Model reference type-2 fuzzy sliding mode control for a novel uncertain hyperchaotic system

Abstract

Hyperchaotic systems have more complex dynamics than the low-dimensional chaotic system. Since hyperchaotic systems have many applications, controlling these systems for engineering applications is a new and attractive field. This paper investigates a design scheme for a class of nonlinear systems with uncertainties and unknown disturbances. In this regard, the idea of classical model reference control, feedback linearization, sliding mode control technique and interval type-2 fuzzy systems (IT2FLS) are combined to suggest a novel hybrid control scheme to resolve the model reference control problem and address the tracking problem for the novel uncertain hyperchaotic Lü system. The IT2FLS is used to approximate the unknown nonlinear terms in control law. The interval type-2 fuzzy adaptation law adjusts the consequent parameters of the rules based on a Lyapunov synthesis approach. It is further equipped with a novel PI type switching structure to attenuate the chattering of the switching law resulting from the fast and large bounded unknown disturbances. There are three major contributions worthy of emphasis. Firstly, the dynamics of the system need not to be known and just the relative degree should be known in advance, which is more flexible in the real implementations. Secondly, it can be applied to a wide range of chaotic or hyperchaotic systems, which is its unique feature. Finally, having applied the proposed approach, both the transient and steady state behavior are improved. Using the Lyapunov theory, the stability of the proposed controller is proved. Compared to conventional sliding mode control and type-1 fuzzy controller, simulation results illustrate that although the proposed approach is highly effective in providing a good tracking performance even in the face of external disturbance and uncertainty; however, it is more computational compared to other mentioned methods, as expected.

Keywords

Hyperchaotic systems type-2 fuzzy systems Lü system PI type switching structure

1 Introduction

A chaotic system is a nonlinear deterministic system that displays complex and unpredictable behavior. Chaos can be found in many engineering systems such as electronic circuits, power converters, chemical systems, etc. [1, 2]. The main characteristic of a chaotic system is that it is very sensitivite to initial conditions. That is, small differences in the initial state can result in extraordinary differences in the system state [2]. So far, different techniques and methods have been proposed to address chaos control. For instance, OGY method [3], differential geometric method [4], inverse optimal control [5], adaptive control [6], backstepping design technique [7, 8] and intelligent control [9, 10]. Besides, many chaos control strategies have been proposed based on feedback control technologies as well as sliding mode control (SMC) [2 , 12]. Recently, many hyperchaotic systems such as the hyperchaotic Chen system [13], the hyperchaotic Lorenz system [14] and hyperchaotic Lü system [15] have been proposed and studied. Hyperchaotic system has more complex dynamics than the low-dimensional chaotic system. Thus, hyperchaotic system has a much wider application than the low dimensional chaotic system [16]. How to realize control and synchronization of hyperchaotic systems is an interesting and challenging work [17]. Recently, some authors have studied synchronizations of hyperchaotic systems [15 , 18]. However, it is worth pointing out that unlike the existing literature in this field, this work is not in the class of synchronization problems. The aim of this paper is to design an effective controller on the basis of the Lyapunov stability theory to directly control the hayperchaotic Lü system as a case study. Besides, most of the papers here mentioned such as [17] are mainly concerned with stabilization problem of the hyperchaotic systems. To the best of our knowledge, this is the first time that the tracking problem of this system using type-2 FNN is investigated thus making this the main novelty of this paper.

In most of the published papers [17, 19], it is often under the assumption of no external disturbance for the successful derivation of a controller. In practice, however, chaotic and hyperchaotic systems contain various types of uncertainties, including unmodeled dynamics, parameter variations, and external disturbances [20, 21]. Their presence may lead to serious degradation of system performance. Therefore, considering the existence of disturbance in chaotic systems is an important concept [6, 22]. Thus, adaptive control techniques such as adaptive neural network [23] may be considered to enhance control performance. On the other hand, model reference adaptive control is one of the most important schemes of adaptive control theory. It has been successfully used to solve many control tasks. However, one shortcoming is that it requires the controlled process to have a known structure which may not be realistic in many cases [24]. Another shortcoming is that the transient responses are usually not good. Motivated by the aforementioned researches and to overcome these issues, this paper presents a model reference adaptive control based on sliding mode for controlling a class of hyperchaotic systems with uncertainties and external disturbance. In addition, we deal with the proposed method under the matching condition; That is, when uncertain terms enter the system at the same point as the control input [25].

So far, fuzzy logic systems (FLS) have been widely used to design model-free controllers. It has been proven that fuzzy-neural network (FNN) can approximate any nonlinear function to any desired accuracy based on universal approximation theory [26]. In recent years, some approaches for controlling chaos systems based on fuzzy systems have been proposed [27, 28]. Although, type-1 FNN (T1FNN) controllers have been successfully employed to control different systems [29], for many real-world applications and dynamic unstructured environments, there is a need to cope with large amount uncertainties such as linguistic uncertainties and uncertainties associated with the use of noisy training data. The conventional type-1 FNN control using type-1 fuzzy sets cannot directly handle such uncertainties [30]. A type-2 FNN (T2FNN) control using type-2 fuzzy sets can deal with such uncertainties to obtain a better performance. The most frequently used T2FLS are interval T2FLS (IT2FLS) for their reduced computational cost [31]. Though the T1FLS is the most widely used application of fuzzy set theory, the T2FLS have been used in a few control applications such as nonlinear control and mobile robot navigation [32], decision making [33], robotic control [34] and sliding mode control design [35]. Recently, type-2 FNN has also been employed for controlling chaotic systems [21 , 36]. It is interesting to note that one of the main concerns with using type-1 fuzzy controllers is the existence of approximation error [37, 38]. To reduce the approximation error, the control designers have to increase the number of rules in the type-1 fuzzy systems [38, 39]. However, this causes drawbacks, such as the complexity of control law, computational burden and imprecise results. To overcome this problem, in the proposed controller IT2FNN is employed.

As mentioned earlier, the dynamics of a chaotic or hyperchaotic system are highly time varying and nonlinear. Motivated from published papers, the objective of this paper is to prepare a novel controller by combining the merit of model reference adaptive control, sliding mode and type-2 FNN to address the tracking problem of the hyperchaotic Lü system. One of the advantages of the proposed approach is that it is equipped with a PI type switching structure to attenuate the chattering of the switching law resulting from the fast and large bounded unknown disturbances. As a result, the chattering phenomenon, due to this control law is attenuated. Consequently, the goal of tracking a reference signal is achieved by considering uncertainties and unknown disturbance without chattering, as a major novelty of this paper. The proposed approach can be applied to any system in the form of (3) which is its unique feature. Using the Lyapunov theory, the stability of the proposed adaptive controller is proven. Theoretical analysis and numerical simulation verify the effectiveness of the proposed method.

The rest of the paper is organized as follows: Section 2 describes system description. Section 3 presents Problem formulation. Section 4 develops the proposed method. In Section 5 the stability analysis is given. In Section 6 benefits of the proposed approach are given. Section 7 presents the simulation results and finally, Section 8 concludes the paper.

2 System description

By using a feedback controller, a novel hyperchaotic Lü system was constructed based on the original three dimensional Lü system which is given by the following Equations [15 –17]: $\begin{matrix} \dot{x} & = & a (y_{1} - x) + w \\ {\dot{y}}_{1} & = & - xz + {cy}_{1} \\ \dot{z} & = & {xy}_{1} - bz \\ \dot{w} & = & xz + lw \end{matrix}$ (1) where x; y₁; z and w are state variables and a; b; c and l are real constant parameters. When a = 36, b = 3, c = 20, - 1.03 ≤ l ≤ -0.46, system (1) has a periodic orbit; when a = 36, b = 3, c = 20, - 0.46 ≤ l ≤ -0.35, system (1) has a chaotic attractor; When a = 36, b = 3, c = 20, - 0.35 ≤ l ≤ -1.3, system (1) exhibits hyperchaotic behavior (Fig. 1).

Taking the consideration of system uncertainties and control input (u), the system (1) can be expressed by $\begin{matrix} \dot{x} & = & a (y_{1} - x) + w \\ {\dot{y}}_{1} & = & - xz + {cy}_{1} \\ + Δ f (x, y_{1}, z, w) + d (t) + u \\ \dot{z} & = & {xy}_{1} - bz \\ \dot{w} & = & xz + lw \end{matrix}$ (2) where d (t) is external disturbance with an unknown constant upper bound and Δf (x, y₁, z, w) is an uncertainty term representing the unmodeled dynamics or structural variation of the system. In general, the uncertain term Δf (x, y₁, z, w) is assumed to be bounded as |Δf (x, y₁, z, w) | ≤ α. In the meantime α is positive constant. In addition, d (t) is matched disturbance.

3 Problem formulation, model reference sliding mode control

The control objective is to design a controller for system (2) such that the output follows a desired reference signal y_r, while all signals in the closed-loop system are bounded. To begin with, suppose that the relative degree of the system (2) is m. Then, for control purpose the state equations of system (2) can be rewritten as $y^{(m)} = f (x, t) + g (x, t) u + D (x, t)$ (3) where D (x, t) includes external disturbance and unmodeled dynamics. In the meantime, y is the state of the system to be controlled (output), x = [x y₁ z w] is the state of the system which is assumed to be available for measurement, f (x, t) and g (x, t) are unknown but bounded functions. For designing the model reference sliding control, there are two basic steps. First, we should define a model reference and sliding surface (Equations 4-5) and second, determine the control law (Equation 14). In this regard, a stable linear model reference can be defined as follows:

$\begin{matrix} p_{m} (s) & = & \frac{Y_{r} (s)}{R (s)} \\ = & \frac{a_{0}}{s^{m} + a_{m - 1} s^{m - 1} + a_{m - 2} s^{m - 2} + \dots + a_{0}} \end{matrix}$ (4) where y_ris output, r is input and a_i (i = 0, 1, m - 1) are positive constants to be chosen such that the desired model reference be stable. At this point it should be noted that the degree of model reference should be identical to relative degree of the system to be controlled. Now, the sliding surface is defined as

$\begin{matrix} s & = & e^{(m - 1)} + a_{m - 1} e^{(m - 2)} + a_{m - 2} e^{(m - 3)} + \dots \\ + a_{1} e + a_{0} \int_{0}^{t} e dt \end{matrix}$ (5)

Taking the derivative of the sliding with respective to time, we get

$\begin{matrix} \dot{s} & = & e^{(m)} + a_{m - 1} e^{(m - 1)} + a_{m - 2} e^{(m - 2)} + \dots \\ + a_{1} \dot{e} + a_{0} e \end{matrix}$ (6)

One can obtain

$\begin{matrix} \dot{s} & = & y^{(m)} + a_{m - 1} y^{(m - 1)} + a_{m - 2} y^{(m - 2)} + \dots \\ + a_{1} \dot{y} + a_{0} y - (y_{r}^{(m)} + a_{m - 1} y_{r}^{(m - 1)} \\ + a_{m - 2} y_{r}^{(m - 2)} + \dots + a_{1} {\dot{y}}_{r} + a_{0} y_{r}) \end{matrix}$ (7) where a_i (i = 0, 1, m - 1)are the same as that (5) and e = y - y_r.

Notice that

$\begin{matrix} y_{r}^{(m)} + a_{m - 1} y_{r}^{(m - 1)} + a_{m - 2} y_{r}^{(m - 2)} \\ + \dots + a_{1} {\dot{y}}_{r} + a_{0} y_{r} = a_{0} r \end{matrix}$ (8)

Considering (7) and (8), after a some manipulation, we have

$\begin{matrix} \dot{s} & = & f (x, t) + g (x, t) u + a_{m - 1} y^{(m - 1)} \\ + a_{m - 2} y^{(m - 2)} + \dots + a_{1} \dot{y} + a_{0} y - a_{0} r \\ + D (x, t) \end{matrix}$ (9)

For simplicity let F = [a₀ a₁ … a_m-1] and $Z = [y \dot{y} \dots y^{(m - 1)}]^{T}$ . Thus $\dot{s} = f (x, t) + g (x, t) u + FZ - a_{0} r + D (x, t)$ (10)

At this point, we have to make an assumption.

Assumption 1. we assume that Z vector is measurable.

In this paper, Lyapunov’s direct method is used to derive an appropriate control law that can force and keep the error trajectory on the sliding surface such that the error will asymptotically reach zero. To achieve this goal the Lyapunov function is defined as $V (s) = \frac{1}{2} s^{2}$ (11)

In order to guarantee that the trajectory of the state error vector will translate from the approaching phase (s ≠ 0) to the sliding phase (s = 0), the sufficient condition $\dot{V} (s) = s \dot{s} \leq - η | s |, η > 0$ (12) must be satisfied.

Consider the control problem of the system (3), if f (x, t) and g (x, t) are known in advance and free of external disturbance, i.e., d (t) =0, then, the equivalent control can be obtained as $u_{eq} = \frac{1}{g (x, t)} (- f (x, t) - FZ + a_{0} r)$ (13)

It can be noted that in order to satisfy the sliding condition, a switching control term u_s is added in the overall control action. As a result the complete sliding mode control may be expressed as $u = u_{eq} + u_{s}, u_{s} = - g (x, t)^{- 1} η_{Δ} sgn (s)$ (14) where η_Δ ≥ η > 0 . To implement the sliding mode control (14), the system functions f (x, t) , g (x, t) and the switching parameter η_Δ must be known in advance. However, f (x, t) and g (x, t) are usually unknown in practice and it is difficult to apply the control law (14) for an unknown nonlinear plant. Moreover, the switching control term u_s will cause chattering problem. To overcome the aforementioned difficulties, the purpose of this paper is to propose a novel approach including the interval T2FNN to approximate the nonlinear functions f (x, t) and g (x, t) and a PI type switching structure to attenuate the chattering of the switching law caused by u_s. As a result the control purpose is achieved.

4 Proposed scheme

As mentioned earlier, since f (x, t) and g (x, t) are usually unknown, the ideal controller (14) cannot be implemented. However, we have the fuzzy IF-THEN rules (16) that describe the input-output behavior of f (x, t) and g (x, t). Therefore, a reasonable idea is to replace the f (x, t) and g (x, t) in (14) by two fuzzy systems, which are constructed from the rules (16) [38]. Thus, our objective is to use an interval type-2 fuzzy neural network (IT2FNN) to approximate the nonlinear functions f (x, t) and g (x, t) to develop adaptive laws to adjust the parameters of the FNN in order to attenuate the approximation error and external disturbance. Therefore, the unknown nonlinear functions f (x, t) and g (x, t) can be approximated by the fuzzy-neural approximates $\hat{f} (x | θ_{f}, t)$ and $\hat{g} (x | θ_{g}, t)$ , respectively. In the meantime, θ_f and θ_g are adjustable vectors of parameters, to improve accuracy. To save apace, regarding to Equation (20) in [40], one has $\hat{f} (x | θ_{f}, t) = θ_{f}^{T} ξ_{f} = 0.5 θ_{r, f}^{T} ξ_{f} + 0.5 θ_{l, f}^{T} ξ_{f}$ (15) $\hat{g} (x | θ_{g}, t) = θ_{g}^{T} ξ_{g} = 0.5 θ_{r, g}^{T} ξ_{g} + 0.5 θ_{l, g}^{T} ξ_{g}$ (16)

In addition, to derive adaptive laws the optimal parameters are defined as $θ_{f}^{*} = arg min_{θ_{f} \in Ω_{f}} (sup_{x \in R^{n}} | \hat{f} (x | θ_{f}, t) - f (x, t) |$ (17) $θ_{g}^{*} = arg min_{θ g \in Ω_{g}} (sup_{x \in R^{n}} | \hat{f} (x | θ_{g}, t) - f (x, t) |$ (18) where Ω_f and Ω_g are compact sets of suitable bounds on θ_f and θ_g, respectively. In the meantime Ω_f ={ θ_f ∈ Rⁿ||θ_f| < M_f } and Ω_g ={ θ_g ∈ Rⁿ||θ_g| < M_g }, in which, M_f and M_g are pre-specified parameters for estimated parameters’ bound. In this regard, to construct $\hat{f} (x | θ_{f}, t)$ and $\hat{g} (x | θ_{g}, t)$ suppose that they are the outputs of two adaptive IT2FNN in the normalized form with the inputs of x₁ and x₂. If three fuzzy sets are given to each fuzzy input, the whole control space will be covered by nine fuzzy rules. The linguistic fuzzy rules are proposed as

$\begin{matrix} R^{i} : if x_{1} is {\tilde{X}}_{1}^{i} and x_{2} is {\tilde{X}}_{2}^{i} then \\ y^{i} is [y_{l}^{i} y_{r}^{i}] \end{matrix}$ (19)

where Rⁱ denotes the ith fuzzy rule for i = 1, …, 9 . In the ith rule, ${\tilde{X}}_{1}^{i}$ and ${\tilde{X}}_{2}^{i}$ are type-2 fuzzy membership functions belonging to the fuzzy variables x₁ and x₂, respectively. Three Gaussian membership functions with uncertain variance, $μ_{{\tilde{X}}_{1}^{i}} (x_{1})$ named as Positive (P), Zero (Z), and Negative (N) are defined by trial and error method for input x₁ as shown in Fig. 2. Three Gaussian membership functions with uncertain variance, $μ_{{\tilde{X}}_{2}^{i}} (x_{2})$ , named as P, Z, and N in the same shape as Fig. 2, are used for input x₂. In this paper, the consequent of the rules namely, $[y_{l}^{i} y_{r}^{i}] (i = 1, 2, \dots, 9)$ are adaptively tuned using some adaptive rules afterward. Using the input scaling factors k₁ and k₂ to scale x₁ and x₂, we have x₁ = k₁e and x₂ = k₂e.

Therefore, control law (14) can be rewritten in of the form

$\begin{matrix} u & = & \frac{1}{\hat{g} (x | θ_{g}, t)} (- \hat{f} (x | θ_{f}, t) - FZ + a_{0} r \\ - η_{Δ} sgn (s)) \end{matrix}$ (20)

It is worthy to note that, there is a difficulty to determine the switching feedback control gain (η_Δ) of a sliding mode controller to achieve desired performance in which a trial and error method is commonly used [41]. Since the sliding control law (20) is discontinuous across the sliding surface and can lead to chattering, a novel PI type switching structure to attenuate the chattering of the switching law resulting from the fast and large bounded unknown disturbances is proposed as:

$\begin{matrix} u_{p} & = & \frac{k_{1}}{1 + | s |} \int_{0}^{t} s dt + k_{2} s = [k_{1} k_{2}] [\begin{matrix} \frac{1}{1 + | s |} \int_{0}^{t} s dt \\ s \end{matrix}] \\ = & ϑ^{T} φ (s) = ϑ_{1}^{T} φ_{1} (s) + ϑ_{2}^{T} φ_{2} (s) \end{matrix}$ (21) where k₁ and k₂ are adaptively tuned. In the meantime, we have ϑ^T = [k₁ k₂] and $φ (s)^{T} = [φ_{1} (s) φ_{2} (s)] = [\begin{matrix} \frac{1}{1 + | s |} \int_{0}^{t} s dt & s \end{matrix}]$ , $ϑ_{1}^{T} = k_{1}, ϑ_{2}^{T} = k_{2} .$ It should be noted that when the system is operating on the sliding surface the second term on the right hand side of Equation (21) is zero. Moreover, when the system is far from sliding surface the first term is small and as a result it doesn’t degrade the transient performance very much. To tradeoff between the fast tracking time and reducing undesirable chattering according the error state trajectories and considering the external disturbance, the reaching-type control u_r (s)is defined as $u_{r} (s) = {\begin{matrix} ϑ^{T} φ (s) & if | s | \leq η_{r} \\ η_{Δ} sgn (s) & if | s | > η_{r} \end{matrix}$ (22) where η_Δ is a suitable sliding boundary to be determined.

Therefore, by incorporating the switching law (22), resulting control law can be expressed as

$\begin{matrix} u & = & \frac{1}{\hat{g} (x | θ_{g}, t)} (- \hat{f} (x | θ_{f}, t) \\ - FZ + a_{0} r - u_{r} (s)) \end{matrix}$ (23)

5 Stability analysis

To begin with, by adding and subtracting the term $\hat{g} (x, t) u$ , Equation (10) can be rewritten as

$\begin{matrix} \dot{s} & = & f (x, t) + g (x, t) u + FZ - a_{0} r \\ + \hat{g} (x, t) u - \hat{g} (x, t) u + D (x, t) \end{matrix}$ (24)

Substituting (23) into (24), yields

$\begin{matrix} \dot{s} & = & (f (x, t) - \hat{f} (x | θ_{f}, t)) + (g (x, t) \\ - \hat{g} (x | θ_{g}, t)) u - u_{r} (s) \\ + D (x, t) \end{matrix}$ (25)

Approximation error can be defined as:

$\begin{matrix} w & = & (f (x, t) - \hat{f} (x | θ_{f}^{*}, t)) + (g (x, t) \\ - \hat{g} (x | θ_{g}^{*}, t)) u + D (x, t) \end{matrix}$ (26)

Therefore, (25) can be rewritten as

$\begin{matrix} \dot{s} & = & (\hat{f} (x | θ_{f}^{*}, t) - \hat{f} (x | θ_{f}, t)) + ((\hat{g} (x | θ_{g}^{*}, t) \\ - \hat{g} (x | θ_{g}, t)) u + (u_{r}^{*} (s) - u_{r} (s)) + w \end{matrix}$ (27)

Substituting (15–16) into (27), we have

$\begin{matrix} \dot{s} & = & (0.5 {\tilde{θ}}_{r, f}^{T} ξ_{f} + 0.5 {\tilde{θ}}_{l, f}^{T} ξ_{f}) + (0.5 {\tilde{θ}}_{r, g}^{T} ξ_{g} \\ + 0.5 {\tilde{θ}}_{l, g}^{T} ξ_{g}) u + {\tilde{ϑ}}_{1}^{T} φ_{1} (s) + {\tilde{ϑ}}_{2}^{T} φ_{2} (s) + w \end{matrix}$ (28) where ${\tilde{θ}}_{r, f}^{T} = {θ_{r, f}^{*}}^{T} - θ_{r, f}^{T}, {\tilde{θ}}_{l, f}^{T} = {θ_{l, f}^{*}}^{T} - θ_{l, f}^{T}, {\tilde{θ}}_{r, g}^{T} = {θ_{r, g}^{*}}^{T} - θ_{r, g}^{T}, {\tilde{θ}}_{l, g}^{T} = {θ_{l, g}^{*}}^{T} - θ_{l, g}^{T}, {\tilde{ϑ}}_{1} = ϑ_{1}^{*} - {\hat{ϑ}}_{1}$ and ${\tilde{ϑ}}_{2} = ϑ_{2}^{*} - {\hat{ϑ}}_{2}$ . In the meantime, $θ_{r, f}^{*}$ is the optimal vector of θ_r,f, $θ_{l, f}^{*}$ is the optimal vector of θ_l,f, $θ_{r, g}^{*}$ is the optimal vector of θ_l,g, $θ_{l, g}^{*}$ is the optimal vector of θ_l,g, ϑ^* is the optimal vector of $\hat{ϑ}$ , $u_{r}^{*} (s) = η_{Δ} sgn (s)$ .

In order to analyze the closed-loop stability, the Lyapunov function candidate is chosen as

$\begin{matrix} V & = & \frac{1}{2} (s^{2} + \frac{1}{2 γ_{1}} {\tilde{θ}}_{r, f}^{T} {\tilde{θ}}_{r, f} + \frac{1}{2 γ_{2}} {\tilde{θ}}_{l, f}^{T} {\tilde{θ}}_{l, f} \\ + \frac{1}{2 γ_{3}} {\tilde{θ}}_{r, g}^{T} {\tilde{θ}}_{r, g} + \frac{1}{2 γ_{4}} {\tilde{θ}}_{l, g}^{T} {\tilde{θ}}_{l, g} + \frac{1}{γ_{5}} {\tilde{ϑ}}_{1}^{2} + \frac{1}{γ_{6}} {\tilde{ϑ}}_{2}^{2}) \end{matrix}$ (29) where γ₁, γ₂, γ₃, γ₄, γ₅ and γ₆ are positive constants. Note that the positive definite function V in (29) is chosen in order to satisfy the control goal. Here, the goal is e → 0 by tuning the parameters $θ_{r, f} \to θ_{r, f}^{*}, θ_{l, f} \to θ_{l, f}^{*}, θ_{r, g} \to θ_{r, g}^{*}, θ_{l, g} \to θ_{l, g}^{*}, {\hat{ϑ}}_{1} \to ϑ_{1}^{*}$ , and ${\hat{ϑ}}_{2} \to ϑ_{2}^{*}$ . Therefore, a quadratic function of variables $s, {\tilde{θ}}_{r, f}, {\tilde{θ}}_{l, f}, {\tilde{θ}}_{l, g}, {\tilde{θ}}_{r, g}, {\tilde{ϑ}}_{1}, {\tilde{ϑ}}_{2}$ are defined as the positive definite. If $\dot{V} < 0$ then e → 0.

Taking the derivative of Equation (29) with respect to time, we get

$\begin{matrix} \dot{V} = s \dot{s} + \frac{1}{2 γ_{1}} {\tilde{θ}}_{r, f}^{T} {\dot{\tilde{θ}}}_{r, f} + \frac{1}{2 γ_{2}} {\tilde{θ}}_{l, f}^{T} {\dot{\tilde{θ}}}_{l, f} + \frac{1}{2 γ_{3}} {\tilde{θ}}_{r, g}^{T} {\dot{\tilde{θ}}}_{r, g} \\ + \frac{1}{2 γ_{4}} {\tilde{θ}}_{l, g}^{T} {\dot{\tilde{θ}}}_{l, g} + \frac{1}{γ_{5}} {\tilde{ϑ}}_{1} {\tilde{ϑ}}_{1} + \frac{1}{γ_{6}} {\tilde{ϑ}}_{2} {\dot{\tilde{ϑ}}}_{2} \end{matrix}$ (30)

Substituting (28) into (30) and after a some simple manipulation, one has

$\begin{array}{l} \dot{V} = {\tilde{θ}}_{r, f}^{T} (\frac{1}{2} s ξ_{f} + \frac{1}{2 γ_{1}} {\dot{\tilde{θ}}}_{r, f}) + {\tilde{θ}}_{l, f}^{T} (\frac{1}{2} s ξ_{f} \\ + \frac{1}{2 γ_{2}} {\dot{\tilde{θ}}}_{l, f}) + {\tilde{θ}}_{r, g}^{T} (\frac{1}{2} s ξ_{g} + \frac{1}{2 γ_{3}} {\dot{\tilde{θ}}}_{r, g}) \\ + {\tilde{θ}}_{l, g}^{T} (\frac{1}{2} s ξ_{g} + \frac{1}{2 γ_{4}} {\dot{\tilde{θ}}}_{l, g}) + {\tilde{ϑ}}_{1} (s ϕ_{1} (s) \\ + \frac{1}{γ_{5}} {\dot{\tilde{ϑ}}}_{1}) + {\tilde{ϑ}}_{2} (s ϕ_{2} (s) + \frac{1}{γ_{6}} {\dot{\tilde{ϑ}}}_{2}) \\ + s w - s u_{r}^{*} (s) \end{array}$ (31)

The adaptation laws can be chosen as ${\dot{θ}}_{r, f} = γ_{1} s ξ_{f}$ (32) ${\dot{θ}}_{l, f} = γ_{2} s ξ_{f}$ (33) ${\dot{θ}}_{r, g} = γ_{3} s ξ_{g} u$ (34) ${\dot{θ}}_{l, g} = γ_{4} s ξ_{g} u$ (35) ${\dot{\tilde{ϑ}}}_{1} = γ_{5} s ϕ_{1} (s)$ (36) ${\dot{\tilde{ϑ}}}_{2} = γ_{6} s ϕ_{2} (s)$ (37)

Substituting (32–37) into (31), one has $\dot{V} = sw - s η_{Δ} sgn (s) = sw - η_{Δ} | s |$ (38)

To prove the stability of control system, it is required that $\dot{V} \leq sw - η_{Δ} | s | \leq 0$ (39)

From the universal approximation theorem, it can be expected that the term sw should be very small in the adaptive fuzzy system. Therefore, all signals in the system are bounded. Using Barbalat’s theorem, we have $lim_{t \to \infty} | e (t) | = 0$ .

6 Benefits of the proposed controller

In some cases, the fuzzy type-1 systems will encounter limitation in the presence of uncertainty. Here, the idea of fuzzy type-2 FNN will give a more effective approach to capture uncertainty effects. In this paper, we tried to make use of the positive characteristics of the type-2 FNN to overcome uncertainty effects in the proposed controller. However, using type-2 FNN is computationally more complicated than using type-1 fuzzy systems. It is the price one must pay for achieving better performance in the face of uncertainties [31].

System (2) is not restrictive. Several nonlinear chaotic and hyperchaotic systems can be transformed into the controllable canonical form (3) [6, 42]. As a result, the proposed approach can easily be applied to a wide range of chaotic and hyperchaotic systems.

In many exiting studies such as [19, 42] external disturbances have been ignored. They have only considered stabilization of equilibrium points while in this study disturbances have been included and the tracking problem handled. Moreover, in many existing works on chaos control such as [42, 43], g in (1) is considered as an identity matrix. However, as it seen in simulation section this may not always be the case.

To clarify the proposed control scheme, a block diagram of the control system is depicted in Fig. 3. In addition, to summarize the above analysis, the design algorithm for the proposed approach is as follows:

Step 1. Define the membership function type-1 and type-2 membership functions (MFs) for x_i (i = 1, 2).

Step 2. Specify the desired coefficients γ₁, γ₂, γ₃, γ₄, γ₅, γ₆ and also specify all appropriate adaptation parameters to obtain the adaptive laws in (32–37).

Step 3. Obtain the control law in (23) and then apply it to plant.

7 Simulation results

This section of the paper presents an illustrative example to verify and demonstrate the effectiveness of the proposed control scheme. The simulation results are carried out using the MATLAB software. The fourth order Runge-Kutta is used to solve the system (2) with time step size 0.001. For the tracking control of the novel hypechaotic Lü system, we consider the controlled system (2). If x is considered as the state of the system to be controlled, namely, y = x, then the hyperchaotic Lü system (2) can be written in the following form $\ddot{y} = f (x, t) + g (x, t) u + D (x, t)$ (40)

where

$\begin{matrix} f (x, t) & = & a (- xz + {cy}_{1}) - a^{2} (y_{1} - x) \\ - aw + xz + bw \\ g (x, t) & = & a \\ D (x, t) & = & ad (t) + a Δ f (x, y_{1}, z, w) \end{matrix}$ (41)

Thus, as seen from (40), since the relative degree is two the reference mode is given as $P_{m} (s) = \frac{1}{s^{2} + 20 s + 100}$ . All the initial conditions of adaptation laws are set to zero except that ${\hat{θ}}_{r, g} (0) = {[1 1 \dots 1]}_{18 \times 1}^{T}, {\hat{θ}}_{l, g} (0) = {[1 1 \dots 1]}_{18 \times 1}^{T} .$ Beside this, the adaptive laws parameters are set to γ₁ = 300, γ₂ = 300, γ₃ = 0.1, γ₄ = 0.1, γ₅ = 100, γ₆ = 200 and ϑ₁ (0) =10, ϑ₂ (0) =20 . Moreover, M_f and M_g all are set to 300. Simulation times t_f = 8s and the step size h = 0.001 in the adaptive law Equations (32–37).

Simulation 1: At this stage of the simulation, the uncertainty term Δf (x, y₁, z, w)is set to zero. The tracking performance of the proposed controller is shown in Fig. 4. It is obvious that the tracking performance is very well. Moreover, the control effort is shown in Fig. 5. As seen, there is no sign of chattering in the control effort.

Simulation 2: At this stage of the simulation, the system is perturbed by Δf (x, y₁, z, ’ w) = sin(πx) sin(2πy₁) sin(3πz) sin(4πw), where |Δf (x, y₁, z, w) |≤1. Besides, external disturbance is considered to be d (t) =0.2 sin(2t) . From the simulation results, it shows that the obtained theoretic results are feasible and efficient for controlling new uncertain hyperchaotic dynamical system, Lu system. See Figs. 6–7.

Simulation 3: At this stage of the simulation, conventional sliding mode (SMC) and type-1 fuzzy controller are also applied to the system. To save space, the results are given in Table 1. From the simulation results shown in Table 1, we can observe that in terms of tracking performance the tracking error of SMC is smaller than the others. However, in the face of uncertainty and external disturbance the proposed interval T2FNN controller gives better performance compared with type-1 fuzzy controller and SMC. This is the main advantage of the results over other methods.

Simulation 4: At this stage of the simulation, the computation times for the implemented controllers are compared. The platform used is windows 7, Intel (R) Core (TM) i5-2410M CPU @ 2.30 GHz, 4 GB RAM. The algorithms were programmed with Matlab 2009b using tic and toc function in seconds. The results are given in Table 2. Note that the results are not independent of the hardware and software environments. From simulation 3, we have observed that the proposed approach gives better performance compared with type-1 fuzzy controller and SMC. Now, it is evident from Table 2 that the computation time of the proposed approach using type-2 FNN is more than the others, as expected. It is the price one must pay for achieving better performance in the face of uncertainties [31].

Simulation 5: At this stage of the simulation, in order to create a more difficult challenge for the proposed control, the non-zero-initial conditions are considered. The initial conditions are chosen as x (0) =0.5, y₁ (0) =0.1, z (0) = -0.5, w (0) =0.4 . After applying the proposed control, according to Fig. 8, it can be seen that controller shows good performance. In Fig. 9 it can be seen that the control input does not have chattering. From Figs. 8 and 9, it can be concluded that the proposed approach in this paper can be applied to the non-zero-initial conditions, as well.

Remarks:

From Fig. 4 and Table 1, it is obvious that the tracking performance of the proposed approach is worse than SMC. However, in the face of uncertainties the tracking performance of the proposed approach is better than SMC. Meanwhile, the computation time of the proposed approach is larger than SMC and proposed approach using type-1 FNN. It is the price one must pay for achieving better performance in the face of uncertainties [31].

As shown in Figs. 10 and 11, there is no sign of chattering. From Fig. 4, it can also be seen that the transient response is good. Thus, the proposed method doesn’t degrade the transient performance very much.

In the implementation of the proposed controller, the condition is not zero should be guaranteed, because the proposed controller (23) have the inverse term of $\hat{g} (x | θ_{g}, t)$ Therefore, we choose the initial parameter vector values ${\hat{θ}}_{g} (0)$ are small positive constants in simulations.

Due to this fat that Markovian jump linear systems (MJLSs) are stochastically varying systems and in them mode switches are governed by a stochastic process [44 –46], the proposed approached in the presented form cannot be directly applied to MJLSs. To deal with this problem, the proposed approach should be extended. It is noted that the extensions of the proposed method to the controller design for continuous-time MJLSs with defective mode information, deserve further investigation.

The proposed approach can be applied to any system in the form of (3) which is its unique feature.

The proposed approach is robust to uncertainties of the parameters of the system and there is no sign of chattering in the control effort. This means that the PI type switching structure has been worked well.

The implementation of the proposed control scheme on a hardware setup will be performed in the future work.

The dynamic of the system to be controlled need not to be known and just the relative degree should be known in advance, which is more flexible in the real implementations.

To sum up, theoretic results obtained have potential in applications.

8 Conclusion

This paper has addressed the problem of control a new hyperchaotic Lü system. To address the tracking problem, by integrating classical model reference control, feedback linearization, sliding mode control technique and type-2 fuzzy systems a novel control approach has been proposed. Based on the Laypunov stability criterion, the stability of the whole system can be achieved and free parameters of the adaptive fuzzy system can be tuned on-line adaptive laws. Moreover, by introducing a novel PI type switching structure the chattering phenomena in the control efforts can be reduced. Simulation results confirm that compared to SMC and Type1 FLS the proposed approach shows better performance in the face of uncertainty and external disturbance. One of the advantages of the proposed approach is that it also can be used to control other chaos or hyperchaotic systems. The computation time of the proposed approach has also been evaluated. We have observed that although the proposed approach gives better performance compared with type-1 fuzzy controller and SMC but it has more computational time compared to other methods. Using general type-2 fuzzy system for controlling of chaotic systems is a good idea for future work. The implementation of the proposed control scheme on a hardware setup will be performed in the future work, as well.

Footnotes

Acknowledgments

The authors gratefully appreciate the support of the Behbahan Khatam Alanbia University of Technology.

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