

5. Detectability and the reciprocal theorem

The notion paired with NUCO is nonuniform exponential detectability (NUED). For a linear system

$\dot{x}(t)=A(t)x(t), \qquad y(t)=C(t)x(t),$6

nonuniform exponential stability forward (NUES forward) means that there exist $\dot{x}(t)=A(t)x(t), \qquad y(t)=C(t)x(t),$7, $\dot{x}(t)=A(t)x(t), \qquad y(t)=C(t)x(t),$8, $\dot{x}(t)=A(t)x(t), \qquad y(t)=C(t)x(t),$9 such that

$\mathrm{NUCO}\Rightarrow\mathrm{NUED}$00

NUES backward means that there exist $\mathrm{NUCO}\Rightarrow\mathrm{NUED}$01, $\mathrm{NUCO}\Rightarrow\mathrm{NUED}$02, $\mathrm{NUCO}\Rightarrow\mathrm{NUED}$03 such that

$\mathrm{NUCO}\Rightarrow\mathrm{NUED}$04

(Huerta et al., 13 Aug 2025).

The system $\mathrm{NUCO}\Rightarrow\mathrm{NUED}$05 is NUED if there exists an output injection $\mathrm{NUCO}\Rightarrow\mathrm{NUED}$06 such that the observer

$\mathrm{NUCO}\Rightarrow\mathrm{NUED}$07

induces the error dynamics

$\mathrm{NUCO}\Rightarrow\mathrm{NUED}$08

and the error system is NUES forward (Huerta et al., 15 Dec 2025).

The first direction is now standard in this framework: NUCO implies NUED. The proof in the 2025 observability paper proceeds by duality. NUCO of $\mathrm{NUCO}\Rightarrow\mathrm{NUED}$09 implies NUCC of the dual system; a Riccati-based stabilizability theorem provides a feedback $\mathrm{NUCO}\Rightarrow\mathrm{NUED}$10 that makes the dual closed loop NUES forward; the equivalence of stability under original, adjoint, and dual systems then yields forward NUES of the observer error system with $\mathrm{NUCO}\Rightarrow\mathrm{NUED}$11 (Huerta et al., 13 Aug 2025).

The converse is false in general. A counterexample uses the planar system

$\mathrm{NUCO}\Rightarrow\mathrm{NUED}$12

for which the nonuniform Kalman condition fails, so NUCO fails, but the observer injection

$\mathrm{NUCO}\Rightarrow\mathrm{NUED}$13

gives

$\mathrm{NUCO}\Rightarrow\mathrm{NUED}$14

which yields NUED (Huerta et al., 13 Aug 2025).

The 2025 reciprocal theorem identifies hypotheses under which the converse does hold. Suppose the plant admits NUBG

$\mathrm{NUCO}\Rightarrow\mathrm{NUED}$15

and also verifies NUES backward with constants $\mathrm{NUCO}\Rightarrow\mathrm{NUED}$16 satisfying $\mathrm{NUCO}\Rightarrow\mathrm{NUED}$17. Assume NUED holds, and assume

$\mathrm{NUCO}\Rightarrow\mathrm{NUED}$18

$\mathrm{NUCO}\Rightarrow\mathrm{NUED}$19

together with

$\mathrm{NUCO}\Rightarrow\mathrm{NUED}$20

Then the system is non-uniformly completely observable (Huerta et al., 15 Dec 2025).

The proof is by contradiction. One assumes the error system is NUES forward but the original system is not NUCO, then uses output-feedback preservation, backward NUES of the plant, variation-of-constants, and the bounds on $\mathrm{NUCO}\Rightarrow\mathrm{NUED}$21 and $\mathrm{NUCO}\Rightarrow\mathrm{NUED}$22 to force NUCO-type lower bounds on the observability Gramian of the error system. The contradiction is summarized by an inequality of the form

$\mathrm{NUCO}\Rightarrow\mathrm{NUED}$23

(Huerta et al., 15 Dec 2025).

6. Scope, design implications, and related usages

The immediate design consequence is that observer synthesis in nonuniform time-varying settings can proceed in two directions. One may certify detectability from NUCO, or, under the additional hypotheses of backward NUES of the plant, NUBG, growth control of $\mathrm{NUCO}\Rightarrow\mathrm{NUED}$24, and sufficiently fast decay of $\mathrm{NUCO}\Rightarrow\mathrm{NUED}$25, certify NUCO from detectability. The 2025 paper states these conditions explicitly: verify backward NUES of the plant with parameters $\mathrm{NUCO}\Rightarrow\mathrm{NUED}$26 satisfying $\mathrm{NUCO}\Rightarrow\mathrm{NUED}$27, ensure NUBG, and choose an output injection satisfying

$\mathrm{NUCO}\Rightarrow\mathrm{NUED}$28

(Huerta et al., 15 Dec 2025).

By duality, analogous statements hold for controllability and stabilizability. The same line of work proves that state-feedback preserves NUCC under analogous bounds on $\mathrm{NUCO}\Rightarrow\mathrm{NUED}$29 and $\mathrm{NUCO}\Rightarrow\mathrm{NUED}$30, and that non-uniform exponential stabilizability implies NUCC under dual conditions (Huerta et al., 15 Dec 2025).

The scope of the recent NUCO theory is explicit: the results are developed for continuous-time linear time-varying systems. Discrete-time and nonlinear generalizations are not addressed there, although the papers note that the techniques based on Gramians, growth bounds, and duality suggest possible extensions (Huerta et al., 15 Dec 2025).

A terminological distinction is therefore needed. In older work on nonuniformly sampled discrete systems derived from continuous-time LTI SISO models, complete $\mathrm{NUCO}\Rightarrow\mathrm{NUED}$31-observability under nonuniform sampling is characterized by the determinant condition

$\mathrm{NUCO}\Rightarrow\mathrm{NUED}$32

with $\mathrm{NUCO}\Rightarrow\mathrm{NUED}$33 and $\mathrm{NUCO}\Rightarrow\mathrm{NUED}$34 the characteristic modes. In that setting, the same determinant gives a joint criterion for $\mathrm{NUCO}\Rightarrow\mathrm{NUED}$35-reachability and $\mathrm{NUCO}\Rightarrow\mathrm{NUED}$36-observability (Fúster-Sabater, 2010).

A second distinct usage appears in Banach-space non-autonomous observation systems, where “nonuniform” refers to the dependence of observability constants on the measurable time set $\mathrm{NUCO}\Rightarrow\mathrm{NUED}$37, on the geometry of the observation sets $\mathrm{NUCO}\Rightarrow\mathrm{NUED}$38, and on operator-family parameters. There the main object is a final-state observability estimate in $\mathrm{NUCO}\Rightarrow\mathrm{NUED}$39, rather than the finite-dimensional NUCO Gramian bounds used in the recent linear time-varying control literature (Bombach et al., 2022).

Within the finite-dimensional LTV theory, however, the current meaning of nonuniform complete observability is precise: it is a Gramian-based property adapted to systems with nonuniform growth, strictly weaker than UCO and stronger than CO, dual to NUCC, preserved under suitable feedback transformations, sufficient for NUED, and, under additional hypotheses, equivalent to NUED in the reciprocal sense established in 2025 (Huerta et al., 13 Aug 2025, Huerta et al., 15 Dec 2025).