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arxiv: 2604.23142 · v1 · submitted 2026-04-25 · 📡 eess.SY · cs.SY

An Algebraic State Observer for a Class of Physical Systems

Alexey Bobtsov , Jose Guadalupe Romero , Romeo Ortega , Anton Pyrkin This is my paper

Pith reviewed 2026-05-08 07:36 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords algebraic observersstate estimationnonlinear systemsphysical systemsSwapping Lemmaobservers without convergencemeasurable derivatives
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The pith

An algebraic relation equates the unmeasurable state to filtered inputs and outputs for all t greater than or equal to zero in certain nonlinear physical systems.

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

The paper introduces a design procedure for state observers that derives an exact algebraic identity connecting the unknown state components directly to filtered versions of the measured inputs and outputs. This identity is required to hold from the initial time onward rather than only asymptotically. The method begins from the assumption that selected state signals have measurable derivatives, which is satisfied in the physical examples considered, and proceeds by applying the Swapping Lemma to isolate the relevant derivative before substituting the available measurements and rearranging terms. The resulting observer avoids any requirement for observability of the system or persistent excitation of the signals.

Core claim

We obtain an algebraic relation between the unmeasurable part of the state and filtered versions of the system inputs and outputs that holds true for all t greater than or equal to zero. The construction starts from the availability of state components whose derivatives are measurable, applies the Swapping Lemma to remove the derivative of one such signal from the dynamics, replaces that derivative by the corresponding measurable quantities, and arranges the remaining terms into the desired identity. No observability or excitation condition is imposed.

What carries the argument

Application of the Swapping Lemma to filtered dynamics to isolate and replace the derivative of a measurable state component, thereby producing a static algebraic equation for the unknown state.

If this is right

  • The state estimate is available without any transient period, allowing immediate use in feedback laws.
  • The same construction applies to electrical motors and mechanical systems that have previously required asymptotic observers.
  • Observer design no longer depends on verifying system observability or signal richness.

Where Pith is reading between the lines

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

  • The algebraic identity could be substituted directly into control laws to eliminate the need for separate observer dynamics.
  • Coordinate transformations that create measurable derivatives might be found systematically for additional classes of physical systems.
  • Because no excitation condition appears, the relation may remain valid even when inputs are constant or zero after an initial interval.

Load-bearing premise

Some state components must have derivatives that are directly measurable or can be made measurable by a coordinate transformation.

What would settle it

A concrete nonlinear physical system satisfying the measurable-derivative assumption in which the constructed algebraic expression differs from the true unmeasurable state value at some positive time.

Figures

Figures reproduced from arXiv: 2604.23142 by Alexey Bobtsov, Anton Pyrkin, Jose Guadalupe Romero, Romeo Ortega.

Figure 1
Figure 1. Figure 1: Transient behavior of ˜x2 0 2 4 6 8 10 -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 7 8 9 10 -5 0 5 10-4 view at source ↗
Figure 2
Figure 2. Figure 2: Transient behavior of ˜x2 for ASLO and A-ASLO with white noise added to u 9 view at source ↗
Figure 3
Figure 3. Figure 3: Transient behavior of ˜x2 for ASLO and A-ASLO with input unmod￾elled dynamics with different time constant τ 3.1 Surface-mount permanent magnet synchronous motor (PMSM) In this subsection we design an ASLO for the flux of the PMSM measuring only voltages and currents. The solution we will present here is radically different from all existing solutions reported in the literature. Assessing its performance o… view at source ↗
Figure 4
Figure 4. Figure 4: Flux observation errors for the case of time-varying des view at source ↗
read the original abstract

In this paper we present a radically new approach to design state observers for nonlinear systems, with particular emphasis on physical ones. Our objective is to obtain an algebraic relation between the unmeasurable part of the state and filtered versions of the systems inputs and outputs, which holds true for all $t \geq 0$. The latter qualifier should be contrasted with the usual asymptotic (or fixed/finite time) objective. The standing assumption for our design is the availability -- or possibility of constructing, via coordinate change -- state components with measurable derivatives. In the physical systems studied in the paper this condition is naturally satisfied. The next step in the design is the application of the Swapping Lemma to pull out from the dynamics the derivative of one of these signals. The design is completed replacing the latter by the measurable signals and arranging the remaining terms. The algebraic observer constitutes a refreshing major departure from classical asymptotic observer designs, even in the case of electrical motors and mechanical systems that have been exhaustively studied. Particularly notable is the fact that no observability or excitation condition is imposed for the construction of the algebraic observer.

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.

Referee Report

2 major / 1 minor

Summary. The manuscript proposes a new algebraic state observer design for nonlinear physical systems. It derives an exact algebraic relation (valid for all t ≥ 0) between unmeasurable state components and filtered inputs/outputs by applying the Swapping Lemma to dynamics involving state components whose derivatives are assumed measurable (or obtainable via coordinate change). The design requires no observability or excitation conditions and is illustrated on physical systems such as electrical motors and mechanical systems where the standing assumption holds naturally.

Significance. If the central construction is made rigorous with explicit coordinate-change procedures and verification that the algebraic relation holds without hidden conditions, the result would constitute a meaningful departure from classical asymptotic observers. The approach leverages the Swapping Lemma to obtain an exact, non-asymptotic relation and could be useful for real-time estimation in systems where the measurability assumption is satisfied by the physics. Credit is due for emphasizing the contrast with finite-time or asymptotic convergence and for targeting a broad class of physical systems.

major comments (2)
  1. [Abstract] Abstract (and presumably §2 or §3 where the design is formalized): The claim that 'no observability or excitation condition is imposed' rests entirely on the standing assumption that 'state components with measurable derivatives' are available or can be constructed via coordinate change. No explicit construction, algorithm, or proof is supplied showing that the coordinate transformation can be obtained without already solving an equivalent observation problem or presupposing knowledge of the unmeasurable states. This makes the assumption load-bearing and risks rendering the algebraic relation true by fiat rather than by the Swapping-Lemma step.
  2. [Abstract] The abstract states that the algebraic relation 'holds true for all t ≥ 0' after replacing the derivative by measurable signals. Without the full derivation (including the precise statement of the Swapping Lemma application and any domain restrictions on the signals), it is impossible to verify whether hidden regularity or boundedness conditions are implicitly required for the equality to hold identically from t=0 onward.
minor comments (1)
  1. [Abstract] The phrasing 'the systems inputs and outputs' should be corrected to 'the system's inputs and outputs' for grammatical consistency.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments and recommendation for major revision. The concerns regarding the standing assumption and the precise conditions for the algebraic relation are well-taken, and we will strengthen the manuscript by adding explicit details on coordinate changes for the examples and clarifying the derivation and regularity assumptions. Our point-by-point responses to the major comments are provided below.

read point-by-point responses

  1. Referee: [Abstract] Abstract (and presumably §2 or §3 where the design is formalized): The claim that 'no observability or excitation condition is imposed' rests entirely on the standing assumption that 'state components with measurable derivatives' are available or can be constructed via coordinate change. No explicit construction, algorithm, or proof is supplied showing that the coordinate transformation can be obtained without already solving an equivalent observation problem or presupposing knowledge of the unmeasurable states. This makes the assumption load-bearing and risks rendering the algebraic relation true by fiat rather than by the Swapping-Lemma step.

    Authors: We agree that the standing assumption is central and load-bearing for the approach, as it permits the direct application of the Swapping Lemma without observability conditions. For the specific class of physical systems addressed (electrical motors and mechanical systems), the coordinate changes are system-specific redefinitions that exploit the physics: measurable quantities such as positions or currents are used to define new coordinates whose derivatives are accessible via direct measurement or the known system equations, without requiring knowledge of the unmeasurable states (e.g., fluxes or velocities in certain frames). These changes do not presuppose an observer or solve an equivalent observation problem but follow from the model structure. We do not provide a general algorithm for arbitrary nonlinear systems outside this class. In the revision we will add a new subsection with explicit coordinate-change procedures and verifications for each example. revision: yes

  2. Referee: [Abstract] The abstract states that the algebraic relation 'holds true for all t ≥ 0' after replacing the derivative by measurable signals. Without the full derivation (including the precise statement of the Swapping Lemma application and any domain restrictions on the signals), it is impossible to verify whether hidden regularity or boundedness conditions are implicitly required for the equality to hold identically from t=0 onward.

    Authors: The full derivation, including the exact statement and application of the Swapping Lemma, is given in Section 3: the dynamics are expressed in the new coordinates, the lemma interchanges the derivative operator on the measurable state component, and substitution yields the algebraic relation. The equality holds for all t ≥ 0 under the standard assumption that the involved signals are sufficiently differentiable (as required for the existence and uniqueness of solutions to the underlying differential equations in physical models). No additional boundedness or domain restrictions beyond those implicit in the system class are needed, since the final relation is purely algebraic. We will expand the abstract slightly for clarity and insert a remark in Section 3 explicitly stating the regularity assumptions on the signals. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation is self-contained under explicit standing assumption

full rationale

The paper states its standing assumption upfront (measurable derivatives or constructible via coordinate change) as a precondition for the algebraic relation, then applies the Swapping Lemma to rearrange dynamics and substitutes measurable signals. This produces the claimed relation for all t ≥ 0 without reducing the result to a fitted parameter, self-definition, or unverified self-citation chain. No observability condition is claimed to be derived; the design is restricted to the subclass satisfying the assumption, which is presented as naturally met for the physical systems considered. The central claim therefore remains independent of its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests primarily on the domain assumption of measurable state derivatives and the use of the Swapping Lemma as a standard tool; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Availability or constructibility via coordinate change of state components with measurable derivatives
    Explicitly stated as the standing assumption required for the design, noted as natural for the physical systems considered.

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Reference graph

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