Higher-Order Super-Twisting (HOST)

• Higher-Order Super-Twisting (HOST) is a continuous sliding mode control algorithm that generalizes classical super-twisting for stabilizing perturbed high-order integrator systems.
• It leverages homogeneity theory, Lyapunov-based design, and barrier-based adaptive schemes to achieve finite-time convergence even under nonlinear uncertainties and perturbations.
• Empirical studies demonstrate that HOST controllers offer smooth transient responses, robust disturbance rejection, and bounded gain adaptation in complex control scenarios.

Higher-Order Super-Twisting (HOST) algorithms are a class of continuous higher-order sliding mode controllers designed for perturbed chains of integrators, particularly in systems with nonlinearities, uncertainties, and bounded or Lipschitz-type perturbations. HOST controllers generalize the classical super-twisting algorithm to arbitrary integrator order, providing homogeneous, finite-time stabilization with continuous control signals, robust disturbance rejection, and, in adaptive schemes, bounded gain adaptation under suitable perturbation assumptions.

1. System Model and Problem Setting

The HOST framework primarily addresses stabilization and robust tracking in perturbed chains of integrators of order $r$ with scalar input $u$ and state $z=[z_1, \ldots, z_r]^T \in \mathbb{R}^r$. The plant model is

$$\begin{aligned} \dot z_i &= z_{i+1}, \quad i=1,\ldots,r-1, \ \dot z_r &= \gamma(t)\,u + \phi(t), \end{aligned}$$

where $\gamma(t)>0$ is a possibly time-varying control gain satisfying $0<\gamma_m\le\gamma(t)\le\gamma_M$, and $\phi(t)$ is a time-varying disturbance with $|\dot\phi(t)|$ bounded almost everywhere. HOST controllers are applicable both when the perturbation and control channel gain are constant and when they are time-varying and only bounded. The extension to general nonlinear plants with matched uncertainties is addressed with appropriate adaptive approximators and Lyapunov-based design (Chitour et al., 2022, Chitour et al., 2015, Hendel et al., 2015).

2. Homogeneity, Dilations, and Sliding Surfaces

HOST design crucially exploits the theory of homogeneity and dilations. For a chosen negative homogeneity degree $\kappa<0$ and base weight $p>0$, define weights $u$0, $u$1, and associated dilations

$u$2

Feedback laws and Lyapunov functions are constructed to be homogeneous with respect to this family:

• A Feedback $u$3 homogeneous of degree $u$4 is selected to stabilize the pure chain $u$5.
• A positive-definite homogeneous Lyapunov function $u$6 (or $u$7), of degree $u$8, is designed to satisfy

$u$9

ensuring finite-time convergence properties (Chitour et al., 2015).

For error feedback systems and output tracking, a standard sliding variable $z=[z_1, \ldots, z_r]^T \in \mathbb{R}^r$0 is used, whose construction depends on the system relative degree (e.g., $z=[z_1, \ldots, z_r]^T \in \mathbb{R}^r$1 for second-order systems (Hendel et al., 2015)).

3. HOST Law Structure and Barrier-Based Adaptation

The core HOST algorithm for the pure (unperturbed) chain employs a generalized super-twisting structure: $z=[z_1, \ldots, z_r]^T \in \mathbb{R}^r$2 where $z=[z_1, \ldots, z_r]^T \in \mathbb{R}^r$3, $z=[z_1, \ldots, z_r]^T \in \mathbb{R}^r$4 are positive gains, $z=[z_1, \ldots, z_r]^T \in \mathbb{R}^r$5 is the nominal stabilizing feedback, and $z=[z_1, \ldots, z_r]^T \in \mathbb{R}^r$6 is the partial derivative of the Lyapunov function with respect to $z=[z_1, \ldots, z_r]^T \in \mathbb{R}^r$7. For perturbed systems with bounded disturbance derivatives and known $z=[z_1, \ldots, z_r]^T \in \mathbb{R}^r$8, this law guarantees finite-time stabilization (Chitour et al., 2022).

To address unknown or time-varying uncertainties, particularly when bounds on perturbations are unavailable, a barrier-based adaptive HOST (also termed Barrier-based Super-Twisting, BST) law is implemented: $z=[z_1, \ldots, z_r]^T \in \mathbb{R}^r$9 with dynamically adapted gains $$\begin{aligned} \dot z_i &= z_{i+1}, \quad i=1,\ldots,r-1, \ \dot z_r &= \gamma(t)\,u + \phi(t), \end{aligned}$$0, $$\begin{aligned} \dot z_i &= z_{i+1}, \quad i=1,\ldots,r-1, \ \dot z_r &= \gamma(t)\,u + \phi(t), \end{aligned}$$1 constructed as follows: $$\begin{aligned} \dot z_i &= z_{i+1}, \quad i=1,\ldots,r-1, \ \dot z_r &= \gamma(t)\,u + \phi(t), \end{aligned}$$2 where $$\begin{aligned} \dot z_i &= z_{i+1}, \quad i=1,\ldots,r-1, \ \dot z_r &= \gamma(t)\,u + \phi(t), \end{aligned}$$3 is the barrier-induced gain modifier and $$\begin{aligned} \dot z_i &= z_{i+1}, \quad i=1,\ldots,r-1, \ \dot z_r &= \gamma(t)\,u + \phi(t), \end{aligned}$$4 is an auxiliary, nondecreasing unbounded function. The hitting time $$\begin{aligned} \dot z_i &= z_{i+1}, \quad i=1,\ldots,r-1, \ \dot z_r &= \gamma(t)\,u + \phi(t), \end{aligned}$$5 is when $$\begin{aligned} \dot z_i &= z_{i+1}, \quad i=1,\ldots,r-1, \ \dot z_r &= \gamma(t)\,u + \phi(t), \end{aligned}$$6. Continuity in gain scheduling is enforced via a suitable constant $$\begin{aligned} \dot z_i &= z_{i+1}, \quad i=1,\ldots,r-1, \ \dot z_r &= \gamma(t)\,u + \phi(t), \end{aligned}$$7 (Chitour et al., 2022).

4. Finite-Time Stability and Lyapunov Analysis

HOST stability is grounded in homogeneous finite-time Lyapunov methodologies. For the HOST law, the extended system (including the integral action) is analyzed with a strict homogeneous Lyapunov function of the form: $$\begin{aligned} \dot z_i &= z_{i+1}, \quad i=1,\ldots,r-1, \ \dot z_r &= \gamma(t)\,u + \phi(t), \end{aligned}$$8 which is positive definite and homogeneous. Along system trajectories, $$\begin{aligned} \dot z_i &= z_{i+1}, \quad i=1,\ldots,r-1, \ \dot z_r &= \gamma(t)\,u + \phi(t), \end{aligned}$$9 holds, with $\gamma(t)>0$0, establishing finite-time convergence (Chitour et al., 2015).

In barrier-adaptive HOST, composite Lyapunov functionals of the form

$\gamma(t)>0$1

are constructed, ensuring all closed-loop variables enter a compact invariant set in finite time and stay there, yielding bounded states and adaptation signals (Chitour et al., 2022).

5. Design Process and Parameter Selection

HOST algorithms are synthesized through a systematic procedure:

1. Select Chain Order and Weights: Choose $\gamma(t)>0$2; set $\gamma(t)>0$3, $\gamma(t)>0$4 (for canonical design), yielding weights $\gamma(t)>0$5.
2. Homogeneous Stabilizer and Lyapunov Pair: Construct a nominal homogeneous stabilizer $\gamma(t)>0$6 (e.g., via Hong’s recursive controller) and compatible Lyapunov function $\gamma(t)>0$7.
3. Gain Selection: Assign $\gamma(t)>0$8 and $\gamma(t)>0$9 (or their adaptive analogs). Adaptation is achieved via the $0<\gamma_m\le\gamma(t)\le\gamma_M$0 and $0<\gamma_m\le\gamma(t)\le\gamma_M$1 scheduling above.
4. Time-Scaling for Perturbed Systems: If disturbances or parameter variations are significant, select time-scaling and scaling matrices $0<\gamma_m\le\gamma(t)\le\gamma_M$2 to ensure gain dominance.
5. Barrier Parameter: Choose $0<\gamma_m\le\gamma(t)\le\gamma_M$3 to trade accuracy for gain magnitude.
6. Barrier Function: Implement $0<\gamma_m\le\gamma(t)\le\gamma_M$4, diverging as $0<\gamma_m\le\gamma(t)\le\gamma_M$5, and ensure gain continuity at $0<\gamma_m\le\gamma(t)\le\gamma_M$6.
7. Integral Action: Employ integral terms to ensure second-order (or higher) sliding, enforcing robust convergence (Chitour et al., 2022, Chitour et al., 2015).

A schematic HOST controller design process, as directly outlined in the cited works, is provided in the table below.

Step	Action	Reference
1	Fix order $0<\gamma_m\le\gamma(t)\le\gamma_M$7, compute weights	(Chitour et al., 2015)
2	Design $0<\gamma_m\le\gamma(t)\le\gamma_M$8, $0<\gamma_m\le\gamma(t)\le\gamma_M$9	(Chitour et al., 2015, Chitour et al., 2022)
3	Select or adapt gains	(Chitour et al., 2015, Chitour et al., 2022)
4	Set time- and gain-scaling	(Chitour et al., 2015)
5	Implement barrier/adaptive $\phi(t)$0	(Chitour et al., 2022)
6	Analyze via Lyapunov functions	(Chitour et al., 2015, Chitour et al., 2022)

6. Variants, Adaptation, and Applications

HOST concepts extend to systems with unknown nonlinearities, internal uncertainties, and external disturbances through adaptive and intelligent approximators. A prominent example is the use of adaptive interval Type-2 fuzzy logical systems (IT2FLS) for online estimation of both system uncertainties and the super-twisting injection terms, as applied to nonlinear chaotic systems. In such contexts:

• The equivalent sliding-mode control is augmented by fuzzy approximators for plant nonlinearity, super-twisting discontinuous and integral terms.
• Parameter adaptation laws are derived via Lyapunov analysis to guarantee stability in the presence of fuzzy approximation errors and uncertainties.
• Gains are selected to satisfy inequalities of the form $\phi(t)$1, $\phi(t)$2, where $\phi(t)$3 upper-bounds model and disturbance uncertainties, inclusive of the fuzzy residuals (Hendel et al., 2015).

HOST controllers are thus realized as continuous, finite-time stabilizers with inherent rejection of disturbances and practical reduction of chattering, suitable for high-precision tracking, robust regulation, and high-order systems encountered in control engineering and nonlinear signal processing.

7. Illustrative Performance and Empirical Results

Simulation results, as reported in HOST literature, confirm that:

• For pure chains, HOST enforces finite-time stabilization with continuous control and smoother transient (especially with modified Hong’s stabilizer) compared to nonsliding or lower-order solutions.
• In perturbed cases, HOST enforces rapid convergence and, after transient, control signals asymptotically track $\phi(t)$4.
• The adaptive HOST (BST) algorithm robustly constrains adaptive gains within finite bounds under Lipschitz-type perturbations, outperforming classical adaptive sliding mode laws that may yield unbounded gain escalation (Chitour et al., 2022).
• Application to nonlinear and chaotic plants, with adaptive fuzzy super-twisting, yields finite-time tracking to unstable periodic orbits, robust rejection of bounded disturbances, and elimination of chattering while preserving global stability (Hendel et al., 2015).

These empirical findings validate the theoretical properties of the HOST architecture and support its application across a broad class of uncertain, high-order control systems.