Adaptive nonsingular barrier function-based terminal sliding mode control for disturbed nonlinear systems with application to chaos-based secure communication

Abstract

The adaptive finite-time sliding-mode control (SMC) scheme with a nonsingular barrier function is presented in this paper as a novel approach to path tracking and stabilization in nonlinear systems that are subject to external disturbances. In order to achieve rapid convergence in finite time without the singularities that are commonly associated with terminal sliding-mode approaches, the control strategy incorporates a nonsingular terminal sliding surface that was designed under Hurwitz stability conditions. Without overestimating control gains or requiring prior knowledge of the disturbance boundaries, robustness is improved by using an adaptive barrier function. A secure chaotic communication system employing controlled chaotic dynamics for data encryption and transmission is created in order to verify the efficacy and applicability of the suggested approach. The method’s potential in advanced control and soft computing applications is highlighted by extensive simulation results that show it can achieve high-accuracy tracking, effective disturbance rejection, low chatter behavior, and secure information encryption.

Keywords

sliding mode control finite time control adaptive control barrier function nonlinear system

Introduction

Most engineering and physical systems are inherently nonlinear and often subjected to high levels of uncertainties. However, modeling and controlling nonlinear systems, especially over a wide range of operation, is a challenging problem. This is because nonlinear systems give rise to various interesting phenomena such as dead-zones, hysteresis, bifurcation, and chaos, which do not permit the use of linear approximations (Mohadeszadeh and Pariz, 2022). Nonlinear equations are also difficult to solve analytically. Uncertainties stemming from the lack of knowledge about system parameters and the unavailability of all system states further complicate the control problem (Cheng et al., 2023). Numerous control designs, ranging from classical robust and adaptive controls, to Lyapunov-based methods, gain-assignment methods, fuzzy controls, backstepping controls, and intelligent controls have been proposed in the literature for nonlinear uncertain systems (Wu et al., 2024; Zuo et al., 2024). Recent advancements in control systems and encryption methodologies have addressed critical challenges in high-order system stabilization and secure data transmission. In the domain of control systems, Guo and Hu proposed a time-base generator (TBG) approach for predefined-time stabilization of high-order systems under unknown disturbances, achieving robust convergence within a user-specified time by leveraging a novel time-varying gain (Guo and Hu, 2023). Similarly, Xiong and Chen introduced a radial basis function neural network (RBFNN)-based adaptive sliding mode control for nonlinear systems, effectively compensating for parameter uncertainties and external disturbances through real-time adaptation (Xiong and Chen, 2025). Both studies emphasize robust control strategies but differ in their approaches: Guo et al. focus on deterministic time convergence, while Xiong et al. prioritize adaptability to dynamic uncertainties. Meanwhile, Zhao et al. developed a supervised nonlinear dimensionality-reduction surrogate modeling method. Such a method combines the supervised kernel principal component analysis (SKPCA) and polynomial chaos-Kriging (PCK) techniques into the jointly surrogate modeling process and adaptively establishes the accurate mapping from the high-dimensional inputs to the output of the system (Zhao et al., 2024). In the encryption domain, Xie et al. present a new chaotic plotting technique, termed 2D-LMSM, which integrates improvements from logistic and sine mappings. The proposed method demonstrates significant advancements in the encryption process, showcasing its applicability in securing speech communications effectively (Huang et al., 2023). Herbadji et al. proposed an improved chaotic map for speech encryption, improving the chaotic behavior of traditional systems. According to the authors of the article proposed algorithm preprocesses speech signals to eliminate silent segments, allowing for efficient encryption while maintaining security. The tweakable aspect enables multiple encrypted outputs from the same voice using a single secret key, reducing computational resources and time (Herbadji et al., 2024). Liu et al. presented a multi-receiver certificateless searchable public key encryption (mCLSPE) scheme that leverages proxy re-encryption. This scheme addresses the inherent issues in searchable public key encryption (SPE) and also allows for more flexible addition of receivers and enhances data-sharing flexibility (Liu et al., 2025).

Sliding mode control (SMC) is renowned for being one of the main approaches to control uncertain nonlinear systems (Bartolini and Punta, 2010; Chen et al., 2018a; Cruz-Ortiz et al., 2022). Among SMC’s remarkable properties are its ability to convert complex systems into simple one, robustness, accuracy and ease of implementation (Luo et al., 2018; Yang and Ma, 2024). Additionally, combining SMC with other techniques such as predictive control (Li et al., 2022a; Ma and Huang, 2021), event-triggered control (Yang et al., 2020; Yao et al., 2020, 2022), backstepping control (Song et al., 2020), adaptive control (Fan and Yang, 2016; Guo et al., 2022; Jiang et al., 2020), and neural networks control (Lian et al., 2019; Park et al., 2019; Wang and Zhao, 2021) was shown to further simplify the control strategy (He et al., 2018).

Because of its resilience to perturbations, SMC finds extensive use in a wide range of industries, including robotics, aerospace, automotive systems, and secure communication networks. In dynamic environments, like autonomous vehicles traveling on constantly shifting terrain or communication systems with channel noise, SMC’s ability to maintain stability and performance is crucial. Traditional SMC methods, however, frequently call for exact knowledge of system parameters or disturbance boundaries, which may not be achievable in real-world scenarios. The adaptive mechanisms suggested in this work get around these restrictions and increase the applicability of SMC for systems with time-varying uncertainties by dynamically altering the control gains. Because it guarantees dependable performance while lowering the computational load related to control gains, this adaptive approach is appropriate for embedded systems with constrained resources.

The convergence time of controlled system is a critical factor in the system’s overall performance and operation. This is especially important for nonlinear systems with disturbances. Finite time control was a breakthrough in control system design and has played a significant role in control engineering field (In-Soo et al., 2006). Unlike linear feedback control, which achieves exponential convergence with infinite time and control gain, finite-time control ensures convergence within a fixed time. This convergence time is set in advance and depends on the initial states as well as the chosen control gain parameters (Li et al., 2022b). In Chen et al. (2018b) and Zuo and Wang (2022), the strict-feedback systems with unknown parameters finite time control problem was examined. In general, finite time control is necessary for SMC-based approaches because the systems’ dynamic behaviors, which are determined by the sliding mode, do not act until the system states reach the sliding mode surface (Wu and Huang, 2022; Zhang et al., 2020; Zhu et al., 2022). Hence, in contrast to the asymptotic stabilization that only ensures convergence in the infinite settling time, finite time control ensures that the states reach the sliding surface in the finite time.

In recent years, the application and the development of SMC for uncertain nonlinear systems has seen significant growth (Geronel and Bueno, 2025; Vaseghi et al., 2023). To address the challenge of state and parameter estimation in nonlinear systems, researchers have introduced sliding mode adaptive observers, as detailed in He et al. (2018), Na et al. (2013), and Rios et al. (2021). A high order sliding-mode observer has been proposed in Rios et al. (2021) for a nonlinear system. Global and uniform asymptotic stability was established using the small-gain theorem and the Lyapunov theory. A new approach to the adaptive control design for the pure-feedback systems is introduced in Na et al. (2013), which avoids the usage of backstepping. This technique employs a set of alternative states and their corresponding transform to achieve the desired control objectives. After system transformation, a high-order finite-time observer is employed to estimate the unknown states of the derived canonical system. Two adaptive neural control schemes have been proposed to achieve the tracking control. Finite time SMC control is proposed in Chen et al. (2018b) and Li et al. (2017, 2018). In Li et al. (2018), a formation SMC design is proposed for under-actuated ships. To realize the sliding mode control and avoid the singularity problem in SMC, terminal sliding mode control with linear sliding mode surface is designed. A terminal sliding mode approach was proposed in Chen et al. (2013) to deal with nonlinear systems with external disturbances and input saturations. However, the approach required large control inputs to properly stabilize the system. Sliding mode approaches were equipped with and adaptation mechanism in Edwards and Shtessel (2016) and Negrete-Chávez and Moreno (2016) to overcome this problem. In Li et al. (2017), the Markov jump nonlinear systems adaptive SMC is planned. Focuses on the actuator faults and system uncertainty, the reduced-order equivalent sliding motion, upon which the SMC surface is constructed, is stochastically stable. The system’s state trajectories are then controlled onto the sliding mode surface in a finite amount of time by the adaptive sliding mode controller. Because of its high application demands, the problem of tracking the trajectory of high order nonlinear systems with uncertainties or disturbances has sparked a lot of effort (Zhang et al., 2020). Current related outcomes, like backstepping design, higher order sliding mode integration, and increased virtual control complexity in each backstepping step, manifest alongside the system controller’s growing complexity and result in an expensive implementation process.

Barrier function (BF)-based SMC approaches were recently proposed as an adaptive strategy capable of ensuring a control gain that is not overestimated, guaranteeing the convergence of output variables to the specified neighborhood of zero, and not requiring any knowledge of the upper bound of the disturbances (Obeid et al., 2018). A barrier adaptive TSMC tactic was proposed in Mobayen et al. (2021) for the control of a disturbed Robotic Manipulator. An adaptive nonsingular TSMC technique was proposed in Mobayen et al. (2022) for robust stabilization of disturbed nonlinear systems. Using the linear matrix inequalities and the Lyapunov stability theorem, the state trajectories’ overall asymptotic stability was determined. Two chaotic systems were successfully controlled using the method when nonlinear dynamics and external disturbances were present. Chaos, an element of nonlinear systems theory, exhibits a fascinating nonlinear phenomenon that has been extensively studied, simulated, and implemented in circuits for the previous 40 years (Strogatz, 2018). From a communications standpoint, chaotic signals display remarkable attributes. Numerous investigations have characterized these signals as having broad bandwidth, extreme sensitivity to initial conditions, noise-resembling properties, impulsive correlation, and minimal cross-correlation between signals originating from different starting points (Dmitriev et al., 2023; Martinez-Gost et al., 2024; Vaseghi et al., 2017). The practical applications of chaos extend to various communication domains, including radar systems (Warjri et al., 2023), spread-spectrum technologies (Liu et al., 2024), secure communications (Suchit et al., 2024), ultra-wide-band transmission (Efremova and Kuzmin, 2024), and signal encryption (Alanazi et al., 2023).

The proposed adaptive nonsingular barrier function-based terminal sliding mode control (SMC) approach builds upon and advances existing barrier function-based SMC methods, such as those presented in Shao et al. (2021) and Shao et al. (2022). The work in Shao et al. (2022) develops a barrier function adaptive sliding mode (BFASM) control for uncertain nonlinear systems with actuator saturation, focusing on asymptotic convergence of tracking errors into a prespecified region. It employs a reaching control input to adapt to time-varying disturbances without requiring disturbance bound information, addressing chattering issues caused by overly aggressive control gains. Similarly, Shao et al. (2022) propose a BFASM controller for linear motor positioners, introducing a modified barrier function (MBF) to handle actuator saturation and ensure finite-time convergence of tracking errors, with adaptive gain adjustment to minimize control input when disturbances decrease. Our approach introduces several key improvements over these methods. First, we incorporate a nonsingular terminal sliding manifold, designed under Hurwitz stability conditions, which guarantees finite-time convergence of the sliding variable to the origin, avoiding singularities inherent in traditional terminal SMC approaches. This contrasts with Shao et al. (2022), which achieve asymptotic convergence, and Shao et al. (2021), which focus on finite-time error convergence but does not address singularity issues in the sliding manifold design. Second, our adaptive barrier function dynamically estimates unknown disturbance bounds without overestimation, similar to the referenced works, but is tailored for a broader class of disturbed nonlinear systems, including those exhibiting chaotic dynamics. This enables our method to handle complex nonlinearities and external disturbances more effectively, as demonstrated in the synchronization of a hyper-jerk chaotic system (sub-3 seconds, Section 5). Third, while Shao et al. (2021, 2022) focus on actuator saturation in mechatronic systems, our method extends the application to chaos-based secure communication, enabling dynamic cryptographic key evolution alongside rapid synchronization. This dual capability addresses critical challenges in secure communication systems, where both security and real-time performance are paramount, a feature not explored in the referenced studies. Finally, our approach further reduces chattering through the nonsingular terminal sliding manifold and adaptive gain tuning, offering smoother control signals (Section 5, Figure 7) compared to the conventional SMC methods discussed in the referenced works. These advancements make our proposed method particularly suitable for applications requiring robust control under significant uncertainties and rapid, secure synchronization, such as in chaos-based communication systems, while maintaining computational efficiency and practical implement ability for resource-constrained systems.

Inspired by the discussion above, we present in this paper a nonsingular finite-time stabilizing controller for a class of disturbance-containing nonlinear systems. The following are the paper’s principal contributions:

(1) A design that combines finite time control with Hurwitz criterion to ensure the stabilization of disturbed nonlinear systems in finite time.

(2) A design that guarantees the convergence of the sliding surface to the origin in finite time.

(3) Successful implementation of the proposed design for chaotic encryption with the aim of ensuring secure communication.

The remainder of the document is structured as follows: The problem formulation is explained in Section Problem Formulation. The stability analysis and control design for the sliding model are highlighted in Section Main Results. The use of the suggested design for chaotic encryption to guarantee secure communications is described in Section Application to Secure Communications. The simulation results are presented later. The paper is finally concluded in Section Conclusions.

Problem formulation

Examine the nonlinear systems of nth order that have disturbances that are described by:

{\begin{cases} {\dot{x}}_{i} = x_{i + 1}, 1 \leq i \leq n - 1 \\ {\dot{x}}_{n} = f (x (t), t) + b (x (t), t) u (t) + d (x (t), t) \end{cases}

(1)

where

x = [x_{1}, x_{2}, \dots, x_{n}] \in R^{n}

is the state vector,

u

is the control input,

b (x (t), t)

and

f (x (t), t)

are bounded nonlinear functions,

b (x (t), t)

is invertible, and

d (x (t), t) \in R

is the bounded external disturbance with unknown bound

D \in R

, that is,

{| d (x (t), t) |}_{\max}

The nonlinear system described in (1) assumes that the functions $f (x (t), t)$ and $b (x (t), t)$ are finite and the control input matrix $b (x (t), t)$ is invertible, which are common assumptions in control theory to ensure controllability. These conditions are met in many practical systems, such as robotic manipulators, where $f (x (t), t)$ represents nonlinear dynamics such as friction or Coriolis forces, and $b (x (t), t)$ models the actuator dynamics. The external disturbance $d (x (t), t)$ is finite but with an unknown upper bound, reflecting real-world scenarios where disturbances such as wind gusts in air vehicles or electromagnetic interference in communication systems are unpredictable. This formulation allows the proposed control strategy to be applied to a wide range of engineering systems, including systems exhibiting chaotic behavior, where accurate modeling is challenging due to sensitivity to initial conditions and parameter variations.

Lemma 1

(Wu and Huang, 2022): Consider the candidate positive-definite Lyapunov function $V (t)$ with

\dot{V} {(t) \leq - α V (t)}^{η},

(2)

where

t \geq t_{0}, V (t_{0}) \geq 0

α > 0

, and

η

is a fraction of two odd positive integers (

1 > η > 0

). The Lyapunov function

V (t)

approaches the origin in the finite time

t_{s}

defined by:

t_{s} \geq t_{0} + \frac{V {(t_{0})}^{1 - η}}{α} .

(3)

Main results

Define the linear sliding surface $s (t)$ as:

s (t) = C_{1} x_{1} + C_{2} x_{2} + \dots C_{n - 1} x_{n - 1} + x_{n},

(4)

where the constants

C_{1}, C_{2}, \dots, C_{n - 1}

are chosen to satisfy the sliding surface’s requirements, which are assigned as the coefficients of Hurwitz polynomial

c_{1} + c_{2} λ + \dots + c_{n - 1} λ^{n - 2} + λ^{n - 1}

. It can be resulted that the system states

X = {[x_{1}, x_{2}, \dots x_{n}]}^{T} \in R^{n}

converge to origin while

s = 0

, that is to confirm that determining a controller to drive the system trajectories into the sliding surface

s = 0

became the new control objective.

When $s (t) = 0$ , one obtains $x_{n} = - C_{1} x_{1} - C_{2} x_{2} - \dots C_{n - 1} x_{n - 1}$ . From (1), one has:

{\begin{cases} {\dot{x}}_{i} = x_{i + 1}, 1 \leq i \leq n - 1 \\ {\dot{x}}_{n - 1} = - C_{1} x_{1} - C_{2} x_{2} - \dots C_{n - 1} x_{n - 1} \end{cases}

(5)

The equivalent control input is found from (5) as

u_{e q} = - b {(x (t), t)}^{- 1} (\sum_{i = 1}^{n - 1} C_{i} x_{i + 1} (t) + f (x (t), t)) .

(6)

The nonsingular terminal sliding manifold has the following definition for the sliding surface s(t)’s finite-time convergence to zero:

σ (t) = α \int_{0}^{t} s {(τ)}^{p / q} d τ + s (t),

(7)

where

p

and

q

are two odd integers, with

α > 0

0 < p / q < 1

Theorem 1

Examine the nonlinear system (1) with external disturbances. If the control input is designed as:

u (t) = u_{e q} (t) + u_{N} (t),

(8)

with the equivalent input (6) and the discontinuous control input as

u_{N} (t) = - b {(x (t), t)}^{- 1} ((D + μ) s i g n (σ (t)) + α s {(t)}^{p / q}),

(9)

where

μ > 0

and D>

{| d (x (t), t) |}_{\max}

, then, the nonsingular terminal sliding manifold (7) converges to the origin in finite time.

Proof

Applying the time-derivative of the nonsingular terminal sliding manifold (7) to the positive-definite Lyapunov function $V (σ (t)) = {σ (t)}^{2} / 2$ , we obtain

\dot{V} (σ (t)) = σ (t) (\dot{s} {(t) + α s (t)}^{p / q}) = σ (t) {\sum_{i = 1}^{n - 1} c_{i} {\dot{x}}_{i} (t) + {\dot{x}}_{n} {(t) + α s (t)}^{p / q}},

(10)

Substituting (1) into (10) yields:

\dot{V} (σ (t)) = σ (t) (C_{i} x_{i + 1} (t) + f (x (t), t) + b (x (t), t) u (t) + d (x (t), t) + α s {(t)}^{p / q}),

(11)

where by substituting (8) with (6) and (9) in the above equation, one obtains:

\dot{V} (σ (t)) = - σ (t) ((D + μ) s i g n (σ (t)) - d (x (t), t)) \leq - (D - | d (x (t), t) |) | σ (t) | - μ | σ (t) | \leq - μ | σ (t) | \leq - 2^{0.5} μ {V (σ (t))}^{0.5} .

(12)

Then, the nonsingular terminal sliding manifold (7) reaches zero in the finite time. □

If $σ (t) = 0$ is reached, one obtains from (7) that $s (t) = - α \int_{0}^{t} s {(τ)}^{p / q} d τ$ , therefore,

\dot{s} (t) = - α s^{\frac{p}{q}} (t), 0 < \frac{p}{q} < 1 .

(13)

Based on Lemma 1, the state $s (t)$ is convergent to origin in the finite time. In practice, to overcome the effect of disturbances, adaptive controllers are designed, where the gains are adjusted as the system states approach to the convergence region. In what follows, a barrier function adaptive finite time control technique is designed. Applying the barrier function adaptive TSMC scheme, the external disturbance is estimated efficiently, and the disturbed nonlinear system is stabilized. When the state trajectories get close to the origin’s boundary, the barrier function’s output goes to infinity. It is a smooth function that is defined in a small, open region surrounding the origin. This feature is presented to realize the control of the state/output constraint. By employing a barrier function in the adaptive SMC, without knowing the upper bounds of external disturbances, the sliding variable is kept within a predetermined neighborhood of the origin (Obeid et al., 2018). Additionally, the barrier function-based adaptive control method is employed on the higher-order sliding mode control systems (Laghrouche et al., 2021). Employing the barrier function, the nonlinear control input is proposed by:

u_{N} (t) = - {b (x (t), t)}^{- 1} ((\hat{D} (t) + μ) sign (σ (t)) + α s {(t)}^{p / q}),

(14)

with

\hat{D} (t) = {\begin{cases} D_{a} (t), if 0 < t \leq \bar{t} \\ D_{p s b} (t), if t > \bar{t} \end{cases}

(15)

where

\bar{t}

is the time that the states converge to the neighborhood

ε

of the terminal sliding mode manifold

σ (t)

. The adaptive law

{\dot{D}}_{a} (t)

and positive-semi-definite barrier function

D_{p s b}

are proposed as

{\dot{D}}_{a} (t) = l | σ (t) |,

(16)

D_{p s b} (t) = \frac{| σ (t) |}{ε - | σ (t) |},

(17)

where

ε

and

l

are two positive parameters. Employing the adaptive law (16), the control gain

\hat{D} (t)

is tuned to increase until the states reach neighborhood

ε

of terminal sliding manifold

σ (t)

at time

\bar{t}

. For

\bar{t} > 0

, the control gain

\hat{D} (t)

switches to the positive-semi-definite barrier function

D_{p s b}

that decreases the convergence region and maintains the states in that region. The stability of the nonlinear disturbed system (1) is studied in the following conditions:

C o n d i t i o n (I) : 0 < t \leq \bar{t}

Theorem 2

Consider the nonlinear system in the presence of external disturbance as equation (1). For the time $0 < t \leq \bar{t}$ , using the total control law (combining the equivalent control input $u_{e q} (t)$ (equation (6)) and the discontinuous control input (14) with $\hat{D} (t) = D_{a} (t)$ ) as

u (t) = - {b (x, t)}^{- 1} [C_{1} x_{2} (t) + C_{2} x_{3} (t) + f (x, t) + (D_{a} (t) + μ) s i g n {(σ) + α s (t)}^{p / q}],

(18)

then the system states reach the neighborhood

ε

of the terminal sliding manifold in the finite time.

Proof

Construct the following positive-definite Lyapunov function:

W (σ (t), D_{a} (t)) = 0.5 ({σ (t)}^{2} + φ^{- 1} {(D_{a} (t) - D)}^{2}),

(19)

where

φ

is a positive scalar and

D

is a positive unknown constant. Differentiating the above function gives

\dot{W} (σ (t), D_{a} (t)) = σ (t) \dot{σ} (t) + φ^{- 1} (D_{a} (t) - D) {\dot{D}}_{a} (t),

(20)

where using the adaptive law

{\dot{D}}_{a} (t)

, one achieves

\dot{W} (σ (t), D_{a} (t)) = σ (t) (C_{1} x_{2} (t) + C_{2} x_{3} (t) + f (x (t), t) + b (x (t), t) u (t) + d (x (t), t) + α s {(t)}^{\frac{p}{q}}) + l φ^{- 1} (D_{a} (t) - D) | σ (t) | .

(21)

Substituting the adaptive controller (18) in (21), one attains

\dot{W} (σ (t), D_{a} (t)) = - (D_{a} (t) + μ) σ (t) s i g n (σ (t)) + d (x (t), t) σ (t) + l φ^{- 1} (D_{a} (t) - D) | σ (t) | \leq - (D - | d (x (t), t) |) | σ (t) | - (1 - l φ^{- 1}) (D_{a} (t) - D) | σ (t) |,

(22)

where since

D > | d (x (t), t) |

and

l φ^{- 1} < 1

, then the above equation can be represented as

\dot{W} (σ (t), D_{a} (t)) \leq - 2^{1 / 2} (D - | d (x (t), t) |) \frac{| σ (t) |}{2^{1 / 2}} - {(2 φ)}^{1 / 2} (1 - l φ^{- 1}) | σ (t) | \frac{D_{a} (t) - D}{{(2 φ)}^{1 / 2}} \leq - \min {2^{1 / 2} (D - | d (x (t), t) |), {(2 φ)}^{1 / 2} (1 - l φ^{- 1}) | σ (t) |} (\frac{| σ (t) |}{2^{1 / 2}} + \frac{| D_{a} (t) - D |}{{(2 φ)}^{1 / 2}}) \leq - Ω {W (σ (t), D_{a} (t))}^{1 / 2},

(23)

where

Ω = \min {2^{1 / 2} (D - | d (x (t), t) |), {(2 φ)}^{1 / 2} (1 - l φ^{- 1}) | σ (t) |}

C o n d i t i o n (I I) : t > \bar{t}

Theorem 3

Consider the nonlinear system with external disturbance as equation (1). For the time $t > \bar{t}$ , if the adaptive control law (containing the equivalent controller (6) and discontinuous control input (14) with $\hat{D} (t) = D_{p s b} (t)$ ) is designed as

u (t) = - {b (x (t), t)}^{- 1} [C_{1} x_{2} (t) + C_{2} x_{3} (t) + f (x (t), t) + (D_{p s b} (t) + μ) s i g n (σ (t)) + α s {(t)}^{p / q}],

(24)

then the system states reach the convergence region

| σ (t) | \leq ε

in finite time.

Proof

Construct the positive-definite Lyapunov function by

Z (σ (t), D_{p s b} (t)) = 0.5 ({σ (t)}^{2} + {D_{p s b} (t)}^{2}),

(25)

where its time-derivate is obtained as

\dot{Z} (σ (t), D_{psb} (t)) = σ (t) (C_{1} x_{2} (t) + C_{2} x_{3} (t) + f (x (t), t) + b (x (t), t) u (t) + d (x (t), t) + α s {(t)}^{p / q}) + D_{psb} (t) {\dot{D}}_{psb} (t),

(26)

where substituting the controller (24) into the above equation, one has

\dot{Z} (σ (t), D_{p s b} (t)) = - (D_{p s b} (t) + μ) σ (t) s i g n (σ (t)) + d (x (t), t) σ (t) + D_{p s b} (t) {\dot{D}}_{p s b} (t) \leq - D_{p s b} (t) | σ (t) | - μ | σ (t) | + | d (x (t), t) | | σ (t) | + D_{p s b} (t) {\dot{D}}_{p s b} (t) \leq - D_{p s b} (t) | σ (t) | - μ | σ (t) | + | d (x (t), t) | | σ (t) | + D_{p s b} (t) \frac{ε}{{(ε - | σ (t) |)}^{2}} s i g n (σ (t)) \dot{σ} (t) \leq - (D_{psb} (t) - | d (x (t), t) |) | σ (t) | - μ | σ (t) | + D_{psb} (t) \frac{ε}{{(ε - | σ (t) |)}^{2}} sign (σ (t) [d (x (t), t) - (D_{psb} (t) + μ) sign (σ (t))] \leq - (D_{p s b} (t) - | d (x (t), t) |) | σ (t) | - \frac{ε}{{(ε - | σ (t) |)}^{2}} [D_{p s b} (t) - | d (x (t), t) |] D_{p s b} (t)

(27)

where since

D_{p s b} (t) > | d (x (t), t) |

and

\frac{ε}{{(ε - | σ (t) |)}^{2}} > 0

, we achieve

\dot{Z} (σ (t), D_{p s b} (t)) \leq - 2^{1 / 2} (D_{p s b} (t) - | d (x (t), t) |) \frac{| σ (t) |}{2^{1 / 2}} - \frac{2^{1 / 2} ε}{{(ε - | σ (t) |)}^{2}} [D_{p s b} (t) - | d (x (t), t) |] \frac{D_{p s b} (t)}{2^{1 / 2}} \leq - 2^{1 / 2} (D_{p s b} (t) - | d (x (t), t) |) \min {1, \frac{ε}{{(ε - | σ (t) |)}^{2}}} (\frac{| σ (t) |}{2^{1 / 2}} + \frac{D_{p s b} (t)}{2^{1 / 2}}) \leq - £ {Z (σ (t), D_{p s b} (t))}^{1 / 2}

(28)

where

£ = 2^{1 / 2} (D_{p s b} (t) - | d (x (t), t) |) \min {1, \frac{ε}{{(ε - | σ (t) |)}^{2}}}

. □

Application to secure communications

This section outlines the implementation of the proposed approach for a chaos-based secure communication system utilizing synchronized chaotic oscillators. In a secure chaotic communication setup, chaotic oscillators are positioned at both the transmitter and receiver ends. The transmitter’s chaotic system generates chaotic signals that, along with scrambling algorithms, blend and transmit the scrambled message. To extract the received signal, the receiver requires identical chaotic signals used in the scrambling process. Consequently, the receiver’s chaotic system must be in sync with the transmitter’s chaotic system. This synchronization is comparable to aligning the receiver’s local oscillator with the sender’s carrier in traditional telecommunications for message detection. However, due to chaotic signals’ extreme sensitivity to initial conditions, synchronizing these chaotic oscillators is considerably more challenging, demanding sophisticated and precise algorithms. As a result, without knowledge of the chaotic oscillator’s dynamics underlying the scrambling process and access to the synchronizing and control signal, message retrieval becomes extremely difficult. Figure 1 illustrates the proposed chaos-based secure communication system. As shown in this figure, the model consists of synchronized chaotic oscillators deployed at both the transmitter and receiver ends. At the transmitter, the master chaotic oscillator generates chaotic signals $x_{m} (t)$ that are used in conjunction with a continuous nonlinear encryption function $Φ : R^{3} \times R \to R$ to encrypt the message $m (t)$ into $m_{e} (t)$ , which is transmitted via one channel. Simultaneously, the master oscillator’s states are transmitted through a second channel to enable synchronization. At the receiver, the proposed adaptive nonsingular barrier function-based terminal sliding mode controller synchronizes the slave chaotic oscillator with the master using the transmitted synchronization signals. The decryption function $Θ : R^{3} \to R$ then recovers the original message $m_{d} (t)$ from $m_{e} (t)$ using the synchronized slave states.

Because chaotic systems are extremely sensitive to initial conditions and parameter changes, synchronizing chaotic oscillators for secure communications is a significant challenge. Out-of-synch and inaccurate message recovery can result from even a small oscillator mismatch between the transmitter and receiver. When there are outside disruptions or channel noise present, traditional synchronization methods like linear feedback control frequently fail to produce quick and reliable synchronization. These problems are addressed by the suggested adaptive nonsingular TSMC technique, which ensures that the synchronization error will converge to zero in finite time even in the presence of disturbances. The technique reduces chatter and ensures robust performance by implementing an adaptive barrier function that dynamically modifies the control gains without requiring prior knowledge of the system parameters or disturbance boundaries. For real-time secure communication systems, where low latency, high reliability, and noise resilience are essential, this makes the suggested approach especially appropriate.

In the proposed chaos-based secure communication method, we use two channels to speed up the synchronization process. The encryption/decryption mechanism can be summarized as follows:

Encryption: A nonlinear function $Φ : R^{3} \times R \to R$ , which is continuous in its first argument $X \in R^{3}$ , is used to encrypt the message $m (t)$ . This function has a unique property: for every fixed pair of $(X, m) \in R^{3} \to R$ , there exists a distinct function $Θ : R^{3} \to R$ that is also continuous in its first argument $X \in R^{3}$ and satisfies $Θ (X, Φ (X, m)) = m$ . The encryption function $Φ$ is constructed using master chaotic signals, resulting in a signal $m_{e} (t)$ that contains the message and is transmitted through one channel.

Synchronization: The receiver side of the communication system incorporates a synchronization block to recover the chaotic oscillator signals and obtain the required data for decryption. The transmitter oscillator signals are transmitted through one channel for synchronization purposes. These signals are adequate to generate receiver oscillator signals which are exact estimates of the master oscillator signals. Retrieving this signal is crucial for decrypting the message received on the second channel. Section 3 will provide detailed information about the synchronizer.

Decryption: After recovering the oscillator signals, the decryption function $Θ$ is applied in conjunction with the signal $m_{e} (t)$ to retrieve the decrypted message $m_{d} (t)$ .

Remark 1

The proposed method offers advantages over existing approaches, including adaptive robustness without over-estimation (dynamically adjusting gains via barrier functions, eliminating the need for prior disturbance bounds), and reduced chattering (smoother control signals compared to conventional SMC). Its nonsingular terminal sliding manifold avoids instability issues, while the application to chaos-based secure communication (Section 4 and 5) demonstrates superior disturbance rejection and synchronization performance (e.g., error convergence in <3s, Figures 2 –6). Though minor limitations exist (e.g., parameter tuning), the method’s faster convergence, lower chattering, and adaptability make it highly effective for real-world nonlinear systems. Also, the proposed adaptive nonsingular barrier function-based finite-time sliding mode control delivers crucial advantages for chaotic secure communications by simultaneously enabling dynamic cryptographic key evolution and ultra-fast synchronization. Since the control proofs are independent of the specific chaotic dynamics (Theorems 1–3), the transmitter can continuously modify oscillator parameters to generate fresh encryption keys for each transmission while maintaining sub-3-second synchronization (Figures 2 –6)—a critical combination that ensures both enhanced security through dynamic key generation and practical usability in real-time applications. This dual capability fundamentally addresses the core challenges of maintaining cryptographic security while achieving the rapid synchronization essential for practical implementation in chaos-based communication systems.

Simulation results

This section employs numerical simulations to assess the effectiveness and relevance of the proposed adaptive nonsingular barrier function-based finite-time sliding mode control for disturbed nonlinear systems with application to chaos-based secure communication. The simulation trials were directed on a computer prepared with an Intel(R) Core (TM) i7-6820HQ CPU running at 2.70 GHz, 16 GB of RAM, and a 750 GB hard drive. The system operated on 64-bit Microsoft Windows 11, with MATLAB R2023a serving as the computational environment.

Consider the following 4-D master chaotic hyper-jerk (Vaidyanathan et al., 2018):

{\begin{cases} {\dot{x}}_{1 m} = x_{2 m} \\ {\dot{x}}_{2 m} = x_{3 m} \\ {\dot{x}}_{3 m} = x_{4 m} \\ {\dot{x}}_{4 m} = f (x_{m}, t) \end{cases}

(29)

where

f (x_{m}, t) = 2.2 x_{4 m} + 5 x_{1 m} x_{2 m} + x_{1 m} x_{4 m} + 2 (x_{1 m}^{2} + x_{1 m} x_{3 m} + 1)

. Furthermore, the slave chaotic system which includes control inputs and external disturbances can be modeled as follows:

{\begin{cases} {\dot{y}}_{1 s} = y_{2 s} \\ {\dot{y}}_{2 s} = y_{3 s} \\ {\dot{y}}_{3 s} = y_{4 s} \\ {\dot{y}}_{4 s} = f (y_{s}, t) + b u (t) + d (t) \end{cases},

(30)

where

f (y_{s}, t) = 2.2 y_{4 s} + 5 y_{1 s} y_{2 s} + y_{1 s} y_{4 s} + 2 (y_{1 s}^{2} + y_{1 s} y_{3 s} + 1)

b = 1

d (t) = 0.1 \sin (t)

Remark 2

The system model in equation (1) includes external disturbances $d (x (t), t)$ , representing bounded exogenous inputs (e.g., sensor noise) with an unknown bound ( $| d (x (t), t) | < D$ , $D > 0$ ). This choice reflects real-world systems, such as robotic manipulators or chaotic oscillators, where disturbances are bounded but lack a known form or bound. The barrier-function adaptive SMC (Equations (14)–(17)) estimates the disturbance bound $\hat{D} (t)$ dynamically, avoiding overconservative control gains and accommodating various disturbance types (e.g., $d (t) = 0.1 \sin (t)$ ).

By subtracting equation (29) from (30), the synchronization error between master and slave chaotic systems is obtained in the following form:

{\begin{cases} {\dot{e}}_{1} = e_{2} \\ {\dot{e}}_{2} = e_{3} \\ {\dot{e}}_{3} = e_{4} \\ {\dot{e}}_{4} = f (E, t) + b u (t) + d (t) \end{cases},

(31)

finally, we reach the disturbed nonlinear system considered in equation (1) as

{\begin{cases} {\dot{e}}_{i} = e_{i + 1}, 1 \leq i \leq 4 - 1 \\ {\dot{e}}_{n} = f (e (t), t) + b (e (t), t) u (t) + d (e (t), t) \end{cases}

(32)

For the implementation, the initial states of the master and slave chaotic oscillators are certain as

x_{1 m} (0) = - 1.154875148512458

x_{2 m} (0) = - 0.621547895124587

x_{3 m} (0) = - 4.551245873254871

x_{4 m} (0) = 1.351245873254871

y_{1 s} (0) = 0.649825487542168

y_{2 s} (0) = - 1.115987025491573

y_{3 s} (0) = - 6.315987025491573

y_{4 s} (0) = - 0.615987025491573

. Now, by choosing the sliding surface as

S (t) = 60 e_{1} + 47 e_{2} + 12 e_{3} + e_{4}

(33)

and design the controller inputs based on Theorems 2 and 3, we reach the synchronization goal between the master and slave chaotic systems. The numerical simulations validate the effectiveness of the proposed barrier-function adaptive nonsingular terminal sliding mode control (SMC) for synchronizing chaotic systems and enabling secure communication under external disturbances. The results, illustrated in Figures 2 –9, demonstrate fast convergence, robustness, and practical applicability in chaotic cryptosystems.

Remark 3

In the proposed barrier-function adaptive sliding mode control (SMC) methodology, key parameters such as $α$ , $μ$ , $l$ , $ε$ , and the coefficients $C_{1}, C_{2}, \dots, C_{n - 1}$ of the sliding surface (Equation (4)) are tuned using a trial-and-error approach to ensure stability and performance. Specifically, $α > 0$ and $0 < p / q < 1$ (Equation (7)) are selected to achieve finite-time convergence, while $μ > 0$ (Equations (9) and (14)) ensures robustness against disturbances. The adaptive gain parameters $l$ and $ε$ (Equations (16) and (17)) are adjusted to balance convergence speed and constraint satisfaction. While systematic tuning methods (e.g., optimization-based or Lyapunov-based approaches) could enhance efficiency, the trial-and-error method was adopted for simplicity and practicality in simulations (e.g., Section 5 with $d (t) = 0.1 \sin (t)$ ). A detailed tuning procedure will be included in future work to provide systematic steps for gain selection, ensuring optimal performance across various systems.

Figures 2 –5 depict the time trajectories of the master chaotic states $x_{1 m}, x_{2 m}, x_{3 m}, x_{4 m}$ and their corresponding slave states $y_{1 s}, y_{2 s}, y_{3 s}, y_{4 s}$ . The replotted figures now use distinct colors (blue for master, red for slave) and higher resolution to clearly distinguish the trajectories. Synchronization is achieved in less than 3 seconds across all state pairs, despite the presence of bounded external disturbances (e.g., $d (t) = 0.1 \sin (t)$ ). This rapid synchronization is critical for real-world applications like secure communication, where delays can compromise data integrity. For instance, in chaotic oscillators used in cryptography, sub-3-second synchronization ensures minimal latency, enabling real-time data transmission. The consistent convergence across all states highlights the robustness of the proposed control strategy against disturbances such as electromagnetic interference or mechanical vibrations, which are common in robotic manipulators or communication systems.

Figure 6 illustrates the temporal progression of the synchronization error signals, defined as $e_{i} (t) = x_{i m} (t) - y_{i s} (t)$ for $i = 1, 2, 3, 4$ . The replotted figure uses a logarithmic scale for the error magnitude to emphasize the rapid decay to near-zero values within approximately 3 seconds. This fast error convergence underscores the finite-time stability guaranteed by Theorems 1–3, where the nonsingular terminal sliding manifold (Equation (7)) and adaptive control law (Equations (14)–(17)) ensure robust tracking despite unknown disturbance bounds. In practical scenarios, such as chaos-based communication systems, this rapid error convergence minimizes data loss during the initial synchronization phase, ensuring reliable message recovery. The absence of significant oscillations in the error signals further indicates the method’s ability to mitigate chattering, a common issue in conventional SMC that can degrade performance in sensitive systems like sensors or actuators.

Figure 7 presents the time histories of the control input $u (t)$ and the sliding surface $s (t)$ . Figure 8 presents the time histories of the adaptive gain $\hat{D} (t)$ . The replotted figures use distinct line styles and a zoomed-in view to highlight the smooth behavior of these signals. The control input $u (t)$ exhibits suitable magnitudes without excessive peaks, indicating reduced chattering compared to traditional SMC approaches. This smoothness is crucial for practical systems, such as robotic manipulators, where abrupt control actions could cause mechanical wear or instability. The sliding surface $s (t)$ converges to zero rapidly, aligning with the theoretical finite-time convergence (Equation (13)), while the adaptive gain $\hat{D} (t)$ dynamically adjusts to estimate the unknown disturbance bound, as defined in equations (15)–(17). This adaptive mechanism eliminates the need for prior knowledge of the disturbance bound $D$ , making the method highly practical for systems with unpredictable disturbances, such as wind gusts in aerial vehicles or noise in communication channels. The smooth evolution of $\hat{D} (t)$ further ensures that control gains are not overconservative, optimizing energy efficiency in real-world applications.

To robustly validate the efficacy of the proposed adaptive nonsingular barrier function-based terminal sliding mode control (ANBFSMC), we conducted a rigorous comparative analysis against the state-of-the-art adaptive terminal sliding-mode control (ATSMC) method by Wang (2024) under identical conditions synchronizing the 4-D hyper-jerk chaotic system (Equations (29) and (30)) while subjected to disturbances $d (t) = 0.1 \sin (t)$ and matching initial conditions. The results, illustrated in Figure 9, unequivocally demonstrate that the proposed ANBFSMC achieves synchronization in under 3 seconds, outperforming the ATSMC approach, which requires over 6 seconds to reach the same error tolerance. This significant acceleration (>50% faster) stems from two core innovations: (1) the nonsingular terminal sliding manifold (Equation (7)) ensures time convergence without instability near the equilibrium, and (2) the adaptive barrier function (Equations (15)–(17)) dynamically optimizes disturbance rejection by tightening the gain $\hat{D} (t)$ as states approach the sliding surface, eliminating over conservatism. Crucially, this rapid synchronization directly enhances the practicality of chaos-based secure communication where delays exceeding 3 seconds can compromise real-time data transmission while the absence of chattering in ANBFSMC (Figure 7(a)) further ensures signal integrity in noise-sensitive applications like medical or military encryption.

The following section demonstrates the feasibility and implementation of the proposed chaotic cryptosystem through numerical simulations. The original message $m (t)$ is composed of a sequence of sine waves with frequencies $ω = 2 π$ , $π$ and $3 π$ rad/s. The encryption and decryption functions were chosen as

E n c r y p t i o n : Φ : m_{e} (t) = {k_{1} | x_{1 m} |}^{\frac{5}{2}} - k_{2} x_{2 m} + (k_{3} {x_{3 m}}^{\frac{2}{3}} - 2.5) + (k_{4} x_{4 m} - 1.2) m (t) D e c r y p t i o n : Θ : m_{d} (t) = \frac{{{m_{e} (t) - k}_{1} | y_{1 s} |}^{\frac{5}{2}} + k_{2} y_{2 s} - (k_{3} {y_{3 s}}^{\frac{2}{3}} - 2.5)}{(k_{4} y_{4 s} - 1.2)}

(34)

where

k_{i}

is chosen as

k_{i} = \mod (s o r t (x_{i m} (u)), f l o o r (x_{i m} (v))), i = 1, 2, 3, 4

(35)

where the function

\mod (f, g)

returns the remainder of

f

divided by

g

, the function

s o r t (x)

sorts the vector elements of chaotic sequence

x

in accordance with the ascending order and

f l o o r (x)

rounds the elements of chaotic sequence

x

to the nearest integer. It should be noted that the values of

u

can be changed based on the coordination between the transmitter and receiver in each data transmission. Also, to facilitate simulation, noise was added to each transmitted signal, necessitating the implementation of a filter bank at the receiver’s input to purify the signals prior to processing. The system incorporates Butterworth filters with cutoff frequencies set at 40 rad/s and 100 rad/s. These filters are characterized by the following transfer functions:

c h a n n e l 1 (s y n c h r o n i z a t i o n s i g n a l) : H_{1} (s) = \frac{1600}{s^{2} + 56.6 s + 1600} c h a n n e l 2 (e n c r y p t e d m e s s a g e) : H_{2} (s) = \frac{10000}{s^{2} + 141 s + 10000}

(36)

Figure 8(a) shows the encrypted signal, and Figure 8(b) compares the original message $m (t)$ (a composite of sine waves with frequencies $ω = 2 π, π, 3 π$ rad/s) and the decrypted signal $m_{d} (t)$ . The replotted figures use contrasting colors and clear labeling to distinguish the signals. The encrypted signal in Figure 8(a) appears chaotic, ensuring security by masking the original message, while Figure 8(b) demonstrates successful decryption with the recovered signal closely matching the original after approximately 3 seconds. The transient errors during the initial decryption phase, visible in Figure 8(b), are mitigated by transmitting non-essential data at the start, as suggested in the original text. This strategy is practical for secure communication systems, where initial synchronization delays are common, and non-critical data can be used to stabilize the system before transmitting sensitive information. The use of Butterworth filters (Equation (36)) with cutoff frequencies of 40 rad/s and 100 rad/s effectively removes noise from the transmitted signals, ensuring high-fidelity decryption in noisy environments, such as wireless communication networks (Figure 10).

Remark 2. The proposed method offers advantages over existing approaches, including finite-time convergence (ensuring stability within a fixed time, as proven in Theorems 1–3), adaptive robustness without overestimation (dynamically adjusting gains via barrier functions, eliminating the need for prior disturbance bounds), and reduced chattering (smoother control signals compared to conventional SMC). Its nonsingular terminal sliding manifold avoids instability issues, while the application to chaos-based secure communication (Sections 4–5) demonstrates superior disturbance rejection and synchronization performance (e.g., error convergence in <3s, Figures 2 –6). Though minor limitations exist (e.g., parameter tuning), the method’s faster convergence, lower chattering, and adaptability make it highly effective for real-world nonlinear systems. Also, the proposed adaptive nonsingular barrier function-based finite-time sliding mode control delivers crucial advantages for chaotic secure communications by simultaneously enabling dynamic cryptographic key evolution and ultra-fast synchronization. Since the control proofs are independent of the specific chaotic dynamics (Theorems 1–3), the transmitter can continuously modify oscillator parameters to generate fresh encryption keys for each transmission while maintaining sub-3-second synchronization (Figures 2 –6)—a critical combination that ensures both enhanced security through dynamic key generation and practical usability in real-time applications. This dual capability fundamentally addresses the core challenges of maintaining cryptographic security while achieving the rapid synchronization essential for practical implementation in chaos-based communication systems.

Conclusions

This study presents a nonsingular barrier function-based SMC for disturbed nonlinear systems, applied to secure chaotic communication. Motivated by the need to control nonlinear systems under disturbances, the proposed method uses an adaptive barrier function to estimate disturbances without prior knowledge of their bounds. A Hurwitz-based sliding surface ensures exponential state convergence, while a nonsingular terminal sliding manifold achieves finite-time convergence to the origin. Applied to a hyper-jerk chaotic system, the method demonstrated zero error signals in finite time, robust to unmodeled dynamics and disturbances, with no chattering in the control signal. Its application to data encryption highlights its potential for secure communication. Future research could explore: (a) extending the method to multi-agent systems, (b) integrating machine learning for enhanced adaptability, and (c) evaluating robustness against cyber-attacks in chaotic encryption. These directions invite further investigation into the method’s scalability and security, advancing control and communication technologies.

Footnotes

ORCID iDs

Saleh Mobayen

Jianhua Zhang

Quanmin Zhu

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

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