Keywords: First Order Sliding Mode, Adaptive Sliding Mode, Lyapunov Methods for Variable Structure Systems & Sliding Mode Control
Abstract: This paper addresses the problem of adaptive stabilization of MIMO systems whose constant high-frequency gain matrix is nonsingular but has completely unknown sign structure. Unlike most existing approaches, no positivity, symmetry, or diagonal dominance assumptions on the gain matrix are required. Building upon the unit-vector framework and the concept of unmixing sets, a modified discontinuous adaptive control strategy is proposed. The key novelty consists in decoupling the mechanism that cycles through the elements of the unmixing set from the adaptation of the control input magnitude. In particular, the adaptive gain is used exclusively to govern the switching among candidate gain matrices, while the control input magnitude is fixed a priori. A Lyapunov-based analysis proves global asymptotic stabilization for driftless systems. The results are further extended to systems affected by matched bounded disturbances under mild additional assumptions. Simulation studies illustrate the effectiveness of the proposed approach and highlight its advantages in terms of reduced control authority and alleviated transient peaking.

Keywords: First Order Sliding Mode, Adaptive Sliding Mode, Discrete Time Sliding Mode
Abstract: Sliding-mode control is robust against external disturbances but degrades in the presence of measurement noise. While prior work has addressed this downside using smoothing, boundary layers, or heuristic gain adaptation, such approaches lack statistical optimality. Here, we characterize the first and second moments of the closed-loop sliding variable for first-order sliding-mode control to analytically derive a statistically optimal control law. The first moment shows that, in the presence of a constant disturbance, a bias remains. Therefore, we propose a modified, unbiased control law. Using the second moment, we derive a mean-squared-error-optimal adaptive gain. The result is an unbiased and adaptive first-order sliding-mode control law that explicitly accounts for noise statistics. Simulations confirm that the adaptive controller is more robust against measurement noise than a fixed-gain sliding-mode controller. We find that statistical optimization provides a generalizable framework for analytic sliding-mode controller design.

Keywords: Higher Order Sliding Mode, Adaptive Sliding Mode
Abstract: This paper proposes a robust and predictive control approach based on optimal predictive control (OPC) and adaptive super-twisting (ASTW) sliding mode control. In this approach, the OPC is integrated with the ASTW algorithm within a unified framework, leading to improved transient response, mitigated chattering effects, and enhanced robustness. In this framework, the ASTW algorithm provides the primary control action, while the OPC is continuously blended through a weighting factor, ensuring a smooth and adaptive contribution to the control input for achieving the desired control objectives. Real finite-time stability is rigorously proven using Lyapunov theory. Finally, an academic example is presented to demonstrate the effectiveness of the proposed approach through simulation results.

Keywords: Adaptive Sliding Mode, Chattering Analysis, Mobile Robots
Abstract: Chattering is an inherent feature of sliding mode control, and its complete elimination is not realistic in practical systems. This paper presents an adaptive sliding mode control scheme for underwater vehicle heave control, where chattering occurs at relatively low frequencies due to the slow plant dynamics. An adaptive law is designed to regulate the relay amplitude to maintain periodic oscillation (chattering) at the plant output, while minimizing its amplitude. External disturbances introduce asymmetry in the relay duty cycle, which is captured through an estimate of the averaged control. This averaged control estimate is obtained using higher-order low-pass filtering of the relay output, with relatively large filter’s time constant. The estimated averaged control is then used to adapt the relay amplitude, maintaining near-symmetric switching without breaking the chattering motion. The proposed method is validated through numerical simulations on an underwater vehicle model, demonstrating that periodic chattering is preserved under the influence of varying external disturbances, while the oscillation amplitude is minimized for a given disturbance, showing enhanced disturbance rejection with minimal chattering amplitude.

Keywords: Adaptive Sliding Mode, First Order Sliding Mode, Application of Sliding Mode Control to other theoretical problems
Abstract: This paper investigates the robust boundary control of a diffusion process subject to matched and unmatched disturbances. The bounds of the matched disturbance are unknown, motivating the use of an adaptive sliding mode control law. It is applied at one boundary, where a matched disturbance acts alongside the control input. Additionally, an unmatched disturbance affects the opposite boundary. While the matched disturbance can be fully rejected, the unmatched disturbance fundamentally limits the achievable performance. It is shown that the adaptive gain remains bounded, and an ultimate bound on the L2-norm of the state is derived in terms of the unmatched disturbance magnitude. The bound is established via a transfer function analysis and an input-to-state stability argument. Numerical simulations illustrate the theoretical findings.

Keywords: Adaptive Sliding Mode, Higher Order Sliding Mode, Lyapunov Methods for Variable Structure Systems & Sliding Mode Control
Abstract: This paper addresses the stabilization problem for a perturbed double–integrator affected by a bounded perturbation featuring a constant upperbound which is a priori unknown. We propose an adaptive feedback law, composed by the interconnection between a linear state-feedback controller and the twisting controller with two automatically tuned nondecreasing time-varying gains adjusted by gradient-based laws. Lyapunov analysis,complemented by Barbalat-based argument, is developed to certify the asymptotical stability of the closed loop system. The analysis provides a closed-form bound for the adaptive gains and it constructively yields simple design conditions for the controller parameters. We prove boundedness of all closed-loop signals and establish asymptotic convergence of both state variables to the origin without requiring prior knowledge of the disturbance bound. Simulation results are presented to corroborate the theoretical findings.

Keywords: Application of Sliding Mode Control to other theoretical problems, First Order Sliding Mode
Abstract: This paper develops a robust fixed-time optimization framework for constrained problems that guarantees exact constraint satisfaction and convergence to Karush–Kuhn–Tucker (KKT) points within a fixed time, independent of initial conditions. The approach treats the Lagrange multipliers as control inputs, composed of an equivalent control and a switching control, with the system states representing the decision variables. An equivalent control steers the gradient flow to a local KKT point asymptotically for nonconvex objectives and to the unique global optimum in fixed time for convex objectives. Constraint enforcement is achieved by embedding the equality constraints directly as a sliding manifold, with a fixed-time switching control ensuring rapid and reliable feasibility. The framework further accounts for matched disturbances, providing robustness guarantees that are theoretically characterized and illustrated using spherical constraints. Numerical studies on a 3-bus AC optimal power flow problem and a distributed consensus-based parameter estimation problem demonstrate the effectiveness, scalability, and robustness of the proposed approach.

Keywords: Application of Sliding Mode Control to other theoretical problems, Higher Order Sliding Mode, Chattering Analysis
Abstract: This paper investigates a reaction wheel pendulum (RWP) system regulated by a PID-like continuous sliding mode controller. For this class of systems, chattering inevitably arises, manifesting as high-frequency oscillations around the equilibrium point due to unmodeled dynamics, as well as actuator and sensor effects. Suboptimal controller gains are obtained using the harmonic balance method in order to minimize the chattering amplitude, considering an actuator model characterized by two time constants. The effectiveness of the proposed methodology is validated both numerically and experimentally.

Keywords: Sliding Mode based Observation, Application of Sliding Mode Control to other theoretical problems, Networked Control Systems
Abstract: The problem of joint state and parameter estimation for adaptively coupled Kuramoto oscillators consisting of the oscillator phases and coupling strengths is addressed. Extending previous work this paper focuses on the implementation of a sliding mode based switching observer, thus dealing with the fact that trajectories typically cross through an observable region several times. Convergence conditions are derived combining bounds for maximum divergence and convergence between switching intervals. A sliding mode differentiator is used to design the observer due to the observability matrix of the chosen system being close to singular for most configurations. The observer performance is illustrated using numerical simulation.

Keywords: Application of Sliding Mode Control to other theoretical problems, Lyapunov Methods for Variable Structure Systems & Sliding Mode Control, Higher Order Sliding Mode
Abstract: This paper presents generalized results of finite-time (FT) and fixed-time (FxT) convergence of trajectories with respect to each other for nonlinear dynamical systems through the framework of contraction theory. While classical contraction theory provides an elegant incremental framework for asymptotic convergence of trajectories, however, a gap exists in connecting this framework to non-asymptotic, finite-time convergence properties. We bridge this gap by introducing the concept of transient contraction metrics. We show that a system can exhibits a time-varying contraction behavior that drives any two trajectories to each other in finite time, and even in fixed time independent of initial conditions. In particular, we provide a novel contraction framework for analyzing sliding mode controllers with guaranteed FT/FxT convergence properties. Later on, we relate these convergences to stability around equilibrium points. We provide a simulation example that illustrates the significant performance of the proposed framework.

Keywords: Higher Order Sliding Mode, Power Electronics, Application of Sliding Mode Control to other theoretical problems
Abstract: This paper presents an energy-consistent super- twisting (ST) sliding-mode control design for a single-phase grid-forming inverter with an output LC filter, motivated by inverter-dominated low-inertia power-electronic applications. Throughout, energy-consistent refers to a sliding surface whose closed-loop motion is shaped to avoid large transient excursions of the filter Hamiltonian. The surface couples the voltage error, its integral, and the inductor current, and its design is informed by the port-Hamiltonian (PH) representation of the LC filter; the Hamiltonian is employed as a physically meaningful diag- nostic rather than as a strict control structure. A super-twisting reaching law is used to provide continuous control action and finite-time convergence under bounded matched perturbations, with explicit gain conditions stated. An actuator saturation constraint is included in the model and its effect is reported. The proposed controller is benchmarked against a classical first- order SMC with boundary layer on the same sliding surface, isolating the contribution of the ST reaching law. Simulation studies on a grid-forming LC inverter subject to renewable- like disturbances and parameter uncertainties show that the ST controller reduces steady-state tracking error, control chatter, and Hamiltonian variation by a factor of two or more across the tested operating envelope, while the gain-sensitivity coefficient of variation of ∆Hstays below 3 over a [0.5, 1.5]×nominal gain box.

Keywords: First Order Sliding Mode, Higher Order Sliding Mode, Lyapunov Methods for Variable Structure Systems & Sliding Mode Control
Abstract: The controller design for a disturbed integrator, as is the nominal case for developing a first-order sliding mode controller (1-SMC) or a super-twisting controller (STC), is considered. We investigate a class of unbounded discontinuous disturbances, that can not be rejected by 1-SMC or STC. Specifically, we consider the superposition of a Lipschitz-continuous component and a bounded, possibly discontinuous, component as the disturbance. We present two dynamic controllers for the stated control problem. The first one is a sliding mode controller that is based on the internal model control principle and incorporates the disturbance model. The second one is an observer-based controller that utilizes a super-twisting observer, i.e. first-order sliding mode differentiator, as well as two 1-SMCs. The closed loops with these controllers are shown to be globally asymptotically stable via Lyapunov functions. Finally, the efficacy of the presented controllers, and the benefits compared to 1-SMC and STC are demonstrated in simulation examples.

Keywords: First Order Sliding Mode, Higher Order Sliding Mode, Chattering Analysis
Abstract: We analyze actuator chattering in a scalar integrator system subject to second-order actuator dynamics with an unknown time constant and first-order sliding-mode control, using both a conventional static sliding manifold and a dynamic sliding manifold. Using the harmonic balance method, we prove that it is possible to adjust the parameters of the dynamic sliding manifold for the specified system class so as to reduce the amplitude of the chattering in comparison to the static manifold. We illustrate our results with a simulation example. This contribution serves as a proof of concept to motivate further investigations in chattering reduction via dynamic sliding manifolds.

Keywords: Higher Order Sliding Mode, Adaptive Sliding Mode, Hydraulic/Pneumatic Systems
Abstract: This study presents a structured control strategy for tidal barrage energy extraction with variable-speed operation. A two-level control architecture is proposed, comprising a supervisory high-level controller and an inner-loop low-level speed control. The former computes optimal operating references using a moment-based optimal control formulation, that exploits the harmonic nature of tides. The latter ensures robust finite-time turbine speed tracking through an Adaptive Fast Terminal Sliding Mode Controller (AFTSMC) with barrier functions, ensuring fast convergence, robustness to uncertainties, and reduced chattering. The effectiveness of the proposed approach is demonstrated through simulations based on the La Rance tidal barrage, highlighting the potential of variable-structure control techniques to enhance energy extraction in tidal power systems.

Keywords: Power Electronics, Lyapunov Methods for Variable Structure Systems & Sliding Mode Control, Higher Order Sliding Mode
Abstract: Robust and efficient control of DC–DC boost converters is crucial for applications subject to severe load variations and reference changes. The Super-Twisting (ST) algorithm theoretically provides a robust and continuous control law; however, its convergence speed and robustness strongly depend on the choice of constant gains. This paper proposes a Power-Tower Super-Twisting (PTST) controller and a modified variant for use in the current control loop of a DC–DC boost converter. The proposed approach replaces the sign function in the classical ST algorithm with a power-tower function, introducing a state-dependent gain that accelerates convergence when the tracking error is large. The modified PTST, based on a normalized sliding surface, further bounds this gain to prevent numerical overflow and reduce overshoot while preserving finite-time stability. Simulation results under reference change and load disturbance scenarios confirm that the proposed controllers achieve faster convergence than the conventional ST controller.

Keywords: Application of Sliding Mode Control to other theoretical problems, Higher Order Sliding Mode, Lyapunov Methods for Variable Structure Systems & Sliding Mode Control
Abstract: This paper presents a finite-time optimization framework for constrained variational problems based on sliding mode control of Lagrange multipliers. By interpreting the Lagrange multipliers as control inputs and the constraint functions as sliding manifolds, the Euler–Lagrange optimality conditions are embedded into a second-order prediction– correction dynamical system. The multiplier input is decomposed into equivalent and switching components, where the equivalent term recovers the Euler–Lagrange equilibrium and the switching term enforces finite-time reachability of the constraint manifold. A Lyapunov-based analysis establishes explicit bounds on the constraint-reaching time and guarantees invariance thereafter. Unlike classical primal–dual or interior-point flows, the proposed method achieves exact constraint satisfaction in finite time while simultaneously driving the primal variables toward the optimal trajectory. Numerical examples illustrate the effectiveness of the approach for infinite-horizon variational problems with both time-invariant and time-varying equality constraints.

Keywords: Lyapunov Methods for Variable Structure Systems & Sliding Mode Control, Higher Order Sliding Mode, First Order Sliding Mode
Abstract: Stability proofs for sliding mode controllers traditionally require Lyapunov functions tailored to each algorithm's structure. Neural networks offer an alternative by learning Lyapunov candidates directly, but a smooth parameterization may fail a strict decrease test near the switching manifold. We present a framework that pairs a neural Lyapunov function with an oriented ellipsoidal gauge marking the part of the training domain that the dense-grid decrease test does not cover. Outside the gauge, a worst-case Filippov derivative is checked deterministically on a dense grid, and the gauge is shrunk only when this test continues to pass. An empty gauge means the decrease test passed at all nonzero grid points in the training box; a non-empty gauge reports the unresolved region together with a grid-validated decrease test on its complement. The same trainer was demonstrated on multiple controller structures without controller-specific analytical derivations.

Keywords: First Order Sliding Mode, Lyapunov Methods for Variable Structure Systems & Sliding Mode Control, Mobile Robots
Abstract: Autonomous underwater vehicles (AUVs) require control strategies that ensure stable, efficient depth regulation and controlled transient behavior under varying operating conditions. Conventional state-feedback control and linear sliding surface designs typically involve negotiating between settling time and overshoot. Sliding mode control (SMC) with nonlinear sliding surfaces has been shown to address this trade-off, although such approaches have largely been demonstrated on linear system models. This paper investigates the use of a nonlinear sliding surface in an SMC framework for systematic shaping of the transient response of an AUV described by nonlinear state equations. The nonlinear sliding surface is employed to influence the transient response, resulting in reduced settling time while limiting overshoot compared to linear sliding surface designs. Simulation results are presented for linear(underdamped and overdamped) and nonlinear sliding surfaces to illustrate their response characteristics.

Keywords: First Order Sliding Mode, Application of Sliding Mode Control to other theoretical problems, Lyapunov Methods for Variable Structure Systems & Sliding Mode Control
Abstract: This paper proposes a robust prescribed-time sliding mode controller for a linear time-invariant system. The proposed controller induces the sliding motion in the system within a prescribed time, which is often user-defined, rather than the prescribed convergence of the plant trajectories. It may be noted that the prescribed-time controllers have a high (infinite) gain at the desired convergence time, leading to high sensitivity to measurement imperfections, such as noise, sampling error, etc. In this paper, we propose a novel algorithm by incorporating the time-varying high-gain term with an exponential factor into the classical control law. Since infinite gain is induced by the time-varying term, we reset it to zero before the prescribed time. Therefore, the prescribed term guarantees the convergence of the sliding trajectory to the vicinity of the sliding manifold, whereas the classical switching function ensures the sliding motion in the system. Thus, the proposed controller not only enforces the sliding mode at any desired time but also shows the robustness to the disturbance. A numerical demonstration of the proposed controller is presented with an example under various scenarios.

Keywords: Application of Sliding Mode Control to other theoretical problems, First Order Sliding Mode, Lyapunov Methods for Variable Structure Systems & Sliding Mode Control
Abstract: This paper proposes a navigation function-based predefined-time sliding mode control framework for spacecraft attitude regulation under strict pointing constraints while preserving the robustness of sliding mode control. Unit quaternions are used to provide a globally nonsingular attitude representation. A novel nonsingular predefined-time sliding manifold is developed to enforce constrained attitude reorientation and guarantee convergence within a user-prescribed finite time, independent of initial conditions. Based on this manifold, a nonsingular and chattering-free predefined-time sliding mode control law is designed to achieve accurate attitude regulation while satisfying pointing constraints. Lyapunov-based analysis establishes predefined-time stability of the closed-loop system. Numerical simulations demonstrate fast convergence, robustness, and effective constraint handling in spacecraft attitude maneuvers in the presence of external disturbances.

Keywords: Adaptive Sliding Mode, Lyapunov Methods for Variable Structure Systems & Sliding Mode Control, Hydraulic/Pneumatic Systems
Abstract: This paper investigates the robust stabilization of a platform’s roll motion using a U-tube-based anti-roll tank system. Exploiting the structural properties of the system, formulated as a nonlinear model, an adaptive fuzzy sliding-mode control (AFSMC) strategy is designed to achieve robust stabilization in the presence of system uncertainties and external disturbances. In the proposed approach, the control input consists of a fuzzy term that approximates an unknown, state-dependent scalar determined by the system’s dynamics and a robust term that compensates for fuzzy approximation errors and uncertainties. The two terms are updated online through the designed adaptation laws while the velocity is estimated using Levant’s second-order differentiator. The effectiveness of the proposed control scheme is verified through numerical simulations, which confirm its capability in enhancing roll motion suppression.

Keywords: Automotive Systems
Abstract: This paper develops a nonlinear impact-time guidance law for intercepting stationary, constant-velocity, and maneuvering targets. The proposed approach augments true proportional navigation (TPN) with a trigonometric modulation function that drives the impact-time error to zero exactly at a user-specified convergence time, thereby enabling precise temporal control over time when the interceptor aligns with the collision course. Unlike existing approaches, the guidance law is formulated directly within a fully nonlinear engagement framework without any linearization, ensuring robust performance under large initial heading errors and arbitrary geometries. Following the convergence of the impact-time error, the guidance command reduces to nominal TPN behavior, thereby significantly reducing control effort. Comprehensive numerical simulations demonstrate the efficacy of the proposed guidance strategy across a wide range of engagement scenarios.

Keywords: Mobile Robots, First Order Sliding Mode, Application of Sliding Mode Control to other theoretical problems
Abstract: This paper presents fault tolerant control design for linear parameter varying system with redundant input. Sliding mode methodology is used with fixed control allocation to exploit the redundancy available in the actuator, which alleviates the need for a fault detection system. A rigorous stability analysis using the small gain theorem is given to ensure the stability in the presence of unmatched term induced by faults/failures. The control scheme was then applied to a non linear longitudinal model of a HALE UAV, which shows no degradation in performance between the fault-free and the failure cases.

Keywords: Mobile Robots, Application of Sliding Mode Control to other theoretical problems, First Order Sliding Mode
Abstract: This paper presents an LPV-based fault-tolerant controller for a variable-sweep wing bio-inspired UAV. The UAV is equipped with independently actuated left and right sweep wings, a centrally mounted propeller, and six independently operated control surfaces for attitude control. The UAV has no conventional horizontal tail, and instead, its left and right main wings can sweep independently to provide attitude control and provide redundancy. The proposed controller employs an LPV-based sliding mode control allocation scheme, ensuring robustness against actuator faults and failures. Nonlinear simulation results verify the effectiveness of the approach.

Keywords: First Order Sliding Mode, Fault Detection
Abstract: This paper describes the design and testing of a fault-tolerant control system for overactuated aircraft that combines a high-level sliding-mode controller with a null-space-enhanced control allocation scheme. The null-space term is treated as a design variable and updated by applying an adaptive scheme to minimise saturation error while preserving the commanded virtual control. The approach is evaluated by performing an alpha-roll manoeuvre on the ADMIRE aircraft model with actuator faults and failures. Simulation results indicate that the proposed null-space-optimised method eliminates saturation and more effectively exploits the available control authority by redistributing control effort to actuators with remaining capacity.

Keywords: Application of Sliding Mode Control to other theoretical problems, Electric Drives And Actuators, Higher Order Sliding Mode
Abstract: Wave propelled platforms represent a promising technology for sustainable ocean monitoring and marine energy harvesting, yet achieving precise position regulation under persistent and time-varying wave disturbances remains a significant control challenge. This paper presents a Super-Twisting Algorithm (STA) controller for position regulation of multi-blade wave-propelled platforms, requiring only platform position measurements and eliminating the need for blade angle sensing or full state feedback. A comprehensive dynamic model captures blade rotation, platform translation, and wave--structure interactions for an N-blade system. The proposed controller achieves robust regulation while eliminating chattering, supported by Lyapunov-based stability analysis. Comparative simulations against conventional PID and standard SMC with boundary layer are conducted under three scenarios: constant wave forcing, target position switching, and time-varying wave disturbances. Results confirm that the STA approach achieves superior steady-state accuracy and tracking performance while maintaining robust regulation under time-varying wave conditions.