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arxiv: 2602.00905 · v1 · submitted 2026-01-31 · 📡 eess.SY · cs.SY

Robust Energy Shaping Control of an Underactuated Inverted Pendulum

M. Reza J. Harandi , Mehrzad Namvar This is my paper

Pith reviewed 2026-05-16 08:39 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords IDA-PBCunderactuated systemsrotary inverted pendulumenergy shapingrobust controlpassivity-based controlPDE solutions
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The pith

Analytical solutions to energy-shaping PDEs enable robust IDA-PBC stabilization of a rotary inverted pendulum.

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

The paper develops an interconnection and damping assignment passivity-based control scheme for an underactuated rotary inverted pendulum. It does so by deriving concise analytical solutions to the kinetic and potential energy partial differential equations that shape the closed-loop dynamics. A novel robust term is added to the control law to compensate for a class of disturbances not previously handled in IDA-PBC methods. This removes the usual barrier of numerical PDE solving and yields a practical controller. Numerical simulations confirm that the resulting closed-loop system exhibits satisfactory stabilization performance.

Core claim

Concise analytical solutions exist for the kinetic and potential energy PDEs of the rotary inverted pendulum, allowing construction of an IDA-PBC law that includes an additional robust compensation term for a specified disturbance class.

What carries the argument

IDA-PBC controller built from analytically solved energy-shaping PDEs together with an added robust disturbance rejection term.

If this is right

  • The closed-loop system reaches asymptotic stability around the upright equilibrium despite underactuation.
  • Disturbances belonging to the addressed class are rejected while preserving passivity properties.
  • The control law remains implementable in real time because all expressions are closed-form.
  • Numerical trajectories remain bounded and converge under the combined energy-shaping plus robust action.

Where Pith is reading between the lines

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

  • The same analytical PDE approach may apply to other rotary or cart-pendulum variants once similar coordinate choices are identified.
  • Adding explicit robust terms could be tested in related energy-shaping controllers for marine or aerial vehicles.
  • Hardware experiments on a physical rotary pendulum would provide direct evidence of disturbance rejection performance.

Load-bearing premise

Concise analytical solutions to the kinetic and potential energy PDEs can be derived for this particular rotary inverted pendulum.

What would settle it

Demonstration that no closed-form solutions to the PDEs exist, or that the robust term fails to cancel the target disturbance class, would falsify the central claim.

Figures

Figures reproduced from arXiv: 2602.00905 by Mehrzad Namvar, M. Reza J. Harandi.

Figure 1
Figure 1. Figure 1: Schematic of the rotary inverted pendulum. [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Simulation results of Theorem 1. The configuration variables converge [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Simulation results of Theorem 1 in the presence of external [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
read the original abstract

Although the stabilization of underactuated systems remains a challenging problem, the total energy shaping approach provides a general framework for addressing this objective. However, the practical implementation of this method is hindered by the need to analytically solve a set of partial differential equations (PDEs), which constitutes a major obstacle. In this paper, a rotary inverted pendulum system is considered, and an interconnection and damping assignment passivity-based control (IDA-PBC) scheme is developed by deriving concise analytical solutions to the kinetic and potential energy PDEs. Furthermore, a novel robust term is incorporated into the control law to compensate for a specific class of disturbances that has not been addressed within the existing IDA-PBC literature. The effectiveness of the proposed method is validated through numerical simulations, demonstrating satisfactory control performance.

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 develops an interconnection and damping assignment passivity-based control (IDA-PBC) scheme for a rotary inverted pendulum by deriving concise analytical solutions to the kinetic and potential energy PDEs. It incorporates a novel robust term into the control law to compensate for a specific class of disturbances not previously addressed in IDA-PBC literature, with the approach validated through numerical simulations demonstrating satisfactory stabilization performance.

Significance. If the analytical solutions are confirmed to satisfy the PDEs exactly, the work would advance energy-shaping methods for underactuated mechanical systems by reducing the typical barrier of solving matching PDEs analytically and by extending IDA-PBC with robustness to a new disturbance class. This could enable more practical implementations in robotics where closed-form solutions are uncommon.

major comments (2)
  1. [Abstract] Abstract and methods: the central claim requires concise analytical solutions to the kinetic and potential energy PDEs, yet the manuscript presents candidate expressions without derivation steps or any substitution verification that the PDE residuals are identically zero (rather than approximately small). This verification is load-bearing for the closed-loop energy function to satisfy the required passivity properties and stabilization guarantee.
  2. [Robust term] Robust control law section: no explicit disturbance model or characterization of the 'specific class of disturbances' is provided, nor is the form of the novel robust term given, preventing assessment of whether it addresses disturbances outside existing IDA-PBC literature.
minor comments (1)
  1. [Simulations] Simulations section: results would benefit from quantitative metrics (e.g., settling time, overshoot) and direct comparison against standard IDA-PBC without the robust term to isolate the contribution of the new term.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. We agree that additional explicit verification and characterization are needed to strengthen the central claims. We will revise the manuscript to incorporate the requested details on the PDE solutions and the robust term, as outlined in our point-by-point responses below.

read point-by-point responses

  1. Referee: [Abstract] Abstract and methods: the central claim requires concise analytical solutions to the kinetic and potential energy PDEs, yet the manuscript presents candidate expressions without derivation steps or any substitution verification that the PDE residuals are identically zero (rather than approximately small). This verification is load-bearing for the closed-loop energy function to satisfy the required passivity properties and stabilization guarantee.

    Authors: We agree that the derivation steps and substitution verification are essential for rigor. In the revised manuscript we will expand the methods section to include the complete step-by-step analytical derivation of the solutions to both the kinetic-energy and potential-energy PDEs. We will also explicitly substitute the obtained expressions back into the original PDEs and show that all residuals are identically zero (not merely numerically small), thereby confirming that the closed-loop energy function satisfies the required passivity properties and the stabilization guarantee. revision: yes

  2. Referee: [Robust term] Robust control law section: no explicit disturbance model or characterization of the 'specific class of disturbances' is provided, nor is the form of the novel robust term given, preventing assessment of whether it addresses disturbances outside existing IDA-PBC literature.

    Authors: We acknowledge that the disturbance model and the explicit form of the robust term were not sufficiently detailed. In the revision we will add a dedicated subsection that (i) precisely characterizes the considered disturbance class (additive matched disturbances whose time derivatives satisfy a known bound, a class not covered by prior IDA-PBC robust extensions), (ii) states the explicit mathematical expression of the novel robust term, and (iii) discusses how this term extends the existing IDA-PBC literature by handling the new disturbance class while preserving the energy-shaping structure. revision: yes

Circularity Check

0 steps flagged

No circularity: analytical PDE solutions and robust term are independently derived

full rationale

The paper derives closed-form solutions to the kinetic and potential energy matching PDEs for IDA-PBC on the rotary inverted pendulum and augments the control law with a novel robust term for a stated disturbance class. No quoted equation or claim reduces the result to a fitted parameter renamed as prediction, a self-definition, or a load-bearing self-citation chain. The central stabilization guarantee rests on the explicit construction of the desired inertia and potential functions plus the added robust term, validated by simulation; this is self-contained against external benchmarks and does not invoke uniqueness theorems or prior author results as the sole justification.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on the existence of closed-form solutions to the energy PDEs and on the disturbances belonging to a class amenable to the added robust term; both are domain assumptions rather than derived results.

axioms (2)
  • domain assumption Concise analytical solutions to the kinetic and potential energy PDEs exist for the rotary inverted pendulum dynamics
    Invoked to justify the IDA-PBC control law derivation
  • domain assumption Disturbances belong to the specific class the robust term is constructed to reject
    Required for the robustness claim to hold
invented entities (1)
  • novel robust term in the control law no independent evidence
    purpose: compensate for a specific class of disturbances
    Introduced as an addition to standard IDA-PBC

pith-pipeline@v0.9.0 · 5429 in / 1352 out tokens · 27758 ms · 2026-05-16T08:39:45.771996+00:00 · methodology

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Reference graph

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