{"id":"a2c15a60-0c15-4a53-a6b4-e0b1e33cbcdb","arxiv_id":"2505.05918","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"The energy-saving sub-optimal sliding mode control achieves comparable tracking to SOSMC in a simulated rough-surface scanning task while reducing fuel consumption.","lead":"This chapter summarizes the energy-saving sub-optimal sliding mode control (ES-SOSMC) and demonstrates it in a simulated scanning and machining task where a tool stays in contact with a moving rough surface. It claims the same tracking and stabilization performance as conventional SOSMC while consuming less control energy.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Given m = 0.0005 kg and U = 0.2 N, Eq. (43) yields Delta/U ≈ 1.5e-4, not 0.3; the Section 4.2 parameterization and energy-saving comparison are therefore not grounded.","rationale":"The reader's conditional verdict identified the unverified disturbance bound as the weakest point. My stress-test sharpens this: even accepting the stated bound, the numbers do not produce the claimed Delta/U = 0.3 regime used to select controller parameters. Eq. (43) defines Delta as an acceleration (N/kg), while U = 0.2 N is a force; the ratio Delta/U is not dimensionless. The correct normalized ratio is (G+Phi)/(m U) = 0.00003/(0.0005*0.2) = 0.3 only in force terms, but the analysis of Section 3 requires accelerations. This means the application example does not actually instantiate the optimization problem whose guarantee it claims, so the central claim 'lower fuel consumption than SOSMC' is not supported by the simulation as reported. The fix is straightforward (correct units/scaling and rerun), so the paper need not be rejected outright; a conditional verdict requiring these corrections is appropriate. The reader's concern about the bound is real but secondary: the parameterization inconsistency fails first.","tokens_in":18751,"tokens_out":11998,"duration_ms":110971,"concrete_test":"Recompute the normalized disturbance-to-control ratio from the stated parameters: with Eq. (43), Delta = (G+Phi)/m = 0.06 m/s^2; with U = 0.2 N and m = 0.0005 kg, the normalized control amplitude is U/m = 400 m/s^2, so Delta/U = 1.5e-4. Verify whether the authors intended U = 0.2 m/s^2 (implying U_phys = m U = 1e-4 N) or Delta = 0.06 N. Then re-solve (29)-(31) for the correct Delta/U and rerun the Section 4.2 simulation with the corrected parameterization; check that the beta pairs still satisfy (22)-(24) and that Fig. 11(b) still shows E_ES < E_SOSMC below the U t bound.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Section 4.2 the paper states G+Phi = 0.00003 N, m = 0.0005 kg, U = 0.2 N, and that 'the latter corresponds to the Delta/U = 0.3 ratio.' With Eq. (43), Delta = (G+Phi)/m = 0.06 m/s^2. In the normalized plant of Section 2.2 (g = 1), the control amplitude is U_norm = U_phys/m = 400 m/s^2, so Delta/U_norm = 1.5e-4. Using the physical force directly gives the same ratio: (G+Phi)/U_phys = 1.5e-4. The number 0.3 is recovered only from (G+Phi)/(m U_phys), which is dimensionally 1/kg, not a valid disturbance-to-control ratio. Consequently the beta pairs (0.85,0.27), (0.97,0.05) and the SOSMC benchmark beta1=0.65 are taken from the Delta/U = 0.3 panels of Figs. 4-5, but the simulated plant with the stated parameters is not in that regime. The constrained minimization (29)-(31) and the energy-saving guarantee (30) are therefore not demonstrated for the example. Independently, the bound G+Phi is asserted without derivation, measurement, or sensitivity analysis, so even if the ratio were corrected, robustness of the guarantee to bound uncertainty remains unaddressed.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript presents the energy-saving sub-optimal sliding mode control (ES-SOSMC) introduced in [35], summarizing its control law, finite-time convergence conditions, energy-cost parameterization via a constrained minimization, and describing-function based chattering analysis. It then applies the controller to a stiff position-control problem in which a tool scans a moving rough surface, comparing tracking performance and a fuel-consumption norm against conventional SOSMC. A second machining scenario is described but not simulated in detail.","tokens_in":19079,"tokens_out":5515,"duration_ms":54423,"significance":"If the validation were sound, the paper would be a useful consolidation of ES-SOSMC and a relevant application to AFM-like scanning tasks. The explicit fuel metric (1), the constrained minimization (29)-(31), and the describing-function estimates (37)-(40) are concrete and potentially transferable tools. However, the paper does not provide machine-checked proofs, code, or a quantitative statistical validation, and the main application example contains a parameterization inconsistency that currently prevents the claimed energy-saving guarantee from being demonstrated.","major_comments":[{"comment":"The parameterization of the simulation is inconsistent with the stated disturbance-to-control ratio. With m=0.0005 kg, G+Phi=0.00003 N and U=0.2 N, Eq. (43) gives Delta=0.06 m/s^2, while the normalized control amplitude in plant (8) is U/m=400 m/s^2, so Delta/(U/m)=1.5e-4, not 0.3. The pairs (0.85,0.27), (0.97,0.05) and the benchmark beta1=0.65 are taken from the Delta/U=0.3 panels of Figs. 4-5, so the simulated plant is not in the regime for which these parameters and the energy-saving guarantee (30) were computed. Please correct the normalization or recompute the parameterization for the actual ratio.","section":"§4.2, Eq. (43), Figs. 4-5"},{"comment":"The bound G+Phi=0.00003 N is asserted without derivation, measurement, or sensitivity analysis. Since the convergence conditions (22)-(24), the cost functions (27)-(28), and the guarantee (30) all depend on Delta, an underestimate of this bound would invalidate the energy-saving claim and could also violate the finite-time convergence conditions. Provide a derivation or measurement of the bound and a sensitivity study showing the behavior for larger Delta.","section":"§4.2, Eqs. (42)-(43)"},{"comment":"The hard constraint (30) enforces J - Jhat < 0 by construction, so the energy-saving result is not an empirical finding unless the simulated comparison is validated quantitatively. Fig. 11(b) shows only single trajectories of a stochastic excitation; no energy-saving percentage, no multiple realizations, and no error bars are reported. Please add a quantitative statistic such as the final E_ES/E_SOSMC ratio with confidence intervals, and state the number of realizations used.","section":"§3.3, §4.2, Fig. 11"},{"comment":"The machining scenario is described only qualitatively and the section explicitly states that a detailed presentation is omitted. Since the abstract and introduction claim a demonstration for both scanning and machining, the machining application claim is not supported by any simulation or experimental result. Either add a simulation for the machining case or restrict the claim to the scanning application.","section":"§4.3"},{"comment":"The central theoretical assertions are cited to [35] and not derived in this manuscript: the convergence conditions (22)-(24), the convergence-time bound (25), the reaching and contraction factors Omega and eta, and the cost functions (27)-(28) are defined only by reference. Because the energy-saving guarantee (30) and the example parameter choices rest on these formulas, the paper would be more convincing if the derivations were reproduced in an appendix or if the manuscript explicitly framed itself as a survey whose original analysis is in [35].","section":"§3.2-3.3, Eqs. (22)-(28)"}],"minor_comments":[{"comment":"The caption of Fig. 10 gives the reference distance as B=2 um, while the text states B=0.2 um; please make these consistent.","section":"§4.2, Fig. 10"},{"comment":"Equation (40) appears to have a units or typographical problem: as written, the right-hand side has dimension 1/s^2 rather than the dimension of sigma_M. Please check against [35] and correct.","section":"§3.4, Eq. (40)"},{"comment":"In the sentence comparing parameters to the scanning case, the cross-reference 'section 4.3' should presumably read 'section 4.2'; please correct this and any similar internal cross-reference errors.","section":"§4.3"}],"recommendation":"major_revision","confidential_remarks":"The core algorithm and theoretical analysis have already appeared in [35]; the contribution of this manuscript is mainly the application study, which is currently under-validated and contains a parameterization inconsistency in §4.2. The issue is fixable, but the energy-saving demonstration should not be relied upon until the normalization is corrected and a sensitivity analysis for the disturbance bound is provided. If the journal's scope includes tutorial-style chapters, the presentation may be acceptable after revision; if it requires original research, the novelty may be thin."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this chapter is a tidy overview of the author's own ES-SOSMC work with a new simulation study of scanning/machining a rough surface. The cleanest thing about it is the exposition; the weakest thing is the application example's parameterization, where the stated Delta/U ratio appears to be off by three orders of magnitude.\n\nWhat's actually new: the application scenario is new relative to [35] and [34], and the fuel-norm comparison in Section 4.2 is a reasonable way to demonstrate energy saving. The paper also does an honest job of signaling what is borrowed: it says plainly that the convergence analysis, cost functions, describing function and chattering formulas are in [35]. That's fine for a consolidated chapter.\n\nThe soft spot is real. Eq. (43) defines Delta = (G+Phi)/m. With m=0.0005 kg and G+Phi=0.00003 N, Delta = 0.06 m/s^2. The control amplitude is U=0.2 N. In the normalized plant the control amplitude is U/m = 400 m/s^2, so Delta/(U/m) = 1.5e-4, not 0.3. The paper writes that U=0.2 N 'corresponds to the Delta/U = 0.3 ratio', but that mixes physical force with normalized acceleration. The beta pairs (0.85,0.27), (0.97,0.05) and the SOSMC benchmark beta1=0.65 are taken from the Delta/U=0.3 panels in Figs. 4-5, so the optimization is not actually being applied to the plant that is simulated. This makes the quantitative energy-saving comparison ungrounded. The bound G+Phi is also asserted with no derivation, measurement, or sensitivity analysis, so even if the ratio were corrected we wouldn't know how robust the guarantee is.\n\nA secondary concern: the constrained minimization (29)-(31) is formulated to force the model-based ES-SOSMC energy to be lower than SOSMC, so the simulation partly confirms a conclusion built into the parameter selection. That's a structural limitation of the approach rather than a fatal flaw, and it would matter less if the parameter mapping were correct and the simulation had more than one realization.\n\nWho is this for? Readers interested in a compact exposition of ES-SOSMC and a plausible application narrative. I would not cite it for the theoretical guarantees, but I would cite it as an application demonstration once the dimensional error is fixed. As written, it needs correction before archival publication. I'd send it to peer review with a request for major revision, and I'd ask for the simulation code, the derivation or measurement of the disturbance bound, and a corrected Delta/U mapping.","headline":"A cleanly written consolidation of prior ES-SOSMC results with a new simulation application, but a dimensional slip in the example's disturbance-to-control ratio undermines the parameterization.","tokens_in":19598,"tokens_out":3691,"would_cite":false,"duration_ms":34811,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A two-threshold sliding-mode controller can track a moving rough surface as accurately as the standard one while consuming less fuel.","keywords":["energy-saving sub-optimal sliding mode control","second-order sliding mode control","fuel-optimal control","control-off mode","chattering","describing function analysis","stiff position control","moving rough surface"],"falsifier":"Measure or simulate the fuel norm $E=\\int_0^{T_f}|u|\\,dt$ for the scanning benchmark with a surface roughness large enough that $G+\\Phi>0.00003$ N, or with intermittent tool-surface separation, and compare with the SOSMC benchmark; observing $E$ meet or exceed $U t$, or worse tracking error than SOSMC, would falsify the energy-saving guarantee (30).","tokens_in":18529,"feed_emoji":"⚙️","tokens_out":10901,"duration_ms":103033,"temperature":0.7,"pith_summary":"This paper is trying to establish that the energy-saving sub-optimal sliding-mode controller (ES-SOSMC), a second-order sliding-mode law that inserts control-off intervals during finite-time convergence, can be tuned so that its total fuel consumption is guaranteed to be lower than that of the conventional sub-optimal sliding-mode controller (SOSMC) for the same task. The tuning is a constrained minimization of an energy-cost function over two anticipation parameters, $\\beta_1$ and $\\beta_2$, whose admissible values form a triangle defined by the convergence conditions (22)--(24). The paper then puts the controller to a stiff position-control application: an actuator-held tool scanning or machining a moving rough surface, with a pre-computed disturbance bound. In simulation, ES-SOSMC matches the tracking and stabilization performance of SOSMC while its fuel norm stays below the linear upper bound $U t$ and the saving grows over time. The payoff is energy saving without the usual robust-tracking performance loss.","feed_headline":"Control-off mode cuts fuel use in sliding-mode tracking","feed_subtitle":"It matches standard sliding-mode accuracy while staying below the linear fuel upper bound","key_machinery":"The central object is the ES-SOSMC control law $u(t)=-0.5U\\,\\mathrm{sign}(\\sigma-\\beta_1\\sigma_M)-0.5U\\,\\mathrm{sign}(\\sigma-\\beta_2\\sigma_M)$ for $t>t_{M1}$, with the initial phase $\\bar u(t)=-U\\,\\mathrm{sign}(\\sigma(t)-\\sigma(0))$ on $[0,t_{M1}]$, where $\\sigma_M$ is the most recent extremum of the sliding variable. The two thresholds $\\beta_1,\\beta_2$ create a three-state relay: control is positive, negative, or zero depending on where $\\sigma$ sits relative to the two threshold lines, and the zero zone is the energy-saving mechanism. The parameterization is carried by the cost functions (27)--(28) and the constrained minimization (29) subject to the hard constraints (30)--(31), which select threshold pairs inside the admissible triangle (22)--(24). A separate harmonic-balance analysis, using the describing function (37) of the three-state relay against the double-integrator-plus-actuator transfer function, predicts the chattering frequency and amplitude in equations (39)--(40).","core_discovery":"The central claim is that the ES-SOSMC defined by equations (20)--(21) and parameterized by the constrained minimization (29)--(31) is globally finite-time convergent for relative-degree-two systems with bounded matched perturbations, and that its fuel consumption is strictly lower than that of the conventional SOSMC while tracking and stabilization performance are equivalent. The control law is the parallel connection of two SOSMC terms with the same authority $0.5U$ but different anticipation factors $\\beta_1>\\beta_2$; when the two sign terms have opposite signs, the control output is zero, creating the energy-saving control-off phase. The paper's energy-cost functions, equations (27)--(28), express fuel consumed during convergence, and the hard constraint (30) enforces the saving. In the application, a tool in non-separating contact with a randomly rough moving surface must hold a constant scanning distance or a fixed machining position; with $\\Delta/U=0.3$, the pairs $(\\beta_1,\\beta_2)=(0.85,0.27)$ and $(0.97,0.05)$ yield the same target-distance behavior as SOSMC while keeping the fuel norm below $U t$. The describing-function analysis adds that residual chattering caused by parasitic actuator dynamics has a larger amplitude and a lower frequency for ES-SOSMC than for SOSMC.","pith_inferences":["Because the saving accumulates with time, the method pays off most in long-duration contact operations such as scanning, grinding, and milling rather than in short transients; the paper's own examples are of the long-duration type.","The same two-threshold construction could in principle be layered onto other second-order sliding-mode algorithms, such as twisting or super-twisting, when a relative-degree-two sliding variable and bounded actuation are available; the paper does not analyze those variants.","The practical guarantee depends on the pre-set disturbance bound $G+\\Phi=0.00003$ N. A natural test is to vary the surface roughness amplitude until the bound is exceeded and measure when the fuel curve touches $U t$, which would mark the limit of the energy-saving regime.","Since chattering amplitude is larger for ES-SOSMC, applications with tight positioning noise tolerances may need to trade some energy saving for a smaller $\\beta_1-\\beta_2$ separation."],"forward_implications":["The energy-saving guarantee is quantitative: for any fixed $\\beta_1$ obeying the upper bound (31), the constrained minimization (29) delivers a $\\beta_2$ for which the ES-SOSMC cost (27) is strictly smaller than the SOSMC cost (28).","In the scanning benchmark, both locally optimal pairs $(\\beta_1,\\beta_2)=(0.85,0.27)$ and $(0.97,0.05)$ keep the same relative-distance accuracy as SOSMC with $\\beta_1=0.65$, while their fuel norms stay below the linear upper bound $U t$ and the gap widens over time.","The same design applies to the machining scenario because that task reduces to stabilization and the plant and control parameters scale in the same way; the chapter reports the same design procedure and similar energy-saving performance.","Chattering is not eliminated but is characterized: with parasitic actuator dynamics, ES-SOSMC oscillates at a lower frequency and higher amplitude than SOSMC, with $\\omega_o$ and $\\sigma_A$ given by (39)--(40)."],"supporting_citations":[{"why":"introduces the ES-SOSMC algorithm and supplies the convergence-time bound, energy-cost functions (27)--(28), and the constrained minimization (29)--(31) that the chapter applies.","marker":"[35]"},{"why":"defines the conventional SOSMC baseline, its tuning conditions, and the local-extrema detection that the energy-saving version extends.","marker":"[9]"},{"why":"formulates the fuel-optimal double-integrator problem with response-time constraint, including the auxiliary curves whose control-off segments motivate ES-SOSMC.","marker":"[4]"},{"why":"gives the original sub-optimal second-order SMC and the measurement-based extremum detection used by both controllers.","marker":"[7]"},{"why":"provides the describing function of the conventional SOSMC, the limiting case from which the ES-SOSMC describing function (37) is built.","marker":"[11]"},{"why":"supplies the harmonic-balance framework and the double-integrator-with-actuator transfer function used to obtain chattering frequency and amplitude (39)--(40).","marker":"[10]"},{"why":"demonstrates experimental feasibility of ES-SOSMC and reports measured energy saving on a laboratory setup.","marker":"[34]"},{"why":"provides the high-speed AFM scanning speed and surface-profile scale used to set the simulation parameters.","marker":"[15]"},{"why":"supplies the lumped actuator mass (0.0005 kg) used in the AFM model.","marker":"[45]"},{"why":"supplies the cantilever stiffness (0.73 N/m) used in the AFM model.","marker":"[14]"}],"fun_headline_variants":["Control-off mode cuts fuel in sliding-mode tracking","Sliding mode saves fuel with zero-output phases","Energy-saving sliding mode matches standard accuracy","Lower fuel, same tracking: sliding mode with off periods","ES-SOSMC: fuel-efficient sliding mode with finite-time convergence"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The energy-saving guarantee rests on the pre-calculated disturbance bound $G+\\Phi=0.00003$ N being valid and on the tool never separating from the moving surface; if the true disturbance exceeds that bound or contact is lost, the tuning ratio $\\Delta/U=0.3$ no longer holds and the promised saving is not guaranteed.","fun_headline_variants_meta":{"raw":{"variants":["Control-off mode cuts fuel in sliding-mode tracking","Sliding mode saves fuel with zero-output phases","Energy-saving sliding mode matches standard accuracy","Lower fuel, same tracking: sliding mode with off periods","ES-SOSMC: fuel-efficient sliding mode with finite-time convergence"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000271,"raw_usage":{"total_tokens":1669,"prompt_tokens":1028,"completion_tokens":641,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":644,"completion_tokens_details":{"reasoning_tokens":566}},"tokens_in":644,"tokens_out":641,"duration_ms":6609,"temperature":1.0,"reasoning_tokens":566,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T22:53:35.637226+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure or simulate the fuel norm $E=\\int_0^{T_f}|u|\\,dt$ for the scanning benchmark with a surface roughness large enough that $G+\\Phi>0.00003$ N, or with intermittent tool-surface separation, and compare with the SOSMC benchmark; observing $E$ meet or exceed $U t$, or worse tracking error than SOSMC, would falsify the energy-saving guarantee (30).","supporting_citations":[{"cited_title":"Design of an inertially counterbalanced Z-nanopositioner for high-speed atomic force microscopy","cited_arxiv_id":null,"evidence_quote":"supplies the lumped actuator mass (0.0005 kg) used in the AFM model."},{"cited_title":"Calibratio n of AFM cantilever stiffness: a microfabricated array of reflective springs","cited_arxiv_id":null,"evidence_quote":"supplies the cantilever stiffness (0.73 N/m) used in the AFM model."},{"cited_title":"Energy-saving sub-optimal sliding mode control with bounded actuation","cited_arxiv_id":"2305.07891","evidence_quote":"introduces the ES-SOSMC algorithm and supplies the convergence-time bound, energy-cost functions (27)--(28), and the constrained minimization (29)--(31) that the chapter applies."},{"cited_title":"A survey of applications of second-order sliding mode control to mechanical systems","cited_arxiv_id":null,"evidence_quote":"defines the conventional SOSMC baseline, its tuning conditions, and the local-extrema detection that the energy-saving version extends."},{"cited_title":"Fuel-optimal control of a double integra l plant with response time constraints","cited_arxiv_id":null,"evidence_quote":"formulates the fuel-optimal double-integrator problem with response-time constraint, including the auxiliary curves whose control-off segments motivate ES-SOSMC."},{"cited_title":"Out put tracking control of uncertain nonlinear second-order systems","cited_arxiv_id":null,"evidence_quote":"gives the original sub-optimal second-order SMC and the measurement-based extremum detection used by both controllers."},{"cited_title":"Parameter tuning of second-order sliding mode controllers for linear plants with dynamic actuators","cited_arxiv_id":null,"evidence_quote":"provides the describing function of the conventional SOSMC, the limiting case from which the ES-SOSMC describing function (37) is built."},{"cited_title":"Discontinuous control systems: frequency-domain analysi s and design","cited_arxiv_id":null,"evidence_quote":"supplies the harmonic-balance framework and the double-integrator-with-actuator transfer function used to obtain chattering frequency and amplitude (39)--(40)."},{"cited_title":"Experimental benchmarking of energy -saving sub-optimal sliding mode control","cited_arxiv_id":null,"evidence_quote":"demonstrates experimental feasibility of ES-SOSMC and reports measured energy saving on a laboratory setup."},{"cited_title":"Fast and accurate: high- speed metrological large-range AFM for surface and nanomet rology","cited_arxiv_id":null,"evidence_quote":"provides the high-speed AFM scanning speed and surface-profile scale used to set the simulation parameters."}],"review_version":1}