Sliding-Mode Observers: Principles & Applications

Updated 19 January 2026

Sliding-Mode Observers (SMOs) are robust state and unknown input estimators that force error dynamics onto a designed sliding manifold, ensuring insensitivity to many uncertainties.
They employ discontinuous or high-order correction laws to guarantee finite-time convergence and maintain performance even in the presence of significant disturbances.
SMOs are applied across fault detection, sensorless control, and robust state estimation in nonlinear and time-varying systems in both academic and industrial contexts.

A Sliding-Mode Observer (SMO) is a robust state and unknown input estimator constructed by enforcing the estimation error trajectory onto a “sliding manifold,” characterized by a discontinuous or high-order continuous injection, such that the error dynamics become insensitive to a broad class of uncertainties and disturbances once in sliding motion. Since the introduction of SMOs, their core properties—finite-time convergence, non-asymptotic invariance with respect to bounded perturbations, and incorporation of algebraic or high-order correction laws—have catalyzed significant advances in fault detection, unknown input estimation, nonlinear and time-varying system observation, and sensorless control across both academic and industrial control systems.

1. Mathematical Foundations and Canonical SMO Structures

SMOs leverage the fundamental sliding-mode control principle: by injecting a discontinuous (or higher-order continuous) correction term into a system copy, the trajectory of a measurable function of the estimation error is forced onto a designed sliding manifold. For a typical observable linear system $\dot x = A x + B u + D w ,\; y = C x$ , the prototypical first-order SMO takes the form: $\dot{\hat x} = A \hat x + B u + L(y - C \hat x) + G\,\operatorname{sign}(y - C \hat x).$ Here, the injection gain $G$ is chosen based on bounds of the unknowns $w(t)$ or matched disturbances, and the sliding variable is often the output estimation error $s = y - C \hat x$ or, for more complex structures, combinations of error and its derivatives (Chakrabarty et al., 2015, Niederwieser et al., 2021).

Higher-order SMOs, most notably those employing the “super-twisting” or generalized homogeneous injection, combine continuous functions of the error and its derivatives to impose sliding up to a higher order, e.g.,

$\mu(e) = \lambda |e|^{1/2} \operatorname{sign}(e) + \alpha \int_0^t \operatorname{sign}(e) d\tau.$

This design enables not only first-order variable convergence but also the annihilation of the equivalent dynamics' derivative, achieving a so-called “second-order sliding mode" (Mousavi et al., 2017, Liu et al., 2013, Ganguly et al., 2020).

2. Structure of the Sliding Manifold and Reaching Laws

The sliding manifold, $S(e) = 0$ , is typically a codimension- $q$ surface in error space constructed to ensure invariance of the error dynamics transverse to uncertain and unknown inputs. In linear applications, $S(e) = F (y - C \hat x)$ for an appropriately designed $F$ . For higher-order, nonlinear, or fractional systems, step-by-step or chain-of-integrators constructions are required, embedding unknowns as state components and designing successive surfaces: $S_i(t) = e_i(t), \quad i=1,\ldots,n,$ with each layer employing hierarchical correction laws, yielding a “step-by-step super-twisting” observer (Mousavi et al., 2017, Mousavi et al., 2017). Fractional-order SMOs forward these ideas with Caputo derivatives, extending classical convergence analysis to fractional-order error dynamics (Mousavi et al., 2017, Mousavi et al., 2017).

In hybrid or time-varying systems, advanced observability concepts such as $Z(T_N)$ -observability are used. Here, the manifold is piecewise observable under switching sequences, and the observer structure is adaptive to the hybrid regime (Liu et al., 2013).

3. Convergence, Robustness, and Stability Analysis

Finite-time convergence of SMOs, notably in the presence of bounded disturbances and exogenous unknowns, is established by Lyapunov or homogeneity arguments. For classical or boundary-layer SMOs, convergence to a sliding manifold neighborhood is exponential, with the ultimate bound directly determined by the smoothing parameter (boundary-layer width) and observer gains: $\limsup_{t\to\infty} \| e(t) \| \leq \sqrt{\frac{\mu \varepsilon \rho}{\alpha}},$ where $\varepsilon$ is the boundary-layer thickness and $\rho$ the relay’s gain (Chakrabarty et al., 2015).

For higher-order and fractional-order SMOs, Lyapunov functions take the form $V_i = \frac{1}{2} \xi_2^2 + \omega | \xi_1 |^{2-q}$ for appropriately chosen injection exponents $q < 1$ (Mousavi et al., 2017). These ensure convergence rates governed by fractional-order Grönwall-Bellman inequalities.

In set-valued or nonstandard uncertainties, the estimation error $e$ is shown to converge to an interval determined by the projection of the uncertainty onto the observable subspace, and if the unobservable uncertainty vanishes, convergence is exact. For cases where uncertainties do not vanish, interval bounding is used: $\|e(t)\| \leq \frac{\lambda_{\max}\|P\|k}{\lambda_{\min}\epsilon} + \dots$ for an explicit set-bound $k$ on the unobservable uncertainty component (Adly et al., 2024).

Adaptive-gain SMOs automatically tune the discontinuous gain to the current magnitude of the error, preventing overestimation and concomitant chattering, and ensuring reachability without prior knowledge of disturbance bounds (Bahrami et al., 2020).

4. Chattering Mitigation and High-Order/Continuous SMO Advances

Chattering—high-frequency switching inherent to discontinuous injection—has prompted extensive research into both higher-order SMOs and continuous approximations. Super-twisting and similar second-order schemes reduce or even eliminate chattering by replacing discontinuous relay terms with continuous, albeit non-Lipschitz, injections that depend on powers of the error and its integrals. For instance, the fractional super-twisting observer reports dramatic chattering attenuation compared to first-order SMOs while maintaining fast convergence (Mousavi et al., 2017).

Continuous approximations, including smooth boundary-layer relays and fuzzy-inference replacements for sign-functions, permit further reduction of chattering, at a controlled cost in convergence speed. Time-varying continuous approximations such as $\operatorname{Sign}_\delta$ offer joint continuity and exponentially decaying ultimate bounds, preserving rapid convergence while providing implementational smoothness (Adly et al., 2023, Belhadjoudja, 2021).

5. Observer Design under Non-Standard Conditions: Unmatched Disturbances and Structural Limitations

Traditional SMOs are predicated on “matching conditions,” requiring disturbances or unknowns to enter through the same channel as the injection term. Recent advancements extend SMOs to unmatched uncertainty regimes. For high-DOF Euler-Lagrangian systems, SMOs are erected on augmented linearizations in momentum coordinates, enforcing sliding on subspaces corresponding to generalized positions or momenta, decoupling the need for inverse inertia and filtering of noisy velocities (Zhang et al., 2023).

For general nonlinear, descriptor, or set-valued Lur'e systems, the observer employs manifold projections of the uncertainty, with sliding injection tailored to the observable uncertainty projection while concurrently bounding the unobservable component’s impact on state error (Adly et al., 2024, Adly et al., 2023).

SMOs have also been extended to nonlinear triangular, hybrid, and time-varying systems via higher-order observers, stepwise fractional designs, and cascaded HOSM structures, achieving finite-time exact reconstruction despite matched/unmatched unknowns or severely time-varying system matrices (Mousavi et al., 2017, Niederwieser et al., 2021, Tranninger et al., 2018).

6. Application Domains and Recent Developments

SMOs are widely applied in sensorless mechanical and electrical systems, robust battery state estimation, cyber-physical system attack reconstruction, and fault detection. Notably:

In electrochemical battery SOC estimation, adaptive dead-zone dual SMOs integrate a Lyapunov-derived threshold to pause unnecessary parameter updates, balancing convergence with computational efficiency and outperforming extended Kalman filters under aging and parameter drift (Hu et al., 29 Aug 2025).
In robotic systems, SMOs enable accurate real-time torque estimation for collaborative high-DOF manipulators without requiring matching, inverse inertia, or velocity estimation, essential for human-safe performance (Zhang et al., 2023).
For secure state estimation in cyber-physical systems, SMOs combined with adaptive gains reconstruct both malicious sensor and state attacks, enabling real-time attack “clean-up” with rigorous finite-time guarantees (Nateghi et al., 2019).
For nonlinear descriptor systems, boundary-layer SMOs provide ultimate error bounds and, with windowed post-filters, afford arbitrarily accurate state and exogenous input reconstruction, even under non-Lipschitz dynamics (Chakrabarty et al., 2015).

A recent advance proposed the use of a cascade SMO structure: two sliding-mode observer layers with multiple discontinuous functions. This cascade significantly smooths the estimated state and mitigates high-frequency control efforts, reducing chattering and RMS control energy compared to single-layer SMOs, particularly under measurement noise (Sun et al., 2023).

7. Tuning Procedures and Implementation Considerations

Observer gains must be selected to ensure the sliding reachability condition in the presence of worst-case disturbance bounds. Higher-order and adaptive-gain SMOs alleviate the need for conservative gain overestimation. In fractional or stepwise observers, the phase-wise design requires layer-dependent gain tuning, often via simulation or convex optimization (Mousavi et al., 2017, Liu et al., 2013).

Numerical implementation requires discretization of Caputo or Grünwald–Letnikov derivatives for fractional systems. Smoothing parameters and boundary layers are tuned to trade off between chattering and steady-state error. For set-valued or hybrid-system SMOs, LMI-based synthesis provides a computationally feasible framework for finding admissible gain sets and verifying stability criteria (Adly et al., 2023, Adly et al., 2024).

Hardware implementation, as demonstrated in real-time MagLev and robot manipulator experiments, confirms the practical relevance of SMOs, with convergence timescales of milliseconds and maintained robustness under model uncertainties, measurement noise, and abrupt external perturbations (Ganguly et al., 2020, Zhang et al., 2023).

References (by representative arXiv ids): (Mousavi et al., 2017, Liu et al., 2013, Bahrami et al., 2020, Zhang et al., 2023, Sun et al., 2023, Adly et al., 2024, Mousavi et al., 2017, Hu et al., 29 Aug 2025, Belhadjoudja, 2021, Adly et al., 2023, Chakrabarty et al., 2015, Tranninger et al., 2018, Niederwieser et al., 2021, Ganguly et al., 2020, Nateghi et al., 2019)