Fuzzy L 1 adaptive controller for chaos synchronization of uncertain fractional-order chaotic systems with input nonlinearities

Abstract

This research work proposes a fuzzy fractional-order $L_{1}$ adaptive controller to deal with projective chaos synchronization problems for a general class of uncertain fractional-order chaotic systems subject to unknown input nonlinearities (dead-zone and sector nonlinearities). The suggested control architecture includes a fractional-order sliding surface, a fuzzy system, and an $L_{1}$ adaptive controller. The latter consists of a predictor, a control law, and its adaptive mechanism. The fuzzy system takes the role of an online estimator for system nonlinear uncertain functions which helps in the handling of the input nonlinearities. A designed low-pass filter is placed within the input channel of the $L_{1}$ adaptive controller to ensure that the control loop is decoupled from the estimation loop, which preserves response robustness along with improving transient performances. Finally, the closed-loop stability is rigorously proved, and the acquired results are demonstrated by two simulation examples as well as a comparative study.

Keywords

fractional-order chaotic systems input nonlinearities projective chaos synchronization fuzzy system

Introduction

Chaotic behavior is an extremely complex nonlinear phenomenon that can be noticed in many real-world applications (Tlelo-Cuautle et al., 2020). This behavior can be demonstrated by chaotic systems which are recognized by delivering infinite and unstable periodical motions, bounded phase space trajectories, and extreme sensitivity to small variations of initial conditions (Boubakir and Labiod, 2022). Due to this sensitivity, a chaotic system is committed to deliver different behaviors for any variations of initial conditions. Chaotic behavior can be detected using the Lyapunov exponent criteria in which a chaotic system is determined by the presence of a positive Lyapunov exponent. Master–slave synchronization is one of the most important research fields on chaotic systems. The phenomena of chaos synchronization were first discovered by Pecora and Carroll (1990). Two chaotic systems either nonequivalent or equivalent can evolve on separate attractors. However, under a designed control law, these systems may initially follow separate attractors and end up following a similar trajectory (Al-sawalha, 2020).

In recent decades, fractional-order calculus, as an extension of the well-known integer-order calculus, has been given considerable attention by scholars in the fields of mathematics and engineering (Tlelo-Cuautle et al., 2020). Motivated by the beneficial properties of fractional calculus and its ability to precisely model systems (Soukkou et al., 2018), many research efforts have been concerned with fractional-order dynamical systems, primarily focusing on several important problems, among others, the design of an appropriate control law, the discretization process of fractional-order operators, closed-loop stability analysis, and system modeling (Soukkou et al., 2016). Besides, a descent controller must satisfy both high performances and cost-effectiveness, which made the research area relatively difficult especially when considering chaotic behavior of fractional-order dynamical systems (Soukkou et al., 2018). Chaos synchronization of fractional-order systems has been the subject of several studies in recent years (Chen et al., 2021; Ma et al., 2020). This interest is due to its important applications in secure communication, information processing, and robotics (Delavari and Mohadeszadeh, 2016; Tlelo-Cuautle et al., 2020). Furthermore, the current researches consider that chaos is a common phenomenon in fractional-order nonlinear systems. Different chaos synchronization problems have been extended to fractional-order ones, for example, projective synchronization (Zhou et al., 2019), complete synchronization (Bouzeriba et al., 2016), anti-synchronization (Huang and Cao, 2017), and impulsive synchronization (Ma et al., 2020). Up to now, many fractional-order chaotic systems have been designed and studied, such as fractional-order Chua’s system (Ma et al., 2020), fractional-order Rössler’s system (Ha et al., 2019), fractional-order Arneodo’s system (Chen et al., 2021) and fractional-order Duffing-Holmes’ system (Hosseinnia et al., 2010), among other forms.

In the literature, several well-known control methods have been used to deal with fractional-order chaotic systems synchronization. Among them, feedback control (Soukkou et al., 2018), active control (Huang and Cao, 2017), sliding mode control (Razminia and Baleanu, 2013), backstepping control (Shukla et al., 2018), fuzzy logic control (Alassafi et al., 2021), and so forth. These traditional synchronization approaches have provided adequate performance for some fractional-order chaotic systems, but they have numerous drawbacks and limitations that make them unsuitable for large-scale applications. In fact, they are only useful for synchronizing a specific type of fractional-order chaotic systems in which the master–slave system’s model is supposed to be totally or partially known. The parametric uncertainties and disturbances have a significant impact on the performance of a synchronization scheme. Moreover, the presence of unknown input nonlinearities further complicates the achievement of synchronization purposes for fractional-order chaotic systems. Adaptive controllers, which can deal with uncertainties and input nonlinearities, have attracted a lot of interest in recent years as a solution to difficulties with traditional synchronization approaches, for either integer-order systems (Ahmad and Shafiq, 2020; Asadollahi et al., 2020) or fractional-order systems (Bouzeriba et al., 2016; Zhang et al., 2018). Since the real-world applications of fractional-order chaotic systems often involve input nonlinearities, it is more convenient to consider these nonlinearities in synchronization schemes. If not, it may end up in poor performances and eventually system instability (Boubellouta et al., 2019). Newly works commonly consider dead-zones, input hysteresis, unknown input gain, and input saturation in the development phase. Variable-structure adaptive fuzzy control was proposed in Boubellouta et al. (2019) to synchronize two fractional-order uncertain chaotic systems with input nonlinearities. In this approach, a fuzzy system is considered to approximate uncertain system dynamics and unknown disturbances. In Ha et al. (2019), the synchronization of fractional-order chaotic systems subject to external disturbances and input saturation was achieved under a proposed fuzzy adaptive backstepping control scheme. In this work, a fuzzy system is used to overcome the existence of fractional-order derivatives in the virtual control function. In Ha et al. (2021), a command-filtered neural network adaptive controller was proposed for synchronization purposes of fractional-order chaotic systems with an unknown dead-zone. The work in Zhou et al. (2019) presents chaos synchronization of a class of fractional-order incommensurate systems subject to input saturation using an adaptive controller with fractional-order adaptation laws. In Chen et al. (2021), authors proposed an event-triggered-based adaptive backstepping neural network sliding mode to achieve chaos synchronization of fractional-order systems with input delay. The work of Mohammadzadeh and Ghaemi (2017) suggests a robust optimal fractional-order synchronization strategy based on a nonsingleton fuzzy cerebellar model articulation control and fractional-adaptation laws, by considering unknown fractional orders and uncertain input nonlinearities.

To enhance the synchronization performances in the adaptive synchronization schemes, it is important to employ a rapid adaptive controller with good robustness against the effects of uncertainties and input nonlinearities. Unfortunately, with a traditional adaptive controller (Chen et al., 2021; Ha et al., 2019, 2021; Zhou et al., 2019), rapid adaptation can compromise control robustness (Hovakimyan and Cao, 2010). $L_{1}$ adaptive control strategy proposed by Hovakimyan and Cao (2010) as an advanced strategy of model reference adaptive control (MRAC) is known for its ability to preserve response robustness along with improving transient performances. Compared to MRAC, a low-pass filter is designed within the input channel of the $L_{1}$ adaptive control technique, aiming to compensate for the low-frequency component of uncertainties. This architecture ensures that the control loop is decoupled from the estimation loop and proved to bypass control signal high-frequency noise (Boubakir et al., 2016). Many works investigated the $L_{1}$ adaptive strategy in control schemes (Boubakir et al., 2016; Vajpayee et al., 2021). Recently, $L_{1}$ adaptive control method is also extended to control fractional-order systems (Boulham et al., 2022). Motivated by the previous discussions, we develop an alternative control scheme, based on $L_{1}$ adaptive control strategy and fuzzy system approximators, to achieve synchronization for a general class of uncertain fractional-order chaotic systems with input nonlinearities. The proposed controller includes a control law, an adaptive mechanism, and a predictor. In order to assist the manipulation and approximation of system uncertainties and input nonlinearities, a fuzzy system is also introduced into the control structure, which enhances the states prediction and hence the overall synchronization performance.

The main goal of this work is to introduce the $L_{1}$ adaptive control strategy into the fractional-order chaos synchronization problem, aiming to investigate its advantages and improve synchronization performances for the considered class of systems. The novelty and extra values of the proposed control approach are itemized as follows:

Develop a fuzzy $L_{1}$ adaptive controller for chaos synchronization of a general class of uncertain fractional-order chaotic systems subject to unknown input nonlinearities. Compared with conventional adaptive synchronization (Bouzeriba et al., 2016; Chen et al., 2021; Ha et al., 2019, 2021; Zhou et al., 2019), fast adaptation is possible in the suggested synchronization method without sacrificing robustness.

The fuzzy approximator considered in the control architecture aids in the handling of uncertainties and input nonlinearities and improves the estimation of external disturbances. This results in small state prediction errors which improves the synchronization accuracy.

The proposed fuzzy fractional-order $L_{1}$ adaptive synchronization method can synchronize all system states with just one controller. Furthermore, the controller is able to provide a smooth control signal thanks to the integral structure of the resulting control law.

The suggested chaotic synchronization technique is model-free, in contrast to the approaches in Zhang et al. (2018), Zhou et al. (2019), and other works in the literature.

The rest of this paper is devised into seven sections and organized as follows. Section “Preliminary concepts” presents some preliminary concepts. Section “Problem formulation” introduces the synchronization problem formulation and the main control objective. Section “Design of the FFOL1AC” develops and explains the control architecture. Section “Analysis of the fractional-order $L_{1}$ adaptive sliding mode controller” investigates the closed-loop stability of the system. Section “Numerical simulation results” presents two simulation examples and a comparative study. Section “Conclusion” concludes the work.

Preliminary concepts

Fractional-order calculus is an extension of the well-known integer-order calculus to real or even complex orders. In fact, numerous definitions exist involving fractional-order operators with order $α > 0$ . However, we will only deal with the definition given by Caputo (1967), since it permits placing initial conditions at the origin of time $t = 0$ for any smooth function $q (t)$ under consideration.

Definition 1. For a considered smooth function $q (t)$ , the Caputo fractional-derivative of the order $α$ is given on the half axis $R^{+}$ (Caputo, 1967)

_{t_{0}}^{C} D_{t}^{α} q (t) = \frac{1}{Γ (κ - α)} \int_{t_{0}}^{t} \frac{q^{(κ)} (τ)}{{(t - τ)}^{α + 1 - κ}} d τ

(1)

where $α > 0$ , $Γ (α) = \int_{0}^{\infty} q^{α - 1} e^{- q} dq$ is the so-called Gamma function, $t > t_{0}$ , $κ \in N^{+}$ , $κ - 1 < α < κ$ , and $α$ , $t$ , $t_{0} \in R$ .

In what follows, the notation $D^{α}$ is used instead of $_{t_{0}}^{C} D_{t}^{α}$ to stand for the Caputo fractional derivative defined in Definition 1.

Lemma 1. For the linear fractional-order system represented in the state-space form (2) with inner dimension $m$ (Matignon, 1996)

{\begin{matrix} D^{α} x = Ax + Bu \\ y = Cx \end{matrix}

(2)

where $D^{α}$ stands for the Caputo fractional derivative, $α \in (0, 1)$ , $x, y \in R^{m}$ with $x (0) = x_{0}$ , and $A \in R^{m \times m}$ . If one assumes that the triplet $(A, B, C)$ is minimal, then the system described by equation (2) is bounded-input bounded-output (BIBO) stable iff $| \arg (eigA) | > α π / 2$ . All system (2) components behave like $t^{- 1 - α}$ at infinity for an impulse response.

Definition 2. The projection operator $Proj (., .)$ is defined for any given vectors $j$ , $h \in R^{k}$ in the following form (Hovakimyan and Cao, 2010)

\begin{matrix} Proj (j, h) = {\begin{matrix} h - \frac{\nabla H (j) {(\nabla H (j))}^{T}}{∥ \nabla H (j) ∥^{2}} h H (j) & if H (j) > 0 and {(\nabla H (j))}^{T} h > 0 \\ h & otherwise \end{matrix} \end{matrix}

(3)

where $\nabla H (j) = (\partial H (j) / \partial j_{1} \dots \partial H (j) / \partial j_{k})^{T}$ , $H (j) : R^{k} \to R$ is a convex function given by $H (j) = (ϵ_{j} + 1) j^{T} j - j_{\max}^{2} / ϵ_{j} j_{\max}^{2}$ , $j_{\max}$ represents a norm bound that is imposed on $j$ , and $ϵ_{j} \in R^{+}$ is an adjustable tolerance bound.

Property 1. Let us suppose that $j^{*} \in χ_{0} \subset χ_{1} \subset R^{k}$ and $q \in χ_{1}$ , with $χ_{0}$ and $χ_{1}$ being the convex sets. Then, we have (Hovakimyan and Cao, 2010)

(q - q^{*})^{T} (Proj (q, h) - h) \leq 0

(4)

Definition 3. A fuzzy logic system (FLS) is commonly used as an estimator in the control architecture (Bouzeriba et al., 2016). Basically, each FLS includes a fuzzifier, a rule base, a fuzzy inference system, and a defuzzifier. In this work, we consider a zero-order Takagi–Sugeno FLS that employs a set of fuzzy $If - Then$ rules to define a mapping from the input vector $z^{T} = [z_{1}, z_{2}, \dots, z_{n}] \in R^{n}$ to a scalar output $\hat{h} \in R$ , where the $i th If - Then$ fuzzy rule is defined as follows

if z_{i} is N_{1}^{i} and, \dots, and N_{n}^{i}, then \hat{h} is h^{i}

(5)

where $N_{1}^{i} and, \dots, and N_{n}^{i}$ are the fuzzy sets, and $h^{i}$ is a fuzzy singleton of the output in the corresponding rule.

The use of a production inference system and a singleton fuzzifier lead to the equation of the FLS output

\hat{h} (z) = \frac{\sum_{i = 1}^{m} h^{i} (Π_{j = 1}^{n} η_{N_{j}^{i}} (z_{j}))}{\sum_{i = 1}^{m} (Π_{j = 1}^{n} η_{N_{j}^{i}} (z_{j}))} = W^{T} (z) ϕ

(6)

where $η_{N_{j}^{i}} (z_{j})$ represents the membership degree of the corresponding input $z_{j}$ , $m$ is the number of fuzzy rules defined by equation (5), $ϕ$ is a vector containing all adjustable elements in the resultant part of the fuzzy rules, and $W^{T} (z) = [W_{1} (z), \dots, W_{n} (z)]$ is the basis functions vector with $W_{i} (z) = Π_{j = 1}^{n} η_{N_{j}^{i}} (z_{j}) / \sum_{i = 1}^{m} (Π_{j = 1}^{n} η_{N_{j}^{i}} (z_{j}))$ .

Problem formulation

Consider the following class of n-dimensional fractional-order uncertain chaotic systems

{\begin{matrix} D^{α} x_{j} = x_{j + 1}, for j = 1, \dots, n - 1 \\ D^{α} x_{n} = f_{x} (x) \end{matrix}

(7)

where $D^{α}$ is the Caputo derivative operator with $α$ is its order with respect to $α \in (0, 1)$ , $x = [x_{1}, \dots, x_{n}]^{T} \in R^{n}$ is the measurable state vector, and $f_{x} (.)$ is a continuous nonlinear uncertain function.

Model (7) is considered as the master system. Thus, the controlled slave system that is related to equation (7) is given by

{\begin{matrix} D^{α} y_{j} = y_{j + 1}, for j = 1, \dots, n - 1 \\ D^{α} y_{n} = f_{y} (y) + φ (u) + d (y, t) \end{matrix}

(8)

where $y = [y_{1}, \dots, y_{n}]^{T} \in R^{n}$ is the system states vector which is assumed to be measurable, $u \in R$ is the control signal, $φ (u)$ is the input nonlinearity that will be defined later, $d (y, t)$ represents the system perturbations, and $f_{y} (.)$ is a smooth unknown continuous nonlinear function.

Input nonlinearity $φ (u)$ includes both dead-zone and sector nonlinearities which can be described as (Boubellouta et al., 2019)

φ (u) = {\begin{matrix} Υ_{1} (u) (u - u_{m a x}), & u > u_{m a x} \\ 0, & - u_{m i n} \leq u \leq u_{m a x} \\ Υ_{2} (u) (u + u_{m i n}), & u < - u_{m i n} \end{matrix}

(9)

where $Υ_{1} (u) > 0$ and $Υ_{2} (u) > 0$ are the uncertain bounded nonlinear functions, and $u_{\max}$ and $u_{\min}$ are the strictly positive and unknown values.

The objective of this work is to design a fuzzy fractional-order $L_{1}$ adaptive controller (FFOL1AC) to achieve a robust projective synchronization between the fractional-order systems (7) and (8), despite the presence of uncertain terms, unknown perturbations, and input nonlinearities.

Design of the FFO $L_{1}$ AC

Selection of a suitable sliding surface

To begin, let us select a fractional-order sliding surface in the following form

s (t) = D^{α - 1} e_{n} (t) + \int_{0}^{t} β^{T} E (τ) d τ

(10)

where $s (t) \in R$ , $β^{T} = [β_{1}, β_{2}, \dots β_{n}] \in R^{n}$ is a design parameters vector that will be defined later to ensure the stability, and $E (t) = [e_{1}, e_{2}, \dots, e_{n}]^{T} \in R^{n}$ is the vector of projective synchronization errors defined as

E (t) = y (t) - δ x (t)

(11)

where $δ$ is a scaling factor. From equation (11), the synchronization errors dynamics can be described as follows

{\begin{matrix} D^{α} e_{j} (t) = e_{j + 1} (t), for j = 1, \dots, n - 1 \\ D^{α} e_{n} (t) = D^{α} y_{n} (t) - δ D^{α} x_{n} (t) \end{matrix}

(12)

Let us consider the time derivative $\overset{\cdot}{s} (t)$ of equation (10). Then, for a significantly small value of the resulted $\overset{\cdot}{s} (t)$ , one has

D^{α} e_{n} (t) = - β^{T} E (t)

(13)

As a generalization from equations (12) and (13), the state-space representation of the errors dynamics can be written as

\underset{D^{α} E}{\underset{︸}{[\begin{matrix} D^{α} e_{1} (t) \\ D^{α} e_{2} (t) \\ ⋮ \\ D^{α} e_{n - 1} (t) \\ D^{α} e_{n} (t) \end{matrix}]}} = \underset{F}{\underset{︸}{[\begin{matrix} 0 & 1 & 0 & \dots & 0 \\ 0 & 0 & 1 & \dots & 0 \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & 0 & \dots & 1 \\ - β_{1} & - β_{2} & - β_{3} & \dots & - β_{n} \end{matrix}]}} \underset{E}{\underset{︸}{[\begin{matrix} e_{1} (t) \\ e_{2} (t) \\ ⋮ \\ e_{n - 1} (t) \\ e_{n} (t) \end{matrix}]}} = FE

(14)

or $D^{α} E = FE$ , where $F \in R^{n \times n}$ is a constant matrix. The well-known stability condition that is applied to linear fractional-order systems and summarized in Lemma 1, i.e., $| \arg (eigF) | > α π / 2$ , requires that the design parameters $β_{i}, i = 1, \dots, n$ , must be chosen strictly positive. In this case and by considering that the value of $\overset{\cdot}{s} (t)$ is significantly small, the synchronization errors and its fractional-order derivatives will accordingly vanish toward zero. Therefore, the purpose of the developed chaos synchronization can be fulfilled by controlling $s (t)$ and $\overset{\cdot}{s} (t)$ toward zero.

By considering equations (10) and (12), the sliding surface dynamics can be extended as

\overset{\cdot}{s} (t) = D^{α} y_{n} (t) - δ D^{α} x_{n} (t) + β^{T} E (t)

(15)

replacing $D^{α} x_{n} (t)$ and $D^{α} y_{n} (t)$ from equations (7) and (8), respectively, in equation (15), one has

\overset{\cdot}{s} (t) = f_{y} (y) + φ (u) + d (y, t) - δ f_{x} (x) + β^{T} E (t)

(16)

Let us define a function $ϑ$ as

ϑ (t, s) \overset{Δ}{=} β^{T} E (t) + λ s (t)

(17)

where $λ \in R^{+}$ . Subtracting and adding $λ s$ in equation (16) and considering equation (17), we have the following

\overset{\cdot}{s} (t) = - λ s (t) + ϑ (t, s) + f_{y} (y) + φ (u) + d (y, t) - δ f_{x} (x)

(18)

For the input nonlinearity $φ (u)$ , there is always a positive unknown constant $g$ and an unknown function $Δ (y, u)$ , such that

φ (u) = gu (t) - Δ (y, u)

(19)

where $0 < ω_{1} < g < ω_{2}$ , $ω_{1}$ and $ω_{2}$ are some positive constants, and $Δ (y, u)$ represents the mismatch between $φ (u) and gu (t)$ . Hence

\overset{\cdot}{s} (t) = - λ s (t) + ϑ (t, s) + f_{y} (y) + gu (t) - Δ (y, u) + d (y, t) - δ f_{x} (x)

(20)

Now, by introducing a function $h (x, y, ϑ, u)$ as

h (x, y, ϑ, u) = ϑ (t, s) + f_{y} (y) + d (y, t) - Δ (y, u) - δ f_{x} (x)

(21)

one obtains

\overset{\cdot}{s} (t) = - λ s (t) + gu (t) + h (x, y, ϑ, u)

(22)

By the use of an FLS in the form of equation (6), the function $h (x, y, ϑ, u)$ can be approximated as follows

h (z) = W^{T} (z) ϕ + ϵ (z), | ϵ (z) | < ε

(23)

Now, replacing equation (23) in equation (22), one can acquire the following representation of the sliding surface dynamics

\overset{\cdot}{s} (t) = - λ s (t) + gu (t) + W^{T} (z) ϕ + ϵ (z), s (0) = s_{0}

(24)

Remark 1. The control input $u$ is used as an input for the fuzzy system approximator employed to estimate the unknown function $h (x, y, ϑ, u)$ . It is worth noting that there is no algebraic loop problem since the control law has an integral structure as can be observed throughout the rest of this work.

Main results

The following predictor (25) replicates the dynamics described by equation (24) with the unknown parameters $g$ and $ϕ$ replaced with their estimation $\hat{g}$ and $\hat{ϕ}$ , respectively.

\overset{\cdot}{\hat{s}} (t) = - λ \hat{s} (t) + \hat{g} u (t) + W^{T} (z) \hat{ϕ}, \hat{s} (0) = s_{0}

(25)

where $\hat{s} (t) \in R$ represents the prediction of the sliding surface $s (t)$ . The values of $\hat{g}$ and $\hat{ϕ}$ are governed by the following projection-type adaptation laws

\begin{matrix} \overset{\cdot}{\hat{g}} (t) = Γ Proj (\hat{g} (t), - \tilde{s} (t) u (t)) \\ \overset{\cdot}{\hat{ϕ}} (t) = Γ Proj (\hat{ϕ} (t), - \tilde{s} (t) W (z)) \end{matrix}

(26)

where $Γ \in R^{+}$ is the adaptation gain, $Proj (., .)$ is introduced in Definition 2 as the projection operator, and $\tilde{s} (t)$ is the prediction error that is defined as $\tilde{s} (t) \overset{Δ}{=} \hat{s} (t) - s (t)$ .

The control law is defined as

u \overset{Δ}{=} - KD (p) (\hat{μ} (p))

(27)

where $\hat{μ} (p)$ is the Laplace transform of $\hat{μ} (t) \overset{Δ}{=} \hat{g} u (t) + W^{T} (z) \hat{ϕ}$ , $p$ is the complex variable, $K$ is a feedback gain, and $D (p)$ is a strictly proper and BIBO-stable transfer function selected to obtain a stable proper closed-loop filter as

C (p) = \frac{gKD (p)}{1 + gKD (p)}

(28)

Consequently, the resulting form of the filter $C (p)$ in equation (28) can represent a low-pass first-order filter with the choice of the transfer function $D (p)$ as $D (p) = 1 / p$ , which gives

C (p) = \frac{gK}{p + gK}

(29)

where $C (0) = 1$ .

Analysis of the fractional-order $L_{1}$ adaptive sliding mode controller

This section will discuss and analyze the stability and performance of the designed controller. Therefore, we consider that the following assumptions are necessary.

Assumption 1. In the description of the system (8), the perturbation $d (y, t)$ is assumed to accept a bound $d^{*} \in R^{+}$ holding $| d (y, t) | \leq d^{*}$ .

Assumption 2. Assuming that $h (x, y, ϑ, u)$ in equation (21) accepts the following bound

| h (x, y, ϑ, u) | \leq L | s (t) | + L_{0}

(30)

where $L$ and $L_{0}$ are some positive constants.

Assumption 3. Let $\overset{\cdot}{ϕ}$ be the bounded derivative of the continuous function $ϕ$ such as $∥ \overset{\cdot}{ϕ} ∥ \leq d_{ϕ} < \infty, \forall t \geq 0$ .

Remark 2. The fractional-order sliding surface (10) ensures that $y (t)$ shares the same stability behavior with $s (t)$ , using the fact that $ϑ (t, s)$ is known and $d (y, t)$ is bounded as suggested in Assumption 1. We can assume that the uncertain quantity $h (x, y, ϑ, u)$ is upper bounded by a functional bound. Hence, with regard to the bound in equation (30), it is obvious that the latter is not constant and it grows as the sliding surface $s (t)$ grows to infinity, which made it physically acceptable.

The control architecture defined via equations (25)–(27), and explained by the descriptive block diagram in Figure 1 is subject to the $L_{1}$ -norm condition that follows

∥ G (p) ∥_{L_{1}} L < 1

(31)

where $G (p) = H (p) (1 - C (p))$ , $H (p) = (p + λ)^{- 1}$ , and $L$ is given in equation (30). Therefore, the design parameter $λ$ and the filter $C (p)$ are selected to fulfill it.

Remark 3. Using the fact that $λ > 0$ and $C (p)$ is a strictly proper and BIBO-stable transfer function, we can easily conclude that both $G (p)$ and $H (p)$ are BIBO-stable and proper transfer functions.

Closed-loop reference system

We consider now a well-behaving reference system that stands for the closed-loop nonadaptive description of the system (24) with the control architecture defined via equations (25)–(27), this system is defined as

{\begin{matrix} {\overset{\cdot}{s}}_{ref} (t) = - λ s_{ref} (t) + g u_{ref} (t) + W^{T} (z_{ref}) ϕ + ϵ (z_{ref}), \\ s_{ref} (0) = s_{0} \\ u_{ref} (p) = - \frac{C (p)}{g} μ_{ref} (p) \end{matrix}

(32)

where $μ_{ref} (p)$ denotes the Laplace transformation of $μ_{ref} (t) \overset{Δ}{=} W^{T} (z_{ref}) ϕ$ .

Remark 4. Unlike the proposed control law $u$ in equation (27), the representation of $u_{ref}$ relies on the unknown terms $g$ and $ϕ$ which does not reflect the purpose of this work. However, we mention that $u_{ref}$ is used for analytical purposes only.

Remark 5. Using control signal $u_{ref}$ in equation (32) we assume that the unknown terms $g$ and $ϕ$ are partially canceled within the bandwidth of $C (p)$ . This assures the best achievable performances for the system (32) and made from it a descent reference for stability and performance analysis.

Lemma 2. Let us consider that both the design parameter $λ$ and the filter $C (p)$ satisfy the $L_{1} - norm$ condition in equation (31), then with regard to the closed-loop reference system representation in equation (32), the bound that follows holds for all $τ > 0$

∥ s_{ref τ} ∥_{L_{\infty}} \leq σ

(33)

where $σ = \frac{∥ G (p) ∥_{ℒ_{1}}}{1 - L ∥ G (p) ∥_{ℒ_{1}}} (ε + L_{0}) + \frac{∥ H (p) ∥_{ℒ_{1}}}{1 - L ∥ G (p) ∥_{ℒ_{1}}}$ $ε + \frac{1}{1 - L ∥ G (p) ∥_{ℒ_{1}}} ∥ s_{i n τ} ∥_{ℒ_{\infty}}$ with $s_{in} \overset{Δ}{=} (p + λ)^{- 1} s_{0}$ .

Proof. The closed-loop reference system expression in equation (32) can be represented in the frequency domain as

p s_{ref} (p) = - λ s_{ref} (p) + g u_{ref} (p) + W_{ref}^{T} ϕ + ϵ_{ref} + s_{0}

(34)

where $ϵ_{ref}$ and $W_{ref}$ are the Laplace transformations of $ϵ (z_{ref})$ and $W (z_{ref})$ , respectively. Thus, replacing $u_{ref} (p)$ and $W_{ref}^{T} ϕ + ϵ_{ref}$ with their expressions, it results

\begin{matrix} s_{ref} (p) = (p + λ)^{- 1} (1 - C (p)) μ_{ref} (p) + (p + λ)^{- 1} ϵ_{ref} + (p + λ)^{- 1} s_{0} \\ = G (p) μ_{ref} (p) + H (p) ϵ_{ref} + s_{in} \end{matrix}

(35)

since $s_{in}$ is known and the transfer functions $G (p)$ and $H (p)$ are BIBO-stable and proper, it follows from equation (35) that the next upper bound is valid for all $τ > 0$

∥ s_{ref τ} ∥_{L_{\infty}} \leq ∥ G (p) ∥_{L_{1}} ∥ μ_{ref τ} ∥_{L_{\infty}} + ∥ H (p) ∥_{L_{1}} ε + ∥ s_{in τ} ∥_{L_{\infty}}

(36)

Next, based on the expression (23) and replacing $W^{T} (z_{ref}) ϕ$ by $μ_{ref} (t)$ , we can easily obtain

∥ μ_{ref τ} ∥_{L_{\infty}} = ∥ h (z_{ref}) - ϵ (z_{ref}) ∥_{L_{\infty}}

(37)

Furthermore, Assumption 2 and the bound of $ϵ (z)$ in equation (23) suggest the following bound

∥ μ_{ref τ} ∥_{L_{\infty}} \leq L ∥ s_{ref τ} ∥_{L_{\infty}} + ε + L_{0}

(38)

replacing equation (38) in equation (36) yields in

\begin{matrix} ∥ s_{ref τ} ∥_{L_{\infty}} \leq ∥ G (p) ∥_{L_{1}} (L ∥ s_{ref τ} ∥_{L_{\infty}} + ε + L_{0}) \\ + ∥ H (p) ∥_{L_{1}} ε + ∥ s_{in τ} ∥_{L_{\infty}} \end{matrix}

(39)

solving for $∥ s_{ref τ} ∥_{L_{\infty}}$ , we deduce that

\begin{matrix} ∥ s_{ref τ} ∥_{L_{\infty}} \leq \frac{∥ G (p) ∥_{L_{1}}}{1 - L ∥ G (p) ∥_{L_{1}}} (ε + L_{0}) + \frac{∥ H (p) ∥_{L_{1}}}{1 - L ∥ G (p) ∥_{L_{1}}} \\ ε + \frac{1}{1 - L ∥ G (p) ∥_{L_{1}}} ∥ s_{in τ} ∥_{L_{\infty}} \end{matrix}

(40)

It is clear that $s_{in}$ is known and uniformly bounded, also both $L_{0}$ and $ε$ have finite values. By considering that the condition in equation (31) is satisfied, it follows from equation (40) that the boundedness of $∥ s_{ref τ} ∥_{L_{\infty}}$ holds for all $τ > 0$ , which completes the proof.

Transient and steady-state

In this section, we discuss the transient and steady-state performances of the system (8) and the control architecture defined via equations (25)–(27) with respect to the well-behaving reference system represented in equation (32).

Since we have $\tilde{s} (t) = \hat{s} (t) - s (t)$ , the prediction error dynamics can be described by subtracting equation (24) from equation (25) as

\overset{\cdot}{\tilde{s}} (t) = - λ \tilde{s} (t) + \tilde{g} u (t) + W^{T} (z) \tilde{ϕ} - ϵ (z)

(41)

where $\tilde{g} \overset{Δ}{=} \hat{g} - g$ and $\tilde{ϕ} \overset{Δ}{=} \hat{ϕ} - ϕ$ are the estimation errors.

Consequently, expanding equation (41) to the frequency domain, one has

\tilde{s} (p) = H (p) (\tilde{μ} (p) - ϵ (z))

(42)

where $\tilde{μ} (p)$ is the Laplace transform of $\tilde{μ} (t) \overset{Δ}{=} \tilde{g} u (t) + W^{T} (z) \tilde{ϕ}$ .

Lemma 3. Let us consider that both the design parameter $λ$ and the filter $C (p)$ satisfy the $L_{1}$ -norm condition in equation (31), then with regard to the prediction error dynamics representation in equation (42), the following bound is valid for all $τ > 0$

∥ {\tilde{s}}_{τ} ∥_{L_{\infty}} \leq \sqrt{θ}

(43)

where $θ = (\frac{1}{Γ} (4 \max_{ϕ \in Θ} ∥ ϕ ∥^{2} + (ω_{2} - ω_{1})^{2}) + \frac{ρ^{- 1}}{Γ} \underset{ϕ \in Θ}{4 \max} ∥ ϕ ∥$ $d_{ϕ} + ρ^{- 1} ε^{2}$ with $ρ \overset{Δ}{=} (2 λ - 1) and λ > \frac{1}{2}$ .

Proof. In view of studying the stability, we consider the candidate Lyapunov function that follows

V (\tilde{s}, \tilde{ϕ}, \tilde{g}) = {\tilde{s}}^{2} + \frac{1}{Γ} {\tilde{ϕ}}^{T} \tilde{ϕ} + \frac{1}{Γ} {\tilde{g}}^{2}

(44)

where its time derivative $\overset{\cdot}{V}$ is given as

\overset{\cdot}{V} (\tilde{s}, \tilde{ϕ}, \tilde{g}) = 2 \tilde{s} \overset{\cdot}{\tilde{s}} + \frac{2}{Γ} {\tilde{ϕ}}^{T} \overset{\cdot}{\tilde{ϕ}} + \frac{2}{Γ} \tilde{g} \overset{\cdot}{\tilde{g}}

(45)

replacing equation (41), it results in

\begin{matrix} \overset{\cdot}{V} (\tilde{s}, \tilde{ϕ}, \tilde{g}) = 2 \tilde{s} (- λ \tilde{s} + \tilde{g} u + W^{T} \tilde{ϕ} - ϵ) + \frac{2}{Γ} {\tilde{ϕ}}^{T} \overset{\cdot}{\hat{ϕ}} - \frac{2}{Γ} {\tilde{ϕ}}^{T} \overset{\cdot}{ϕ} \\ + \frac{2}{Γ} \tilde{g} \overset{\cdot}{\hat{g}} \end{matrix}

(46)

\begin{matrix} \overset{\cdot}{V} (\tilde{s}, \tilde{ϕ}, \tilde{g}) = - 2 λ {\tilde{s}}^{2} (t) + 2 {\tilde{ϕ}}^{T} (\frac{1}{Γ} \overset{\cdot}{\hat{ϕ}} + \tilde{s} W) + 2 \tilde{g} (\frac{1}{Γ} \overset{\cdot}{\hat{g}} + \tilde{s} u) \\ - \frac{2}{Γ} {\tilde{ϕ}}^{T} \overset{\cdot}{ϕ} - 2 \tilde{s} ϵ \end{matrix}

(47)

consequently, from equation (26), one has

\begin{matrix} \overset{\cdot}{V} (\tilde{s}, \tilde{ϕ}, \tilde{g}) = - 2 λ {\tilde{s}}^{2} (t) + 2 {\tilde{ϕ}}^{T} (Proj (\hat{ϕ}, - \tilde{s} W) + \tilde{s} W) \\ + 2 \tilde{g} (Proj (\hat{g}, - \tilde{s} u) + \tilde{s} u) - \frac{2}{Γ} {\tilde{ϕ}}^{T} \overset{\cdot}{ϕ} - 2 \tilde{s} ϵ \end{matrix}

(48)

It follows from Property 1 that $2 {\tilde{ϕ}}^{T} (Proj (\hat{ϕ}, - \tilde{s} W) + \tilde{s} W) + 2 \tilde{g} (Proj (\hat{g}, - \tilde{s} u) + \tilde{s} u) \leq 0$ . Hence

\overset{\cdot}{V} (\tilde{s}, \tilde{ϕ}, \tilde{g}) \leq - 2 λ {\tilde{s}}^{2} + \frac{2}{Γ} | {\tilde{ϕ}}^{T} \overset{\cdot}{ϕ} | + 2 | \tilde{s} | ε

(49)

It is straightforward to establish that $2 \tilde{s} ε \leq {\tilde{s}}^{2} + ε^{2}$ , that is

\overset{\cdot}{V} (\tilde{s}, \tilde{ϕ}, \tilde{g}) \leq - ρ {\tilde{s}}^{2} + \frac{2}{Γ} | {\tilde{ϕ}}^{T} \overset{\cdot}{ϕ} | + ε^{2}

(50)

Letting $Θ$ be a convex set, with $\hat{ϕ} \in Θ$ follows from the definition of the projection operator. Besides, we have $0 < ω_{1} < g < ω_{2}$ , it follows from it that

\max_{t \geq 0} ({\tilde{ϕ}}^{T} \tilde{ϕ} + {\tilde{g}}^{2}) \leq (4 \max_{ϕ \in Θ} ∥ ϕ ∥^{2} + {(ω_{2} - ω_{1})}^{2})

(51)

Similarly, based on the upper bound in Assumption 3, we get

\frac{2}{Γ} | {\tilde{ϕ}}^{T} \overset{\cdot}{ϕ} | \leq \frac{4}{Γ} (\max_{ϕ \in Θ} ∥ ϕ ∥ d_{ϕ})

(52)

as a result

\overset{\cdot}{V} (\tilde{s}, \tilde{ϕ}, \tilde{g}) \leq - ρ {\tilde{s}}^{2} + \frac{4}{Γ} \max_{ϕ \in Θ} ∥ ϕ ∥ d_{ϕ} + ε^{2}

(53)

adding and substituting $ρ / Γ {\tilde{ϕ}}^{T} \tilde{ϕ} + ρ / Γ {\tilde{g}}^{2}$ in equation (53) yield in

\begin{matrix} \overset{\cdot}{V} (\tilde{s}, \tilde{ϕ}, \tilde{g}) \leq - ρ {\tilde{s}}^{2} - \frac{ρ}{Γ} {\tilde{ϕ}}^{T} \tilde{ϕ} - \frac{ρ}{Γ} {\tilde{g}}^{2} + \frac{ρ}{Γ} {\tilde{ϕ}}^{T} \tilde{ϕ} + \frac{ρ}{Γ} {\tilde{g}}^{2} \\ + \frac{4}{Γ} \max_{ϕ \in Θ} ∥ ϕ ∥ d_{ϕ} + ε^{2} \end{matrix}

(54)

\begin{matrix} \overset{\cdot}{V} (\tilde{s}, \tilde{ϕ}, \tilde{g}) \leq - ρ V (\tilde{s}, \tilde{ϕ}, \tilde{g}) + ρ \\ (\frac{1}{Γ} (4 \max_{ϕ \in Θ} ∥ ϕ ∥^{2} + {(ω_{2} - ω_{1})}^{2}) + \frac{ρ^{- 1}}{Γ} \underset{ϕ \in Θ}{4 \max} ∥ ϕ ∥ d_{ϕ} + ρ^{- 1} ε^{2}) \end{matrix}

(55)

Finally, one can conclude that

\overset{\cdot}{V} (\tilde{s}, \tilde{ϕ}, \tilde{σ}) \leq - ρ V (\tilde{s}, \tilde{ϕ}, \tilde{g}) + ρ θ

(56)

Thus, for any given time $t_{1} > 0$ , if we have $V (t_{1}) > θ$ , it follows from equation (56) that $\overset{\cdot}{V} (t_{1}) < 0$ . Besides, since $s (0) = \hat{s} (0)$ , it is obvious that $V (0) < 1 / Γ (4 \max_{ϕ \in Θ} ∥ ϕ ∥^{2} + (ω_{2} - ω_{1})^{2}) < θ$ . Therefore, we can conclude that $V (t) < θ$ for all $t > 0$ . Finally, the bound $∥ {\tilde{s}}_{τ} ∥_{L_{\infty}} \leq \sqrt{θ}$ can be acquired based on the facts that ${\tilde{s}}^{2} < V (t)$ and $∥ . ∥_{\infty} \leq ∥ . ∥$ , that complete the proof.

Remark 6. Lemma 3 admits that the prediction error can be significantly reduced by increasing the design parameter $λ$ and the adaptation gain $Γ$ .

Proposition 1. Let us consider that both design parameter $λ$ and the filter $C (p)$ satisfy the $L_{1}$ -norm condition in equation (31), then for the system represented in equation (8), we have

\begin{matrix} ∥ {(s_{ref} (p) - s (p))}_{τ} ∥_{L_{\infty}} \leq ϱ_{1} \\ ∥ {(u_{ref} (p) - u (p))}_{τ} ∥_{L_{\infty}} \leq ϱ_{2} \end{matrix}

(57)

and

\begin{matrix} lim_{Γ \to \infty, Γ \to ε} (s_{ref} (t) - s (t)) = 0 \\ lim_{Γ \to \infty, Γ \to ε} (u_{ref} (t) - u (t)) = 0 \end{matrix}

(58)

where $ϱ_{1} = \frac{∥ C (p) ∥_{ℒ_{1}}}{1 - ∥ G (p) ∥_{ℒ_{1}} L} \sqrt{θ} + \frac{3 ∥ G (p) ∥_{ℒ_{1}} + ∥ H (p) ∥_{ℒ_{1}}}{1 - ∥ G (p) ∥_{ℒ_{1}} L} ε$ , $ϱ_{2} = ∥ \frac{C (p)}{g} ∥_{ℒ_{1}} (L ϱ_{1} + 3 ε) +$ $∥ \frac{H_{1} (p)}{g} ∥_{ℒ_{1}} \sqrt{θ}, and H_{1} (p) = \frac{C (p)}{H (p)}$

Proof. First, letting $μ (t) \overset{Δ}{=} W^{T} (z) ϕ$ , which allows us to rewrite the control law in equation (27) as

\begin{matrix} u (p) = - KD (p) (\tilde{g} u (p) + W^{T} (p) \tilde{ϕ} + gu (p) + W^{T} (p) ϕ) \\ = - KD (p) (\tilde{μ} (p) + μ (p) + gu (p)) \\ = - \frac{C (p)}{g} (\tilde{μ} (p) + μ (p)) \end{matrix}

(59)

System representation in equation (24) can be expressed in the frequency domain as

s (p) = (p + λ)^{- 1} (gu (p) + W^{T} (p) ϕ + ϵ (p)) + s_{in}

(60)

replacing equation (59) gives

s (p) = G (p) μ (p) - H (p) C (p) \tilde{μ} (p) + H (p) ϵ (p) + s_{in}

(61)

using the representation of $\tilde{s} (p)$ in equation (42) to deduce that

s (p) = G (p) μ (p) - C (p) \tilde{s} (p) + G (p) ϵ (p) + s_{in}

(62)

Next, letting $μ_{e} (t) \overset{Δ}{=} μ_{ref} (t) - μ (t)$ and subtracting equation (62) from equation (35) results in

s_{ref} (p) - s (p) = G (p) μ_{e} (p) + C (p) \tilde{s} (p) + H (p) ϵ_{ref} - G (p) ϵ (p)

(63)

According to Assumption 2 and using the bound of $ϵ (z)$ in equation (23), $μ_{e}$ can be bounded as

∥ μ_{e τ} ∥_{L_{\infty}} \leq L ∥ {(s_{ref} (p) - s (p))}_{τ} ∥_{L_{\infty}} + 2 ε

(64)

Since the transfer functions $C (p)$ , $G (p)$ , and $H (p)$ are BIBO-stable and proper and based on equation (63), we deduce that the next upper bound holds for all $τ > 0$

\begin{matrix} ∥ {(s_{ref} (p) - s (p))}_{τ} ∥_{L_{\infty}} \leq ∥ G (p) ∥_{L_{1}} (L ∥ {(s_{ref} (p) - s (p))}_{τ} ∥_{L_{\infty}} + 2 ε) \\ + ∥ C (p) ∥_{L_{1}} ∥ {\tilde{s}}_{τ} ∥_{L_{\infty}} + ∥ G (p) + H (p) ∥_{L_{1}} ε \end{matrix}

(65)

solving for $∥ (s_{ref} (p) - s (p))_{τ} ∥_{L_{\infty}}$ results in

\begin{matrix} ∥ {(s_{ref} (p) - s (p))}_{τ} ∥_{L_{\infty}} \leq \frac{∥ C (p) ∥_{L_{1}}}{1 - ∥ G (p) ∥_{L_{1}} L} \sqrt{θ} \\ + \frac{3 ∥ G (p) ∥_{L_{1}} + ∥ H (p) ∥_{L_{1}}}{1 - ∥ G (p) ∥_{L_{1}} L} ε \end{matrix}

(66)

Similarly, in the second part of the proof, from equations (59) and (32), one can have

u_{ref} (p) - u (p) = - \frac{C (p)}{g} μ_{e} (p) + \frac{C (p)}{g} \tilde{μ} (p)

(67)

It follows from Lemma A.12.1 in Hovakimyan and Cao (2010) that $\frac{C (p)}{g} \tilde{μ} (p)$ can be written as

\begin{matrix} \frac{C (p)}{g} (\tilde{μ} (p) - ϵ (p)) = \frac{1}{g} \frac{C (p)}{H (p)} H (p) (\tilde{η} (p) - ϵ (p)) \\ = \frac{1}{g} H_{1} (p) \tilde{s} (p) \end{matrix}

(68)

consequently

\frac{C (p)}{g} \tilde{μ} (p) = \frac{1}{g} H_{1} (p) \tilde{s} (p) + \frac{C (p)}{g} ϵ (p)

(69)

$H_{1} (p)$ is a stable and proper transfer function according to the fact that $C (p)$ is BIBO-stable and strictly proper. Thus, replacing equation (69) in equation (67) leads to conclude that the next upper bound is valid for all $τ > 0$

\begin{matrix} ∥ {(u_{ref} (p) - u (p))}_{τ} ∥_{L_{\infty}} \leq ∥ \frac{C (p)}{g} ∥_{L_{1}} \\ (L ∥ {(s_{ref} (p) - s (p))}_{τ} ∥_{L_{\infty}} + 3 ε) + ∥ \frac{H_{1} (p)}{g} ∥_{L_{1}} \sqrt{θ} \end{matrix}

(70)

∥ {(u_{ref} (p) - u (p))}_{τ} ∥_{L_{\infty}} \leq ∥ \frac{C (p)}{g} ∥_{L_{1}} (L ϱ_{1} + 3 ε) + ∥ \frac{H_{1} (p)}{g} ∥_{L_{1}} \sqrt{θ}

(71)

Next, it is obvious that $θ$ defined in Lemma 3 agrees with the following limit $lim_{Γ \to \infty, ε \to 0} θ = 0$ that leads to easily deduce that $lim_{Γ \to \infty, ε \to 0} ϱ_{1} = 0$ and $lim_{Γ \to \infty, ε \to 0} ϱ_{2} = 0$ . In addition, from Proposition 1, we have $∥ (s_{ref} (p) - s (p))_{τ} ∥_{L_{\infty}} \leq ϱ_{1}$ and $∥ (u_{adref} (p) - u_{ad} (p))_{τ} ∥_{L_{\infty}} \leq ϱ_{2}$ , hence, $lim_{Γ \to \infty, ε \to 0} (s_{ref} (t) - s (t)) = 0$ and $lim_{Γ \to \infty, ε \to 0} (u_{ref} (t) - u (t)) = 0$ , that completes the proof.

Remark 7. Proposition 1 affirms that if the chosen adaptation gain $Γ$ is large enough and the approximation error $ε$ delivered by the fuzzy system is sufficiently small, the proposed controller represented via equations (25)–(27) can drive the signals $s$ and $u$ to track those of the well-behaving reference system (32).

Remark 8. A brief comparison between the state of art and the proposed fuzzy fractional-order $L_{1}$ adaptive controller is summarized in Table 1.

Table 1.

Summary of state of art compared to the proposed method.

References	Synchronization method	System structure	Input function	Uncertainties	Stability analysis
Our paper	Fuzzy $L_{1}$ adaptive control	Single input chaotic systems	Dead-zone and sector nonlinearities	System dynamics, external disturbance, and input nonlinearities	Integer-order Lyapunov method
Ha et al. (2021)	Command-filtered adaptive neural network control	Single input chaotic systems	Dead-zone	System dynamics and input nonlinearities	Fractional-order Lyapunov method
Boubellouta et al. (2019)	Variable-structure adaptive fuzzy control	Multiple input chaotic systems	Dead-zone and sector nonlinearities	System dynamics, external disturbance, and input nonlinearities	Integer-order Lyapunov method
Mohammadzadeh and Ghaemi (2017)	Adaptive fuzzy cerebellar model articulation control	Multiple input chaotic systems	Dead-zone and sector nonlinearities	Fractional orders, system dynamics, external disturbance, and input nonlinearities	Integer-order Lyapunov method
Chen et al. (2021)	Event-triggered-based adaptive backstepping neural network sliding mode control	Single input chaotic systems	Delay	System dynamics, external disturbance, and input delay	Fractional-order Lyapunov method
Ha et al. (2019)	Fuzzy adaptive backstepping control	Single input chaotic systems	Saturation	System dynamics, external disturbance, and input saturation	Fractional-order Lyapunov method
Zhou et al. (2019)	Generalized fuzzy adaptive control	Multiple input chaotic systems	Saturation	Partially system dynamics, external disturbance, and input saturation	Fractional-order Lyapunov method

Numerical simulation results

This section discusses the synchronization of different fractional-order two-dimensional and three-dimensional systems to examine the efficiency of the suggested FFOL1AC. These systems are subject to uncertain dynamic terms, unknown perturbations, and input nonlinearities. Both numerical simulations will be performed for a scaling factor $δ = 1$ and based on the designed controller, including a control law in the form of equation (27) with adaptation laws as in equation (26), a predictor in the form of equation (25) and a fuzzy system as in equation (6). For the second simulation example, the performances of the proposed synchronization approach will be compared to those of the command-filtered adaptive neural network controller (CFANNC) developed in Ha et al. (2021).

Example 1: fractional-order Duffing-Holmes synchronization

This example considers two different fractional-order Duffing-Holmes systems. The master and the slave descriptions can be represented in the forms of equations (72) and (73), respectively

{\begin{matrix} D^{α} x_{1} = x_{2} \\ D^{α} x_{2} (t) = x_{1} - a_{11} x_{2} - x_{1}^{3} + a_{12} \cos (t) \end{matrix}

(72)

According to this work in Hosseinnia et al. (2010), fractional-order Duffing-Holmes system chaotic behavior appears for $α = 0.98$

{\begin{matrix} D^{α} y_{1} = y_{2} \\ D^{α} y_{2} (t) = y_{1} - a_{21} y_{2} - y_{1}^{3} + a_{22} \cos (t) + f_{1 y} (y, t) + d_{1} (t) + φ_{1} (u) \end{matrix}

(73)

where $a_{11} = 0.25$ , $a_{12} = 0.3$ , $a_{21} = 0.3$ , $a_{22} = 0.35$ , $d_{1} (t) = 0.8 \sin (t) + 0.45 \cos (3 t)$ , and $f_{1 y} (y, t) = 0.1 \sin (t) \sqrt{y_{1}^{2} + y_{2}^{2}}$ .

The initial condition for the considered systems is chosen as $[x_{1} (0), x_{2} (0)] = [0.2, 0.1]$ and $[y_{1} (0), y_{2} (0)] = [- 0.15, 0.25]$ . To reach the best performances, the simulation was executed using the following design parameters acquired through trial and error: $β = [6.2, 5.6]^{T}$ , $λ = 10.5$ , and $Γ = 85, 000$ . The parameters of the low-pass filter (28) are selected here as $K = 35$ and $D (p) = 1 / p$ . For the estimates $\hat{g} and \hat{ϕ}$ , we chose their initial values as $\hat{g} (0) = 0.01$ and $\hat{ϕ} (0) = 0$ with projection bounds $\hat{g} \in [0.01, 6]^{T} and \hat{ϕ} \in [- 6, 6]^{T}$ . The used fuzzy system in this simulation is in the form of equation (6) with the input vector $z = [e_{1}, e_{2}, u, s]^{T}$ and three membership functions $η_{N_{j}^{i}} (z_{j})$ set for each input $z_{j};$ $j = 1, \dots, 4$ , as follows

{\begin{matrix} η_{N_{j}^{1}} (z_{j}) = \frac{1}{2} (1 + \tanh (- \frac{z_{j} + ξ_{1 j}}{ς_{j}})) \\ η_{N_{j}^{2}} (z_{j}) = \exp (- \frac{1}{2} (\frac{z_{j} + ξ_{2 j}}{ς_{j}})) \\ η_{N_{j}^{3}} (z_{j}) = \frac{1}{2} (1 + \tanh (\frac{z_{j} + ξ_{3 j}}{ς_{j}})) \end{matrix}

(74)

with the variances $ς_{j} = [0.2, 0.2, 2.9, 0.03]^{T}$ and the centers $ξ_{i 1} = [0.5, 0, - 0.5]$ , $ξ_{i 2} = [0.5, 0, - 0.5]$ , $ξ_{i 3} = [7, 0, - 7]$ , and $ξ_{i 4} = [0.08, 0, - 0.08]$ . Finally, the input nonlinearity $φ_{1} (u)$ is fixed in this case as

φ_{1} (u) = {\begin{matrix} (0.5 - 0.2 e^{0.1 \sin (u)}) (u - 0.7) & u > 0.7 \\ 0, & - 0.7 \leq u \leq 0.7 \\ (0.5 - 0.2 e^{0.2 \cos (u)}) (u + 0.7), & u < - 0.7 \end{matrix}

(75)

Figures 2 –4 show Example 1 simulation results. Figure 2 illustrates the master and slave synchronization trajectories of the states $(x_{1}$ , $y_{1})$ and $(x_{2}, y_{2})$ . The synchronization errors $e_{1}$ and $e_{2}$ are represented in Figure 3. These figures affirm that the slave system (73) states trajectories are synchronized in a short time to those of the master system (72). The control signal $u$ is relatively smooth and bounded as it can be seen in Figure 4 (top). Figure 4 (bottom) shows that the sliding surface $s$ converges to its expected values along with its prediction $\hat{s}$ . The synchronization performances for different fractional-order values $(α \in (0.90, 0.99))$ are summarized in Table 2. These performances were calculated employing mean squared error (MSE) and affirm that the fractional-order does not influence the acquired results. Finally, the overall results confirm that the proposed controller ensures the boundedness of all closed-loop signals with fast convergence of the synchronization errors $e_{1}$ and $e_{2}$ to zero, despite the control complexity due to the considered uncertain models, unknown perturbations, and input nonlinearities.

Table 2.

Synchronization performances for different values of $α$ for the simulation Example 1.

Derivative order	$MSE (e_{1}) \times 10^{- 4}$	$MSE (e_{2}) \times 10^{- 4}$
$α = 0.90$	5.7985	8.1694
$α = 0.91$	5.7869	8.1397
$α = 0.92$	5.7791	8.1154
$α = 0.93$	5.7749	8.0968
$α = 0.94$	5.7742	8.0833
$α = 0.95$	5.7768	8.0756
$α = 0.96$	5.7831	8.0873
$α = 0.97$	5.7915	8.0773
$α = 0.98$	5.8038	8.0864
$α = 0.99$	5.8189	8.0990

Example 2: fractional-order hybrid optical and Genesio–Tesi systems synchronization

To further highlight the advantages of the proposed FFOL1AC, a comparative study has been carried out between the FFOL1AC and the CFANNC (Ha et al., 2021). As a start, let us consider the fractional-order hybrid optical system (Ha et al., 2021) as the master system with its description that takes the form of equation (7) as follows

{\begin{matrix} D^{α} x_{1} = x_{2} \\ D^{α} x_{2} = x_{3} \\ D^{α} x_{3} (t) = - 0.326 x_{3} - x_{2} + 0.761 x_{1} (1 - x_{1}) \end{matrix}

(76)

The corresponding slave system is the fractional-order Genesio–Tesi system (Faieghi and Delavari, 2012). It description is given by equation (77) in the form of equation (8)

{\begin{matrix} D^{α} y_{1} = y_{2} \\ D^{α} y_{2} = y_{3} \\ D^{α} y_{3} (t) = - 6 y_{1} - 2.92 y_{2} - 1.2 y_{3} + y_{1}^{2} + d_{2} (t) + φ_{2} (u) \end{matrix}

(77)

For the fractional-order $α = 0.99$ , the master system (76) and the slave system (77) behave chaotically (Faieghi and Delavari, 2012; Ha et al., 2021).

The simulation was performed for the initial conditions $[x_{1} (0), x_{2} (0), x_{3} (0)] = [0.5, 0.25, 0.12]$ and $[y_{1} (0), y_{2} (0), y_{3} (0)] = [1.6, - 0.05, 0.55]$ . The external perturbations introduced in equation (77) are chosen as $d_{2} (t, y) = 0.45 \sin (π t) y_{1} + 1.1 \cos (2 t)$ . To achieve satisfactory projective synchronization performance, the following design parameters were chosen using the trial and error method: $β = [3.12, 4.6, 2.8]^{T}$ , $λ = 95$ , and $Γ = 16000$ . For the low-pass filter (28), we select $K = 45$ and $D (p) = 1 / p$ . The projection bounds are set as $\hat{g} \in [0.01, 6]^{T} and \hat{ϕ} \in [- 6, 6]^{T}$ , with initial values selected as $\hat{g} (0) = 0.01$ and $\hat{ϕ} (0) = 0$ . The applied fuzzy system in this simulation is in the form of equation (6) with the input vector $z = [e_{1}, e_{2}, e_{3,} u, s]^{T}$ . As in the previous example, three membership functions $η_{N_{j}^{i}} (z_{j})$ are fixed for each input $z_{j},$ $j = 1, \dots, 5$ . Their forms are already presented in equation (74), with the variances $ς_{j} = [0.3, 0.25, 0.6, 1.5, 0.09]^{T}$ and the centers $ξ_{i 1} = [1, 0.25, - 0.5]$ , $ξ_{i 2} = [0.3, 0.25, - 0.8]$ , $ξ_{i 3} = [1, - 0.25, - 1.5]$ , $ξ_{i 4} = [- 3, - 6.5, - 10]$ , and $ξ_{i 5} = [0.2, 0, - 0.2]$ . Finally, this case considers the following dead-zone input nonlinearity

φ_{2} (u) = {\begin{matrix} 2.1 (u - 1), u > 1 \\ 0, - 1 \leq u \leq 1 \\ 2 (u + 1), u < - 1 \end{matrix}

(78)

Figures 5 –7 illustrate Example 2 simulation results. Figure 5 depicts the master and slave synchronization state trajectories ( $(x_{1}$ , $y_{1}),$ $(x_{2}, y_{2})$ , and $(x_{3}, y_{3})$ ) obtained by the proposed FFOL1AC and CFANNC, and Figure 6 shows the synchronization errors $e_{1}$ , $e_{2}$ , and $e_{3}$ for both controllers. Based on the mentioned figures, it is clear that a fast synchronization response is achieved between the slave system (77) and the master system (76). Figure 7 shows the control signals $u$ resulted from the FFOL1AC and CFANNC; this figure reveals that the control signal is also smooth and bounded for this example. Table 3 shows the synchronization performances for different fractional-order values ( $α \in (0.90, 0.99)$ ). As it can be seen, the fractional-order values do not affect the control performances, which means that the proposed controller can also deliver consistent responses for different values of the derivative order. The synchronization precision in this comparative study was inspected based on the errors $e_{1}$ , $e_{2}$ , and $e_{3}$ delivered by the two synchronization methods over a period of time (focusing on steady-state performances). Three main characteristics are presented in Table 4—the range, the standard deviation (STD), and the average absolute value of the synchronization errors. Evidently, Figures 5 –7 affirm the main advantage of the $L_{1}$ adaptive controller which is fast adaptation with decent transient response. Table 4 reveals the very satisfactory results of the steady-state regime.

Table 3.

Synchronization performances for different values of $α$ for the simulation Example 2.

Derivative order	$MSE (e_{1}) \times 10^{- 2}$	$MSE (e_{2}) \times 10^{- 2}$	$MSE (e_{3}) \times 10^{- 2}$
$α = 0.90$	1.16	0.44	0.13
$α = 0.91$	1.15	0.44	0.14
$α = 0.92$	1.15	0.44	0.15
$α = 0.93$	1.14	0.44	0.15
$α = 0.94$	1.14	0.44	0.17
$α = 0.95$	1.14	0.44	0.18
$α = 0.96$	1.14	0.45	0.19
$α = 0.97$	1.14	0.45	0.20
$α = 0.98$	1.14	0.45	0.22
$α = 0.99$	1.15	0.46	0.23

Table 4.

Precision comparisons between the proposed FFOL1AC and the developed CFANNC in Ha et al. (2021).

Synchronization method		Avg (abs( $e_{i}$ )) $\times 10^{- 2}$	Range of $e_{i} \times 10^{- 2}$	STD of $e_{i} \times 10^{- 2}$
FFOL1AC	$e_{1}$	0.2	$(0.16, 0.23)$	0.03
	$e_{2}$	0.12	$(- 0.25, - 0.07)$	0.04
	$e_{3}$	0.11	$(- 0.04, - 0.47)$	0.15
CFANNC	$e_{1}$	1.4	$(- 3.34, 0.82)$	1.5
	$e_{2}$	2.6	$(- 2.84, 3.76)$	2.9
	$e_{3}$	4.53	$(- 6.42, 6.62)$	4.86

In summary, it is obvious that the simulation results agree with the theoretical and analytical results and demonstrate the effectiveness of the designed fuzzy fractional-order $L_{1}$ adaptive synchronization method.

Conclusion

This work investigates the chaos synchronization problem based on a designed FFOL1AC for a general class of fractional-order chaotic systems with uncertain models, unknown perturbations, and input nonlinearities. The controller is derived based on a fractional-order sliding surface and includes a control law, an adaptive mechanism, and a predictor. Besides, a fuzzy system is used in this synchronization method to efficiently handle system uncertainties and input nonlinearities. The estimation loop is decoupled from the control loop thanks to the low-pass filter placed within the input channel. Thereby, the controller can achieve the master–slave synchronization and preserve response robustness along with improving transient performances, which offers an efficient and robust synchronization strategy for the general class of fractional-order chaotic systems considered in this work. The simulation results provided from the two numerical simulation examples demonstrate the applicability and efficacy of the proposed synchronization approach, as well as the theoretical debates. Our future works would focus on extending the suggested control architecture to synchronize uncertain fractional-order time-delayed and incommensurate chaotic systems.

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) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the general directorate of scientific research and technological development of the Ministry of Higher Education and Scientific Research of Algeria.

ORCID iD

Ihab Abderraouf Boulham

References

Ahmad

Shafiq

(2020) Oscillation free robust adaptive synchronization of chaotic systems with parametric uncertainties. Transactions of the Institute of Measurement and Control 42(11): 1977–1996.

Alassafi

Alsaadi

, et al. (2021) Fuzzy synchronization of fractional-order chaotic systems using finite-time command filter. Information Sciences 579: 325–346.

Al-sawalha

(2020) Synchronization of different order fractional-order chaotic systems using modify adaptive sliding mode control. Advances in Difference Equations 2020(1): 417.

Asadollahi

Ghiasi

Badamchizadeh

(2020) Adaptive synchronization of chaotic systems with hysteresis quantizer input. ISA Transactions 98: 137–148.

Boubakir

Labiod

(2022) Observer-based adaptive neural network control design for projective synchronization of uncertain chaotic systems. Journal of Vibration and Control. Epub ahead of print 28 May. DOI: 10.1177/10775463221101935.

Boubakir

Souanef

Labiod

, et al. (2016) An L1 fuzzy adaptive controller for a class of SISO nonaffine nonlinear systems: Application to the control of an electropneumatic actuator. Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering 230(8): 736–748.

Boubellouta

Zouari

Boulkroune

(2019) Intelligent fuzzy controller for chaos synchronization of uncertain fractional-order chaotic systems with input nonlinearities. International Journal of General Systems 48(3): 211–234.

Boulham

Boubakir

Labiod

(2022) Neural network L1 adaptive control for a class of uncertain fractional order nonlinear systems. Integration 83: 1–11.

Bouzeriba

Boulkroune

Bouden

(2016) Fuzzy adaptive synchronization of uncertain fractional-order chaotic systems. International Journal of Machine Learning and Cybernetics 7(5): 893–908.

10.

Caputo

(1967) Linear models of dissipation whose Q is almost frequency independent II. Geophysical Journal International 13(5): 529–539.

11.

Chen

Yang

Yuan

(2021) Event-triggered adaptive neural network backstepping sliding mode control for fractional order chaotic systems synchronization with input delay. IEEE Access 9: 100868–100881.

12.

Delavari

Mohadeszadeh

(2016) Robust finite-time synchronization of nonidentical fractional-order hyperchaotic systems and its application in secure communication. IEEE/CAA Journal of Automatica Sinica 6(1): 228–235.

13.

Faieghi

Delavari

(2012) Chaos in fractional-order Genesio–Tesi system and its synchronization. Communications in Nonlinear Science and Numerical Simulation 17(2): 731–741.

14.

Chen

Liu

(2021) Command filtered adaptive neural network synchronization control of fractional-order chaotic systems subject to unknown dead zones. Journal of the Franklin Institute 358(7): 3376–3402.

15.

Liu

, et al. (2019) Backstepping-based adaptive fuzzy synchronization control for a class of fractional-order chaotic systems with input saturation. International Journal of Fuzzy Systems 21(5): 1571–1584.

16.

Hosseinnia

Ghaderi

Mahmoudian

, et al. (2010) Sliding mode synchronization of an uncertain fractional order chaotic system. Computers & Mathematics with Applications 59(5): 1637–1643.

17.

Hovakimyan

Cao

(2010)

L_{1}

Adaptive Control Theory: Guaranteed Robustness with Fast Adaptation. New Delhi, India: SIAM.

18.

Huang

Cao

(2017) Active control strategy for synchronization and anti-synchronization of a fractional chaotic financial system. Physica A: Statistical Mechanics and Its Applications 473: 262–275.

19.

Luo

Zhang

, et al. (2020) Impulsive synchronization of fractional-order chaotic systems with actuator saturation and control gain error. IEEE Access 8: 36113–36119.

20.

Matignon

(1996) Stability results for fractional differential equations with applications to control processing. Computational Engineering in Systems Applications 2: 963–968.

21.

Mohammadzadeh

Ghaemi

(2017) Optimal synchronization of fractional-order chaotic systems subject to unknown fractional order, input nonlinearities and uncertain dynamic using type-2 fuzzy CMAC. Nonlinear Dynamics 88(4): 2993–3002.

22.

Pecora

Carroll

(1990) Synchronization in chaotic systems. Physical Review Letters 64(8): 821–824.

23.

Razminia

Baleanu

(2013) Complete synchronization of commensurate fractional order chaotic systems using sliding mode control. Mechatronics 23(7): 873–879.

24.

Shukla

Sharma

Azar

(2018) Control and synchronization of a fractional order hyperchaotic system via backstepping and active backstepping approach. In: Azar

Radwan

Vaidyanathan

(eds) Mathematical Techniques of Fractional Order Systems. Amsterdam: Elsevier, pp. 559–595.

25.

Soukkou

Boukabou

Goutas

(2018) Generalized fractional-order time-delayed feedback control and synchronization designs for a class of fractional-order chaotic systems. International Journal of General Systems 47(7): 679–713.

26.

Soukkou

Boukabou

Leulmi

(2016) Prediction-based feedback control and synchronization algorithm of fractional-order chaotic systems. Nonlinear Dynamics 85(4): 2183–2206.

27.

Tlelo-Cuautle

Pano-Azucena

Guilléen-Fernáandez

, et al. (2020) Analog/Digital Implementation of Fractional Order Chaotic Circuits and Applications. Cham: Springer.

28.

Vajpayee

Becerra

Bausch

, et al. (2021) L1-adaptive robust control design for a pressurized water-type nuclear power plant. IEEE Transactions on Nuclear Science 68: 1381–1398.

29.

Zhang

Liu

(2018) Robust adaptive control for fractional-order chaotic systems with system uncertainties and external disturbances. Advances in Difference Equations 2018(1): 412.

30.

Zhou

Wang

Liu

(2019) Generalized function projective synchronization of incommensurate fractional-order chaotic systems with inputs saturation. International Journal of Fuzzy Systems 21(3): 823–836.

Fuzzy L 1 adaptive controller for chaos synchronization of uncertain fractional-order chaotic systems with input nonlinearities

Abstract

Keywords

Introduction

Preliminary concepts

Problem formulation

Design of the FFO L 1 AC

Selection of a suitable sliding surface

Main results

Analysis of the fractional-order L 1 adaptive sliding mode controller

Closed-loop reference system

Transient and steady-state

Numerical simulation results

Example 1: fractional-order Duffing-Holmes synchronization

Example 2: fractional-order hybrid optical and Genesio–Tesi systems synchronization

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References

Design of the FFO $L_{1}$ AC

Analysis of the fractional-order $L_{1}$ adaptive sliding mode controller