Pith. sign in

REVIEW 3 major objections 4 minor 34 references

Sliding Mode Control for Uncertain Systems with Time-Varying Delays via Predictor Feedback and Super-Twisting Observer

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper claims that combining an open-loop predictor with a super-twisting observer lets a sliding mode controller globally stabilize delayed, uncertain, partially unmeasured linear systems and reconstruct actuator faults online.

desk verdict Plausible predictor-observer-SMC architecture with a genuinely derived control law, but Theorem 1's fixed-gain reachability argument does not go through for unstable plants; the paper needs revision, not desk rejection. read the letter →

arxiv 2507.21281 v1 pith:CGNQZJN5 submitted 2025-07-28 math.OC cs.SYeess.SY

Hardy Pinto , Tiago Roux Oliveira , Liu Hsu This is my paper · ORCID
classification math.OCcs.SYeess.SY MSC 93B1293C2393B5393D20
keywords slidingmodecontroltime-varyingdelayspredictorfeedbacksuper-twistingobserveractuatorfaultreconstructionparametricuncertaintiesmeasurementdelayfinite-timestabilization
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 tries to establish that a linear time-invariant system with a known time-varying measurement delay, partially unmeasured state, matched nonlinear disturbances, and parametric uncertainties can be globally stabilized by a three-part control law: an open-loop predictor that advances the delayed output to the present, a super-twisting observer that estimates the unmeasured state and reconstructs the fault, and a sliding mode controller that drives a sliding variable to zero in finite time. If the claim is correct, systems whose actuator faults cannot be modeled by a known exogenous system still admit both stabilization and online fault reconstruction using only the delayed output and the currently measurable part of the state. The benefit is that delay compensation, estimation, and robust stabilization are handled by separate blocks, each with its own tuning.

What carries the argument

The load-bearing mechanism is the observer-predictor pair together with the sliding variable transformation. The predictor (6), derived from the variation-of-constants formula, computes $\hat{x}_2 = e^{A_{22}\tau(t)}x_{\tau,2} + \int_{t-\tau(t)}^t e^{A_{22}(t-\theta)}A_{21}x_1(\theta)\,d\theta$, which propagates the delayed output into a current-time estimate of the unmeasured state and leaves a residual error bounded by (9). The super-twisting observer (11) drives $\tilde{x}_1 = x_1 - \hat{x}_1$ to zero in finite time and produces the signal $\hat{\xi}$ that, on the sliding manifold, equals the combined fault and uncertainty term (14). The sliding variable $s = x_1 + S_2\hat{x}_2$, with $S_2$ chosen so that $\bar{A}_{22} = A_{22} - A_{21}S_2$ has all eigenvalues in the left half-plane, reduces the closed loop to $\dot{\hat{x}}_2 = \bar{A}_{22}\hat{x}_2 + \zeta_2(x,t)$, and a standard time-delay stability argument converts the delayed uncertainty term $\zeta_2 = (1-\dot{\tau})e^{A_{22}\tau}D_2\delta(x,t-\tau)$ into the gain bound (35) and the small-gain condition (46).

What would settle it

Run the closed loop for a plant with unstable $A_{22}$ (for example a scalar system with $A_{22}=1$), a nonzero uncertainty channel $D_2$, and uncertainty $\delta(x,t)=\bar{\delta}x$; if for some initial condition $\|x(t)\|$ diverges before $s(t)=0$, then Theorem 1's gain condition (35), whose right-hand side contains $\|x(t)\|$, cannot be satisfied by the constant gain $\rho$, and the claimed global finite-time stabilization is refuted.

Watch

Extended reading notes

Core claim

The paper's central claim is Theorem 1: under assumptions (A.1)-(A.6), the control law (33) enforces the sliding mode $s = S\bar{x} = 0$ in finite time when the switching gain satisfies $\rho \ge \varphi \bar{\delta}(1+\bar{r})\|e^{A_{22}\tau}D_2\|\|x\| + \eta$, and Corollary 1 then gives asymptotic stability of the reduced closed loop once sliding holds, provided the uncertainty bound obeys the small-gain condition (46). The same construction reconstructs the actuator fault: after sliding, the observer output satisfies $\hat{\xi} = B_1 d + D_1\delta(x,t) + A_{12}\tilde{x}_2$, which reduces to $d = B_1^\dagger \hat{\xi}$ in the uncertainty-free case. The paper argues that this combines predictor-based delay compensation with second-order sliding mode observation, so the fault need not be generated by a known exogenous system and no full-state measurement is required.

Load-bearing premise

The proof assumes, rather than proves, that the state remains bounded during the transient before sliding is reached, even though the required switching gain (35) grows with the state norm; if the state can diverge in that transient, the claimed global finite-time result collapses.

Editorial extensions

If this is right

  • If the theorem holds, the same three-block structure stabilizes any plant satisfying (A.1)-(A.6) without full-state feedback and without an exogenous model for the fault.
  • The signal $\hat{\xi}$ gives an online reconstruction of the actuator fault in the uncertainty-free case and of the combined fault-plus-uncertainty effect otherwise, so the controller doubles as a diagnostic scheme.
  • Finite-time reachability of the sliding surface follows from the differential inequality $\dot{V} \le -\eta\sqrt{V}$, and asymptotic stability on the surface is governed by the small-gain bound (46) on the uncertainty magnitude $\bar{\delta}$.
  • If the uncertainty bound is relaxed to $\|\delta(x,t)\| \le \bar{\delta}_1\|x\| + \bar{\delta}_2$, the result weakens from asymptotic stability to ultimate boundedness, as the paper notes.

Reading between the lines

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

  • Beyond the paper, the gain condition (35) suggests a limitation: since $\rho$ is constant while the right-hand side grows with $\|x(t)\|$, the global claim is only as strong as the ability to keep the state bounded before sliding is reached, and an explicit reaching-phase bound would turn this into a design rule.
  • An untested extension would be to replace the constant gain $\rho$ with an adaptive or state-dependent gain that grows during the reaching phase, which could extend the result to plants whose $A_{22}$ has eigenvalues in the closed right half-plane.
  • Because the predictor error (9) grows with the delay duration, one would expect an explicit trade-off between the admissible uncertainty size $\bar{\delta}$ and the maximum delay length; deriving that trade-off in closed form would tell a designer when the method stops being practicable.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper proposes a predictor-observer sliding-mode control strategy for linear time-invariant systems with known time-varying measurement delay, partially measured state, matched actuator faults, and norm-bounded parametric uncertainties. The design combines an open-loop predictor (6) for the unmeasured delayed component, a super-twisting observer (11) that yields the combined fault/disturbance estimate (14), and a sliding-mode law (33). The main theoretical claims are finite-time reachability of the sliding surface under the gain condition (35), asymptotic stability once sliding is reached under the small-gain condition (46), and actuator fault reconstruction. Numerical simulations are presented for nominal and uncertain cases.

Significance. The architecture is plausible and, if the main theorem were correct, the paper would make a useful contribution: fault reconstruction for unstructured disturbances combined with delay compensation and partial-state output feedback. The algebraic derivations of the predictor, the disturbance cancellation step leading to (29), and the sliding dynamics (34) are explicit and checkable, and no parameters are fitted to data. However, the central global reachability claim is not established. The proof of Theorem 1 uses a gain condition that depends on the full state norm while the implemented gain is constant, and it invokes a Razumikhin bound that is neither assumed nor derived. The numerical example does not instantiate the assumed output structure and retunes the gain, so it cannot compensate for the missing proof.

major comments (3)
  1. [Theorem 1, Eqs. (33), (35), (43)] The switching gain rho in the control law (33) is a fixed constant, but the sufficient condition (35)/(43) requires rho >= phi * delta_bar * (1 + r_bar) * ||exp(A22*tau)*D2|| * ||x|| + eta, whose right-hand side contains the full state norm at the current time. For a plant with at least one eigenvalue in the closed right half-plane, no constant rho can dominate this bound along an a priori unbounded trajectory; proving that ||x|| remains bounded during the reaching phase is exactly what the theorem must establish, and the proof offers no such bound before s = 0. The first equation in (34) does not preclude growth of s, and hence growth of x1 and x2, before sliding is reached. Therefore global finite-time reachability with a constant gain is not proved.
  2. [Theorem 1 proof, Eqs. (40)-(41)] The proof states "recall that the condition ||x(t+theta)|| < phi*||x(t)|| holds" and then uses it to replace ||delta(x,t-tau)|| by phi*delta_bar*||x(t)||. This Razumikhin condition is not listed in assumptions (A.1)-(A.6) and is not derived from the closed-loop equations (34). In a valid Razumikhin argument, the derivative inequality must be shown to hold whenever the Razumikhin bound holds; here the resulting sufficient condition (43) itself involves ||x(t)||, so the chain (40)-(44) is circular. The same issue affects the observer-predictor interconnection: the predictor error bound (9) requires a bound on sup over theta in [t-tau(t), t] of ||x(theta)||, which is not available during the reaching phase, so the finite-time convergence of the super-twisting observer used in the disturbance cancellation (28) is also not established.
  3. [Section 5, Eqs. (54) and the output definition in Section 2] The numerical example does not match the problem formulation. In Section 2 the output is y = x_tau,2 = x2(t - tau(t)), but the example takes C = [1 0], which gives y(t) = x1(t - tau(t)). Moreover, the uncertain case changes the switching gain from rho = 2 to rho = 5, so the simulations do not demonstrate that a fixed gain selected according to (35) achieves global stabilization; at best they show local behavior for a particular trajectory.
minor comments (4)
  1. [Assumption (A.4) and figure captions] Assumption (A.4) misspells "controllable", and the figure axes use "Tem po (s)" instead of "Time (s)".
  2. [Equations (1)-(2)] The relationship between the general output y(t) = C*x(t - tau(t)) in (1) and the specialized output y = x_tau,2 in (2) should be made explicit, for example by stating that after partitioning the state, C = [0 I_p].
  3. [Equation (14) and Section 3.3] In the presence of uncertainties, xi_hat estimates B1*d + D1*delta + A12*x2_tilde, so the statement that d can be directly reconstructed should be carefully qualified to the uncertainty-free case; the qualification appears later in Section 3.3, but it would help to state it at the point of equation (14).
  4. [Corollary 1 proof] The proof uses "Reminding that A22_bar is Hurwitz by design" and a condition (46) involving phi; it would help to state explicitly that phi is the Razumikhin constant and that the resulting condition is only sufficient, not necessary.

Circularity Check

1 steps flagged · score 5.0 of 10

Theorem 1's proof is circular: the state-dependent gain condition (35) and the unproved Razumikhin bound — “recall that ∥x(t+θ)∥ < φ∥x(t)∥ holds” — presuppose the closed-loop boundedness that the theorem must establish.

  1. self definitional [Section 4.2, Theorem 1, Eqs. (35) and (41)–(43)]
    "“The control law (33) applied to (2), under the assumptions (A.1) to (A.6), with ρ≥φ δ̄(1+ r̄)‖eA22τD2‖‖x‖+ η ... guarantees the existence of the sliding mode s =S x̄≡ 0 in finite time. ... To begin with, recall that the condition ‖x(t+θ)‖ < φ‖x(t)‖ holds for some φ > 1 and θ∈[− τ(t), 0].”"

    The theorem’s hypothesis (35) is a condition on the time-varying full state norm ‖x(t)‖, while the control law (33) uses a fixed constant ρ. Satisfying (35) for all t during the reaching phase requires an a priori bound on ‖x‖; but boundedness (or the Razumikhin ratio bound ‖x(t+θ)‖ < φ‖x(t)‖) is exactly what the theorem must prove. The proof invokes this bound as a ‘recall’ rather than deriving it from (34). For plants with A22 having eigenvalues in the closed right half-plane, no fixed ρ can dominate φδ̄(1+r̄)‖e^{A22τ}D2‖‖x‖ unless ‖x‖ is already known to be bounded, so the reachability conclusion is presupposed rather than derived.

full rationale

The constructive parts of the paper are standard and not circular: the predictor (6) is the variation-of-constants formula, the STA observer gains are taken from the external reference [20], and the control law (33) is a standard equivalent-control-plus-switching design. No parameter is fitted to data and renamed a prediction, and the self-citations [26,27] are not load-bearing for the main derivation. The circularity is concentrated in the stability proof of Theorem 1. Condition (35) is written as ρ ≥ φδ̄(1+r̄)‖e^{A22τ}D2‖‖x‖ + η with ρ fixed in (33); hence the theorem’s premise already contains the state norm whose boundedness is the conclusion. The proof then derives V˙ ≤ −‖s‖[ρ − φδ̄(1+r̄)‖e^{A22τ}D2‖‖x(t)‖] using ‖x(t+θ)‖ < φ‖x(t)‖, which is neither an assumption (A.1)–(A.6) nor proved from the closed-loop equations (34). The same unproved boundedness appears in the observer error bound (9), where ‖x̃2‖ is bounded by τδ̄‖e^{A22τ}D2‖ sup_θ‖x(θ)‖. Consequently, the finite-time reachability claim does not follow from the stated assumptions; it is conditional on the very bound the theorem establishes. This is a genuine circular proof step, though it does not make the entire construction a renamed fit or a self-citation artifact, so the score is moderate rather than maximal.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

No invented physical entities, forces, or conserved quantities are introduced. The free parameters are standard control design gains, with rho the most consequential: the reachability theorem demands a state-dependent bound that the constant-gain law cannot meet. The axioms include the stated problem assumptions (A.1)-(A.6) and two implicit premises: B1 square invertible for exact cancellation in (29), and the Razumikhin norm condition assumed along trajectories in the proofs. The STA finite-time convergence is imported from [20] without verifying its perturbation bounds.

free parameters (4)
  • switching gain rho = rho = 2 (nominal case); rho = 5 (uncertain case)
    Constant switching gain in control law (33). Theorem 1 requires rho >= phi*delbar*(1+rbar)*||exp(A22*tau)*D2||*||x|| + eta, a state-dependent bound, but the law uses a fixed rho; in the example rho = 5 was tuned to keep the closed loop stable, with the unstable curves omitted.
  • super-twisting observer gains k1..k4 = k1 = k3 = 5, k2 = k4 = 2
    Observer gains in (11), selected following [20, eq. (25)]; the perturbation bounds required by that gain rule, involving x_tilde_2 and ||x||, are not verified in the paper.
  • sliding surface coefficient S2 = -5
    Sliding surface s = x1 + S2*x_hat_2 in Section 5; chosen so that A_bar_22 = A22 - A21*S2 = -14 is Hurwitz (pole placement).
  • Razumikhin constants phi > 1 and eta > 0 = free constants in (35) and (46)
    Arbitrary constants in the stability conditions; condition (46) can be satisfied for any finite delbar by taking phi close to 1, making the small-gain condition effectively non-restrictive.
assumptions (5)
  • domain assumption Assumptions (A.1)-(A.6): x1 measured, d(t) bounded by alpha, |tau_dot| <= rbar < 1, (A,B) controllable, B full rank, ||delta|| <= delbar*||x||
    Stated in Section 2; they define the problem class and are used throughout the derivations.
  • domain assumption B1 is square and invertible, so (SB) is nonsingular and the exact disturbance cancellation in (29) holds
    Implicit in the design of S in Section 4.1: with S = [I S2], SB = B1 must be nonsingular. Required for equation (29) to hold exactly; never stated explicitly as an assumption.
  • domain assumption The delay tau(t) is exactly known and the variation-of-constants predictor (6) is valid
    Given in the problem statement and Section 3.1; the predictor is exact only when delta = 0, and its residual error is bounded by (9).
  • ad hoc to paper Razumikhin norm condition ||x(t+theta)|| <= phi*||x(t)|| holds along closed-loop trajectories
    Invoked in the proofs of Theorem 1 and Corollary 1 (Section 4.2) to bound the delayed uncertainty; not derived from the system assumptions and dependent on the trajectory.
  • standard math Finite-time convergence of the super-twisting observer with gains from [20, eq. (25)]
    Cited result (Nagesh and Edwards 2014) used in Section 3.2; the required perturbation bounds depend on ||x|| over the delay interval and are not established here.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Sliding Mode Control for Uncertain Systems with Time-Varying Delays via Predictor Feedback and Super-Twisting Observer." pith.science (2026). https://pith.science/paper/CGNQZJN5

@misc{pith2026250721281,
  author       = {Pith},
  title        = {Pith review of: Sliding Mode Control for Uncertain Systems with Time-Varying Delays via Predictor Feedback and Super-Twisting Observer},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CGNQZJN5}},
  note         = {Machine review of arXiv:2507.21281}
}
read the original abstract

This paper introduces a novel stabilization control strategy for linear time-invariant systems affected by known time-varying measurement delays and matched unknown nonlinear disturbances, which may encompass actuator faults. It is considered that part of the state vector is not available for real-time measurement. To address this, the proposed approach combines an open-loop predictor with a state observer designed using the Super-Twisting Algorithm, aiming to compensate for the delays and estimate the unmeasured state components. Specifically, the nonlinear observer-based framework enables the reconstruction of unmodeled fault signals without assuming that they originate from a known exogenous system, offering robustness against parametric uncertainties. Meanwhile, the predictor forwards the delayed output in time. Subsequently, a sliding mode control law is formulated to enforce an ideal sliding mode and ensure global stabilization, even under a broader class of perturbations, unmodeled disturbances, parametric uncertainties, and delays, owing to the integration of the Super-Twisting observer. Numerical simulations illustrate the efficiency of the proposed approach.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

34 extracted references · 34 canonical work pages

  1. [1]

    and Lamnabhi-Lagarrigue, F

    Ahmed-Ali, T., Cherrier, E. and Lamnabhi-Lagarrigue, F . Cascade high gain predictors for a class of nonlinear systems, IEEE T. Automat. Contr. 57(1): 224–229, 2012

    work page 2012

  2. [2]

    Linear systems with delayed controls: A reduction, IEEE T

    Artstein, Z. Linear systems with delayed controls: A reduction, IEEE T. Automat. Contr. 27(4): 869–879, 1982

    work page 1982

  3. [3]

    and Krstic, M

    Bekiaris-Liberis, N. and Krstic, M. Compensation of state-dependent input delay for nonlinear systems, IEEE T. Automat. Contr. 58(2): 275–289, 2013

    work page 2013

  4. [4]

    and Krstic, M

    Bresch-Pietri, D. and Krstic, M. Adaptive trajectory tracking despite unknown input delay and plant parameters, Auto- matica 45(9), 2009

    work page 2009

  5. [5]

    J., Buzzard, G

    Chakrabarty, A., Corless, M. J., Buzzard, G. T., Zak, S. H. and Rundell, A. E. State and unknown input observers for nonlinear systems with bounded exogenous inputs, IEEE Transactions on Automatic Control 62(11): 5497–5510, 2017

    work page 2017

  6. [6]

    and Moreno, J

    Chalanga, A., Kamal, S., Fridman, L., Bandyopadhyay, B. and Moreno, J. A. Implementation of super-twisting control: Super-twisting and higher order sliding-mode observer-based approaches, IEEE Transactions on Industrial Electronics 63(6): 3677–3685, 2016

    work page 2016

  7. [7]

    L., Oliveira, T

    Coutinho, C. L., Oliveira, T. R. and Cunha, J. P . V. S. Output-feedback sliding mode control via cascade observers for global stabilisation of a class of nonlinear systems with output delay, Int. J. Control 87(11): 2327–2337, 2014

    work page 2014

  8. [8]

    and Spurgeon, S

    Edwards, C. and Spurgeon, S. Sliding Mode Control: Theory And Applications , T aylor & Francis, New Y ork, 1998

    work page 1998

Show all 34 references
  1. [9]

    Edwards, C., Spurgeon, S. K. and Patton, R. J. Sliding mode observers for fault detection and isolation, Automatica 36(4): 541–553, 2000

    2000

  2. [10]

    Gu, K., Kharitonov, V. L. and Chen, J. Stability of Time-Delay Systems (Control Engineering) , 1 edn, Birkhauser, Boston, 2003

    2003

  3. [11]

    and Krstic, M

    Holloway, J. and Krstic, M. Predictor observers for proportional navigation systems subjected to seeker delay, IEEE T. Contr. Syst. T. 24(6): 2002–2015, 2016

    2002

  4. [12]

    and Zak, S

    Hui, S. and Zak, S. Observer design for systems with unknown inputs, International Journal of Applied Mathematics and Computer Science 15: 431–446, 2005

    2005

  5. [13]

    Compensation of infinite-dimensional actuator and sensor dynamics, IEEE Contr

    Krstic, M. Compensation of infinite-dimensional actuator and sensor dynamics, IEEE Contr. Syst. Mag. 30(1): 22–41, 2010

    2010

  6. [14]

    Lyapunov stability of linear predictor feedback for time-varying input delay, IEEE T

    Krstic, M. Lyapunov stability of linear predictor feedback for time-varying input delay, IEEE T. Automat. Contr.55(2): 554– 559, 2010

    2010

  7. [15]

    and Plestan, F

    Léchappé, V., Moulay, E. and Plestan, F . Prediction-based control of LTI systems with input and output time-varying delays, Systems & Control Letters 112: 24–30, 2018

    2018

  8. [16]

    and Chriette, A

    Léchappé, V., Moulay, E., Plestan, F ., Glumineau, A. and Chriette, A. New predictive scheme for the control of LTI systems with input delay and unknown disturbances, Automatica 52: 179–184, 2015

    2015

  9. [17]

    Sliding order and sliding accuracy in sliding mode control, International Journal of Control 58(6): 1247–1263, 1993

    Levant, A. Sliding order and sliding accuracy in sliding mode control, International Journal of Control 58(6): 1247–1263, 1993

    1993

  10. [18]

    and Raskin, N

    Mirkin, L. and Raskin, N. Every stabilizing dead-time controller has an observer-predictor structure, Automatica 39(10): 1747–1754, 2003. Hardy Pinto et al. 13

    2003

  11. [19]

    Moreno, J. A. and Osorio, M. A Lyapunov approach to second-order sliding mode controllers and observers, 2008 47th IEEE Conference on Decision and Control, pp. 2856–2861, 2008

    2008

  12. [20]

    and Edwards, C

    Nagesh, I. and Edwards, C. A multivariable super-twisting sliding mode approach, Automatica 50(3): 984–988, 2014

    2014

  13. [21]

    R., Cunha, J

    Oliveira, T. R., Cunha, J. P . V. S. and Hsu, L. Adaptive sliding mode control based on the extended equivalent control concept for disturbances with unknown bounds, Advances in Variable Structure Systems and Sliding Mode Control—Theory and Applications, Springer International...

    2017

  14. [22]

    Oliveira, T. R. and Krstić, M. Extremum Seeking through Delays and PDEs, Society for Industrial and Applied Mathematics, 2022

    2022

  15. [23]

    R., Peixoto, A

    Oliveira, T. R., Peixoto, A. J., Nunes, E. V. L. and Hsu, L. Control of uncertain nonlinear systems with arbitrary relative degree and unknown control direction using sliding modes, Int. J. Adapt. Control Signal Process. 21(8-9): 692–707, 2007

    2007

  16. [24]

    R., Rodrigues, V

    Oliveira, T. R., Rodrigues, V. H. P ., Krstić, M. and Başar, T. Nash equilibrium seeking in quadratic noncooperative games under two delayed information–sharing schemes, J. Optim. Theory Appl. 191: 700–735, 2021

    2021

  17. [25]

    Patton, R. J. and Chen, J. Optimal unknown input distribution matrix selection in robust fault diagnosis, Automatica 29(4): 837–841, 1993

    1993

  18. [26]

    Pinto, H. L. C. P ., Oliveira, T. R., Hsu, L. and Krstic, M. Sliding mode control for disturbance rejection in systems with measurement delay using PDE-backstepping predictor, American Control Conference, pp. 4099–4104, 2018

    2018

  19. [27]

    Pinto, H. L. C. P ., Oliveira, T. R., Rodrigues, V. H. P ., Hsu, L. and Krstic, M. Sliding mode control for disturbance rejection and estimation under measurement delay with PDE-backstepping predictor, Int. J. Robust Nonlin. , 13(5): 2865–2886, 2024

    2024

  20. [28]

    Rodrigues, V. H. P . and Oliveira, T. R. Global adaptive HOSM differentiators via monitoring functions and hybrid state- norm observers for output feedback, International Journal of Control 91(9): 2060–2072, 2018

    2018

  21. [29]

    and Levant, A

    Shtessel, Y., Edwards, C., Fridman, L. and Levant, A. Sliding Mode Control and Observation, Springer, New Y ork, 2014

    2014

  22. [30]

    Smith, O. J. M. Closer control of loops with dead time, Chem. Eng. Prog. 53(5): 217–219, 1957

    1957

  23. [31]

    and Krstic, M

    Smyshlyaev, A. and Krstic, M. Backstepping observers for a class of parabolic PDEs, Syst. Control Lett. 54(7): 613–625, 2005

    2005

  24. [32]

    Introduction to Linear Algebra, 4th edn, Wellesley Cambridge Press, 2009

    Strang, G. Introduction to Linear Algebra, 4th edn, Wellesley Cambridge Press, 2009

    2009

  25. [33]

    and Edwards, C

    T an, C. and Edwards, C. Sliding mode observers for robust detection and reconstruction of actuator and sensor faults, Int. J. Robust Nonlin. 13(5): 443–463, 2003

    2003

  26. [34]

    Utkin, V. I. Sliding Modes in Control and Optimization (Communications and Control Engineering) , Springer, Berlin, 1992. 14 Hardy Pinto et al. 0 1 2 3 4 5 Te mpo (s) −0.2 0.0 0.2 0.4 d B† 1 ˆξ F I G U R E 3 Without Parametric Uncertainties: disturbance d(t) (in blue) and its ...

    1992

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.