Adaptive interval type-2 fuzzy sliding mode control for fractional-order systems based on finite-time scheme

Abstract

In this paper, the adaptive sliding mode control problem of fractional-order nonlinear systems in the presence of model uncertainties and external disturbances is considered. More particularly, an adaptive interval type-2 fuzzy sliding mode controller is designed for achieving finite-time stability of unknown fractional-order nonlinear systems. First, a nonsingular terminal sliding mode surface is designed to reduce chattering and improve the robustness and effectiveness of the controller. Then, the interval type-2 fuzzy logic is used to approximate the unknown parts appearing in the fractional-order nonlinear system model. Following these are augmented by a novel adaptive interval type-2 fuzzy sliding mode controller which can guarantee the stability of the target system in a finite time. In terms of Lyapunov approach, the adaptive laws are designed, meanwhile the proof of the corresponding finite-time property is presented. Finally, three examples are given to illustrate the effectiveness of the proposed strategy.

Keywords

Sliding mode control adaptive interval type-2 fuzzy controller fractional-order systems finite-time scheme

1 Introduction

Fractional calculus has more than 300 years of history, and in recent decades it has been one of the most important fields of sciences in engineering, mathematics and physics, among others, since fractional differential equations have been demonstrated that are more feasible than traditional integer counterparts [1 –4]. This is the reason why fractional-order (FO) systems, which are dynamic systems that involve fractional derivatives and integrals, have received much attention of researchers in the last years (see e.g. [5 –7]).

Recently, the stabilization and synchronization problems of FO systems have been widely studied in the specialized literature. Up to now, many control methods, such as active control [8], active sliding mode control [9], backstepping control [10], T-S fuzzy control [11], sliding mode control [12, 13], have been successfully applied to stabilize FO systems. However, taking into account that the requirements for systems increase in practice, as known, finite-time control of nonlinear systems gives rise to a high-precision performance as well as finite-time convergence to origin [14], therefore finite-time stability of FO systems has become a hot spot, there are many authors have considered the stability analysis and stabilization problems for FO systems based on finite-time scheme [15 –18].

In what concerns the community of sliding mode control (SMC), nonsingular terminal SMC has been widely investigated since it can achieve finite-time fast convergence property without causing any singularity problem in traditional SMC [19]. Over the past few years, some relevant results have been presented in this research area. For example, a nonsingular terminal sliding mode controller was designed for integer-order (IO) chaotic systems with uncertain parameters or disturbances in [20]. Furthermore, a novel FO terminal sliding mode controller was proposed for a class of IO chaotic systems in [21]. And for FO chaotic systems, nonsingular terminal SMC has been discussed in [15, 16].

On the other hand, since Zadeh initiated the fuzzy set theory [22], fuzzy logic control (FLC) scheme has been widely developed and successfully applied to many real world purposes [23 –26]. Thanks to the development of adaptive control theory, adaptive FLC schemes have been used to control FO systems recently [27 –31]. However, for dynamic unstructured environments, there is an urgent need to cope with large amount of uncertainties such as those in inputs to the FLC, in control outputs, linguistic uncertainties and uncertainties associated with noisy training data. Nowadays, it is shown that type-1 fuzzy systems have difficulties in modeling and minimizing the effect of uncertainties [32 –36]. One reason is that a type-1 fuzzy set is certain in the sense that the membership grade for a particular input is a crisp value. Thus, type-2 fuzzy sets, which are characterized by membership functions (MFs), were introduced in order to overcome the limitations of type-1 FLC [37]. For such fuzzy sets, each input has unity secondary membership grade defined by two type-1 MFs: upper MF and lower MF. Recently, owing to its high performance, type-2 FLC has been successfully applied to different engineering systems, such as image processing [38], pattern recognition [39], embedding intelligent agents [40], and mobile robots control [41]. Combing sliding mode control and adaptive type-2 FLC theories, for IO chaotic system, authors in [42] investigated the adaptive interval type-2 fuzzy sliding mode controller design problem, while for synchronization of FO chaotic systems, the adaptive interval type-2 fuzzy sliding mode control and the adaptive interval type-2 fuzzy active sliding mode control were discussed in [43] and [44], respectively. However, based on finite-time scheme, the adaptive interval type-2 fuzzy sliding mode control problem for FO nonlinear systems has never been considered.

Given this context, this paper investigates the stabilization problem of FO systems with model uncertainties and external disturbances based on the finite-time scheme. An adaptive interval type-2 fuzzy control, together with nonsingular terminal SMC, are combined for the design of a novel adaptive fuzzy sliding controller, which will guarantee the finite-time stability of the FO closed-loop system under those circumstances. Meanwhile, the proposed controller is able to reduce the chattering effect and improve robustness of the controlled system.

The rest of the paper is organized as follows. In Section 2, some important definitions and properties related to fractional derivative/integral are addressed. A brief description of interval type-2 fuzzy systems is presented in Section 3. The design of the proposed adaptive type-2 fuzzy sliding mode controller is discussed in Section 4. Numerical examples are given in Section 5 to show the applicability of the proposed strategy. Finally, conclusions are presented in Section 6.

2 Preliminaries of fractional calculus

This section gives some preliminaries of fractional calculus needed to understand the main contribution of this paper.

There are two commonly used definitions for the fractional operator D^α, namely Riemann-Liouville and Caputo definitions (see e.g. [2]). The Riemann-Liouville fractional integral of order q of a function f (t) on $ℝ^{+}$ is defined as

$\begin{matrix} D_{t}^{q} f (t) & = & I^{m - q} f^{(m)} (t) \\ = & \frac{1}{Γ (m - q)} \frac{d^{m}}{{dt}^{m}} \int_{0}^{t} \frac{f (τ)}{(t - τ)^{q + 1 - m}} d τ . \end{matrix}$ (1) where m is the first integer which is not less than q, i.e., m - 1 < q ≤ m and Γ (·) is the well-know Euler’s gamma function $Γ (s) = \int_{0}^{\infty} t^{s - 1} e^{- t} dt$ (2)

The Caputo fractional derivative of order q of a function f (t) on $ℝ^{+}$ is given by $D_{t}^{q} f (t) = {\begin{matrix} \frac{1}{Γ (m - q)} \int_{0}^{t} \frac{f^{(m)} (τ)}{(t - τ)^{q + 1 - m}} d τ, & m - 1 < q < m \\ \frac{d^{m} f (t)}{{dt}^{m}}, & q = m, \end{matrix}$ (3) where m is the first integer which is not less than q, i.e., m - 1 < q ≤ m.

The Laplace transform of the Riemann-Liouville fractional derivative is $L [D_{t}^{q} f (t)] = s^{q} L [f (t)] - \sum_{k = 0}^{n - 1} {s^{k} [\frac{d^{q - 1 - k} f (t)}{{dt}^{q - 1 - k}}]}_{t = 0}$ (4) where L denotes Laplace transform. It shows that the non-integer order derivative of the function at t = 0 is required in the Laplace transform of the Riemann-Liouville fractional derivative. However, this problem does not exist in the Laplace transform of the Caputo definition, which is given by $L [D_{t}^{q} f (t)] = s^{q} L [f (t)] - \sum_{k = 0}^{n - 1} s^{q - 1 - k} f^{(k)} (0)$ (5)

Thus, only integer-order derivative of a function appear in the Laplace transform of the Caputo derivative. Upon considering initial conditions as zero, Equation (5) reduces to $L [D_{t}^{q} f (t)] = s^{q} L [f (t)]$ (6)

On the other hand, the Riemann-Liouville definition is equivalent to a wide class of the function. However, initial conditions of a fractional differential equation (FDE) defined by the Caputo derivative are in the same form for IO derivatives which have well understood physical meaning. This is the reason why the Caputo definition of a fractional derivative is more popular than the Riemann-Liouville when modeling real world phenomena. Hence, the Caputo derivative will be considered in this work.

Now, some important and commonly used properties for the stability analysis of FO systems are listed below. The interested reader can find more details in [45].

Property 1. For Caputo derivative, we can obtain $_{t_{0}}^{C} D_{t}^{1 - α} (_{t_{0}}^{C} D_{t}^{α} f (t)) =_{t_{0}}^{C} D_{t}^{1} f (t) = \frac{df (t)}{dt}$ (7)

Property 2. Similarly to integer derivatives, the following equation holds for fractional derivatives $_{t_{0}}^{C} D_{t}^{1 - α} (_{t_{0}}^{C} D_{t}^{α - 1} f (t)) =_{t_{0}}^{C} D_{t}^{0} f (t) = f (t)$ (8)

Property 3. The following equality holds for the Caputo derivative $_{t_{0}}^{C} D_{t}^{α} (_{t_{0}}^{C} D_{t}^{m} f (t)) =_{t_{0}}^{C} D_{t}^{α + m} f (t)$ (9) where m = 0, 1, 2, …; n - 1 < α ≤ n; n ≥ 1 is a integer.

3 Interval type-2 fuzzy logic system

It is known that fuzzy logic systems (FLS) are universal approximators and have various applications in identification and controller design. A traditional type-1 fuzzy system is composed of four major parts: fuzzifier, rule base, inference engine and defuzzifier. While a type-2 fuzzy system has a similar structure, but one of the main differences arises from the rule base part, where a type-2 rule base has antecedents and consequents using type-2 fuzzy sets. Due to the complexity of the type reduction, the general type-2 FLS becomes computationally intensive. In order to make things simpler and easier to compute, secondary MFs of an interval type-2 FLS (hereafter referred to as IT2FLS) are all equal to the unity, which finally leads to a type reduction. The interval type-2 Gaussian MF with uncertain mean m ∈ [m₁, m₂] and a known standard deviation σ are shown in Fig. 1.

By using singleton fuzzification, the singleton inputs are fed into the inference engine. Combining the fuzzy “IF-THEN” rules, the inference engine maps the singleton input x = [x₁, x₂, ⋯ , x_n] into a type-2 fuzzy system set as the output. A typical form of an “IF-THEN” rule is as follows

R_i: IF x₁ is ${\tilde{F}}_{1}^{i}$ and x₂ is ${\tilde{F}}_{2}^{i}$ and ⋯ and x_n is ${\tilde{F}}_{n}^{i}$

THEN ${\tilde{Z}}^{i}$

where ${\tilde{F}}_{k}^{i}$ s are the antecedents (k = 1, 2, ⋯ , n), and ${\tilde{Z}}^{i}$ is the consequent of the ith rule.

First, the firing set for the ith rule is evaluated as follows $F^{i} (\underline{x}) = [{\underline{f}}^{i} (\underline{x}), {\bar{f}}^{i} (\underline{x})]$ (10) where

$\begin{matrix} {\underline{f}}^{i} (\underline{x}) & = & {\underline{μ}}_{{\tilde{F}}_{1}^{i}} (x_{1}) * {\underline{μ}}_{{\tilde{F}}_{2}^{i}} (x_{2}) * \dots * {\underline{μ}}_{{\tilde{F}}_{n}^{i}} (x_{n}) \\ = & \prod_{k = 1}^{n} {\underline{μ}}_{{\tilde{F}}_{k}^{i}} (x_{k}) \\ {\bar{f}}^{i} (\underline{x}) & = & {\bar{μ}}_{{\tilde{F}}_{1}^{i}} (x_{1}) * {\bar{μ}}_{{\tilde{F}}_{2}^{i}} (x_{2}) * \dots * {\bar{μ}}_{{\tilde{F}}_{n}^{i}} (x_{n}) \\ = & \prod_{k = 1}^{n} {\bar{μ}}_{{\tilde{F}}_{k}^{i}} (x_{k}) \end{matrix}$ (11) being the term ${\underline{μ}}_{{\tilde{F}}_{k}^{i}}$ and ${\bar{μ}}_{{\tilde{F}}_{k}^{i}}$ the lower and upper membership functions, respectively.

In the next step, the firing set $F^{i} (\underline{x})$ is combined with the ith consequent using the product t-norm to produce the type-2 output fuzzy set. The type-2 output fuzzy sets are fed into the type reduction part and the defuzzified crisp output forms an IT2FLS as the average of y_l and y_r as follows

$\begin{matrix} y (\underline{x}) & = & \frac{1}{2} [y_{l} (\underline{x}) + y_{r} (\underline{x})] \\ = & \frac{1}{2} [θ_{l}^{T} ξ_{l} (\underline{x}) + θ_{r}^{T} ξ_{r} (\underline{x})] = \frac{1}{2} θ^{T} ξ (\underline{x}) \end{matrix}$ (12) where $θ = [θ_{l}^{T} θ_{r}^{T}]^{T}$ and $ξ = [ξ_{l}^{T} ξ_{r}^{T}]^{T}$ , y_l and y_r are the left and right centroids of each rule consequent.

Then, the defuzzified output is evaluated by finding the solutions of the following optimization problem:

$\begin{matrix} y_{l} (\underline{x}) & = & min_{\forall f^{k} \in {{\underline{f}}^{k} {\bar{f}}^{k}}} (\frac{\sum_{k = 1}^{N} y_{l}^{k} f^{k} (\underline{x})}{\sum_{k = 1}^{N} f^{k} (\underline{x})}), \\ y_{r} (\underline{x}) & = & min_{\forall f^{k} \in {{\underline{f}}^{k} {\bar{f}}^{k}}} (\frac{\sum_{k = 1}^{N} y_{r}^{k} f^{k} (\underline{x})}{\sum_{k = 1}^{N} f^{k} (\underline{x})}) . \end{matrix}$ (13)

Defining $y_{l}^{k} (\underline{x})$ and $y_{r}^{k} (\underline{x})$ as the functions that fulfill the above-mentioned issues, and let $ξ_{l}^{k} (\underline{x}) = f_{l}^{k} (\underline{x}) / - \sum_{k = 1}^{M} f_{l}^{k} (\underline{x})$ , $ξ_{r}^{k} (\underline{x}) = f_{r}^{k} (\underline{x}) / - \sum_{k = 1}^{M} f_{r}^{k} (\underline{x})$ then the following equation can be obtained $y_{l} (\underline{x}) = \frac{\sum_{k = 1}^{N} y_{l}^{k} f_{l}^{k} (\underline{x})}{\sum_{k = 1}^{N} f_{l}^{k} (\underline{x})} = \sum_{k = 1}^{N} y_{l}^{k} ξ_{l}^{k} (\underline{x}) = θ_{l}^{T} ξ_{l} (\underline{x})$ (14) $y_{r} (\underline{x}) = \frac{\sum_{k = 1}^{N} y_{r}^{k} f_{r}^{k} (\underline{x})}{\sum_{k = 1}^{N} f_{r}^{k} (\underline{x})} = \sum_{k = 1}^{N} y_{r}^{k} ξ_{r}^{k} (\underline{x}) = θ_{r}^{T} ξ_{r} (\underline{x})$ (15) where $\begin{matrix} ξ_{l} (\underline{x}) & = & [ξ_{l}^{1} (\underline{x}) ξ_{l}^{2} (\underline{x}) \dots ξ_{l}^{N} (\underline{x})], \\ ξ_{r} (\underline{x}) & = & [ξ_{r}^{1} (\underline{x}) ξ_{r}^{2} (\underline{x}) \dots ξ_{r}^{N} (\underline{x})] \end{matrix}$ are the fuzzy basis functions; $\begin{matrix} θ_{l} (\underline{x}) & = & [θ_{l}^{1} (\underline{x}) θ_{l}^{2} (\underline{x}) \dots θ_{l}^{N} (\underline{x})], \\ θ_{r} (\underline{x}) & = & [θ_{r}^{1} (\underline{x}) θ_{r}^{2} (\underline{x}) \dots θ_{r}^{N} (\underline{x})] \end{matrix}$ are adjustable parameters.

Figure 2 shows an interval type-2 fuzzy neural network (IT2FNN) system, i.e., the implementation of interval type-2 FLS with some of their parameters and components presented by fuzzy logic terms [43]. As observed, the IT2FNN is a four layers structure. Layers I and II represent input nodes and type-2 fuzzification nodes, respectively, which form the antecedent part of the IT2FNN. Meanwhile, a classic 2-layer neural network with fuzzy rule nodes and output nodes is used to construct layers III and IV, respectively, which represent the consequent part of this IT2FNN.

4 Main result

This section is devoted to the design of a novel adaptive type-2 fuzzy sliding mode controller for fractional-order nonlinear systems.

Consider a fractional-order nonlinear system with the following form ${\begin{matrix} D^{α} x_{1} (t) = A_{1} x (t) + G_{1} (x, t) + F_{1} (x, t) + d_{1} (t) + u_{1} (t) \\ D^{α} x_{2} (t) = A_{2} x (t) + G_{2} (x, t) + F_{2} (x, t) + d_{2} (t) + u_{2} (t) \\ ⋮ \\ D^{α} x_{n} (t) = A_{n} x (t) + G_{n} (x, t) + F_{n} (x, t) + d_{n} (t) + u_{n} (t) \end{matrix}$ (16)

where x (t) = [x₁ (t) , x₂ (t) , ⋯ , x_n (t)] ^T is the state vector, $A_{i} \in ℝ^{1 \times n}$ is the ith line of the system matrix $A \in ℝ^{n \times n}$ , G_i (x, t) (i = 1, 2, ⋯ n) represent bounded nonlinear functions of the system, F_i (x, t) and d_i (t) (i = 1, 2, ⋯ n) are the model uncertainties and external disturbances respectively. u_i (t) (i = 1, 2, ⋯ n) are the control inputs which will be designed later.

Definition 1. [16] Consider the uncertain fractional-order nonlinear system (16), if there exists a constant T = T (x (0)) >0, such that $lim_{t \to T} ∥ x (t) ∥ = 0$ and ∥x (t) ∥ ≡ 0, if t ≥ T, then the states of the system (16) will converge to zero in the finite time T.

Lemma 1. [46] Considering the general equation of fractional-order dynamical system Σ: $D^{β} x (t) = g (x)$ where g (x) represents a smooth nonlinear vector field Rⁿ → Rⁿ, x = [x₁, x₂, ⋯ , x_n] ^T is state vector of the system Σ, β represents fractional order which satisfies 0 < β ≤ 1, and when β = 1, system Σ is an integer order system.

If there exists a positive definite Lyapunov function V (x) such that D^βV (x) <0, for all t ≥ t₀, where t₀ is the initial time. Then the trivial solution of system Σ is asymptotically stable.

The control objective of this paper is to stabilize the FO system (16) by an adaptive type-2 fuzzy sliding mode controller in the sense of Definition 1.

The design procedure of the proposed controller can be divided into three steps as follows:

Firstly, a novel nonsingular terminal sliding mode surface is introduced as: $s_{i} (t) = x_{i} (t) + \int (k_{1 i} x_{i} (t) + k_{2 i} sgn (x_{i} (t)) {| x_{i} (t) |}^{θ})$ (17) where x_i (t) (i = 1, 2, ⋯ n) is the state variable, k_1i , k_2i (i = 1, 2, ⋯ n) are positive constant parameters and θ ∈ (0, 1) is a constant.

Based on the SMC theory, the process can be classified into two phases with s (t) ≠0 and the sliding phase with s (t) =0. In order to guarantee that the trajectory of the system state vector x (t) will move from the approach phase to the sliding phase, sufficient condition to be satisfied once the state trajectory reaches the sliding surface is $s (t) = 0$ (18)

Taking account for Properties 2, 3 and Equations (17–18), we have $D^{α} x_{i} (t) = - D^{α - 1} (k_{1 i} x_{i} (t) + k_{2 i} sgn (x_{i} (t)) {| x_{i} (t) |}^{θ})$ (19)

Here, Equation (19) refers to the sliding mode dynamics proposed in this paper.

The next step is to design a sliding mode controller to force the state trajectories of the system to reach the sliding surface and remain on it for the subsequent time.

Theorem 1. If F_i (x, t) is known and free of external disturbance, i.e., d_i (t) =0, a controller which guarantees asymptotic stability of system (16) can be given by

$\begin{matrix} u_{i} (t) & = & - A_{i} x (t) - G_{i} (x, t) - F_{i} (x, t) \\ - D^{α - 1} (k_{1 i} x_{i} + k_{2 i} sgn (x_{i}) {| x_{i} |}^{θ}) \\ - [k_{3 i} s_{i} + k_{4 i} {| s_{i} |}^{δ} sgn (s_{i}) + η_{i} sgn (s_{i})] \end{matrix}$ (20) where k_1i, k_2i, k_3i, k_4i (i = 1, 2, ⋯ n) are positive constant parameters, δ ∈ (0, 1) is a constant and η_i is a positive constant.

Proof Choose a positive definite Lyapunov function as follows $V_{1} (t) = \frac{1}{2} s_{i}^{2} (t)$ (21)

Taking the derivative of V₁ (t) with respect to time t, we have

$\begin{matrix} D^{α} V_{1} (t) & \leq & s_{i} (t) [D^{α} s_{i} (t)] \\ = & s_{i} (t) [A_{i} x (t) + G_{i} (x, t) + F_{i} (x, t) \\ + u_{i} (t) + D^{α - 1} (k_{1 i} x_{i} + k_{2 i} sgn (x_{i}) {| x_{i} |}^{θ}] \\ = & - [k_{3 i} s_{i}^{2} + k_{4 i} {| s_{i} |}^{δ + 1} + η_{i} | s_{i} |] < 0 \end{matrix}$ (22)

Hence, according to Lemma 1, the state trajectories of the system (16) will converge to s (t) =0 asymptotically.

However, model uncertainty F_i (x, t) in the practical system may be unknown and external disturbance d_i (t) ≠0, so the ideal controller (20) can not be implemented.

Therefore, in the following procedure, we replace F_i (x, t) by the IT2FLS $F_{i} (x | {\underline{θ}}_{f})$ with the form of (10) as:

$\begin{matrix} F_{i} (x | \underline{θ}) & = & \frac{1}{2} ({\underline{θ}}_{r i}^{T} {\underline{ξ}}_{ri} (x_{i}) + {\underline{θ}}_{li}^{T} {\underline{ξ}}_{li} (x_{i})) \\ = & \frac{1}{2} {\underline{θ}}^{T} \underline{ξ} (x) \end{matrix}$ (23) where $\begin{matrix} \underline{θ} & = & [{\underline{θ}}_{l}^{T} {\underline{θ}}_{r}^{T}]^{T}, \underline{ξ} = [{\underline{ξ}}_{l}^{T} {\underline{ξ}}_{r}^{T}]^{T}, \\ {\underline{θ}}_{l} & = & [{\underline{θ}}_{l 1}, {\underline{θ}}_{l 2}, \dots, {\underline{θ}}_{li}], {\underline{θ}}_{r} = [{\underline{θ}}_{r 1}, {\underline{θ}}_{r 2}, \dots, {\underline{θ}}_{ri}], \\ {\underline{ξ}}_{l} & = & [{\underline{ξ}}_{l 1}, {\underline{ξ}}_{l 2}, \dots, {\underline{ξ}}_{li}], {\underline{ξ}}_{r} = [{\underline{ξ}}_{r 1}, {\underline{ξ}}_{r 2}, \dots, {\underline{ξ}}_{ri}] . \end{matrix}$

Here, the parameters ${\underline{ξ}}_{li} (x_{i}), {\underline{ξ}}_{ri} (x_{i}) (i = 1, 2, \dots, n)$ depend on the fuzzy MF and are supposed to be fixed, while ${\underline{θ}}_{l}^{T}, {\underline{θ}}_{r}^{T}$ are adjusted by adaptive laws based on Lyapunov stability criterion. Therefore, the resulting control effort can be obtained as

$\begin{matrix} u_{i} (t) & = & - A_{i} x (t) - G_{i} (x, t) - F_{i} (x | θ) \\ - D^{α - 1} (k_{1 i} x_{i} + k_{2 i} sgn (x_{i}) {| x_{i} |}^{θ}) \\ - [k_{3 i} s_{i} + k_{4 i} {| s_{i} |}^{δ} sgn (s_{i}) + η_{i} sgn (s_{i})] \end{matrix}$ (24)

Then, the approaching condition can be guaranteed for $η_{i} > | d_{i} | + | ω_{i} |$ (25) where ω_i is the minimum approximation error, and |d_i|, |ω_i| are assumed to be bounded.

Theorem 2. Considering FO nonlinear system (16), the control effort of the objective system is given by (24) and the fuzzy-based adaptive laws are designed as

$\begin{matrix} D^{α} {\tilde{\underline{θ}}}_{li} & = & - γ_{i} s_{i} (t) {\underline{ξ}}_{li} (x_{i}) \\ D^{α} {\tilde{\underline{θ}}}_{ri} & = & - ζ_{i} s_{i} (t) {\underline{ξ}}_{ri} (x_{i}) (i = 1, 2, \dots, n) \end{matrix}$ (26) where γ_i and ζ_i are positive scalars, with s_i (t) of the form of (17). Then, the overall adaptive scheme guarantees the global stability of the system (16) in the sense that all signals involved are uniformly bounded and the system states will asymptotically converge to zero.

Proof. Define the optimal parameters ${\underline{θ}}^{*}$ as ${\underline{θ}}^{*} = arg min_{θ \in Ω} [sup_{x \in Ω_{x}} | F (x | θ) - F (x, t) |]$ (27) where θ and x belong to the compact sets Ω and Ω_x, respectively. These sets are defined as $Ω = {\underline{θ} | ∥ \underline{θ} | ∥ \leq m}, Ω_{x} = {x | ∥ x ∥ \leq m_{x}},$ where m and m_x are finite positive constants.

Then, the following Lyapunov function candidate is chosen $V_{2} (t) = \frac{1}{2} s_{i}^{2} (t) + \frac{1}{4 γ_{i}} {\tilde{\underline{θ}}}_{li}^{2} + \frac{1}{4 ζ_{i}} {\tilde{\underline{θ}}}_{ri}^{2}$ (28)

Now, define the minimum approximation error as $ω_{i} = F_{i} (x, t) - F_{i} (x | {\underline{θ}}^{*}),$ (29) and adaptive parameters ${\tilde{\underline{θ}}}_{li}$ and ${\tilde{\underline{θ}}}_{ri}$ as ${\tilde{\underline{θ}}}_{li} = {\underline{θ}}_{li}^{*} - {\underline{θ}}_{li}, {\tilde{\underline{θ}}}_{r i} = {\underline{θ}}_{ri}^{*} - {\underline{θ}}_{ri} .$ (30)

Taking the derivative of V₂ (t) with respect to time, we have

$\begin{matrix} D^{α} V_{2} (t) & \leq & s_{i} (t) D^{α} s_{i} (t) + \frac{1}{2 γ_{i}} {\tilde{\underline{θ}}}_{li} D^{α} {\tilde{\underline{θ}}}_{li} + \frac{1}{2 ζ_{i}} {\tilde{\underline{θ}}}_{ri} D^{α} {\tilde{\underline{θ}}}_{ri} \\ = & s_{i} (t) [A_{i} x (t) + G_{i} (x, t) + F_{i} (x | θ) + d_{i} (t) \\ + u_{i} (t) + D^{α - 1} (k_{1 i} x_{i} + k_{2 i} sgn (x_{i}) {| x_{i} |}^{θ})] \\ + \frac{1}{2 γ_{i}} {\tilde{\underline{θ}}}_{li} D^{α} {\tilde{\underline{θ}}}_{li} + \frac{1}{2 ζ_{i}} {\tilde{\underline{θ}}}_{ri} D^{α} {\tilde{\underline{θ}}}_{ri} \\ = & s_{i} (t) [F_{i} (x, t) - F_{i} (x | θ) + d_{i} (t) \\ - (k_{3 i} s_{i} + k_{4 i} {| s_{i} |}^{δ} sgn (s_{i}) + η_{i} sgn (s_{i}))] \\ + \frac{1}{2 γ_{i}} {\tilde{\underline{θ}}}_{li} D^{α} {\tilde{\underline{θ}}}_{li} + \frac{1}{2 ζ_{i}} {\tilde{\underline{θ}}}_{ri} D^{α} {\tilde{\underline{θ}}}_{ri} \\ = & s_{i} (t) {\frac{1}{2} [({\underline{θ}}_{li}^{*} l - {\underline{θ}}_{li})^{T} {\underline{ξ}}_{li} (x_{i}) \\ + ({\underline{θ}}_{ri}^{*} - {\underline{θ}}_{ri})^{T} {\underline{ξ}}_{ri} (x_{i})] \\ - (k_{3 i} s_{i} + k_{4 i} {| s_{i} |}^{δ} sgn (s_{i})} \\ - (η_{i} | s_{i} (t) | - ω s_{i} (t) - d_{i} (t) s_{i} (t)) \\ + \frac{1}{2 γ_{i}} {\tilde{\underline{θ}}}_{li} D^{α} {\tilde{\underline{θ}}}_{li} + \frac{1}{2 ζ_{i}} {\tilde{\underline{θ}}}_{ri} D^{α} {\tilde{\underline{θ}}}_{ri} \\ = & \frac{1}{2} {\tilde{\underline{θ}}}_{li} [s_{i} (t) {\underline{ξ}}_{li} (x_{i}) + \frac{1}{γ_{i}} D^{α} {\tilde{\underline{θ}}}_{li}] \\ + \frac{1}{2} {\tilde{\underline{θ}}}_{ri} [s_{i} (t) {\underline{ξ}}_{ri} (x_{i}) + \frac{1}{ζ_{i}} D^{α} {\tilde{\underline{θ}}}_{ri}] \\ - (k_{3 i} s_{i}^{2} + k_{4 i} {| s_{i} |}^{δ + 1}) \\ - (η_{i} | s_{i} (t) | - ω_{i} s_{i} (t) - d_{i} (t) s_{i} (t)) \end{matrix}$ (31)

Substituting (26) into (31) and using (25), the following inequality can be obtained

$\begin{matrix} D^{α} V_{2} (t) & \leq & - (η_{i} | s_{i} (t) | - | d_{i} (t) | | s_{i} (t) | - | ω_{i} | | s_{i} (t) |) \\ = & - | s_{i} (t) | (η_{i} - | d_{i} (t) | - | ω_{i} |) < 0 \end{matrix}$ (32)

As a result, the FO nonlinear system (16) is stable, and the system states will converge to zero asymptotically in a finite time. This completes the proof.

Remark 1. In this paper, we consider the state-feedback-based sliding mode controller design problem for FO nonlinear systems. Actually, in practical application, most of the previous methods (including our case) have been developed under the assumption that the full state can be measured directly. If the full-state information is not available and only the output is accessible, it is meaningful to control a system via output feedback controllers. Based on our results, the output-feedback-based controller approach can be incorporated into the proposed method to ensure the stability of the closed-loop FO system, using the method proposed in [47 –49]. Via some matrix inequality linearization procedures combined with two approaches, namely, the convex linearization approach and iterative approach, are developed to solve the sliding-mode output feedback controller synthesis problem, which is our future work.

In the sequence, to show the finite time property of the proposed controller, the following theorem is given.

Theorem 3. The sliding mode dynamics (19) is asymptotically stable and its state trajectories converge to the equilibrium x (t) =0 in a finite time $T \leq \frac{1}{μ (1 - θ)} (ln ({∥ x (t_{r}) ∥}_{1}^{1 - θ} + 1)$ .

Proof. Choose a positive definite Lyapunov function as follows: $V_{3} (t) = {∥ x (t) ∥}_{1} = \sum_{i = 1}^{n} | x_{i} (t) |$ (33)

Taking the time derivative of V₃ (t), we have ${\dot{V}}_{3} (t) = \sum_{i = 1}^{n} [sgn (x_{i} (t)) {\dot{x}}_{i} (t)]$ (34)

According to Properties 2 and 3, we have ${\dot{V}}_{3} (t) = \sum_{i = 1}^{n} [sgn (x_{i} (t)) (D^{1 - α} (D^{α} x_{i} (t)))]$ (35)

Substituting D^αx_i (t) in (19) into the above equation, we obtain

$\begin{matrix} {\dot{V}}_{3} (t) & = & \sum_{i = 1}^{n} [sgn (x_{i} (t)) (D^{1 - α} (D^{α - 1} (- (k_{1 i} x_{i} (t) \\ + k_{2 i} sgn (x_{i} (t)) {| x_{i} (t) |}^{θ})))] \end{matrix}$ (36)

Based on Property 3, we have ${\dot{V}}_{3} (t) = - \sum_{i = 1}^{n} [(k_{1 i} | x_{i} (t) | + k_{2 i} {| x_{i} (t) |}^{θ})]$ (37)

Defining μ = min {k_1i, k_2i} , i = 1, 2, ⋯ , n, we have

$\begin{matrix} {\dot{V}}_{3} (t) & \leq & - μ \sum_{i = 1}^{n} [| x_{i} (t) | + {| x_{i} (t) |}^{θ}] \\ = & - μ (| x_{1} | + | x_{2} | + \dots + | x_{3} | + {| x_{1} |}^{θ} \\ + {| x_{2} |}^{θ} + \dots + {| x_{3} |}^{θ}) \\ \leq & - μ ({∥ x (t) ∥}_{1} + {∥ x (t) ∥}_{1}^{θ}) \leq 0 \end{matrix}$ (38)

Therefore, the states x_i (i = 1, 2, ⋯ , n) will converge to zero asymptotically in a finite time after the system states reach the sliding surface.

In the following, we show that the convergence occurs in a finite time.

From inequality (38), we have ${\dot{V}}_{3} (t) = \frac{d {∥ x (t) ∥}_{1}}{dt} \leq - μ ({∥ x (t) ∥}_{1} + {∥ x (t) ∥}_{1}^{θ})$ (39)

After simple calculations, we get

$\begin{matrix} dt & \leq & - \frac{d {∥ x (t) ∥}_{1}}{μ ({∥ x (t) ∥}_{1} + {∥ x (t) ∥}_{1}^{θ})} \\ = & \frac{{∥ x (t) ∥}_{1}^{- θ} d {∥ x (t) ∥}_{1}}{μ ({∥ x (t) ∥}_{1}^{1 - θ} + 1)} \\ = & \frac{d {∥ x (t) ∥}_{1}^{1 - θ}}{μ (1 - θ) ({∥ x (t) ∥}_{1}^{1 - θ} + 1)} \end{matrix}$ (40)

Taking integral of both sides of the above inequality from t_s to t_r, and knowing that x (t_s) =0, we get

$\begin{matrix} t_{s} - t_{r} & \leq & - \frac{1}{μ (1 - θ)} \sum_{t_{r}}^{t_{s}} \frac{d {∥ x (t) ∥}_{1}^{1 - θ}}{({∥ x (t) ∥}_{1}^{1 - θ} + 1)} \\ = & - \frac{1}{μ (1 - θ)} (ln {({∥ x (t) ∥}_{1}^{1 - θ} + 1)) |}_{x_{t_{r}}}^{x_{t_{s}}} \\ = & \frac{1}{μ (1 - θ)} (ln ({∥ x (t_{r}) ∥}_{1}^{1 - θ} + 1) \end{matrix}$ (41)

By the above inequality, it is clear that the state variables x_i (t) (i = 1, 2, ⋯ , n) will converge to zero asymptotically in a finite time $T \leq \frac{1}{μ (1 - θ)} (ln ({∥ x (t_{r}) ∥}_{1}^{1 - θ} + 1)$ once the system states reach the sliding surface. This completes the proof.

Remark 2. Comparisons with previous related works.

First, from the controller design method point of view, the adaptive interval type-2 fuzzy sliding mode controller design problem for IO chaotic systems was investigated by Hwang et al. [42]. While adaptive interval type-2 fuzzy sliding mode control for the synchronization problem of FO chaotic systems was solved in [43, 44]. However, until now there are seldom results considering finite-time synchronization and/or controller design problem using adaptive interval type-2 fuzzy sliding mode control for FO nonlinear systems, which is very important for the application of FO nonlinear systems.

Second, from the FO nonlinear system point of view, the synchronization and/or controller design results have been obtained via various kinds of methods. For example, Aghababa studied the fractional nonsingular terminal sliding mode control [15], the terminal sliding mode control [16], and robust finite-time stabilization for FO nonlinear systems [17], respectively. Lin et al. discussed the adaptive fuzzy sliding mode control for FO nonlinear systems in terms of general FLC [31]. In contrast to the previous works, our work combining the finite-time scheme and adaptive interval type-2 fuzzy sliding mode control technique, in terms of FO Lyapunov method, the effective controller design method is investigated.

5 Numerical example

In this section, three examples are presented to illustrated the efficiency and usefulness of the proposed adaptive interval type-2 fuzzy sliding mode controller to stabilize FO nonlinear dynamics.

Example 1. First, let us consider FO Lorenz chaotic system as follows. ${\begin{matrix} D^{α} x_{1} (t) = a (x_{2} - x_{1}) + F_{1} (x, t) + d_{1} (t) + u_{1} (t) \\ D^{α} x_{2} (t) = {cx}_{1} - x_{1} x_{3} - x_{2} + F_{2} (x, t) + d_{2} (t) + u_{2} (t) \\ D^{α} x_{3} (t) = x_{1} x_{2} - {bx}_{3} + F_{3} (x, t) + d_{3} (t) + u_{3} (t) \end{matrix}$ (42)

where α is the fractional differential order, x_i (t) (i = 1, 2, 3) are state variables, a, b, c are system parameters and u₁, u₂, u₃ are the control inputs. The uncertainties F_i (x, t) (i = 1, 2, 3) are supposed to be as follows ${\begin{matrix} F_{1} (x, t) = 0.1 cos (t) x_{1} \\ F_{2} (x, t) = - 0.05 sin (2 t) x_{2} \\ F_{3} (x, t) = 0.08 cos (3 t) x_{3} \end{matrix}$

Meanwhile, external disturbances d_i (t) (i = 1, 2, 3) are chosen as random Gaussian noise.

The initial conditions and parameters of FO chaotic system (42) are selected as $\begin{matrix} [x_{1} (0), x_{2} (0), x_{3} (0)]^{T} = [2, 5, 3]^{T}, α = 0.95 \\ δ = θ = 0.9, a = 10, b = 8 / 3, c = 28 \\ η_{1} = η_{2} = η_{3} = 1, [γ_{1}, γ_{2}, γ_{3}]^{T} = [2, 3, 2]^{T} \\ [ζ_{1}, ζ_{2}, ζ_{3}]^{T} = [2, 3, 2]^{T} \\ k_{1 i} = k_{2 i} = k_{3 i} = [2, 3, 5]^{T} \\ k_{4 i} = [k_{41}, k_{42}, k_{43}]^{T} = [10, 10, 10]^{T} \end{matrix}$

And the upper and lower MFs of $\underline{ξ}$ are defined as [50] $\begin{matrix} {\underline{ξ}}_{l 1} & = & 0.95 - \frac{0.925}{1 + e^{- \frac{(x_{1} + 4.5)}{8}}}, \\ {\underline{ξ}}_{r 1} & = & 0.95 - \frac{0.925}{1 + e^{- \frac{(x_{3} - 3.5)}{8}}} \\ {\underline{ξ}}_{l 2} & = & 0.025 + \frac{0.925}{1 + e^{- \frac{(x_{2} - 4.5)}{8}}}, \\ {\underline{ξ}}_{r 2} & = & 0.025 + \frac{0.925}{1 + e^{- \frac{(x_{1} - 3.5)}{8}}} \\ {\underline{ξ}}_{l 3} & = & 0.025 + \frac{0.925}{1 + e^{- \frac{(x_{3} + 4.5)}{8}}}, \\ {\underline{ξ}}_{r 3} & = & 0.95 - \frac{0.925}{1 + e^{- \frac{(x_{2} + 3.5)}{8}}} \end{matrix}$

We simulate the system in the Matlab/Simulink environment. The simulation results are shown in Figs. 3–9. Figures 3–4 display the phase trajectories and state trajectories of FO chaotic system (42) in open-loop, before adding control input (24). Figure 5 shows the controlled state trajectories of the system (42) after applying the designed controller (24). It can be observed that the system is asymptotically stable and the outputs are free of chattering. The evolution of the sliding surface for each state is shown in Fig. 6. Likewise, adaptive laws ${\underline{\tilde{θ}}}_{li}, {\underline{\tilde{θ}}}_{ri} (i = 1, 2, 3)$ are illustrated in Figs. 7 and 8, respectively.

Example 2. In this example, based on a chaotic non-autonomous electromechanical transducer model proposed in reference [51] which is schematically depicted in Fig. 9, its fractional-order model can be described by the following equation: ${\begin{matrix} D^{α} x_{1} (t) = x_{2} + F_{1} (x, t) + d_{1} (t) + u_{1} (t) \\ D^{α} x_{2} (t) = - λ_{1} x_{2} - x_{1} - γ x_{1}^{3} - β_{1} x_{4} \\ + E_{m} cos ω_{s} t + F_{2} (x, t) + d_{2} (t) + u_{2} (t) \\ D^{α} x_{3} (t) = x_{4} + F_{3} (x, t) + d_{3} (t) + u_{3} (t) \\ D^{α} x_{4} (t) = - λ_{2} x_{4} - ω^{2} x_{3} + β_{2} x_{2} \\ + F_{4} (x, t) + d_{4} (t) + u_{4} (t) \end{matrix}$ (43)

In this case, F₄ (x, t) = - sin(5t) x₄, the other model uncertainties and external disturbances are similar with ones in Example 1. Besides, initial parameters and other conditions of the system are randomly selected as $\begin{matrix} [x_{1} (0), x_{2} (0), x_{3} (0), x_{4} (0)]^{T} = [0.1, 0.15, 0.2, 0.25]^{T}, \\ α = 0.97, η_{i} = 1, γ_{i} = ζ_{i} = 2, \\ k_{1 i} = k_{2 i} = k_{3 i} = [2, 3, 5, 5]^{T}, \\ k_{4 i} = [10, 10, 10, 10]^{T} (i = 1, 2, 3, 4) . \end{matrix}$

The chaotic attractor of the nonlinear electromechanical system (43) obtained for λ₁ = 0.1,λ₂ = 0.3, γ = 1.32, β₁ = 0.01, β₂ = 0.06, ω = 1.2, ω_s = 1.3 and E_m = 22 is displayed onFig. 10.

Meanwhile, the upper and lower MFs of $\underline{ξ}$ can be defined as $\begin{matrix} {\underline{ξ}}_{l 1} & = & 0.95 - \frac{0.925}{1 + e^{- \frac{(x_{1} + 4.5)}{8}}}, \\ {\underline{ξ}}_{r 1} & = & 0.025 + \frac{0.925}{1 + e^{- \frac{(x_{4} + 3.5)}{8}}} \\ {\underline{ξ}}_{l 2} & = & 0.025 + \frac{0.925}{1 + e^{- \frac{(x_{2} - 4.5)}{8}}}, \\ {\underline{ξ}}_{r 2} & = & 0.025 + \frac{0.925}{1 + e^{- \frac{(x_{1} - 3.5)}{8}}} \\ {\underline{ξ}}_{l 3} & = & 0.025 + \frac{0.925}{1 + e^{- \frac{(x_{3} + 4.5)}{8}}}, \\ {\underline{ξ}}_{r 3} & = & 0.95 - \frac{0.925}{1 + e^{- \frac{(x_{2} + 3.5)}{8}}} \\ {\underline{ξ}}_{l 4} & = & 0.95 - \frac{0.925}{1 + e^{- \frac{(x_{4} - 4.5)}{8}}}, \\ {\underline{ξ}}_{r 4} & = & 0.95 - \frac{0.925}{1 + e^{- \frac{(x_{3} - 3.5)}{8}}} \end{matrix}$

Then, based on the controller Equation (24), the corresponding simulation results are shown in Figs. 11, 12.

Example 3. In this example, to show the advantages of our controller design method, the proposed adaptive type-2 fuzzy sliding mode controller is applied to stabilize an uncertain FO nonautonomous rotational mechanical system, which is investigated in [15].

Consider the following nonautonomous FO chaotic rotational mechanical system with model uncertainties, external disturbances, and control inputs: ${\begin{matrix} D^{α} x_{1} = x_{2} + F_{1} (x, t) + d_{1} (t) + u_{1} (t) \\ D^{α} x_{2} = 0.25 (x_{3} + 2.4)^{2} sin (x_{1} - 0.69) \\ * cos (x_{1} - 0.69) - sin (x_{1} - 0.69) \\ - 0.7 x_{2} + F_{2} (x, t) + d_{2} (t) + u_{2} (t) \\ D^{α} x_{3} = 2.8 cos (x_{1} - 0.69) - 1.942 \\ + F_{3} (x, t) + d_{3} (t) + u_{3} (t) \end{matrix}$ (44) where the following uncertainty terms are considered in the simulation: $\begin{matrix} F_{1} (x, t) + d_{1} (t) = - 0.15 sin (2 t) x_{1} + 0.15 sin (3 t) \\ F_{2} (x, t) + d_{2} (t) = 0.25 cos (4 t) x_{2} + 0.1 cos (t) \\ F_{3} (x, t) + d_{3} (t) = 0.2 sin (3 t) x_{3} + 0.2 cos (3 t) \end{matrix}$ and initial conditions of the system (44) are selected as x₁ (0) =0.1, x₂ (0) = -0.1, and x₃ (0) =0.2. The order α is chosen as 0.99, and other parameters are chosen as in Example 1.

The simulation results using our proposed adaptive type-2 fuzzy sliding mode control method is shown in Fig. 13, while the state trajectories of the chaotic system (44) controlled with the fractional nonsingular terminal sliding mode controller proposed in [15] was depicted in Fig. 4 of the reference [15], which showed that the state trajectories will convergent to zero about 4 s. However, from Fig. 13, we can see that in our case, the setting time is about 0.4 s, which shows the advantages of our proposed method.

6 Conclusions

In this paper, adaptive interval type-2 fuzzy sliding mode control has been proposed to deal with stabilization of a fractional-order nonlinear system with model uncertainties and external disturbances, i.e., of a system with unknown local structure. The considered strategy is able to reduce the chattering problem by applying both nonsingular terminal sliding mode control and interval type-2 fuzzy control. Moreover, the proposed controller not only guarantee the asymptotic stability of the fractional-order system but also can achieve it in a finite time, which fulfills the needs of the practical engineering. Simulation results, related to three different fractional-order chaotic systems, are given to show the effectiveness of the proposed stabilization scheme.

Footnotes

Acknowledgments

This work is partially supported by National Natural Science Foundation of China (Nos. 61203047, U1604146, 61473115), Science and Techno-logy Research Project in Henan Province (Nos. 152102210273, 162102410024) and China Scholarship Council (No. 201408410277).

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