A Systematic LMI Approach to Design Multivariable Sliding Mode Controllers

Iury Bessa; Pedro Henrique Silva Coutinho; Tiago Roux Oliveira; Victor Hugo Pereira Rodrigues

REVIEW 1 major objections 8 minor 21 references

A Systematic LMI Approach to Design Multivariable Sliding Mode Controllers

T0 review · 1 major / 8 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Two LMI theorems certify finite-time sliding-mode stabilization for uncertain multivariable plants.

desk verdict Sound LMI synthesis for multivariable sliding mode with a repairable gap in the UVC proof; worth refereeing after minor fixes. read the letter →

arxiv 2411.10592 v2 pith:Q6MOELN6 submitted 2024-11-15 math.OC cs.SYeess.SY

Pedro Henrique Silva Coutinho , Iury Bessa , Victor Hugo Pereira Rodrigues , Tiago Roux Oliveira This is my paper · ORCID

classification math.OCcs.SYeess.SY MSC 93B1293D3093D05

keywords slidingmodecontrolvariablestructureunit-vectorlinearmatrixinequalitiesfinite-timestabilityrobustconvexoptimizationmultivariablesystems

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

[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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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.

The 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.

The 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.

My 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.

Referee Report

1 major / 8 minor

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.

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 (1)

[§3.1, Eq. (38)] 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.

minor comments (8)

[Theorem 2 and Problem 2] 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.
[Theorems 1 and 2 statements] 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.
[§2.1, Eq. (17)] 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.
[§3.1, Eq. (39)] Equation (39) contains the term 'SBK'; from the preceding derivation it is clear that this should be 'PBK'.
[§4.1, Eq. (47)] 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.
[§3.2, Eq. (44)] 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.
[§2.1, after Eq. (10)] 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.
[§3.2, Eq. (45)] The phrase 'subject to and LMIs in (34), (35), (24), (42)' should read 'subject to the LMIs in (34), (35), (24), (42)'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the LMI design conditions are derived from Lyapunov inequalities by explicit congruence transformations, with no fitted quantity or load-bearing self-citation.

full rationale

The central claims are self-contained. Theorem 1 (Section 2.1) starts from the diagonal Persidskii-type Lyapunov function (7)-(8) and derives LMI (11) by the standard change of variables K=ZX^{-1}, P=X^{-1}WX^{-1}, Q=X^{-1}RX^{-1}; inequality (14) is obtained by multiplying (13) by the convex weights and is then used to prove negative definiteness of dV/dt and the reaching-time bound (23). No parameter is fitted to data, and the reaching-time estimate is a Lyapunov-decay bound rather than a prediction of a measured quantity. Theorem 2 (Section 3.1) follows the same pattern: LMI (35) is transformed by congruence to (36), convex combination and Schur complement give (37), and the projection property ||Pi_sigma||=1 justifies the cross-term bound (38); the final finite-time bound (41) is again derived from the Lyapunov decay rate. The self-citations [13,14] appear only in the motivating visual-servo example, not in the proofs, and [12] is background; none is load-bearing. The disturbance-free assumption is stated explicitly in (1) and acknowledged in Section 5; this is a scope limitation, not circularity. The algebraic issue noted in the reviewer materials concerning H^T H = (mu/4) P Pi_sigma P versus (mu/4) P^2 in (38) is a proof-correctness point, not an input-output circularity, and the displayed inequality is still recoverable since P Pi_sigma P <= P^2 in the Loewner order. Overall, the derivation chain does not reduce to its own inputs by construction.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central claim rests on standard LMI and Lyapunov background, the specific disturbance-free plant model, and the user-chosen scalars ξ, µ and φ. No new physical entities are introduced.

free parameters (3)

ξ (Theorem 1 VSC) = 0.001 in Example 1; 0.2395 in Example 2
Scalar tuning parameter in the LMI (11). The proof's final Lyapunov inequality is independent of ξ, but feasibility and the optimized reaching time depend on it; the paper gives no systematic selection rule.
µ (Theorem 2 UVC) = 1000 in Example 1; 32.9034 in Example 2
Scalar tuning parameter in the LMI (35). Same status as ξ: needed to shape the LMI, with no selection guidance provided.
φ_vsc and φ_uvc = 0.1 in Example 1; 0.4 in Example 2
Given scalars that define the desired guaranteed initial-condition set via constraints (25) and (42). Chosen by the designer to include the specified initial condition.

assumptions (4)

domain assumption The system is exactly σdot = Bu with no additive disturbances; the only uncertainty is the constant polytopic input matrix B.
This is the problem setup in Section 2 (equation (1)) and Section 3. The Lyapunov inequalities and finite-time bounds are derived for this disturbance-free family; the authors explicitly defer disturbances to future work.
domain assumption Filippov solutions of the discontinuous closed-loop systems exist and are absolutely continuous, and the Lyapunov chain-rule computation with the extended equivalent control holds almost everywhere.
Used in the proofs of Theorem 1 (equations (17)-(21)) and Theorem 2 (around (40)). The argument is cited from [7] and is not reproved.
standard math Standard convex analysis results: Schur complement, congruence transformations preserve definiteness, and PΠP ≤ ||Π|| P^2 for P>0, ||Π||=1.
Invoked in equations (24)-(25), (37)-(38), and (42).
standard math The matrix [I; B_iK] in the congruence step has full column rank, so negativity of the block LMI is preserved.
Needed to pass from (12) to (13) and from (36) to (37); I is invertible and B_iK is square, so the congruence by [I; B_iK] is injective.

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Cite this review

Pith. "Pith review of A Systematic LMI Approach to Design Multivariable Sliding Mode Controllers." pith.science (2026). https://pith.science/paper/Q6MOELN6

@misc{pith2026241110592,
  author       = {Pith},
  title        = {Pith review of: A Systematic LMI Approach to Design Multivariable Sliding Mode Controllers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Q6MOELN6}},
  note         = {Machine review of arXiv:2411.10592}
}

read the original abstract

This paper deals with sliding mode control for multivariable polytopic uncertain systems. We provide systematic procedures to design variable structure controllers (VSCs) and unit-vector controllers (UVCs). Based on suitable representations for the closed-loop system, we derive sufficient conditions in the form of linear matrix inequalities (LMIs) to design the robust sliding mode controllers such that the origin of the closed-loop system is globally stable in finite time. Moreover, by noticing that the reaching time depends on the initial condition and the decay rate, we provide convex optimization problems to design robust controllers by considering the minimization of the reaching time associated with a given set of initial conditions. Two examples illustrate the effectiveness of the proposed approaches.

Figures

Figures reproduced from arXiv: 2411.10592 by the authors.

**Figure 2.** States of the closed-loop robotics visual servo syst [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 4.** States of the closed-loop over-actuated ROV system [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [7]

L. Hsu, E. Kaszkurewicz, and A. Bhaya. Matrix-theoretic conditions for the realizability of sliding manifolds. Systems & control letters , 40(3):145–152, 2000

work page 2000

[2] [1]

W.-H. Chen, W. Xu, and W. X. Zheng. Sliding-mode-based impulsive control for a class of time-delay systems with inp ut disturbance. Automatica, 164:111633, 2024

work page 2024

[3] [2]

Chuei, Y

R. Chuei, Y. Zheng, Z. Cao, and Z. Man. Improved sliding mode observer based repetitive control for linear systems w ith time-varying input and state delays. International Journal of Adaptive Control and Signal Processing , 38(2):393–414, Oct. 2023

work page 2023

[4] [3]

Cucuzzella and A

M. Cucuzzella and A. Ferrara. Practical second order sli ding modes in single-loop networked control of nonlinear system s. Automatica, 89:235–240, 2018

work page 2018

[5] [4]

Geromel, E

J. Geromel, E. Nunes, and L. Hsu. LMI-based robust multivariable super-twisting algorithm design. IEEE Transactions on Automatic Control, 69(7):4844 – 4850, 2024

work page 2024

[6] [5]

J. C. Geromel, L. Hsu, and E. V. L. Nunes. On multivariable super-twisting algorithm reaching time assessment. IEEE Transactions on Automatic Control , 69(11):7972 – 7979, 2024

work page 2024

[7] [6]

L. Hsu, J. P. V. S. da Cunha, R. R. Costa, and F. Lizarralde. Multivariable output-feedback sliding mode control. Variable Structure Systems: Towards the 21 st Century , pages 283– 313, 2002

work page 2002

[8] [8]

Lian and C

J. Lian and C. Li. Event-triggered sliding mode control o f uncertain switched systems via hybrid quantized feedback. IEEE Transactions on Automatic Control , 66(6):2809–2816, 2020

work page 2020