Super-Twisting Algorithm in Control Systems

Updated 2 August 2025

The Super-Twisting Algorithm is a second-order sliding mode control technique that ensures finite-time convergence and minimizes chattering using only measurable system outputs.
It employs a fractional power control law and homogeneous Lyapunov functions to guarantee robust performance against uncertainties, disturbances, and noise.
Extensions, including adaptive, variable-gain, and multivariable forms, enable effective application in complex systems, fault-tolerant control, and high-precision observer designs.

The super-twisting algorithm (STA) is a second-order sliding mode control technique designed for systems with relative degree one, enabling continuous control action and finite-time convergence even in the presence of bounded uncertainties and perturbations. It is distinguished from classical sliding mode controllers by its ability to suppress chattering—the high-frequency oscillations arising from discontinuous control—while maintaining robustness against model uncertainties, external disturbances, and, in certain extensions, measurement noise. The STA's governing principle is to drive both the sliding variable and its derivative to zero in finite time using only the measured output, without the need for output derivatives.

1. Foundations and Mathematical Structure

The canonical STA is formulated for a scalar system: $\dot{s}(t) = u(t) + d(t)$ where $s$ is the sliding variable, $u$ is the control input, and $d(t)$ is a bounded disturbance. The STA employs a control law: $\begin{cases} u(t) = -\alpha |s|^{1/2} \operatorname{sign}(s) + \nu(t) \ \dot{\nu}(t) = -\beta \operatorname{sign}(s) \end{cases}$ where $\alpha, \beta > 0$ are design gains. The fractional power and continuous integral action ensure that $s(t)$ and $\dot{s}(t)$ both reach zero in finite time—critical for achieving exact sliding motion in the presence of Lipschitz disturbances.

The algorithm requires only the measurement of $s$ (not $\dot{s}$ ), providing an advantage over classical first-order sliding modes and enabling practical implementations under noisy measurements or low-rate sampling.

2. Higher-Order Extensions and Homogeneity

The classic STA is limited to relative degree one systems, i.e. single/double integrator chains. To generalize for perturbed chains of arbitrary order, the higher-order super-twisting (HOST) framework introduces integral actions on Lyapunov function derivatives: $u_{HOST}(z, t) = k_p u_0(z) - k_i \int_0^t \partial_r V_1(z(s))\, ds$ with system dynamics: $z_1^\cdot = z_2, \dots, z_r^\cdot = \gamma(t) u + \varphi(t)$ where $u_0(z)$ is a stabilizing feedback for the pure integrator, $V_1$ a homogeneous, positive-definite Lyapunov function, and $\partial_r V_1$ its partial derivative with respect to the last state. The geometric condition ensures that the integral term aligns with the decay direction of the Lyapunov function, facilitating a Lyapunov-based finite-time stability proof for the extended system (Chitour et al., 2015).

Homogeneity with respect to dilations is central: all objects (controller, Lyapunov function, closed-loop dynamics) are homogeneous with negative degree, which guarantees finite-time convergence. This structure also enables systematic gain tuning, scaling laws, and ensures controller continuity at the origin—even for higher order systems.

3. Observer Design and Fault Estimation

The STA's robustness and chattering-suppression qualities motivate its use in observer design. For instance, a fractional-order extension has been developed for observers estimating both system states and faults in nonlinear fractional order systems: $\begin{cases} D^{\alpha} \hat{x}_1 = x_2 + k_1 |e_1|^{1/2} \operatorname{sign}(e_1) \ D^{\alpha} \hat{x}_2 = a_1 \operatorname{sign}(e_1) \end{cases}$ with additional steps for higher state/fault estimates (Mousavi et al., 2017). Embedding the fault as a separate observer state greatly reduces chattering in reconstruction and directly yields bounded finite-time estimation errors even in the presence of measurement noise and system perturbations. This stepwise or recursive observer architecture allows successive error terms to enter finite-time convergence using the STA at each step, yielding superior performance over classical sliding mode or high-gain observers.

4. Adaptive and Variable-Gain Super-Twisting Algorithms

Robustness to time-varying and a priori unknown disturbances motivates the use of adaptive and variable-gain super-twisting algorithms:

Adaptive STA: Uses disturbance estimates (from, e.g., third-order sliding mode observers) to update gains $\alpha(t), \beta(t)$ in real time:

$\beta(t) = \max\{\beta_m, |\rho_0(z,t)| / \eta\}$

where $|\rho_0(z,t)|$ is the estimated perturbation magnitude, $\beta_m$ is a minimal robustifying gain, and $\eta \in (0,1)$ is a tunable parameter. $\alpha(t)$ is then geometry-driven based on $\beta(t)$ (Xiong et al., 2018). This approach ensures the smallest possible gains for required robustness, reducing chattering while maintaining finite-time convergence.

Barrier Function-Based Variable Gain: Adjusts controller gain using a barrier function that increases the gain when the output is far from zero and enforces a continuous decrease as the output approaches a small prescribed band, ensuring non-overestimation and maintaining the sliding variable within a desired error neighborhood (Obeid et al., 2019).

Adaptive and variable gain strategies enable the STA to maintain robust performance with minimal chattering across a variety of uncertain and time-varying operating conditions, simplifying tuning and improving efficiency compared to standard fixed-gain implementations.

5. Application to Complex and Multivariable Systems

Extending STA techniques to multi-input, multi-output (MIMO) and perturbed systems involves novel frameworks:

Multivariable Generalized STA (MGSTA): Utilizes full matrix gains (not merely scalars) and strict Lyapunov analysis to accommodate uncertainties in both the input distribution matrix and external perturbations (Moreno et al., 2021). The MGSTA control law is formulated as

$u = -k_1 \phi_1(x) + b G_0^{-1}(t, x) v,$

with $\phi_1, \phi_2$ generalized homogenous terms and guarantees global finite-time stability under explicit gain design conditions.

Passivity-Based MIMO STA: Employs a storage function (Lyapunov) motivated by passivity, leveraging the interconnection of passive elements and full-matrix gain design to handle uncertain, coupled multivariable dynamics, including time- and state-varying uncertain input matrices (Garcia-Mathey et al., 2022). Homogeneity and piecewise approximations enable a full family of controllers from discontinuous to fully continuous (including PI), providing a flexible and scalable framework.

6. Implementation: Discretization and Noise Robustness

Robust discrete-time implementations require careful treatment to preserve the continuous-time STA's finite-time convergence and chattering-suppression properties:

Implicit and Eigenvalue Mapping-Based Discretizations: Methods using eigenvalue mapping maintain the eigenstructure of the continuous-time closed-loop system, avoid the chattering induced by explicit Euler discretization, and provide insensitivity to gain overestimation (Ding, 2022). Modified implicit discretizations achieve finite-time convergence to an invariant set whose size scales with $h^2$ (sampling period squared) and is independent of gains, thus decoupling steady-state accuracy from aggressive tuning (Andritsch et al., 2023).
STA Differentiators: The super-twisting differentiator achieves robust exact differentiation for signals with bounded second derivative. In the presence of measurement noise, a worst-case error bound of the form $2\sqrt{\alpha(\lambda_2+1)}\sqrt{N L}$ (with $N$ the noise bound and $L$ the signal curvature bound) is provably tight and can be approached with optimal parameter selection (Seeber, 2023). The bound is rigorously derived via a Lipschitz Lyapunov function and confirmed by simulation.

7. Control, Observer, and Fault-Tolerant Applications

The super-twisting algorithm and its generalizations are widely used in:

Attitude control of quadcopters/UAVs: ASTSM controllers provide robust performance under strong nonlinearities, uncertainties, and faults, achieving fast convergence with low chattering. Fault-tolerant extensions integrate control allocation algorithms to handle actuator loss by redistributing control commands among remaining actuators without sacrificing trajectory tracking (Hoang et al., 2018, Karahan et al., 2023).
Motion control of wafer stages and photolithography systems: Fractional-order STA with adaptive neural network compensation achieves nanometer-scale tracking accuracy under high-acceleration and disturbance regimes, outperforming integer-order and classic SMC methods (Kuang et al., 2021, Kuang et al., 2021).
Aerial manipulators: Proxy-based STA provides finite-time convergence for both position and attitude, canceling manipulator-induced disturbances in aerial robots (Hua et al., 2023).
State and fault estimation: Fractional-order and generalized STAs yield smooth, finite-time convergent observers for nonlinear and interconnected systems, with rigorous Lyapunov analysis showing convergence in the presence of strong nonlinearities and uncertainties, outperforming high-gain observers (Mousavi et al., 2017, Tafat et al., 2 Jun 2025).
Time-delay systems: Super-twisting based Lyapunov redesigns reach sliding regimes in finite time and suppress chattering in systems with delays, uncertainties, and even partial state measurement, as in predictor-based feedback configurations (Gomez et al., 2021, Pinto et al., 28 Jul 2025).

Controllers designed via MGSTA and LMI synthesis systematically handle parameter uncertainties and Lipschitz disturbances, as shown in fault-tolerant control for multibody mechanical systems (e.g., chains of trailers) (Geromel et al., 26 Feb 2025).

The super-twisting algorithm represents a significant advancement in higher-order sliding mode control, offering mathematically guaranteed finite-time convergence, reduction of chattering, and robustness to broad classes of uncertainties. Its extensions—fractional order, adaptive, multivariable, and discrete-time—enable its application to a wide span of demanding modern control and estimation problems with structured methodologies for synthesis, gain tuning, and rigorous performance verification.