{"id":"5956c2b5-7ac3-4c17-ae53-03da531c9688","arxiv_id":"2411.10592","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"New sufficient LMI conditions design variable structure and unit vector sliding mode controllers that guarantee global finite-time stabilization of polytopic uncertain systems, with convex optimization of reaching time bounds.","lead":"This paper gives LMI-based rules for designing sliding mode controllers for uncertain systems where the input gain matrix is unknown but belongs to a known polytope. The proposed controllers guarantee the error state reaches the origin in finite time, with an optimization step that minimizes a guaranteed reaching time bound for a set of initial conditions.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2's proof (inequality (38)) expands H^T H as μ/4 P^2, but the actual expansion is μ/4 P Πσ P; the stated equality requires P and Πσ to commute, which is not guaranteed for arbitrary P>0 and rank-one Πσ.","rationale":"The reader's weakest assumption was the absence of additive disturbances, which is a scope limitation explicitly stated in the problem formulation (1) and acknowledged in the conclusion; it does not threaten the correctness of the theorems for the disturbance-free family. My stress-test pass instead found a concrete internal proof gap in the central UVC theorem: inequality (38) is justified by an algebraic expansion that is incorrect as written because it treats P Πσ P as P^2 without commutativity. This is load-bearing because (38) is exactly what removes the indefinite Πσ terms and produces the negative-definite derivative estimate (40). However, the gap is fixable with the observation P Πσ P ≤ P^2, so the theorem itself is likely true. The reader did not identify this concern, hence agreement_with_reader is 'disagree'. The verdict remains CONDITIONAL: the authors should correct and justify (38), along with the presentation issues already noted by the reader. I did not find a more fundamental flaw in the VSC proof (Theorem 1): the congruence and elimination steps check out, the ξ terms cancel, the finite-time bound is valid because u_eq is a sign vector whenever σ ≠ 0, and the reaching-time optimization inequalities (24)–(25), (42) are correct Schur complements. Example 1's vertex list (47) is garbled, but that is an example-level presentation error rather than a flaw in the central sufficiency argument.","tokens_in":9942,"tokens_out":27555,"duration_ms":264835,"concrete_test":"Re-derive (38) without assuming that P and Πσ commute. For arbitrary P>0, Πσ = σσ^T/‖σ‖^2, and A = BK, expand (1/√μ A + √μ/2 ΠσP)^T(1/√μ A + √μ/2 ΠσP) to obtain the exact RHS 1/μ A^T A + μ/4 P Πσ P, then verify the Loewner inequality P Πσ P ≤ P^2 by writing v^T(P Πσ P)v = ‖Πσ P v‖^2 ≤ ‖P v‖^2. If this succeeds, Theorem 2 stands after replacing the displayed equality in (38) with the corrected bound; if the corrected bound fails for some P and σ, Theorem 2 is unsupported and the UVC design condition must be revised.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step in the proof of Theorem 2 is inequality (38), which is used to eliminate the cross terms involving Πσ and to conclude that the UVC closed-loop derivative is negative definite. The proof justifies (38) by writing the nonnegative quadratic form (A+H)^T(A+H) with A = (1/√μ)BK and H = (√μ/2)ΠσP. Expanding gives A^T A + A^T H + H^T A + H^T H, with H^T H = (μ/4) P Πσ P. The paper replaces this by (μ/4) P^2. That replacement is not an equality unless P and Πσ commute, which is false for a generic symmetric positive definite P and the rank-one projection Πσ = σσ^T/‖σ‖^2. This is not merely a cosmetic typo: (38) is the exact inequality that cancels the indefinite terms in (37) and yields the derivative bound (40). If the displayed RHS were actually too large or too small in the wrong direction, the UVC theorem would not follow. Fortunately, the gap is repairable: since ‖Πσ‖ ≤ 1, one has P Πσ P ≤ P^2 in the Loewner order, so A^T A + (μ/4) P Πσ P ≤ A^T A + (μ/4) P^2, and the desired (38) can be recovered with one extra argument. Thus the central claim is likely correct, but the proof as printed contains an unjustified algebraic identity in a key lemma, and a referee should require the corrected justification before accepting Theorem 2.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies MIMO polytopic uncertain systems of the form \\dot{\\sigma}=Bu with B in the convex hull of known vertices, and proposes LMI-based sufficient conditions for designing variable-structure controllers (VSC, u=K sgn(\\sigma)) and unit-vector controllers (UVC, u=K\\sigma/\\|\\sigma\\|). Theorem 1 gives an LMI feasibility condition for the VSC gain and Theorem 2 gives an analogous condition for the UVC gain, with K=ZX^{-1}, so that the origin of the closed-loop system is finite-time stable. The paper also formulates convex optimization problems to minimize upper bounds on the reaching time over prescribed sets of initial conditions, and illustrates the approach on a visual-servo example and an over-actuated ROV example. I verified the congruence and Schur-complement algebra in The proofs of Theorems 1 and 2; the central derivations are mostly sound, but the proof of Theorem 2 contains a load-bearing inequality whose printed justification is an incorrect algebraic identity.","tokens_in":10314,"tokens_out":18944,"duration_ms":173144,"significance":"If the proof gap in Theorem 2 is repaired, the paper makes a useful contribution: it provides a systematic, genuinely LMI-based synthesis procedure for multivariable sliding-mode control under polytopic uncertainty, with explicit reaching-time estimates that can be optimized in a convex manner. The conditions are sufficient Lyapunov conditions derived from first principles rather than fitted to data, and the reaching-time bounds are falsifiable predictions that can be checked in simulation. The main limitation, the absence of matched or unmatched disturbances, is explicitly acknowledged in Section 5 and does not undermine the stated contribution within the declared scope. The two examples demonstrate feasibility, although they do not include code or Monte-Carlo validation.","major_comments":[{"comment":"The proof of Theorem 2 uses inequality (38) to eliminate the cross terms involving \\Pi_\\sigma, and this step is load-bearing for the negative-definiteness conclusion in (39)-(40). The printed justification expands (1/\\sqrt{\\mu}BK + \\sqrt{\\mu}/2\\,\\Pi_\\sigma P)^\\top(\\cdots)\\ge 0, but the H^\\top H term in that expansion equals (\\mu/4)P\\Pi_\\sigma P, not (\\mu/4)P^2, and P and \\Pi_\\sigma do not commute in general. The gap is repairable: because \\Pi_\\sigma is an orthogonal projection with \\|\\Pi_\\sigma\\|\\le 1 and P>0, one has P\\Pi_\\sigma P\\le P^2 in the Loewner order, since v^\\top P\\Pi_\\sigma P v = \\|\\Pi_\\sigma P v\\|^2 \\le \\|P v\\|^2 = v^\\top P^2 v. Thus the desired inequality follows from the displayed expansion together with this extra bound. This argument should be inserted explicitly; as printed, the algebraic identity used to justify (38) is not valid.","section":"§3.1, Eq. (38)"}],"minor_comments":[{"comment":"Theorem 2 states that it considers the sliding-mode controller (4), but the actual controller is the unit-vector controller (29); Problem 2 similarly refers to the closed-loop system (5) instead of (30). These should be corrected.","section":"Theorem 2 and Problem 2"},{"comment":"Both theorems conclude 'globally asymptotically stable', while the proofs establish finite-time stability. The statements should be aligned with the abstract and with Problems 1 and 2.","section":"Theorems 1 and 2 statements"},{"comment":"Equation (17) is dimensionally incorrect: the expression should read dV/dt = u_\\mathrm{eq}^\\top P B K u_\\mathrm{eq}(t), not PBKu_\\mathrm{eq}(t). The subsequent equations use the correct form.","section":"§2.1, Eq. (17)"},{"comment":"Equation (39) contains the term 'SBK'; from the preceding derivation it is clear that this should be 'PBK'.","section":"§3.1, Eq. (39)"},{"comment":"Equation (47) lists the same vector [cos(\\Delta\\varphi); sin(\\Delta\\varphi)] four times, which would make the uncertainty set a singleton. If the intent is a four-vertex polytopic description of the rotation uncertainty, the four vertices must be written explicitly.","section":"§4.1, Eq. (47)"},{"comment":"The reaching-time bound in (44) uses V_0, but the quantity defined in (41) is U_0 = U(\\sigma(0)); the notation should be made consistent.","section":"§3.2, Eq. (44)"},{"comment":"The proof observes that X>0 follows from the negative definiteness of the (2,2) block in (11). This implication is valid but deserves to be stated explicitly, since X>0 is not listed among the hypotheses of Theorem 1.","section":"§2.1, after Eq. (10)"},{"comment":"The phrase 'subject to and LMIs in (34), (35), (24), (42)' should read 'subject to the LMIs in (34), (35), (24), (42)'.","section":"§3.2, Eq. (45)"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the scope of mathematical optimization and control, and the central LMI-based synthesis idea is sound. The load-bearing issue is the unjustified identity in the proof of Theorem 2, which is repairable with a short Loewner-order argument; I do not see circular reasoning or data-fitting concerns. The examples are illustrative only and contain a typographical error in the vertex description of the uncertainty set. With the proof patch and the typographical corrections listed above, the paper would be suitable for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The short version: this is a legitimate, modest advance. The paper supplies LMI-based synthesis conditions for variable structure and unit vector controllers for polytopic uncertain systems of the form σdot = Bu, plus a convex way to trade off the initial-condition set against a guaranteed finite reaching-time bound. The cited analysis-only results in [7] and the super-twisting designs in [4,5] leave this gap, and the paper fills it cleanly. I checked the congruence algebra in both theorems and the ξ/μ cancellations; the central derivations hold up. The design conditions are sufficient, the reaching-time bounds follow from the Lyapunov decay rate, and the examples give concrete gains with credible simulations. That is real work, and the paper deserves a serious referee.\n\nThe soft spots are real but mostly presentation-level. The most substantive one is in the proof of Theorem 2: inequality (38) is justified by expanding a squared norm, but the printed justification silently replaces P Πσ P with P^2. Strictly, the expansion gives (μ/4) P Πσ P, not (μ/4) P^2. The inequality still holds because Πσ ≤ I implies P Πσ P ≤ P^2 in the Loewner order, so the proof is repairable with one extra argument. But as printed, the reader cannot follow the step, and a referee should require the corrected justification. Theorem 2's statement also references the wrong controller equation (it says (4) instead of (29)), and Example 1's vertex list for the polytope is garbled—the same expression appears four times in (47). Those are easy fixes but they matter because the example is the main evidence that the conditions are usable. No code or data is shipped, so reproducibility is limited to the stated numbers, which is a minor negative for a numerical paper.\n\nThe bigger conceptual caveat is the one the authors acknowledge in Section 5: the plant has no additive disturbances. All guarantees are for the disturbance-free family σdot = Bu with B unknown in a polytope. That is a legitimate starting class, and the paper does not oversell it, but it limits the practical range compared with the disturbance-robust super-twisting literature they cite.\n\nMy verdict: conditional accept. The central claim is very likely correct, the paper is honest about its scope, and the flaws I found are fixable without changing the method. I would send this to a competent reviewer, not desk-reject it, and I would expect a revised version to be publishable. If you are working on LMI-based sliding mode synthesis, it is worth a read; otherwise, a skim of Theorem 1 and Section 3.1 is enough.","headline":"Sound LMI synthesis for multivariable sliding mode with a repairable gap in the UVC proof; worth refereeing after minor fixes.","tokens_in":10847,"tokens_out":1386,"would_cite":false,"duration_ms":14896,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["93B12","93D30","93D05"],"pacs":[],"model":"deepseek-v4-flash","headline":"Two LMI theorems certify finite-time sliding-mode stabilization for uncertain multivariable plants.","keywords":["sliding mode control","variable structure control","unit-vector control","linear matrix inequalities","finite-time stability","robust control","convex optimization","multivariable systems"],"falsifier":"Simulate the closed-loop relay system with the gain from Theorem 1 (or the unit-vector system with the gain from Theorem 2) for every vertex $B_i$ and for convex combinations of the vertices, starting from initial conditions inside the sets guaranteed by constraints (25) and (42); a single trajectory that has not reached the origin by the promised time $T_{\\mathrm{vsc}} = 2\\rho_{\\mathrm{vsc}}$ or $T_{\\mathrm{uvc}} = \\rho_{\\mathrm{uvc}}$ would refute the central claim.","tokens_in":9749,"feed_emoji":"🎛️","tokens_out":11655,"duration_ms":102813,"temperature":0.7,"pith_summary":"This paper aims to turn multivariable sliding-mode controller design into a convex feasibility problem. For plants whose unknown-but-constant input matrix $B$ lies in a known polytope, it proposes two laws—a relay-type variable structure controller and a unit-vector controller—and derives linear matrix inequalities whose feasible solutions produce a gain $K = ZX^{-1}$. When the inequalities are feasible, the origin of the closed loop is globally finite-time stable for every $B$ in the polytope, and the proof gives an explicit upper bound on the reaching time. The same inequalities are augmented with convex constraints that relate the guaranteed reaching time to the set of initial conditions, letting the designer minimize that time. A sympathetic reader should care because the result replaces case-by-case tuning or simulation-based checking with a certificate: solve the LMIs, and the gain is guaranteed to work.","feed_headline":"LMI recipe gives sliding-mode gains with finite-time convergence","feed_subtitle":"Feasible inequalities certify the gain and also bound the time to reach the origin.","key_machinery":"The load-bearing object is the Lyapunov-function certificate encoded as LMIs. The diagonal Lyapunov function $V(\\sigma)=\\sum_{i=1}^n p_i|\\sigma_i|$ is written as $x^\\top P x$ in the $x$-coordinates, and the unit-vector Lyapunov function $U(\\sigma)=\\sigma^\\top P \\sigma/\\|\\sigma\\|$ is written as $z^\\top P z$ in the $z$-coordinates. The matrix variables in the LMIs have direct meanings: $P$ is the Lyapunov matrix, $K = ZX^{-1}$ is the control gain, and $Q = X^{-1}RX^{-1}$ is a positive-definite decay-rate matrix that appears explicitly in the reaching-time bounds. The scalar $\\xi$ or $\\mu$ is a slack variable that makes the nonlinear cross-coupling between $P$ and $K$ expressible as a linear matrix inequality—for the unit-vector case it enters a Young-type inequality that absorbs the projection term $\\Pi_\\sigma$. Feasibility of the inequalities implies $PBK + K^\\top B^\\top P + Q < 0$ at every vertex, hence on the whole polytope, and that inequality is exactly what forces $V$ or $U$ to decrease at a rate fast enough for finite-time convergence.","core_discovery":"The central claim, stated on the paper's own terms, is that both standard multivariable sliding-mode laws admit coordinate changes that make their closed loops amenable to LMI synthesis. With $x_i = \\sqrt{|\\sigma_i|}$, the relay loop $\\dot{\\sigma} = B K \\operatorname{sgn}(\\sigma)$ becomes $\\dot{x} = \\tfrac{1}{2}L(\\sigma)BKL(\\sigma)x$ and the diagonal Lyapunov function $V = \\sum_i p_i|\\sigma_i|$ becomes quadratic; with $z = \\sigma/\\sqrt{\\|\\sigma\\|}$, the unit-vector loop $\\dot{\\sigma} = BK\\sigma/\\|\\sigma\\|$ becomes a form whose derivative is bounded using the projection $\\Pi_\\sigma = \\sigma\\sigma^\\top/\\|\\sigma\\|^2$. Theorem 1 proves that feasibility of (10)-(11) makes $K = ZX^{-1}$ globally finite-time stabilizing for the relay loop for all $B$ in the polytope, with reaching time $t_{\\mathrm{vsc}} \\le 2V_0/\\lambda_{\\min}(Q)$. Theorem 2 proves the analogue for the unit-vector loop from (34)-(35), with $t_{\\mathrm{uvc}} \\le U_0/\\lambda_{\\min}(Q)$. The optimization problems (28) and (45) then use $\\rho$ and $\\phi$ to trade off convergence rate against the guaranteed set of initial conditions.","pith_inferences":["[Editorial inference] The same LMI structure is the natural starting point for plants with additive disturbances, but the theorems in this paper do not cover that case; the paper itself defers it to future work.","[Editorial inference] Sweeping the scalar $\\phi$ while re-solving (28) or (45) would trace a design curve trading guaranteed initial-condition set size against guaranteed reaching time, which the paper presents as a single trade-off point.","[Editorial inference] Because the conditions are sufficient and tied to one Lyapunov structure, there may exist stabilizable plants for which (11) or (35) is infeasible; constructing such an example would map the conservatism of the certificate.","[Editorial inference] The VSC's elementwise sign lets each component $\\sigma_i$ reach zero independently, while the UVC reaches the origin as a whole; this behavioral difference suggests the two laws will have different chattering and discretization properties in digital implementation."],"forward_implications":["Solving the LMIs in Theorem 1 returns a gain $K$ that certifies global finite-time stability of $\\dot{\\sigma} = BK\\operatorname{sgn}(\\sigma)$ for every admissible $B$, with no gridding over the uncertainty.","Solving the LMIs in Theorem 2 gives the same certificate for $\\dot{\\sigma} = BK\\sigma/\\|\\sigma\\|$, with the sliding mode occurring only at the origin.","The upper bounds $t_{\\mathrm{vsc}} \\le 2V_0/\\lambda_{\\min}(Q)$ and $t_{\\mathrm{uvc}} \\le U_0/\\lambda_{\\min}(Q)$ make reaching time a design objective: maximizing the smallest eigenvalue of $Q$ through constraint (24) minimizes the bound.","Constraints (25) and (42) enlarge the estimated set of initial conditions for a fixed reaching time, and the optimization problems (28) and (45) combine both objectives in one convex program.","The examples indicate that smaller $\\xi$ improves the VSC reaching-time bound while larger $\\mu$ improves the UVC bound, giving tuning rules for the slack parameters."],"supporting_citations":[{"why":"Supplies the equivalent-control definition and the diagonal-type Lyapunov analysis for relay systems that Theorem 1 extends to synthesis under polytopic uncertainty.","marker":"[7]"},{"why":"Introduces the multivariable unit-vector control law whose design and origin-reaching behavior Theorem 2 formalizes.","marker":"[6]"},{"why":"Provides the over-actuated ROV model used in Example 2 and the LMI-based reaching-time assessment for multivariable super-twisting that motivates the design conditions.","marker":"[5]"},{"why":"Shows how LMI conditions can synthesize multivariable sliding-mode controllers for uncertain systems, the technique that this paper applies to VSC and UVC.","marker":"[4]"},{"why":"Documents that relay systems $\\dot{\\sigma} = A\\operatorname{sgn}(\\sigma)$ can be unstable even with Hurwitz $A$, motivating the need for the certificate-based design conditions.","marker":"[20]"}],"fun_headline_variants":["LMI recipe for sliding-mode with finite-time reach","Systematic LMI design for multivariable sliding mode","Finite-time sliding-mode via LMIs","Relay and unit-vector sliding-mode via LMIs"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the plant has the exact form $\\dot{\\sigma} = B u$, with $B$ constant but unknown inside a known polytope and nothing else on the right-hand side; if additive disturbances are present, the finite-time guarantees of both theorems cease to apply.","fun_headline_variants_meta":{"raw":{"variants":["LMI recipe for sliding-mode with finite-time reach","Systematic LMI design for multivariable sliding mode","Finite-time sliding-mode via LMIs","Relay and unit-vector sliding-mode via LMIs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000544,"raw_usage":{"total_tokens":2606,"prompt_tokens":952,"completion_tokens":1654,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":568,"completion_tokens_details":{"reasoning_tokens":1592}},"tokens_in":568,"tokens_out":1654,"duration_ms":12411,"temperature":1.0,"reasoning_tokens":1592,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T19:34:55.558672+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate the closed-loop relay system with the gain from Theorem 1 (or the unit-vector system with the gain from Theorem 2) for every vertex $B_i$ and for convex combinations of the vertices, starting from initial conditions inside the sets guaranteed by constraints (25) and (42); a single trajectory that has not reached the origin by the promised time $T_{\\mathrm{vsc}} = 2\\rho_{\\mathrm{vsc}}$ or $T_{\\mathrm{uvc}} = \\rho_{\\mathrm{uvc}}$ would refute the central claim.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the equivalent-control definition and the diagonal-type Lyapunov analysis for relay systems that Theorem 1 extends to synthesis under polytopic uncertainty."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the multivariable unit-vector control law whose design and origin-reaching behavior Theorem 2 formalizes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the over-actuated ROV model used in Example 2 and the LMI-based reaching-time assessment for multivariable super-twisting that motivates the design conditions."},{"cited_title":"Geromel, E","cited_arxiv_id":null,"evidence_quote":"Shows how LMI conditions can synthesize multivariable sliding-mode controllers for uncertain systems, the technique that this paper applies to VSC and UVC."},{"cited_title":"Utkin, A","cited_arxiv_id":null,"evidence_quote":"Documents that relay systems $\\dot{\\sigma} = A\\operatorname{sgn}(\\sigma)$ can be unstable even with Hurwitz $A$, motivating the need for the certificate-based design conditions."}],"review_version":1}