Adaptive nonsingular integral-type dynamic terminal sliding mode synchronizer for disturbed nonlinear systems and its application to secure communication systems

Abstract

This study proposes an adaptive nonsingular integral dynamic terminal sliding mode tracker/synchronizer for disturbed nonlinear systems along with its usage in safe communication systems. The convergence of the closed-loop structure under unknown uncertainty and disturbances is guaranteed via Lyapunov analysis. Furthermore, a parameter-tuning method is planned to approximate the upper bound of uncertainty and disturbance terms, since this latter is typically unknown in practice. The proposed approach is used to design a digital secure transmission scheme according to the chaotic systems. The effectiveness of the suggested approach is validated using computer simulations on a benchmark example of chaotic system. The obtained outcomes clearly confirm the ability of the planned control approach enables to attain the desired tracking/synchronizing performance despite the disturbances. Additionally, when implemented to the data encryption of a communication system, the proposed control and secure communication techniques enabled the complete and secure retrieval of the original digital sequences.

Keywords

secure communication terminal sliding mode synchronization adaptation nonsingular control

Highlights

This article recommends a finite-time chaos synchronization method based on the nonsingular adaptive second-order integral-type sliding mode control for nonlinear systems subject to external disturbances and parametric uncertainties. The proposed approach is used to design a digital secure transmission scheme based on the chaotic systems. Its chief innovations are listed as follows: a design scheme is proposed to combine the robustness of SMC with the low cost and security features of chaotic systems and a design method is presented that is robust to uncertainties and parameter variation, simple to be constructed, and there is no necessity to the information of upper bounds of uncertainties.

1. Introduction

Nonlinear systems such as chaotic systems were shown to have many useful applications in many engineering systems such as chemical reactors (Berezowski, 2020), genetic control systems (Azar and Vaidyanathan, 2015), power converters (Lv et al., 2019), lasers (Bao et al., 2020), biological systems (Justin et al., 2020), permanent-magnet motors (Lu and Wang, 2020), and secure communication systems (Chan et al., 2020; Xiu et al., 2021b). The control (Ouannas et al., 2020), tracking, or synchronization (Godavarthi et al., 2020; Ma et al., 2021) of these systems has a vital role in the engineering applications. However, disturbances and system uncertainties are ubiquitous in any engineering system (Li et al., 2021b; Mirrezapour et al., 2021). Hence, their effect should be taken into account when analyzing the system efficiency. A number of approaches were proposed in the former decades for control of the systems with disturbances and uncertainties (Koronovskii et al., 2020; Trikha et al., 2021; Yadav et al., 2019; Yan et al., 2021). Sliding mode control (SMC) is known to be one of the most efficient robust controller design techniques (Hu et al., 2021; Li et al., 2021a; Modiri and Mobayen, 2020; Mofid and Mobayen, 2022; Roopaei and Zolghadri Jahromi, 2008; Wang et al., 2021a). It is widely used in tracking/controlling nonlinear uncertain systems. Its robustness against uncertainties, ability to completely reject matched disturbances, as well as its simple design and simplicity of execution are amongst its main characteristics (Mobayen et al., 2020; Goel and Mobayen, 2021). The design of SMC consists of two phases (Wang et al., 2021b; Zhu et al., 2021): the design of switching manifold and the control input design. The sliding surface is frequently planned to attain the wanted dynamics of control system. The control input is designed to force the sliding mode, that is, direct the system states to the sliding surface in the finite time (Fu and Ma, 2020; Perruquetti and Barbot, 2002; Wang et al., 2021c). Chattering, however, is recognized as the main problem when implementing first-order SMC. Second-order SMC was shown to alleviate this problem by using an integrator term in the input of conventional first-order SMC, thus making the real controller input and its derivative clearly appear. As an alternative to acting on the first-order SMC surface time derivative, the discontinuous term incomes on its second-order derivative. Therefore, the real controller signal is a continuous integration term of its derivative, and the chattering problem is removed. This process refers to the second-order SMC design (Kchaou et al., 2017; Matraji et al., 2018; Muñoz et al., 2017).

Chaotic systems have recently received great attention in data encryption and secure communication systems due to their nonlinearity features and interesting behaviors (Asgari-Chenaghlu et al., 2021; Balamash et al., 2020; Chen et al., 2020; Hanif et al., 2020; Wu et al., 2021). Among the most relevant features of chaotic sequences is the close similarity of their behavior to that of noise. This characteristic can be applied to design digital data encryption systems (Nguyen et al., 2020; Moradi Zirkohi and Shoja-Majidabad, 2021; Zarebnia et al., 2019; Zong et al., 2020). Chaotic-based encryption techniques use chaotic sequences and signals made by nonlinear chaotic systems to encrypt systems (Hsiao and Lee, 2015; Liu et al., 2016; Mazloom and Eftekhari-Moghadam, 2009). It is worth noting that though ordinary encryption methods can meet the security of digital message information through transmission, there are some restrictions to protecting the content of all digital messages and barricading illegal access. Additionally, researches have shown that these techniques cannot display good performance against attacks, in some applications, due to lower key space (Vaseghi et al., 2017). Moreover, conventional encryption methods entail long computational time and require high computing power (Lv et al., 2019). Additionally, they lead to significant latency in practice as a result of low speed of encryption/decryption. Consequently, cryptography based on chaotic signals and sequences provide better alternative than the traditional encryption methods, specifically in terms of ensuring a suitable mixture of security, capability, and speed. According to the above discussion, designing secure digital communication system by means of SMC-based synchronized nonlinear systems can combine the low cost and security of chaotic sequences with the noteworthy features of SMC and properly solve the security problem in digital data transmission systems.

This article recommends a chaos tracking/synchronization method based on nonsingular adaptive integral sliding control for nonlinear systems subject to disturbances and uncertainties with unknown upper bounds. The proposed approach is used to design a digital secure transmission scheme based on chaotic systems. Its chief innovations are listed as follows: a design to combine the robustness of SMC with the low cost and security features of chaotic systems; a design that is robust to uncertainties and parameter variation and simple to be constructed; and a design that does not need any information for the upper bounds of uncertainties.

The paper remainder is set as follows: The explanation of the system and required preliminaries are given in Section 2. The main outcomes including the integral sliding surface, terminal sliding mode tracker design, and estimation-based control for synchronizing the nonlinear chaotic systems are detailed in Section 3. Section 4 highlights the successful implementation of the planned controller approach to secure communication systems. The numerical results are given in section 5. Concluding remarks are finally provided in section 6.

2. System definition and preliminaries

Consider the subsequent nonlinear system with disturbance (Mondal and Mahanta, 2014)

\dot{x} (t) = f (x (t)) + Δ f (x (t)) + d (t) + g (x (t)) u (t)

(1)

where

x (t) \in R^{n}

is the vector of the states, and

u (t) \in R^{n}

is the controller vector.

f (x (t)) \in R^{n}

and

g (x (t)) \in R^{n \times n}

signify the known nonlinear functions where

| g (x (t)) | \neq 0

(determinant of the function

g (x (t))

is non-zero).

Δ f (x (t)) \in R^{n}

d (t) \in R^{n}

are model uncertainty and external disturbance of the system, respectively.

Assumption 1

The nonlinear functions $f (x (t))$ , $Δ f (x (t))$ , $g (x (t))$ , $u (t)$ , $d (t)$ , $x (t)$ and $\dot{x} (t)$ are supposed to be differentiable with respect to time.

Assumption 2

The system uncertainty $Δ f (x (t))$ is assumed to be bounded as follows:

Δ f (x (t)) \leq α x

(2)

where

.

denotes the Euclidean norm in

R^{n}

and

α

is a given positive constant. Also, it is assumed that the external disturbance

d (t)

is norm-bounded, that is,

d (t) \leq D

(3)

where

D

is a known positive constant.

Assumption 3

(Razzaghian et al. (2021)): The matrices $f (x (t))$ and $g (x (t))$ are known, smooth nonlinear functions and $| g (x (t)) | \neq 0$ for all possible $x$ and $t \geq 0$ .

The main control aim is to guarantee that the system output follows a desirable reference trajectory $x_{d} (t) \in R^{n}$ in the presence of disturbances. The reference signal $x_{d} (t)$ denotes a differentiable function of time as

{\dot{x}}_{d} (t) = f (x_{d} (t))

(4)

The tracking error signal is defined as

e (t) = x (t) - x_{d} (t)

(5)

To fulfill the above control objectives for system (1), we propose in the paper to design a nonsingular second-order terminal sliding tracker control signal with a rapid reaching condition.

Definition 1

Consider the subsequent time-invariant nonlinear dynamics:

\dot{x} (t) = f (x)

(6)

where

f (x) : D \to R^{n}

is continuous and expressed on an open neighborhood D of the equilibrium point.

Assume that the subsequent circumstances are fulfilled:

(I) The equilibrium point is asymptotically stable in the subset $\hat{D} \subseteq D$ .

(II) The equilibrium is finite-time convergent in $\hat{D}$ . That is, there exists a convergence time $t_{1} (x_{0})$ so that $x (t, x_{0}) \to 0$ as $t \to t_{1} (x_{0})$ and remains there thereafter.

Then, the equilibrium is locally finite-time stable. In addition, if $\hat{D} = R^{n}$ , then the equilibrium point is finite-time stable globally.

Definition 2

The affine nonlinear dynamics is considered as:

\dot{x} (t) = f (x) + g (x) u (t)

(7)

where

x (t) \in R^{n}

represents the state,

u (t) \in R^{n}

denotes the control signal,

f (0) = 0

, and

| g (x) | \neq 0

. The feedback control input

u (t) = Π (x, t)

is dubbed a finite-time stabilizer input if the equilibrium point of (7) becomes finite-time stable when one uses the controller.

Lemma 1

(Xiu and Guo, 2018) Let $x \in ℵ \subset R^{n}$ , $\dot{x} = ℑ (x)$ , $ℑ : R^{n} \to R^{n}$ is the continuous functional on open neighborhood $ℵ$ of origin and locally Lipschitz on $ℵ / {0}$ and $ℑ (0) = 0$ . Assuming there exists a continuous functional $V : ℵ \to R$ where (a) $V$ represents a positive-definite function; (b) the time derivative of the Lyapunov function with respect to time is negative on $ℵ / {0}$ ; (c) there exist the real scalars $m > 0$ and $0 < α < 1$ , and a neighborhood $N \subset ℵ$ of the origin so that $\dot{V} + m V^{α} \leq 0$ on $N \ {0}$ . The origin is then finite-time stable of the system $\dot{x} = ℑ (x)$ . Thus, for any given $t_{0}$ , the Lyapunov function $V$ is convergent to the origin in finite time $t_{s}$ denoted by

t_{s} = \frac{V {(t_{0})}^{1 - α}}{m (1 - α)}

(8)

3. Main results

3.1. Design of the sliding manifold

To ensure the tracking control design purpose, define the subsequent integral terminal sliding surface

s (t) = k_{p} e (t) + k_{i} \int_{0}^{t} e {(τ)}^{\frac{q}{p}} d τ + k_{d} \dot{e}

(9)

where

q

and

p

denote odd positive integers such that

q < p

k_{p}, k_{i}, k_{d}

are positive constants. If the initial error

e_{0}

is zero, as a result, the tracking problem aims at maintaining the errors on

s (t) = 0

. Once the state reaches the sliding surface

s (t) = 0

, it will remain on it while sliding to

e (t) = 0

and

\dot{e} (t) = 0

When the tracking error reaches the terminal switching surface $s = 0$ , we have

k_{p} e (t) + k_{i} \int_{0}^{t} e {(τ)}^{\frac{q}{p}} d τ + k_{d} \dot{e} = 0

(10)

Thus, we have, $\dot{s} = 0$ , which yields

\ddot{e} = - \frac{k_{p}}{k_{d}} \dot{e} - \frac{k_{i}}{k_{d}} e^{\frac{q}{p}}

(11)

Define the following Lyapunov function

V_{0} = 0.5 {\dot{e}}^{2} + \frac{k_{i} p}{k_{d} (q + p)} e^{1 + \frac{q}{p}}

(12)

Computing the time derivative of $V_{0}$ and using equation (11) yield

{\dot{V}}_{0} = \dot{e} \ddot{e} + \frac{k_{i} p}{k_{d} (q + p)} (\frac{q}{p} + 1) e^{\frac{q}{p}} \dot{e} = \dot{e} (- \frac{k_{p}}{k_{d}} \dot{e} - \frac{k_{i}}{k_{d}} e^{\frac{q}{p}}) + \frac{k_{i}}{k_{d}} e^{\frac{q}{p}} \dot{e} = - \frac{k_{p}}{k_{d}} {\dot{e}}^{2} \leq 0

(13)

This implies the uniform bounded of the error signal and its asymptotic convergence to the origin, once the error trajectory reaches the sliding surface (9). Additionally, $\underset{t \to \infty}{l i m} V_{0} = V_{0} (\infty)$ exists for $V_{0} (\infty) \in R^{+}$ since $V_{0}$ is positive-definite, and time derivative of it is negative semi-definite. Further, the boundedness of the error signal implies that ${\dot{V}}_{0}$ is uniformly continuous. Hence, applying Barbalat lemma produces $\underset{t \to \infty}{l i m} \dot{e} (t) = 0$ . Equation (9), yields $\underset{t \to \infty}{l i m} e (t) = 0$ .

3.2. Tracking controller design

To ensure that the sliding surfaces are converged to the origin, consider the subsequent surface

σ (t) = s (t) - (ℑ (t) + ℜ (t)) s (0)

(14)

where

ℑ (t) \in R

and

ℜ (t) \in R

are two exponential functions defined as

ℑ (t) = ℓ_{1} \exp (- ϕ_{1} t) + ℓ_{2} \exp (- ϕ_{2} t) + ℓ_{3}

(15)

ℜ (t) = ℓ_{4} \exp (- ϕ_{3} t) - ℓ_{3}

(16)

where

ℓ_{i}

signify the gain coefficients, and

ϕ_{1}, ϕ_{2}, ϕ_{3} > 0

The sliding surface (14) results in a faster decay rate than (7) and hence speeds up the dynamic response of the system. Assuming the subsequent initial conditions for functions (15) and (16)

ℑ (0) + ℜ (0) = 1

(17)

where it is simplified as

ℓ_{1} + ℓ_{2} + ℓ_{4} = 1

(18)

Further, the boundary condition of (15) and (16) is given by

ℑ (t_{s}) + ℜ (t_{s}) = 0

(19)

where it is simplified as

ℓ_{1} \exp (- ϕ_{1} t_{s}) + ℓ_{2} \exp (- ϕ_{2} t_{s}) + ℓ_{4} \exp (- ϕ_{3} t_{s}) = 0

(20)

The parameters $ℓ_{1}$ and $ℓ_{2}$ are calculated from (18) and (20) as

ℓ_{1} = \frac{(1 - ℓ_{4}) \exp (- φ_{2} t_{s}) + ℓ_{4} \exp (- φ_{3} t_{s})}{\exp (- φ_{2} t_{s}) - \exp (- φ_{1} t_{s})}, t \leq t_{s}

(21)

ℓ_{2} = \frac{- (1 - ℓ_{4}) \exp (- φ_{1} t_{s}) - ℓ_{4} \exp (- φ_{3} t_{s})}{\exp (- φ_{2} t_{s}) - \exp (- φ_{1} t_{s})} . t \leq t_{s}

(22)

Remark 1

Using the sliding surface (14), the reaching interval of the sliding surface is removed. Since $σ (0) = s (0) - (ℑ (0) + ℜ (0)) s (0)$ and $ℑ (0) + ℜ (0) = 1$ , then $σ (0) = s (0) - s (0) = 0$ ; it means that the sliding surface in the initial time is equal to zero and the reaching phase is removed from the initial condition. Moreover, it is stated in (20) that the term $ℑ (t) + ℜ (t)$ is zero. As a result, the sliding surface (14) is stabilized; that is, $σ (t_{s}) = s (t_{s}) - (ℑ (t_{s}) + ℜ (t_{s})) s (0) = s (t_{s}) = 0$ . Since the terms $ℑ (t)$ and $ℜ (t)$ are two exponential functions in the sliding surface (14), they can cause a fast decay rate via suitable adjustment of gain coefficients $ℓ_{i}$ .

Deriving (14) with respect to time gives

\dot{σ} (t) = \dot{s} (t) - (\dot{ℑ} (t) + \dot{ℜ} (t)) s (0)

(23)

where using (9), one has

\dot{σ} (t) = k_{p} \dot{e} (t) + k_{i} e {(t)}^{\frac{q}{p}} + k_{d} \ddot{e} (t) - (\dot{ℑ} (t) + \dot{ℜ} (t)) s (0)

(24)

In light of (1) and (5), we can re-write equation (24) as

\begin{array}{l} \dot{σ} (t) = k_{p} (f (x (t)) + Δ f (x (t)) + g (x (t)) u (t) + d (t) - {\dot{x}}_{d} (t)) + k_{i} e {(t)}^{q / p} \\ + k_{d} (\dot{f} (x (t)) + Δ \dot{f} (x (t)) + \dot{g} (x (t)) u (t) + g (x (t)) \dot{u} (t) + \dot{d} (t) - {\ddot{x}}_{d} (t)) - (\dot{ℑ} (t) + \dot{ℜ} (t)) s (0) \end{array}

(25)

Theorem 1

Considering system (1) and forming the control input as

\dot{u} (t) = - k_{d}^{- 1} g {(x (t))}^{- 1} {\begin{cases} k_{p} (f (x (t)) - {\dot{x}}_{d} (t) + g (x (t)) u (t)) + k_{i} e {(t)}^{\frac{q}{p}} \\ + (ϕ_{1} ℓ_{1} \exp (- ϕ_{1} t) + ϕ_{2} ℓ_{2} \exp (- ϕ_{2} t) + ϕ_{3} ℓ_{4} \exp (- ϕ_{3} t)) s (0) + k_{d} \dot{g} (x (t)) u (t) \\ + k_{d} \dot{f} (x (t)) - k_{d} {\ddot{x}}_{d} (t) + κ sgn (σ (t)) {| σ (t) |}^{η} + γ σ (t) + δ sgn (σ (t)) \end{cases}}

(26)

where

κ

and

γ

denote two arbitrary positive coefficients, and

δ

is a positive constant, so that

δ \geq k_{p} (Δ f (x (t)) + d (t)) + k_{d} (Δ \dot{f} (x (t)) + \dot{d} (t))

(27)

forces the sliding manifold (14) to zero.

Proof

The Lyapunov functional is constructed by

V_{1} (σ (t)) = \frac{1}{2} σ {(t)}^{T} σ (t)

(28)

From (15), (16), and (25), the term $\dot{σ} (t)$ is rewritten as

\begin{array}{l} \dot{σ} (t) = k_{p} (f (x (t)) + Δ f (x (t)) + g (x (t)) u (t) + d (t) - {\dot{x}}_{d} (t)) (t) + k_{i} e {(t)}^{\frac{q}{p}} \\ + k_{d} (\dot{f} (x (t)) + Δ \dot{f} (x (t)) + \dot{g} (x (t)) u (t) + g (x (t)) \dot{u} (t) + \dot{d} (t) - {\ddot{x}}_{d} (t)) \\ + (ϕ_{1} ℓ_{1} \exp (- ϕ_{1} t) + ϕ_{2} ℓ_{2} \exp (- ϕ_{2} t) + ϕ_{3} ℓ_{4} \exp (- ϕ_{3} t)) s (0) \end{array}

(29)

Calculating the time derivative of (29) gives

{\dot{V}}_{1} (σ (t)) = σ {(t)}^{T} {\begin{cases} k_{p} (f (x (t)) + Δ f (x (t)) + g (x (t)) u (t) + d (t) - {\dot{x}}_{d} (t)) + k_{i} e {(t)}^{\frac{q}{p}} \\ + k_{d} (\dot{f} (x (t)) + Δ \dot{f} (x (t)) + \dot{g} (x (t)) u (t) + g (x (t)) \dot{u} (t) + \dot{d} (t) - {\ddot{x}}_{d} (t)) \\ + (ϕ_{1} ℓ_{1} \exp (- ϕ_{1} t) + ϕ_{2} ℓ_{2} \exp (- ϕ_{2} t) + ϕ_{3} ℓ_{4} \exp (- ϕ_{3} t)) s (0) \end{cases}}

(30)

substituting (26) into (30) yields

\begin{array}{l} {\dot{V}}_{1} (σ (t)) = k_{p} σ {(t)}^{T} (Δ f (x (t)) + d (t)) + k_{d} σ {(t)}^{T} (Δ \dot{f} (x (t)) + \dot{d} (t)) \\ - σ {(t)}^{T} (κ sgn (σ (t)) {| σ (t) |}^{η} + γ σ (t) + δ sgn (σ (t))) \end{array}

(31)

where

| σ (t) |

is the absolute value of

σ (t)

. In light of (27), we can simplify equation (31) as

\begin{array}{l} {\dot{V}}_{1} (σ (t)) \leq k_{p} σ {(t)}^{T} (Δ f (x (t)) + d (t)) + k_{d} σ {(t)}^{T} (Δ \dot{f} (x (t)) + \dot{d} (t)) \\ - κ σ {(t)}^{η + 1} - γ σ {(t)}^{T} σ (t) - δ σ {(t)}^{T} sgn (σ (t)) \\ \leq - γ σ {(t)}^{2} - κ σ {(t)}^{η + 1} \leq - α V_{1} (σ (t)) - β V_{1} {(σ (t))}^{\bar{η}} \end{array}

(32)

where

\bar{η} = (η + 1) / 2 < 1

α = 2 γ > 0

β = 2^{\bar{η}} (κ) > 0

and

σ (t)

is the Euclidean norm of

σ (t)

. Hence, the gradual decrease of the Lyapunov function (28) and the convergence of the switching surface (14) to the origin are confirmed.

3.3. Adaptive tracker design

Practically, the upper bound of uncertainty and disturbances $Δ f (x (t))$ and $d (t)$ , and their time derivatives $Δ \dot{f} (x (t))$ and $\dot{d} (t)$ are not available. Thus, in this section, we propose the design method of an estimator to estimate the unknown upper bounds.

Theorem 2

The disturbed nonlinear system (1) and the dynamic terminal sliding manifold (14) are considered. Assume that the perturbation terms are bounded with unknown upper bound. Defining the following adaptive terminal sliding tracker:

\dot{u} (t) = - k_{d}^{- 1} g {(x (t))}^{- 1} {\begin{cases} k_{p} (f (x (t)) + g (x (t)) u (t) - {\dot{x}}_{d} (t)) + k_{i} e {(t)}^{\frac{q}{p}} \\ + (ϕ_{1} ℓ_{1} \exp (- ϕ_{1} t) + ϕ_{2} ℓ_{2} \exp (- ϕ_{2} t) + ϕ_{3} ℓ_{4} \exp (- ϕ_{3} t)) s (0) + k_{d} \dot{g} (x (t)) u (t) \\ + k_{d} \dot{f} (x (t)) - k_{d} {\ddot{x}}_{d} (t) + κ sgn (σ (t)) {| σ (t) |}^{η} + γ σ (t) + \hat{δ} (t) sgn (σ (t)) \end{cases}}

(33)

with

\hat{δ} (t)

as the estimator of

δ

, and adaptation law as

\dot{\hat{δ}} (t) = ℓ^{- 1} σ (t)

(34)

with

ℓ > 0

, makes the sliding manifold (14) converge to origin and achieves the tracker design objective.

Proof

Consider the estimator error defined by

\tilde{δ} (t) = \hat{δ} (t) - δ

(35)

From time derivative of (3) and employing (34) yields

\dot{\tilde{δ}} (t) = \dot{\hat{δ}} (t) = ℓ^{- 1} σ (t)

(36)

The Lyapunov function is constructed by

V_{2} (σ (t)) = \frac{1}{2} (σ {(t)}^{T} σ (t) + ℓ \tilde{δ} {(t)}^{2})

(37)

where taking time-derivative of (37) gives

{\dot{V}}_{2} (σ (t)) = σ {(t)}^{T} \dot{σ} (t) + ℓ \tilde{δ} (t) \dot{\tilde{δ}} (t)

(38)

Using (25), (36), and (38), one obtains

\begin{array}{l} {\dot{V}}_{2} (σ (t)) = k_{p} σ {(t)}^{T} (Δ f (x (t)) + d (t)) + k_{d} σ {(t)}^{T} (Δ \dot{f} (x (t)) + \dot{d} (t)) \\ - κ σ {(t)}^{T} sgn (σ (t)) {| σ (t) |}^{η} - γ σ {(t)}^{T} σ (t) - \hat{δ} (t) σ {(t)}^{T} sgn (σ (t)) + (\hat{δ} (t) - δ) σ (t) \end{array}

(39)

Because $σ {(t)}^{T} {k_{p} (Δ f (x (t)) + d (t)) + k_{d} (Δ \dot{f} (x (t)) + \dot{d} (t))} \leq σ (t) k_{p} (Δ f (x (t)) + d (t)) + k_{d} (Δ \dot{f} (x (t)) + \dot{d} (t))$ , equation (39) can be written as

{\dot{V}}_{2} (σ (t)) \leq σ (t) k_{p} (Δ f (x (t)) + d (t)) + k_{d} (Δ \dot{f} (x (t)) + \dot{d} (t)) - κ σ {(t)}^{η + 1} - γ σ {(t)}^{2} - \hat{δ} (t) σ (t) + (\hat{δ} (t) - δ) σ (t)

(40)

where by addition and subtraction of

δ σ (t)

to it, one can obtain

\begin{array}{l} {\dot{V}}_{2} (σ (t)) \leq σ (t) k_{p} (Δ f (x (t)) + d (t)) + k_{d} (Δ \dot{f} (x (t)) + \dot{d} (t)) - κ σ {(t)}^{η + 1} \\ - γ σ {(t)}^{2} - \hat{δ} (t) σ (t) + (\hat{δ} (t) - δ) σ (t) + δ σ (t) - δ σ (t) \leq - γ σ {(t)}^{2} - κ σ {(t)}^{η + 1} < 0 \end{array}

(41)

Thus, confirming the gradual decrease of the Lyapunov function (37) and the convergence of terminal switching manifold to origin. □

Remark 2

By employing the dynamic terminal sliding mode control law (33) for the disturbed nonlinear system (1), the finite-time convergence of terminal switching manifold (14) is not fulfilled. Hence, via the adaptive control input (33) in Theorem 2, the Lyapunov function (37) is decreased gradually and the convergence of dynamic terminal sliding surface (14) is guaranteed.

4. Application to secure communication

Chaotic secure communication systems contain a master nonlinear chaotic system in the transmitter and a slave nonlinear chaotic system in the receiver. Chaotic Shift, Parametric Modulation, Chaotic Masking, Keying, and the Inclusion methods are among the most popular techniques applied for chaotic cryptosystems. In the chaotic masking, the message data $m (t)$ at the transmitter is added to the outputs of the master chaotic system creates a noise-like signal prior to transmission. The produced signal is sent to the receiver. At the receiver side, chaotic synchronization is performed such that the master/slave systems are synchronized. Then, the message is completely recovered by subtracting the output of the slave chaotic system from the received signal. It should be noted that the controller/tracker should be chosen in such a way that it can overcome uncertainties and external disturbances.

In this regard, the nonlinear chaotic system in the transmitter side can be considered as

\begin{array}{l} {\dot{x}}_{d} (t) = f (x_{d} (t)) \\ y_{T} (t) = C . x_{d} (t) \end{array}

(42)

where

y_{T} (t)

is the combination of the outputs of the transmitter chaotic system. Moreover, the chaotic system in the receiver with uncertainties, external disturbances, and control input can be considered as

\begin{array}{l} \dot{x} (t) = f (x (t)) + Δ f (x (t)) + d (t) + g (x (t)) u (t) \\ y_{R} (t) = C . x (t) \end{array}

(43)

where

y_{R} (t)

is the combination of the outputs of the receiver chaotic system. The block diagram of the proposed chaotic cryptosystem is demonstrated in Figure 1. The diagram includes five main sections: data modulation, chaotic encryption, chaotic tracker/synchronization, chaotic decryption, and data demodulation, respectively.

Figure 1.

Block diagram of planned chaotic cryptosystem.

In the proposed chaotic cryptosystem, $A [k], ⥂ k = 0,1,2, ...,$ is a digital data that will be sent. $m (t)$ is a carrier signal obtained from the data modulation method as

m (t) = {\begin{matrix} \frac{1}{2} + \frac{1}{2} \sin \frac{2 π}{T} t, i f A [k] = 1 \\ \frac{1}{2} - \frac{1}{2} \sin \frac{2 π}{T} t, i f A [k] = 0 \end{matrix}

(44)

where

k N T \leq t < (k + 1) N T, f o r k = 0,1,2, ...

and T is the period of the carrier signal. In the chaotic encryption block, modulated signal

m (t)

and the output of the master chaotic system

y_{T} (t)

are added and masked signal

m_{e} (t) = m (t) + y_{T} (t)

is obtained and sent to the receiver. At the receiver side, when the synchronization/tracking goal is attained and the chaotic signals at receiver are synchronized by chaotic signals at transmitter side, by subtracting

y_{R} (t)

from

m_{e} (t)

, the modulated signal

\tilde{m} (t)

can be obtained as

\tilde{m} (t) = m_{e} (t) - y_{R} (t)

and the recovered digital data

\tilde{A} [k], ⥂ k = 0,1,2, ...,

will finally be obtained after same certain demodulation method.

5. Numerical simulation

This portion illustrates the performance of recommended scheme. To this end, consider the differentiable function $f (x (t)) = [\begin{matrix} 0 \\ 0 \\ - 6 x_{1} - 2.92 x_{2} - 1.2 x_{3} + x_{1}^{2} \end{matrix}]$ and the nonlinear reference trajectory (4) as the transmitter chaotic system

\begin{array}{l} {\dot{x}}_{1 d} (t) = x_{2 d} (t) \\ {\dot{x}}_{2 d} (t) = x_{3 d} (t) \\ {\dot{x}}_{3 d} (t) = - 6 x_{1 d} - 2.92 x_{2 d} - 1.2 x_{3 d} + x_{1 d}^{2} \end{array}

(45)

Moreover, the desired system (45) with uncertainties, external disturbances, and control inputs can be consider as the receiver chaotic system as follows

\begin{array}{l} {\dot{x}}_{1} (t) = x_{2} (t) \\ {\dot{x}}_{2} (t) = x_{3} (t) \\ {\dot{x}}_{3} (t) = - 6 x_{1} - 2.92 x_{2} - 1.2 x_{3} + x_{1}^{2} + Δ f (x (t)) + d (t) + g (x (t)) u (t) \end{array}

(46)

The exterior disturbance and uncertainty terms of receiver chaotic system are chosen as follows

\begin{array}{l} Δ f (x (t)) = [\begin{matrix} 0 \\ 0 \\ 5 t + 1.5 \sin (x_{1} (t)) + \cos (x_{2} (t)) + 1.2 \sin (x_{3} (t)) \end{matrix}] \\ d (t) = [\begin{matrix} 0 \\ 0 \\ 2 t + 0.2 \sin (t) + \cos (0.5 t) + 0.15 \sin (3 \sqrt{t}) \end{matrix}] \end{array}

(47)

Also, the nonlinear function $g (x (t))$ is chosen as

g (x (t)) = [\begin{matrix} 0 \\ 0 \\ 1.5 + 2 cos (x_{1} (t)) \end{matrix}]

(48)

In this simulation, the initial conditions of transmitter/receiver chaotic systems are chosen as $x_{1 d} (0) = 3, x_{2 d} (0) = - 0.4, x_{3 d} (0) = 0.2$ and $x_{1} (0) = 0.1, x_{2} (0) = 0.3, x_{3} (0) = 0.3$ , respectively. The states of chaotic systems are demonstrated in Figures 2–4, when the proposed control signal (33) is used. It is understood that the nonlinear chaotic signals $x_{1} (t)$ and $x_{1 d} (t)$ are synchronized in 1 s. Moreover, the chaotic signals $x_{2} (t)$ and $x_{3} (t)$ are synchronized with $x_{2 d} (t)$ and $x_{3 d} (t)$ in 0.8 s. Figure 5 demonstrations the error trajectories. It is concluded from Figure 5 that the error dynamics converge to the origin in less than 1 s. Hence, the proposed technique ensures appropriate synchronization performance while properly mitigating parametric uncertainties. Figure 6 exhibits the dynamics of controller $u (t)$ and FTSM surface $σ (t)$ . Note that both variables display satisfactory amplitudes have chattering-free dynamics.

Figure 2.

State responses $x_{1} (t), x_{1 d} (t)$ .

Figure 3.

State responses $x_{2} (t), x_{2 d} (t)$ .

Figure 4.

State trajectories $x_{3} (t), x_{3 d} (t)$ .

Figure 5.

Error signals $e_{1}, e_{2}, e_{3}$ .

Figure 6.

Control signal $u (t)$ and FTSM surface $σ (t)$ .

In what follows, the practicality and application of suggested scheme for chaotic cryptosystem is confirmed using numerical simulation. Binary digital input with data rate 3 bit/s is applied in this simulation as the original digital sequences that will be delivered. Also, the combination matrix $C$ is chosen as $C = [\begin{matrix} 2.375 & 1.971 & 0.254 \end{matrix}]$ . The simulation outcomes of the chaotic cryptosystem are demonstrated in Figures 7 and 8. In Figure 7, the upper subfigures compare original modulated signal $m (t)$ and decrypted modulated signal $\tilde{m} (t) = m (t) - y_{R} (t)$ , whereas the bottom subfigures display the encrypted modulated signal $m_{e} (t) = m (t) + y_{T} (t)$ . Figure 8 compares the original and reiterative digital sequences. Note that the encrypted message signals are recovered in approximately 1 s.

Figure 7.

(a) Original and decrypted modulated signal. (b) Encrypted modulated signal.

Figure 8.

Original and reiterative digital sequences.

The conventional pseudorandom signals are used for encryption of digital signals (“0”, “1”) in wireless communication. Due to the increase of gain of the system, the code word length is required to be increased; but, due to the practical limitation and increase in complexity, the chaotic signals can be used as alternative to pseudorandom codes for encryption of digital signals. In this regard, a signal encryption technique based on the chaotic systems is presented. By using the functions (45) and (46), the chaotic signals are generated and used as the encryption signals by which the digital modulated signal function (44) is scrambled at the transmitter system. The proposed scheme recovers the digital information signal exactly via synchronization by using the control input (33) in the receiver portion.

Remark 3

The SMC approach is a robust control method which has many powerful characteristics, such as low sensitivity to external disturbances and robustness to the uncertainties due to the structural variations and unmodeled dynamics. Furthermore, the finite-time control strategies have demonstrated better robustness and disturbance rejection properties (Yu, 2010). However, due to the application of the proposed nonlinear dynamic system in the implementation of secure communication system, and also due to the characteristics of the communication channel such as noise and parametric uncertainty, the adaptive nonsingular integral-type sliding mode controller introduced in this paper can overcome time-varying parametric uncertainties and time-varying external disturbances (Xiu et al., 2021a; Zhang et al., 2020).

Remark 4

Various nonsingular TSMC approaches have been investigated in recent years to mitigate singularity problem (Han et al., 2015; Jing et al., 2019; Pukdeboon and Siricharuanun, 2014; Wang et al., 2018; Yi and Zhai, 2019). On the other hand, in order to have a reliable secure chaotic communication system, the synchronization time should be as short as possible, the transmitter must be able to change the chaotic function of the dynamic system ( $f (x (t))$ in equation (1)) in the specified time intervals, or be able to overcome the channel effects and unknown uncertainties, and the received signal must be free of chattering. These researches do not meet all the requirements of the synchronized nonlinear chaotic systems in secure communications at the same time (Alattas et al., 2021). By using a general form of the nonlinear system and a chattering-free adaptive control method, the adaptive nonsingular integral-type sliding mode controller introduced in this paper can overcome the above-mentioned problems simultaneously.

Remark 5

In comparison with the common encryption schemes, due to high time consumption, key distribution problem, and low-efficiency level in the traditional encryption schemes, various new research works based on chaos encryption algorithms have been investigated. The chaos-based encryption is a fast and advanced security algorithm with high sensitivity to the initial conditions, pseudo-randomness property, no periodicity, and the system parameter dependency. These properties enable to support permutation-diffusion requirement in the cryptosystem establishment (Thein et al., 2017). Specially, unlike the models introduced in other chaotic secure communication methods reported in the literature, such as (Chen et al., 2020), our proposed method as multi-shift cipher encryption and chaotic modulation presents the following main aspects: (a) using the advantages of the chaotic masking method such as simplicity, low time consumption, and very easily implementation in electronic circuits and (b) eliminating the disadvantages of chaos masking such as weakness against the conventional attack methods by using data modulation algorithm and improvement of the security of masking method.

6. Conclusions

This article proposed an adaptive second-order SMC tracking control law with fast reaching condition for tracking/synchronization of nonlinear dynamical systems with unknown upper bounds of uncertainties and disturbances. The proposed approach was also used to design a digital secure transmission scheme according to the chaotic systems. Computer simulations and analytical study confirmed the robustness, ease of implementation, and fast convergence rate of recommended tactic. Furthermore, its implementation to a digital data transmission scheme showed that chaotic secure communication improves the security of the communication system and results in the recovery of the encrypted signals in approximately 1s.

Footnotes

Declaration of conflicting interests

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

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Data Availability Statement

The data that support the findings of this research paper are available from the corresponding author upon reasonable request.

ORCID iD

Saleh Mobayen

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