---
title: Higher-Order Super-Twisting (HOST)
url: https://www.emergentmind.com/topics/higher-order-super-twisting-host
type: topic
---

# Higher-Order Super-Twisting (HOST)

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 [2206.10904, 1507.07893, 1601.04245].

## 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 $p_i=p+(i-1)\kappa$, $i=1,\ldots,r+1$, and associated dilations
\[
\delta_\varepsilon(z_1,\ldots,z_r) = (\varepsilon^{p_1}z_1, \ldots, \varepsilon^{p_r}z_r).
\]
Feedback laws and Lyapunov functions are constructed to be homogeneous with respect to this family:
- A Feedback $u_0(z)$ homogeneous of degree $p_{r+1}$ is selected to stabilize the pure chain $\dot z = J_r z + u e_r$.
- A positive-definite homogeneous Lyapunov function $V(z)$ (or $V_1(z)$), of degree $2p_{r+1}$, is designed to satisfy
  \[
  \dot V(z) \le -c V(z)^{1+\frac{\kappa}{2}},
  \]
  ensuring finite-time convergence properties [1507.07893].

For error feedback systems and output tracking, a standard sliding variable $s(\cdot)$ is used, whose construction depends on the system relative degree (e.g., $s = e_2 + \lambda e_1$ for second-order systems [1601.04245]).

## 3. HOST Law Structure and Barrier-Based Adaptation

The core HOST algorithm for the pure (unperturbed) chain employs a generalized super-twisting structure:
\[
\begin{aligned}
u_{ST}(t) &= k_P u_r(z(t)) + \bar{\xi}(t), \\
\dot{\bar{\xi}}(t) &= -k_I\,\partial_r V(z(t)),\quad \bar{\xi}(0)=0,
\end{aligned}
\]
where $k_P$, $k_I$ are positive gains, $u_r(z)$ is the nominal stabilizing feedback, and $\partial_r V(z)$ is the partial derivative of the Lyapunov function with respect to $z_r$. For perturbed systems with bounded disturbance derivatives and known $\gamma_m$, this law guarantees finite-time stabilization [2206.10904].

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:
\[
\begin{aligned}
u_{BST}(t) &= L_1(t, z(t)) u_r(z(t)) + \xi(t), \\
\dot{\xi}(t) &= -L_2(t, z(t))\,\partial_r V(z(t)),\quad \xi(0)=0,
\end{aligned}
\]
with dynamically adapted gains $L_1$, $L_2$ constructed as follows:
\[
\begin{aligned}
L_1(t,z) = \begin{cases}
l(t), & 0\leq t < \bar t,\\
\bar c\,[L_\varepsilon(z)]^{-\kappa/2}, & t\geq \bar t,
\end{cases}
\qquad
L_2(t,z) = \begin{cases}
0, & 0\leq t < \bar t,\\
L_\varepsilon(z), & t\geq \bar t,
\end{cases}
\end{aligned}
\]
where $L_\varepsilon(z) = \frac{\varepsilon}{\varepsilon - V(z)}$ is the barrier-induced gain modifier and $l(t)$ is an auxiliary, nondecreasing unbounded function. The hitting time $\bar t$ is when $V(z(\bar t))\leq \varepsilon/2$. Continuity in gain scheduling is enforced via a suitable constant $\bar c$ [2206.10904].

## 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:
\[
W(z, \xi) = A [V_1(z) + \xi^2/(2k_I)]^{2-\alpha} - z_r \xi,
\]
which is positive definite and homogeneous. Along system trajectories, $\dot W \le -d W^{1/(2-\alpha)}$ holds, with $\alpha = 1 + \kappa/(2p_{r+1}) \in (0,1)$, establishing finite-time convergence [1507.07893].

In barrier-adaptive HOST, composite Lyapunov functionals of the form
\[
G(Y, \phi) = \varepsilon\ln(\varepsilon+Y)+\frac{\gamma_m}{2}\phi^2-2\gamma_m\psi_M s(\phi)z_r,
\]
are constructed, ensuring all closed-loop variables enter a compact invariant set in finite time and stay there, yielding bounded states and adaptation signals [2206.10904].

## 5. Design Process and Parameter Selection

HOST algorithms are synthesized through a systematic procedure:
1. **Select Chain Order and Weights**: Choose $r$; set $\kappa=-1/(r+1)$, $p=1$ (for canonical design), yielding weights $p_i=1+(i-1)\kappa$.
2. **Homogeneous Stabilizer and Lyapunov Pair**: Construct a nominal homogeneous stabilizer $u_0(z)$ (e.g., via Hong’s recursive controller) and compatible Lyapunov function $V_1(z)$.
3. **Gain Selection**: Assign $k_P\geq 1$ and $k_I>0$ (or their adaptive analogs). Adaptation is achieved via the $L_1(t,z)$ and $L_2(t,z)$ scheduling above.
4. **Time-Scaling for Perturbed Systems**: If disturbances or parameter variations are significant, select time-scaling and scaling matrices $D_\lambda$ to ensure gain dominance.
5. **Barrier Parameter**: Choose $\varepsilon>0$ to trade accuracy for gain magnitude.
6. **Barrier Function**: Implement $l(t)\geq 1$, diverging as $t\to\infty$, and ensure gain continuity at $\bar t$.
7. **Integral Action**: Employ integral terms to ensure second-order (or higher) sliding, enforcing robust convergence [2206.10904, 1507.07893].

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 $r$, compute weights| [1507.07893]         |
| 2    | Design $u_0(z)$, $V_1(z)$     | [1507.07893, 2206.10904] |
| 3    | Select or adapt gains         | [1507.07893, 2206.10904] |
| 4    | Set time- and gain-scaling    | [1507.07893]         |
| 5    | Implement barrier/adaptive $L$| [2206.10904]         |
| 6    | Analyze via Lyapunov functions| [1507.07893, 2206.10904] |

## 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 $k_1>\Delta$, $k_2>\Delta^2/(4k_1)$, where $\Delta$ upper-bounds model and disturbance uncertainties, inclusive of the fuzzy residuals [1601.04245].

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 $-\phi/\gamma$.
- 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 [2206.10904].
- 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 [1601.04245].

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.

Source: https://www.emergentmind.com/topics/higher-order-super-twisting-host