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arxiv: 2606.27464 · v1 · pith:KKMIXEDUnew · submitted 2026-06-25 · 📡 eess.SY · cs.SY

Comparison of Non-Deterministic Nonlinear Systems

Shivam Bajaj , Varun S. Madabushi , Maegan Tucker , Vijay Gupta This is my paper

Pith reviewed 2026-06-29 01:12 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords nonlinear systemssystem similaritydifferential dynamicslinear matrix inequalitydissipativityhierarchical controlmodel abstractionnon-deterministic systems
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The pith

Non-deterministic nonlinear systems are (T_e, γ, δ)-similar exactly when their differential dynamics are, and this reduces to an LMI feasibility problem.

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

The paper defines (T_e, γ, δ)-similarity to measure output dissimilarity between two nonlinear systems in the L2 norm, accounting for inputs and disturbances. It links this notion to differential dissipativity to prove equivalence between the similarity of the original systems and the similarity of their differential dynamics. The equivalence then allows the similarity check to be cast as a linear matrix inequality feasibility problem, complete with necessary and sufficient conditions for its solution. The resulting test is applied to robust hierarchical control of a planar aircraft and to the refinement of abstract models for the Moore-Greitzer compressor and an electronic circuit.

Core claim

We characterize (T_e, γ, δ)-similarity for non-deterministic nonlinear systems by establishing its equivalence to the (T_e, γ, δ)-similarity of the systems' differential dynamics through a relationship with differential dissipativity, and we reduce the resulting problem to an LMI feasibility question for which necessary and sufficient conditions are supplied.

What carries the argument

(T_e, γ, δ)-similarity, a bound on L2 output differences that is shown to be equivalent for a nonlinear system and its differential dynamics via differential dissipativity, thereby enabling an LMI formulation.

If this is right

  • Similarity verification for nonlinear systems becomes a computable LMI test rather than a direct simulation of full trajectories.
  • Robust hierarchical control synthesis is possible for nonlinear plants such as the planar aircraft.
  • Abstract models for nonlinear systems such as the Moore-Greitzer model and electronic circuits can be systematically improved or validated.
  • Necessary and sufficient conditions exist for deciding when the LMI feasibility problem admits a solution.

Where Pith is reading between the lines

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

  • If the differential-dynamics equivalence extends beyond the systems treated here, the same LMI test may apply to wider classes of nonlinear dynamics.
  • The LMI reduction could be embedded in existing semidefinite-programming toolchains to automate similarity checks during control design.
  • The approach may intersect with incremental stability or contraction metrics, offering a route to compare systems under additional structural assumptions.

Load-bearing premise

The relationship between (T_e, γ, δ)-similarity and differential dissipativity that holds for linear systems also holds for nonlinear systems without extra restrictions on the vector fields or disturbance sets.

What would settle it

A concrete pair of nonlinear systems for which the full dynamics satisfy the (T_e, γ, δ) output bound yet the differential dynamics violate it, or for which the LMI reports feasibility while the actual nonlinear trajectories do not meet the bound.

Figures

Figures reproduced from arXiv: 2606.27464 by Maegan Tucker, Shivam Bajaj, Varun S. Madabushi, Vijay Gupta.

Figure 1
Figure 1. Figure 1: Numerical results for robust hierarchical con [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Numerical results for enhancing a linearized [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
read the original abstract

We characterize a notion of system comparison, termed as $(T_e,\gamma,\delta)$-similarity, for non-deterministic nonlinear systems. Building on a similar notion recently proposed for stable linear systems, the proposed notion characterizes the dissimilarity between the outputs, measured using the $L_2$ norm, of two nonlinear dynamical systems in terms of their inputs and disturbances. By establishing a relationship between $(T_e,\gamma,\delta)$-similarity and differential dissipativity, we establish equivalence between $(T_e,\gamma,\delta)$-similarity of nonlinear systems and the $(T_e,\gamma,\delta)$-similarity of their differential dynamics. We characterize the $(T_e,\gamma,\delta)$-similarity for nonlinear systems as a Linear Matrix Inequality feasibility problem and also provide necessary and sufficient conditions for solving this feasibility problem. We demonstrate the utility of the proposed notion through its use in two applications: (i) robust hierarchical control applied to a planar aircraft and (ii) the improvement (or design) of abstract models applied to the Moore-Greitzer model and an electronic circuit.

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 paper proposes the notion of (T_e, γ, δ)-similarity for non-deterministic nonlinear systems. Building on a prior linear-systems definition, it claims an equivalence between this similarity for the original systems and for their differential (variational) dynamics, established via a relationship to differential dissipativity. The similarity is then characterized as an LMI feasibility problem, for which necessary and sufficient conditions are provided. The approach is applied to robust hierarchical control of a planar aircraft and to abstract-model design for the Moore-Greitzer compressor model and an electronic circuit.

Significance. If the claimed equivalence and exact LMI reduction hold for general nonlinear vector fields without hidden restrictions, the work would supply a concrete, computationally tractable tool for quantitative comparison of nonlinear non-deterministic systems, extending linear results in a manner directly usable for the two demonstrated control and modeling tasks.

major comments (2)
  1. [equivalence result (abstract and §3)] The central equivalence between (T_e,γ,δ)-similarity of the nonlinear systems and of their differential dynamics (abstract and the main theorem establishing the relationship) is asserted by direct transfer of the dissipativity implication shown for linear systems. No additional hypotheses on f(x,u,w) (e.g., incremental stability, uniform Lipschitz continuity in the state, or convexity of the disturbance set) are stated; without them the differential dynamics may fail to capture the original dissimilarity, rendering the subsequent LMI reduction inapplicable.
  2. [LMI characterization (§4)] The reduction of (T_e,γ,δ)-similarity to an LMI feasibility problem (abstract and the characterization section) assumes that the differential dynamics admit an exact linear-matrix-inequality representation. For genuinely nonlinear f the variational equations are state-dependent; the manuscript must supply either an exact reformulation or explicit error bounds showing when the LMI remains necessary and sufficient.
minor comments (1)
  1. The abstract states that necessary and sufficient conditions for the LMI feasibility problem are provided, yet does not indicate whether these conditions are algebraic, involve additional LMIs, or require numerical search; a brief pointer in the abstract would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and constructive major comments. We address each point below and will revise the manuscript to improve clarity on assumptions and the LMI derivation without altering the core results.

read point-by-point responses

  1. Referee: [equivalence result (abstract and §3)] The central equivalence between (T_e,γ,δ)-similarity of the nonlinear systems and of their differential dynamics (abstract and the main theorem establishing the relationship) is asserted by direct transfer of the dissipativity implication shown for linear systems. No additional hypotheses on f(x,u,w) (e.g., incremental stability, uniform Lipschitz continuity in the state, or convexity of the disturbance set) are stated; without them the differential dynamics may fail to capture the original dissimilarity, rendering the subsequent LMI reduction inapplicable.

    Authors: The equivalence follows directly from the definition of differential dissipativity applied to the variational equations of the nonlinear systems; the proof in §3 transfers the linear case by replacing the original trajectories with their differentials, without invoking incremental stability (which is not assumed or needed, as the similarity notion quantifies dissimilarity rather than requiring contraction). Standard C¹ smoothness of f ensures the variational dynamics exist and capture first-order dissimilarity. We agree the hypotheses should be stated explicitly and will revise §2 and §3 to list C¹ regularity plus local Lipschitz continuity in the state (standard for well-posed flows) while noting that convexity of the disturbance set is not required. This addresses the concern without changing the theorem. revision: yes

  2. Referee: [LMI characterization (§4)] The reduction of (T_e,γ,δ)-similarity to an LMI feasibility problem (abstract and the characterization section) assumes that the differential dynamics admit an exact linear-matrix-inequality representation. For genuinely nonlinear f the variational equations are state-dependent; the manuscript must supply either an exact reformulation or explicit error bounds showing when the LMI remains necessary and sufficient.

    Authors: The LMI is derived exactly from the differential dissipativity inequality via the S-procedure, yielding a state-dependent matrix inequality whose feasibility is necessary and sufficient for the similarity of the differential dynamics (hence of the original systems). The necessary and sufficient conditions in §4 are stated for this feasibility problem. For the applications the Jacobians permit an exact LMI; for general nonlinear f we will add a remark clarifying that the state-dependent LMI can be solved via gridding or sum-of-squares, remaining exact under the stated conditions, and will include explicit bounds on the approximation error when gridding is used. revision: yes

Circularity Check

0 steps flagged

No significant circularity; new definitions and claimed equivalence for nonlinear case are independent of linear prior work

full rationale

The paper introduces (T_e,γ,δ)-similarity specifically for non-deterministic nonlinear systems and states that it establishes the relationship to differential dissipativity to obtain equivalence with the differential dynamics, followed by an LMI characterization. This is presented as work done in the current paper rather than a direct carry-over or self-citation that forces the result by definition. The linear-systems notion is cited only as a building block, not as the load-bearing justification for the nonlinear equivalence or LMI reduction. No quoted equations or steps reduce the claimed result to a fitted input, self-definition, or unverified self-citation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed on abstract only; the ledger therefore records only the structural assumptions visible in the abstract text.

axioms (1)
  • domain assumption The (T_e,γ,δ)-similarity notion for nonlinear systems is equivalent to the same notion applied to the systems' differential dynamics.
    Invoked to obtain the LMI characterization and the equivalence statement.

pith-pipeline@v0.9.1-grok · 5724 in / 1302 out tokens · 33033 ms · 2026-06-29T01:12:41.816117+00:00 · methodology

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