Interval Observer Design Using Observability Decomposition for Detectable Linear Systems

Gia Quoc Bao Tran; Thach Ngoc Dinh; Zhenhua Wang

arxiv: 2604.24939 · v1 · submitted 2026-04-27 · 📡 eess.SY · cs.SY· math.DS· math.OC

Interval Observer Design Using Observability Decomposition for Detectable Linear Systems

Gia Quoc Bao Tran , Thach Ngoc Dinh , Zhenhua Wang This is my paper

Pith reviewed 2026-05-08 01:38 UTC · model grok-4.3

classification 📡 eess.SY cs.SYmath.DSmath.OC

keywords interval observerlinear time-invariant systemsobservability decompositiondetectable systemsstate estimationSylvester equationJordan form

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The pith

An observability-based decomposition enables interval observers for detectable linear systems.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a systematic way to design interval observers for detectable linear time-invariant systems in which only part of the state is observable from measurements. It relies on an invertible linear transformation based on observability to split the state into an observable decoupled subsystem and a detectable subsystem that does not show up in the output. Interval observers are designed separately: a linear time-invariant one using Sylvester equations for the observable part and a time-varying one using Jordan canonical form for the detectable part, with the coupling treated as bounded known inputs. The state intervals are recovered in the original coordinates by inverting the transformation or implementing the observer directly. Readers interested in state estimation would care because this extends interval methods beyond fully observable systems to the larger class of detectable ones.

Core claim

We provide a systematic interval observer design method for detectable linear time-invariant (LTI) systems, where a part of the state is observable from the measured output. An observability-based invertible LTI transformation decomposes the state into two parts. The first part is decoupled from the other and observable from the output, while the second is affected by the first, does not appear in the output, but is detectable. A Sylvester-based LTI interval observer is designed for the first part. For the second part, a Jordan-based linear time-varying interval observer is built, treating the interaction from the first part as inputs with known bounds. The intervals in the original for the

What carries the argument

An observability-based invertible LTI transformation that decomposes the state into an observable decoupled part and a detectable interacting part.

If this is right

Interval bounds for the complete state vector are obtained by inverting the decomposition or by direct implementation in original coordinates.
The approach applies to detectable systems that may not be fully observable.
A constant-gain Sylvester interval observer suffices for the observable subsystem.
The detectable subsystem uses a time-varying observer that incorporates the bounded coupling from the first subsystem as known inputs.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This decomposition technique might simplify the design of estimators in systems with structured observability properties, such as in sensor networks.
The method could be extended to time-varying or switched linear systems by adapting the transformation accordingly.
It opens the possibility of using interval observers in applications like model-based fault detection where detectability is the key assumption rather than observability.

Load-bearing premise

An observability-based invertible linear transformation exists that splits the state into an observable decoupled subsystem and a detectable subsystem not appearing in the output.

What would settle it

A concrete detectable linear system for which the designed intervals do not contain the true state trajectory under the proposed observers.

Figures

Figures reproduced from arXiv: 2604.24939 by Gia Quoc Bao Tran, Thach Ngoc Dinh, Zhenhua Wang.

**Figure 1.** Figure 1: Structure of the decomposition-based interval observer view at source ↗

**Figure 2.** Figure 2: Interval estimation for zo,t. Top: Convergence in the absence of (dt, wt); bottom: Intervals in the presence of (dt, wt). Now, we can guarantee interval in the zo-coordinates. However, this is insufficient for recovering the bounds in the x-coordinates, because of zno,t. We must then build another interval observer for zno,t in cascade with (10). This is done in the next section. 3.2 Jordan-Based Design fo… view at source ↗

**Figure 3.** Figure 3: Interval estimation for zno,t. Left: Convergence in the absence of (dt, wt); right: Intervals in the presence of (dt, wt). 3.3 Decomposition-Based Designs for xt Now that we have built interval observers giving the bounds of zo,t and zno,t, we use these to reconstruct the bounds of xt. Naturally, the bounds in the x-coordinates are obtained after time 0 as xt = M⊕ o zo,t − M⊖ o zo,t + M⊕ nozno,t − M⊖ nozno… view at source ↗

**Figure 4.** Figure 4: Interval estimation for xt using observer (10)-(12)- (13). Top: Convergence in the absence of (dt, wt); bottom: Intervals in the presence of (dt, wt). 4. CONCLUSION We propose a systematic interval observer for detectable LTI systems, where a decomposition separates the state into two parts—one observable from the output and one detectable—for which suitable interval observers are designed and concatenate… view at source ↗

read the original abstract

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a workable decomposition method to design interval observers for detectable LTI systems that are not fully observable, using standard tools without major internal problems.

read the letter

The core contribution is a systematic procedure that takes a detectable pair (A, C), applies the usual observability decomposition to split the state into an observable subsystem decoupled from the rest and a stable unobservable subsystem that receives coupling from the first, then builds a Sylvester-equation LTI interval observer on the observable part and a Jordan-form LTV interval observer on the detectable part while treating the coupling as bounded inputs. Intervals are recovered in original coordinates either by inverting the transformation or by writing the observer directly there. Academic examples show the construction in action. This combination is new enough in the interval-observer literature to be worth recording, and the paper executes it cleanly with an invertible LTI change of coordinates and explicit handling of the two subsystems. The logic follows directly from Kalman decomposition plus existing Sylvester and Jordan-based interval techniques, so there are no circular steps or missing existence conditions. The soft spots are modest and expected. Validation stays at simple academic examples, so we lack evidence on numerical conditioning for larger systems or performance under real measurement noise. Jordan canonical form and Sylvester solves can be sensitive, but the paper does not explore mitigation or alternatives. No load-bearing assumptions appear unjustified once the standard decomposition is granted. This is for researchers already working on interval observers or robust state estimation for linear systems. A reader who needs a structured design recipe for the detectable-but-not-observable case will find a clear recipe and can adapt it. I would send it to peer review; the argument stands on its own and the gap it fills is narrow but real.

Referee Report

0 major / 3 minor

Summary. The paper proposes a systematic interval observer design for detectable LTI systems. An observability-based invertible LTI transformation decomposes the state into an observable subsystem (decoupled from the rest and directly observable from the output) and a detectable unobservable subsystem (affected by the first but not appearing in the output). A Sylvester-equation-based LTI interval observer is constructed for the observable part; a Jordan-form LTV interval observer is built for the detectable part, treating the coupling terms as bounded inputs. Intervals in original coordinates are recovered either by online inversion of the transformation or by direct implementation in original coordinates. Academic examples are used to illustrate the approach.

Significance. If the constructions hold, the result meaningfully extends interval-observer design from the observable case to the broader detectable case, which is common in applications. The method reuses standard Kalman decomposition, Sylvester solutions, and Jordan-based interval techniques without introducing new existence conditions, providing a practical, modular design procedure. No machine-checked proofs or open-source code are mentioned, but the reliance on well-established linear-algebra tools reduces the risk of hidden assumptions.

minor comments (3)

[Section 3] §3 (or equivalent): the statement that the decomposition 'always exists' for detectable pairs should explicitly cite the Kalman observability canonical form and note the required rank conditions on the observability matrix.
[Section 4] The error-bound propagation through the Jordan-based LTV observer (Eq. for the time-varying gain or coupling) is only sketched; a short lemma showing that the interval width remains bounded when the unobservable subsystem is stable would strengthen the claim.
[Section 5] Figure captions and example descriptions do not report quantitative metrics (e.g., interval width convergence rate or comparison with a full-order observer); adding these would make the illustrations more informative.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive evaluation of our manuscript, including the recognition that the proposed method extends interval-observer design to the detectable case using standard tools. The recommendation for minor revision is noted; however, the report lists no specific major or minor comments requiring changes. We are therefore prepared to address any editorial or typographical issues that may arise during production.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained using standard linear systems results

full rationale

The paper's central construction begins from the standard Kalman observability decomposition of any detectable pair (A, C), which exists independently as a linear-algebra fact and is not derived or fitted within the paper. The subsequent steps apply well-known tools (Sylvester equations for the observable subsystem observer and Jordan-form LTV dynamics for the stable unobservable subsystem) to the already-decomposed coordinates; these techniques are drawn from the general interval-observer literature and do not reduce to any self-citation chain or redefinition of the target intervals. No equation or existence claim is shown to be equivalent to its own inputs by construction, and the final reconstruction of intervals in original coordinates is a direct, invertible linear transformation that adds no new circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The contribution is a design procedure relying on established concepts in linear systems theory without new parameters or entities.

axioms (2)

domain assumption The considered systems are detectable LTI systems with partial observability from the output.
This is the problem setting stated in the abstract.
domain assumption An observability-based invertible LTI transformation exists to decompose the state as described.
This is invoked as the basis for the design method.

pith-pipeline@v0.9.0 · 5455 in / 1242 out tokens · 57285 ms · 2026-05-08T01:38:19.250626+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

3 extracted references · 3 canonical work pages

[1]

Alcaraz-Gonzalez, V., Harmand, J., Rapaport, A., Steyer, J., Gonzalez-Alvarez, V., and Pelayo-Ortiz, C. (2002). Software sensors for highly uncertain WWTPs: a new approach based on interval observers. Water Research, 36, 2515–2524. Antsaklis, P.J. and Michel, A.N. (1997). Linear systems , volume

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[2]

Bartels, R.H

Springer. Bartels, R.H. and Stewart, G.W. (1972). Algorithm 432 [C2]: Solution of the matrix equation AX + XB = C [F4]. Communications of the ACM , 15(9), 820–826. Brivadis, L., Andrieu, V., Bernard, P., and Serres, U. (2023). Further remarks on KKL observers. Systems and Control Letters , 172, 105429. Cacace, F., Germani, A., and Manes, C. (2015). A new ...

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and Traynor, T

Caron, R. and Traynor, T. (2005). The zero set of a polynomial. WMSR Report number 05-03 . Dinh, T.N. and Ito, H. (2017). Decentralization of interval observers for robust controlling and monitoring a class of nonlinear systems. SICE Journal of Control, Mea- surement, and System Integration , 10, 117–123. Dinh, T.N., Rauh, A., Yong, S.Z., and Wang, Z. (20...

work page 2005

[1] [1]

Alcaraz-Gonzalez, V., Harmand, J., Rapaport, A., Steyer, J., Gonzalez-Alvarez, V., and Pelayo-Ortiz, C. (2002). Software sensors for highly uncertain WWTPs: a new approach based on interval observers. Water Research, 36, 2515–2524. Antsaklis, P.J. and Michel, A.N. (1997). Linear systems , volume

work page 2002

[2] [2]

Bartels, R.H

Springer. Bartels, R.H. and Stewart, G.W. (1972). Algorithm 432 [C2]: Solution of the matrix equation AX + XB = C [F4]. Communications of the ACM , 15(9), 820–826. Brivadis, L., Andrieu, V., Bernard, P., and Serres, U. (2023). Further remarks on KKL observers. Systems and Control Letters , 172, 105429. Cacace, F., Germani, A., and Manes, C. (2015). A new ...

work page 1972

[3] [3]

and Traynor, T

Caron, R. and Traynor, T. (2005). The zero set of a polynomial. WMSR Report number 05-03 . Dinh, T.N. and Ito, H. (2017). Decentralization of interval observers for robust controlling and monitoring a class of nonlinear systems. SICE Journal of Control, Mea- surement, and System Integration , 10, 117–123. Dinh, T.N., Rauh, A., Yong, S.Z., and Wang, Z. (20...

work page 2005