ReactionWheel Pendulum Stabilization Using Various State-Space Representations

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors The manuscript investigates the stabilization of a reaction wheel pendulum (RWP) using several state-space representations, including a generalized nonlinear model, linear approximations, and feedback-linearizable formulations. The paper compares three models (M1–M3) under different control schemes (LQR and pole placement), presents parameter identification via least squares, and validates the theoretical results experimentally. My comments are as follows:
* My main concern is that the study does not yield new theoretical insight into RWP dynamics or control. Its main value is pedagogical, i.e., it shows how different model simplifications affect LQR or pole-placement performance on real hardware. However, the comparison results do not yield general conclusions or guidelines that extend beyond the specific laboratory setup. There is no broader insight that would interest the readers focused on advances in modeling, identification, or control theory. 
* The introduction provides good context but lacks a critical gap statement. Clarify the novelty beyond summarizing prior works. Specify why comparing different state-space representations offers new insights.
* References [2]–[14] should be expanded to include more recent studies on energy-based and nonlinear control of RWPs (e.g., energy shaping, or robust IDA-PBC approaches, adaptive backstepping). For example, energy-based control and LMI-based control for a quadrotor transporting a payload, math; adaptive backstepping control of uncertain nonlinear systems with input and state quantization, tac.
* The contributions are listed but not linked to the experimental results. Indicate which sections demonstrate each contribution. The bullet “rigorous stability analysis…using the Lyapunov approach” should explicitly state whether the analysis is new or based on standard feedback linearization results.
* The reduction to three states (x = [\phi, \dot{\phi}, \omega]^\top) is not fully justified; please clarify how neglecting \theta affects controllability.
* Clarify the physical interpretation of parameters p_1–p_3 and explain why b_1\dot{u} appears (non-causal input term).
* Clarify which equilibrium point each model is linearized around.
* Compare control performance of M1–M3 quantitatively (overshoot, settling time, control effort).
* Discuss control saturation and whether compensation was implemented.

Author Response

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Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

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Comments for author File: Comments.pdf

Author Response

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Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

Overall recommendation: Major Revision

1. The keywords and the contribution statements require substantial refinement. First, several keywords are not sufficiently concise or appropriate—for instance, “experimental validation” should not be used as a keyword. Second, the stated contributions are not well distilled. For example, the item “rigorous stability analysis of the state-feedback controlled system based on the proposed model structure, using the Lyapunov approach” does not constitute a meaningful scientific contribution, as Lyapunov-based stability proofs are standard and widely used.Moreover, the introduction does not adequately summarize the existing modeling and control approaches for inverted pendulum systems. Many relevant works in the literature are missing and should be properly reviewed and cited.

2. In Model 3, the second column of the matrix A(x) consists entirely of zeros. The authors should clarify the implications of this structure for system behavior and controllability. In addition, the manuscript claims that Model 3 is equivalent to Model 4. However, no detailed analytical justification is provided. A rigorous derivation or equivalence analysis is needed to support this claim.

3. The authors should explain how the approximation accuracy of Model (15) with respect to the original nonlinear system is evaluated. Is Model (15) only reliable in a neighborhood of the origin due to its linearized nature？ Similarly, does the same limitation apply to the Third-Order Model?
A clear discussion on the validity region of each approximated model is essential.

4. The manuscript does not specify which algorithm or optimization method is employed for parameter identification. A detailed description of the identification procedure, cost function, numerical solver, and convergence considerations should be included.

5. It is unclear whether the designed controllers are applied to the linearized models or to the full original nonlinear system during simulations. This distinction is critical and should be explicitly stated. In addition, the inverted pendulum should ideally converge to the equilibrium point; however, the presented results do not clearly demonstrate this behavior. Finally, I strongly recommend including a real-world experiment. Experimental validation would significantly strengthen the credibility and practical relevance of the proposed modeling and control framework.

Author Response

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Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

I am happy with the revised manuscript. 

Author Response

Comments: I am happy with the revised manuscript.

Response: We thank the Reviewer for the positive evaluation of the revised manuscript and for the time devoted to assessing our previous changes. We greatly appreciate the Reviewer's constructive feedback, which has helped us improve the clarity and quality of the paper.

Reviewer 2 Report

Comments and Suggestions for Authors

Recommend for publication

Author Response

Comments: Recommend for publication

Response: We thank the Reviewer for the recommendation for publication and for the constructive feedback provided during the earlier review rounds. We appreciate the Reviewer’s comments, which contributed to strengthening the overall presentation of the work.

Reviewer 3 Report

Comments and Suggestions for Authors

This round of responses has addressed most of my comments. However, regarding the optimization method for Comment 4, if the objective function is quadratic or strongly convex, there are many new approaches that could be considered and summarized — for example, Distributed Optimization of Nonlinear Multiagent Systems: A Small-Gain Approach, in IEEE Transactions on Automatic Control, and Global Asymptotic Stability Analysis for Autonomous Optimization, in IEEE Transactions on Automatic Control.

Author Response

Comments: This round of responses has addressed most of my comments. However, regarding the optimization method for Comment 4, if the objective function is quadratic or strongly convex, there are many new approaches that could be considered and summarized — for example, Distributed Optimization of Nonlinear Multiagent Systems: A Small-Gain Approach, in IEEE Transactions on Automatic Control, and Global Asymptotic Stability Analysis for Autonomous Optimization, in IEEE Transactions on Automatic Control.

Response: We thank the Reviewer for the positive assessment of our previous revisions and for the constructive additional remark provided in this round. We also thank the Reviewer for bringing attention to recent developments in optimization for convex and strongly convex problems. Following this suggestion, a concise remark has been added in the Introduction to acknowledge that modern optimization approaches may be applicable when the identification or tuning criterion admits a convex structure. Representative examples include distributed small-gain-based optimization for multiagent systems and autonomous optimization schemes with global asymptotic stability guarantees:
- Liu, T., Qin, Z., Hong, Y., \& Jiang, Z. P. (2021). Distributed optimization of nonlinear multiagent systems: A small-gain approach. IEEE Transactions on Automatic Control, 67(2), 676--691.
- Jin, Z. (2025). Global Asymptotic Stability Analysis for Autonomous Optimization. IEEE Transactions on Automatic Control.

For completeness, we note that the method in [Liu et al. 2022] addresses distributed strongly convex optimization under partial gradient information, whereas [Jin 2025] analyzes autonomous optimization dynamics under the assumption that the underlying gradient-flow system admits a {unique globally asymptotically stable equilibrium} corresponding to the optimizer, with validation carried out on numerical examples. These works provide valuable theoretical insights, although their formulations differ substantially from the single-system parameter identification setting considered in the present study.

The added remark further explains that, although advanced optimization tools exist for strongly convex objectives, such methods lie outside the scope of this contribution, which focuses on comparing nonlinear model structures and their impact on closed-loop performance rather than on proposing new identification algorithms.

To avoid any ambiguity, we have also clarified in Sec. 2.4 that the identification procedure used in this work relies on the classical Least Squares (LS) method. The regression structure and the corresponding normal-equation solution are now stated explicitly.

We hope that these additions address the Reviewer’s concern and improve the clarity and completeness of the identification methodology.