arxiv: 2605.06097 · v1 · submitted 2026-05-07 · 📡 eess.SY · cs.SY· math.OC

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Absolute Stability of Nonlinear Negative Imaginary Systems with Application to Potential Energy Shaping

Kanghong Shi , Ian R. Manchester

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Pith reviewed 2026-05-08 06:41 UTC · model grok-4.3

classification 📡 eess.SY cs.SYmath.OC

keywords negative imaginary systemsabsolute stabilitynonlinear feedbackstorage functionspotential energy shapinginterconnected systemsstability conditions

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The pith

Nonlinear negative imaginary systems preserve their property and achieve absolute stability under gradient nonlinear feedback if the composite storage function remains positive definite.

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

The paper shows that interconnecting a nonlinear negative imaginary system with static nonlinear feedback preserves the NI property precisely when the feedback is the gradient of a continuously differentiable function and the composite storage function stays positive definite. This preservation directly links the feedback to storage-function shaping along the measured outputs. Absolute stability of the closed loop then follows under mild assumptions. The linear version of the result extends earlier NI stability theorems by permitting coupled nonlinearities outside slope-restricted or sector-bounded classes. The approach is illustrated on both linear and nonlinear examples, including potential energy shaping.

Core claim

The paper claims that the negative imaginary property is preserved when the feedback nonlinearity can be expressed as the gradient of a continuously differentiable function and the composite storage function of the resulting system remains positive definite. This condition supplies a direct connection between nonlinear static feedback and storage-function shaping along the measured output channels. Absolute stability conditions for the closed-loop system are then derived under mild assumptions. The linear specialization strictly generalizes prior absolute stability results for linear NI systems by allowing coupled nonlinearities not covered by existing slope-restricted or sector-bounded fram

What carries the argument

The negative imaginary property, preserved under gradient interconnection of the feedback nonlinearity with the system output, together with the positive-definiteness requirement on the composite storage function.

If this is right

The closed-loop system is absolutely stable under the stated gradient and positive-storage conditions.
Linear NI absolute stability extends to coupled nonlinearities beyond slope or sector restrictions.
The framework directly supports potential energy shaping along measured outputs.
Stability holds without requiring every nonlinearity to satisfy slope or sector bounds.

Where Pith is reading between the lines

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

The same gradient-preservation argument may compose with multiple NI subsystems while retaining stability.
Designers could choose the potential explicitly to place equilibria at desired locations.
The structure parallels passivity-based control and might combine with it for larger networks.

Load-bearing premise

The feedback nonlinearity must be the gradient of a continuously differentiable function and the composite storage function must remain positive definite.

What would settle it

A concrete nonlinear NI system in feedback with a non-gradient nonlinearity that remains stable, or a gradient nonlinearity whose composite storage fails to be positive definite yet the closed loop is unstable.

Figures

Figures reproduced from arXiv: 2605.06097 by Ian R. Manchester, Kanghong Shi.

**Figure 1.** Figure 1: Closed-loop interconnection of the system view at source ↗

**Figure 2.** Figure 2: Closed-loop interconnection of the system view at source ↗

**Figure 4.** Figure 4: Two pendulums with hinge springs k1, k2, hinge dampers d1, d2, coupling spring kc, coupling damper dc, gravity g, and input torques u1, u2. Such a pendulum system is OSNI as can be verified using Definition 2 with storage function V (θ1, θ2, ω1, ω2) = 1 2 kc(θ1 − θ2) 2 + X i 1 2 kiθ 2 i + 1 2 miℓ 2 i ω 2 i + migℓi(1 − cos θi) . (23) The equilibrium at the origin is asymptotically stable. However, becau… view at source ↗

**Figure 3.** Figure 3: State trajectories of the system (13) with (19) and (2 view at source ↗

**Figure 5.** Figure 5: Angular displacements of the two pendulums and their view at source ↗

**Figure 6.** Figure 6: Potential energy surfaces of the system before and aft view at source ↗

**Figure 7.** Figure 7: State trajectories of the plant under input view at source ↗

read the original abstract

This paper establishes absolute stability conditions for nonlinear negative imaginary (NI) systems interconnected with static nonlinear feedback. We first show that the NI property is preserved when the feedback nonlinearity can be expressed as the gradient of a continuously differentiable function, and the composite storage of the resulting system remains positive definite. This condition provides a direct connection between nonlinear static feedback and storage-function shaping along the measured output channels. Building on this result, conditions are derived for absolute stability of the closed-loop system under mild assumptions. The linear specialization of the results strictly generalizes prior absolute stability results for linear NI systems, allowing coupled nonlinearities not covered by existing slope-restricted or sector-bounded frameworks. Finally, the proposed theory is illustrated through a linear example highlighting this generalization and a nonlinear example that shows the utility of the proposed results in potential energy shaping.

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 clean extension of absolute stability results to nonlinear negative imaginary systems under gradient feedback, but the key positive-definiteness assumption on composite storage lacks general checks or counterexamples.

read the letter

The main thing to know is that the paper shows the negative imaginary property is preserved for the closed-loop system when the static nonlinearity is the gradient of a C1 function and the sum of the plant storage function plus the nonlinearity potential stays positive definite. From that they derive absolute stability conditions, and the linear specialization broadens earlier NI results to coupled nonlinearities outside the usual slope or sector bounds. The potential-energy-shaping example is a direct fit for mechanical systems work. They do this without circular arguments or loose citations; the logic starts from the NI definition and builds the preservation claim independently. The examples are straightforward and show the claimed generalization in action. The soft spot is exactly the one the stress test flagged: positive definiteness of the composite storage is assumed rather than equipped with a general test, Lyapunov construction, or counterexample showing when the sum can fail. That makes the theorems conditional, and users will have to verify the assumption separately for each plant and nonlinearity. No other load-bearing gaps appear in the abstract or stated claims. This is aimed at control theorists already working with negative imaginary systems or energy-shaping methods. A reader in that niche will pick up usable stability tools and see how the linear case widens. The thinking is clear and engages the literature directly, so the paper deserves a serious referee even if the storage condition will need more discussion in review. I would send it out for peer review.

Referee Report

2 major / 2 minor

Summary. The manuscript establishes absolute stability conditions for nonlinear negative-imaginary (NI) systems under static nonlinear feedback. It first proves that the NI property is preserved when the feedback nonlinearity is the gradient of a continuously differentiable function and the composite storage function (sum of the plant storage function and the potential associated with the nonlinearity) remains positive definite. Building on this preservation result, the paper derives absolute-stability conditions for the closed-loop system. The linear specialization is shown to strictly generalize prior absolute-stability results for linear NI systems by accommodating coupled nonlinearities outside existing slope-restricted or sector-bounded frameworks. The theory is illustrated by a linear example demonstrating the generalization and a nonlinear example illustrating its use in potential-energy shaping.

Significance. If the central claims hold, the work is significant for control theory because it directly links nonlinear static feedback to storage-function shaping along measured outputs in the NI setting, thereby extending absolute-stability analysis beyond classical sector or slope restrictions. The explicit connection to potential-energy shaping offers a constructive route for mechanical-system design. The linear specialization supplies a concrete generalization of existing NI absolute-stability theorems, and the two examples provide immediate evidence of applicability.

major comments (2)

[§3] §3 (preservation of the NI property): The central claim that the closed-loop system remains NI rests on the composite storage function V(x) + Φ(y) being positive definite. No general criterion, Lyapunov-function construction, or sufficient condition is supplied to verify this property for arbitrary gradient nonlinearities; the result is therefore conditional on an assumption whose validity must be checked case-by-case. Without such a check the subsequent absolute-stability theorems do not apply.
[§4] §4 (absolute-stability conditions): The derived LMI-type stability criteria inherit the same composite-storage hypothesis. Because no counter-example analysis or boundary case is examined where the sum fails to be positive definite, it is unclear how restrictive the hypothesis is and whether the claimed generalization over slope-restricted frameworks remains valid when the hypothesis is only marginally satisfied.

minor comments (2)

[§3] Notation for the composite storage function is introduced without an explicit equation number; adding a numbered display would improve traceability when the same expression is reused in the stability theorems.
[§5.1] The linear example in §5.1 cites a prior NI absolute-stability result but does not state the precise theorem number or equation from the reference; a direct citation would clarify the claimed generalization.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback and the positive evaluation of the significance of our work. We address the major comments point by point as follows.

read point-by-point responses

Referee: [§3] §3 (preservation of the NI property): The central claim that the closed-loop system remains NI rests on the composite storage function V(x) + Φ(y) being positive definite. No general criterion, Lyapunov-function construction, or sufficient condition is supplied to verify this property for arbitrary gradient nonlinearities; the result is therefore conditional on an assumption whose validity must be checked case-by-case. Without such a check the subsequent absolute-stability theorems do not apply.

Authors: We agree that the preservation result is conditional on the positive definiteness of the composite storage function. This is a fundamental requirement for the NI property to hold for the closed-loop system and is explicitly part of the theorem. No general criterion is supplied for arbitrary nonlinearities because any such criterion would depend on the particular choice of V and Φ, which are problem-specific. The condition is to be verified case-by-case, as is typical in nonlinear systems theory, and our examples show how it can be satisfied in practice. The absolute-stability results apply when this holds, and we do not believe a general construction is needed or possible without further assumptions. revision: no
Referee: [§4] §4 (absolute-stability conditions): The derived LMI-type stability criteria inherit the same composite-storage hypothesis. Because no counter-example analysis or boundary case is examined where the sum fails to be positive definite, it is unclear how restrictive the hypothesis is and whether the claimed generalization over slope-restricted frameworks remains valid when the hypothesis is only marginally satisfied.

Authors: The stability criteria are conditional on the same hypothesis to ensure the closed-loop NI property. This makes the results applicable precisely when the composite storage is positive definite, which is a mild condition in energy-based designs like potential shaping. In the linear case, it is easily checked. We did not analyze cases where it fails because those are outside the scope of the theorems; the generalization holds under the hypothesis, as shown by the example that goes beyond slope-restricted cases. The restrictiveness is thus not a limitation but a precise statement of when the results apply. revision: no

Circularity Check

0 steps flagged

Derivation is self-contained from NI definitions and gradient feedback

full rationale

The paper begins from the standard definition of negative imaginary systems and proves preservation of the NI property when feedback is the gradient of a C¹ function provided the composite storage function (plant storage plus potential) remains positive definite. Absolute stability conditions are then derived from this preservation result under mild assumptions. No step reduces by construction to a fitted parameter, self-citation, or renamed input; the composite-storage hypothesis is an explicit assumption whose verification is left to the user rather than smuggled in. No self-citation is load-bearing for the central claim, and the linear specialization is shown to generalize prior results without circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard domain assumptions from nonlinear control theory regarding the form of the feedback and storage functions; no free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption Feedback nonlinearity is the gradient of a continuously differentiable function
Invoked to preserve the NI property and enable storage shaping
domain assumption Composite storage function remains positive definite
Required for the direct connection to storage-function shaping and stability

pith-pipeline@v0.9.0 · 5439 in / 1345 out tokens · 52610 ms · 2026-05-08T06:41:27.861926+00:00 · methodology

discussion (0)

Reference graph

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