Sliding Mode Controller Overview

• Sliding Mode Controller (SMC) is a robust nonlinear control technique that uses discontinuous feedback to force system trajectories onto a sliding surface despite model uncertainties and external disturbances.
• SMC achieves finite-time convergence and improved performance through carefully designed sliding surfaces and reaching laws that mitigate issues like chattering.
• Advanced SMC variants, including discrete-time, fractional-order, and neuro-adaptive methods, offer efficient digital implementations and enhanced robustness for diverse applications.

Sliding Mode Controller (SMC) is a class of robust nonlinear control schemes that leverage variable structure systems and discontinuous feedback to force system trajectories to a designated manifold (the "sliding surface") and to maintain invariance of this manifold in the face of model uncertainty, external disturbances, and unmodeled dynamics. SMC has been developed in continuous-time, discrete-time, integer- and fractional-order, and hybrid forms, with mathematically rigorous performance guarantees derived via Lyapunov or invariance principles. Its key features include finite-time convergence, insensitivity to matched uncertainties, and amenability to efficient real-time digital implementation. The method is applicable to a wide range of systems, from industrial servos and piezo actuators to high-dimensional multi-agent robotics and stochastic systems.

1. Mathematical Foundations and Sliding Surface Design

The foundation of SMC is the enforcement of sliding behavior: for a system
[
\dot x = f(x) + g(x)u + d(x),
]
the designer prescribes a sliding variable (switching function) ( s(x) ) (scalar or vector), often a combination of state errors and their derivatives, such as
[
s(x) = Cx - r \quad \text{(regulation)}, \quad s(x) = k_1 e + k_2 \dot e \quad \text{(tracking)}.
]
In higher-order systems, sliding variables may involve derivatives up to the system's relative degree. In PID-type SMC, the sliding surface is synthesized as
[
s(t) = K_P e(t) + K_I \int_0^t e(\tau)d\tau + K_D \dot e(t),
]
where the integral term ensures finite-time convergence even for large initial errors, as detailed in [2207.11129], [2508.06567].

Well-posedness and finite-time reachability of the sliding manifold are ensured by appropriate selection of switching gains and possible inclusion of anticipatory terms or multiple sliding phases, as in hybrid or suboptimal SMC schemes [2305.07891], [1706.01951].

For MIMO and affine systems with uncertain high-frequency gain matrices, sliding manifolds are constructed componentwise, and the control law is derived via nonlinear vector equations and convex cone analysis to guarantee solution uniqueness [1808.04627].

2. Reaching Laws, Chattering, and Robustness to Uncertainty

The reaching law governs the convergence of the sliding variable to zero. Classical first-order SMC uses discontinuous switching:
[
\dot s = -\eta\,\mathrm{sgn}(s), \; \eta > 0,
]
enforcing finite-time convergence but potentially generating high-frequency control "chattering." Modern variants employ improved reaching laws—exponential, power-rate, and power-rate exponential forms—to accelerate approach while mitigating chattering:
[
\dot s = -k s - k_{sc}|s|^a \mathrm{sat}(s/\phi), \qquad 0 \le a \le 2,
]
with the boundary-layer function "sat" yielding a continuous control within (|s| < \phi) [2207.11129], [2508.06567].

Second-order and higher-order sliding mode control shifts the discontinuity into higher derivatives:
[
\xi(k) = s(k+1) + \beta s(k),\qquad \text{enforce } \xi(k) = 0, \xi(k+1) = 0,
]
resulting in a smoother control law and significant reduction in chattering, especially in digital implementations [1706.01951], [1805.06800]. Adaptive laws are embedded via Lyapunov analysis to compensate model uncertainties online, ensuring sustained robustness as demonstrated in automotive and piezoelectric systems [1706.01951], [1805.06800], [2007.15462].

3. Extensions: Discrete-Time, Fractional-Order, Data-Driven, and Neuro-Adaptive SMC

Discrete-Time SMC

Digital implementation of SMC requires addressing sampling imprecisions and quantization effects. Second-order discrete sliding mode design, with adaptation, guarantees performance even in low-resolution, high-noise settings, achieving up to 90% improvement in tracking error compared to first-order methods [1805.06800]. Multirate output feedback (MROF) SMC obviates the need for a full state observer by reconstructing state using high-rate measured outputs, offering low-chattering and strong robustness to mismatched disturbances [2110.12363].

Fractional-Order and Hybrid SMC

Fractional-order SMC generalizes the sliding variable and control law through fractional calculus, enabling enhanced tracking precision and chattering minimization, especially for highly nonlinear or distributed-parameter systems such as ultrasonic motors [2103.12327]. Short-memory implementations manage the computational burden of fractional operators in real time, and neural-network compensators are integrated to mitigate residual errors arising from approximate fractional calculations.

Hybrid and sub-optimal SMC laws, such as energy-saving three-level (ON-OFF-ZERO) switching, support constraints including actuator saturation and fuel/power minimization while maintaining finite-time convergence guarantees for double-integrator systems [2305.07891].

Data-Driven and RL-Enhanced SMC

In partially known dynamics or high-uncertainty environments, SMC design is enhanced with data-driven approaches. Semidefinite programming is adopted to compute nominal gains from trajectory data, while the sliding regime is reached and maintained via an SMC law with a robust, data-dependent structure [2403.16136]. For nonlinear plants with unknown or fast-varying dynamics, SMC is combined with deep reinforcement learning (e.g., DDPG), which learns on-policy corrections to the SMC output, resulting in superior tracking under severe model mismatch [2205.02975].

Deep neuro-adaptive SMC architectures embed online-trained deep neural networks to approximate unknown nonlinearities in high-order, multi-agent, or consensus systems. Adaptive weight update laws are derived for both inner and output layers, ensuring robust synchronization and invariance by restricting system trajectories within compact sets, supported by Lyapunov analysis employing set-theoretic restricted potentials [2507.21667].

4. Practical Applications and Performance

SMC has seen widespread application in domains requiring stringent robustness, rapid transition, and uncertainty tolerance:

• Servo drives and electric motors: SMC and its PID/PI hybrids achieve fast convergence (e.g., <1 s settling), negligible steady-state error, and substantial resilience against disturbances and parameter drift, greatly outperforming classical PID controllers [1202.1742], [2508.06567].
• Automotive and engine control: Second-order adaptive discrete SMC reduces chattering and sustains trajectory tracking despite harsh sampling and 25% or more model uncertainties, as validated on SI engine cold-starts and exhaust regulation [1706.01951], [1805.06800].
• Precision actuators: For piezoelectric and ultrasonic motors, SMC with partial model compensation and, when appropriate, fractional order design with neural compensation, yields sub-micrometer MAE/RMSE, reduced voltage chattering, and fast correction after setpoint changes [2007.15462], [2103.12327].
• Structural control and seismically excited buildings: SMC designs using sliding-surface tuning (Ackermann’s formula) meet frequency-domain performance and amplitude constraints, attaining 87% RMS displacement reduction and outperforming LQR/OSMC in experimental shake-table tests [2012.05966].
• Robotics: Tendon-driven robotic wrists and mixed-conventional/braking actuation mobile robots exploit SMC's invariance and discontinuity-handling for precise trajectory tracking, achieving rapid settling and sub-degree accuracy despite abrupt actuator transitions or hybrid ON/OFF dynamics [2601.07009], [1909.06478].
• Distributed and multi-agent networks: Deep neuro-adaptive SMC ensures consensus and synchronization in leader-follower and networked nonlinear systems with unknown or time-varying couplings [2507.21667].

5. Stability, Lyapunov Analysis, and Invariance Principles

Lyapunov theory is central to SMC analysis in both continuous and discrete time. Typical candidate functions include
[
V(s) = \frac{1}{2} s^2, \qquad V(k) = \tfrac{1}{2}\left[s^2(k+1) + \beta s^2(k)\right] + \cdots
]
Finite-time reachability to the manifold is established by ensuring
[
\dot V = s \dot s \leq -\eta|s| < 0, \quad \text{or} \quad \Delta V(k) \leq -c s^2(k) \leq 0,
]
where the dissipativity is enforced via appropriate gain bounds on the discontinuous terms. Lyapunov-difference methods, discrete invariance principles, and LaSalle–Yoshizawa-type arguments extend the analysis to sampled, switched, or non-smooth settings [1706.01951], [1805.06800], [2110.12363], [2507.21667].

In systems with integral sliding, set-theoretic, or barrier-type Lyapunov functions, trajectories are guaranteed to remain within the region of validity for neural/DNN approximators [2507.21667]. For phase-field and PDE control, Lyapunov functional and comparison principles confirm convergence and invariance even in infinite-dimensional settings [1706.03564].

6. Implementation, Tuning, and Limitations

SMC algorithms admit a wide range of digital and embedded realization modalities. Guidelines stress:

• Boundary-layer width ((\phi)) selection to trade-off chattering versus precision.
• Adaptive gain and sliding surface parameter optimization, often leveraging meta-heuristics such as modified PSO to minimize integral-squared error or control effort [2207.11129].
• Multirate sampling and discretization schemes for tight QSM bands [2110.12363].
• Model identification and partial model compensation, especially in systems with hard nonlinearities or input constraints [2007.15462].
• Explicit inclusion of actuator dynamics, as in sub-optimal and energy-efficient SMC [2305.07891].
• Handling of mismatched uncertainties via observer-based or neuro-fuzzy augmentation [2103.11277].

Limitations include residual chattering when boundary-layers are narrow, compromised robustness under severe mismatched or unmodeled dynamics unless observer/learning mechanisms are included, and possible destabilization from excessive gain, actuator quantization, or sampling delays.

7. Comparative Performance and Current Research Trends

Quantitative metrics consistently demonstrate SMC's superiority over classical control techniques in terms of tracking error, settling time, steady-state error, and robustness under both simulation and experimental conditions [2508.06567], [1202.1742], [2007.15462], [2110.12363]. Research focus has shifted towards:

• High-order, adaptive, and fractional variants for improved noise and disturbance rejection [1805.06800], [2103.12327].
• Integration with data-driven, deep learning, and reinforcement learning algorithms for operation under partial knowledge [2205.02975], [2403.16136], [2507.21667].
• Energy saving and constrained SMC formulations, where ON/OFF logic or actuator limitations are essential [2305.07891].
• Applications to high-dimensional, networked, or infinite-dimensional systems (multi-agent, PDE) [1706.03564], [2507.21667].

These directions further consolidate SMC as a versatile, mathematically rigorous, and application-agile control paradigm across scientific and engineering domains.