Adaptive neural network control for unified chaotic systems with dead-zone input

Abstract

An adaptive control scheme is studied for unified chaotic systems with unknown function and dead-zone input. Because uncertain nonlinear property is included in the considered unified chaotic systems, the neural networks are used to approximate the uncertainties. An adaptive technique is employed to construct the neural controllers and compensate for the dead-zone parameters. By using the scheme, the chaotic phenomena for unified chaotic systems are overcome. It is proven that the proposed algorithm can guarantee that all the signals in the closed-loop system are bounded and the system states can converge to a neighborhood of zero based on the Lyapunov analysis method. The simulation example for a unified chaotic system is provided to demonstrate the effectiveness of the proposed method.

Keywords

Adaptive control dead-zone input neural networks the controller design unified chaotic systems

1. Introduction

Over recent years, the adaptive control technique has been quickly developed. Based on fuzzy logic systems and neural networks, many results using the adaptive design were studied for nonlinear systems with unknown functions, (Liu et al., 2009a, 2011a,b,c, 2013; Tong and Li, 2009; Tong et al., 2009, 2010, 2011; Li et al., 2013; Chen et al., 2010, 2011, 2013; Li et al., 2010; Zhou et al., 2011, 2012), and their references. At the same time, the control methods have also been used in some real systems (Li, 2012, Li et al., 2014; Liu et al., 2012). Specifically, the controller design using adaptive control has been widely used in chaotic systems.

A synchronization controller using sliding mode was presented in Chen et al. (2009) for two chaotic systems with Radial Bassi Function (RBF) neural network. The compound disturbance of the synchronization error system consists of nonlinear uncertainties and exterior disturbances of chaotic systems. In Liu and Zheng (2009), the output feedback control was addressed for uncertain chaotic systems. The fuzzy systems were utilized to approximate uncertain nonlinear functions in the chaotic systems. Because only partial information of the system’s states needs to be known, an observer is given to estimate the unmeasured states. In Li (2012), an adaptive output feedback algorithm based on the dynamic surface control was proposed for a class of uncertain chaotic systems. The main advantage of this algorithm can overcome the problem of an “explosion of complexity” inherent in the backstepping design. In Luo and Wang (2013), a novel synchronization scheme was proposed for non-identical hyperchaotic complex systems with uncertainties. But these approaches ignored the effect of the dead-zone input.

To this end, a practical projective synchronization problem was studied in Boulkroune and M’Saad (2011a) to control master–slave chaotic systems with dead-zone nonlinearity by using a fuzzy adaptive method. In Boulkroune and M’Saad (2011b), a fuzzy adaptive controller was designed for a class of uncertain multiple-input multiple-output (MIMO) chaotic systems with both sector nonlinearities and dead-zones. However, a lot of adjustable parameters were required to be used in these results. This will result in a heavy computation burden online. The problem of controlling chaotic dead-zone systems with fewer parameters needs to be investigated.

Based on the above presentations, this paper is to explore an adaptive neural control strategy to control a unified chaotic system. The neural networks are utilized to approximate the unknown functions in the chaotic systems. Based on the Lyapunov function, it is guaranteed that all the signals are bounded. Compared with the results in chaotic systems with the dead-zone, fewer parameters are required in the design. Accordingly, the online computation burden is reduced. Simulation results are given to verify the effectiveness of the proposed approach.

2. System description and preliminaries

The controlled chaotic system can be expressed as

{x \cdot 1 = (25 α + 10) (x 2 - x 1) x \cdot 2 = (28 - 35 α) x 1 - x 1 x 3 + (29 α - 1) x 2 + u 2 x \cdot 3 = x 1 x 2 - (α + 8) x 3 / 3 + u 3

(1)

where

x i, i = 1, 2, 3

are the states of the systems, u₂ and u₃ are the control inputs of the systems. In this paper, the control input

u i (t) = P (w i (t))

is considered to be a non-symmetric dead-zone. The non-symmetric dead-zone

P (w i (t))

is defined as

u i = P (w i (t)) = n i (t) w i + d i (t)

(2)

where

n i (t) = {n r i, if w i > 0 n l i, if w i \leq 0

and

d i (t) = {- n r i d r i, if w i \geq d r i - n i w i, if - d l i < w i < d r i n l i d l i, if w i \leq - d l i .

The parameters $n r i$ and $n l i$ stand for the right and left slope of the dead-zone characteristic and are known. The parameters $d r i$ and $d l i$ represent the breakpoints of dead-zone to be unknown where $n r i, d r i, n l i$ and $d l i$ are the positive constants.

Let $n ¯ i = max (n r i, n l i)$ and $\underline{n} i = min (n r i, n l i)$ . It is easy to know that $\underline{n} i \leq n i \leq n ¯ i$ and $| d i (t) | \leq d ¯ i =$ $max (n r i d r i, n l i d l i)$ .

The aim of this paper is to design controllers u₂ and u₃ to stabilize the system (1), i.e., the system states can converge to a small neighborhood of zero and all the signals in the closed-loop system are bounded.

Because the system (1) contains the unknown parameters, they can be used in the controller. Due to the approximation property of the neural networks, they have been used in the control problem and modeling of the systems. Using neural networks, unknown functions $f (y) : R \to R$ can be denoted as

f (y) = ψ T Q (y) + ϕ (y)

where ψ is the optimal weight vector,

Q (y) = [q 1 (y), \dots, q N (y)] T

are basis function vector and

ϕ (y)

is the approximation error.

Assumption 1 (Boulkroune and M’Saad, 2011a): ψ and $ϕ (y)$ are bounded, i.e., $‖ ψ ‖ \leq ψ ¯$ and $| ϕ (y) | \leq ϕ ¯$ .

3. Adaptive controller design

In this section, a stable controller and adaptation laws are designed and the corresponding theorem is explained. The detailed design procedure is given in the following.

Consider the Lyapunov function candidate as

L = \frac{1}{2} \sum_{i = 1}^{3} x_{i}^{2} + \sum_{i = 2}^{3} \frac{1}{2} h_{i}^{- 1} {β ~}_{i}^{2} + \sum_{i = 2}^{3} \frac{1}{2} γ_{i}^{- 1} ¯ ~ d_{i}^{2}

(3)

where

β ~ i = β \land i - β i

d ¯ i ~ = d ¯ i \land - d ¯ i

;

β \land i

and

d ¯ i \land

are the estimation of

β i

and

d ¯ i

, respectively;

β i

are unknown parameters to be specified later on.

Differentiating (3) along the trajectory of the system in (1) yields

L \cdot = - (25 α + 10) x_{1}^{2} + \sum_{i = 2}^{3} x i (f i (x) + u i) + \sum_{i = 2}^{3} h_{i}^{- 1} β ~ i \land \cdot β i + \sum_{i = 2}^{3} γ_{i}^{- 1} d ¯ i ~ \land \cdot d ¯ i

(4)

where

f 2 (x) = (38 - 10 α) x 1 - x 1 x 3 + (29 α - 1) x 2

, and

f 3 (x) = x 1 x 2 - x 3 (α + 8) / 3

Because α is unknown, the nonlinear functions $f i (x)$ for $i = 2, 3$ are unknown. By using the neural networks, there follows

f i (x) = ψ_{i}^{T} Q i (x) + ϕ i (x)

(5)

where

x = [x 1, x 2, x 3] T

and

ψ_{i}^{*}

i = 2, 3

are the neural weight vectors, and

‖ ϕ i (x) ‖ \leq ϕ ¯ i

. Then, it follows from substituting (5) into (4) that

L \cdot = - (25 α + 10) x_{1}^{2} + \sum_{i = 2}^{3} h_{i}^{- 1} β ~ i \land \cdot β i + \sum_{i = 2}^{3} γ_{i}^{- 1} d ¯ i ~ \land \cdot d ¯ i + \sum_{i = 2}^{3} x i [ψ_{i}^{T} Q i (x) + ϕ i (x) + u i]

(6)

The following fact can be obtained

x i [ψ_{i}^{T} Q i (x) + ϕ i (x) + u i] = x i [\frac{‖ ψ i ‖}{‖ ψ i ‖} ψ_{i}^{T} Q i (x) + ϕ i (x) + u i] \leq \frac{1}{2 m i} x_{i}^{2} β i Q_{i}^{T} (x) Q i (x) + \frac{1}{2} m i + \frac{1}{2 l i} x_{i}^{2} + \frac{1}{2} l i {ϕ ¯}_{i}^{2} + x i u i = x i [\frac{1}{2 m i} x i β i Q_{i}^{T} (x) Q i (x) + \frac{1}{2 l i} x i + u i] + \frac{1}{2} (m i + l i {ϕ ¯}_{i}^{2})

(7)

where

β i = ‖ ψ i ‖ 2, i = 2, 3

are unknown constants,

m i > 0

and

l i > 0

are the design parameters to be chosen by the designer.

The adaptive neural network controllers are constructed as follows

w i = \frac{1}{n i (t)} [- (λ i + \frac{1}{2 l i}) x i - sign (x i) d ¯ i \land - \frac{1}{2 m i} x i β \land i Q_{i}^{T} (x) Q i (x)]

(8)

where

λ i > 0, i = 2, 3

are the design parameters. By taking (7) and (8) into account, the following inequality is obtained:

x i [ψ_{i}^{T} Q (x) + ϕ i (x) + u i] \leq - λ i x_{i}^{2} - \frac{1}{2 m i} x_{i}^{2} β ~ Q_{i}^{T} (x) Q i (x) - | x i | ¯ \land d + x i d i (t) + \frac{1}{2} (m i + l i {ϕ ¯}_{i}^{2})

(9)

Substituting (9) into (6) gives

L \cdot \leq - (25 α + 10) x_{1}^{2} - \sum_{i = 2}^{3} λ i x_{i}^{2} + \sum_{i = 2}^{3} x i d i (t) + \sum_{i = 2}^{3} \frac{1}{h i} (- \frac{h i}{2 m i} x_{i}^{2} Q_{i}^{T} (x) Q i (x) + \land \cdot β i) β ~ i + \sum_{i = 2}^{3} \frac{1}{2} (m i + l i {ϕ ¯}_{i}^{2}) + \sum_{i = 2}^{3} γ_{i}^{- 1} d ¯ i ~ \land \cdot d ¯ i - \sum_{i = 2}^{3} | x i | ¯ \land d

(10)

Further, choose adaptive laws $\land \cdot β i$ and $\land \cdot d ¯ i$ in the following form:

\land \cdot β i = \frac{h i}{2 m i} x_{i}^{2} Q_{i}^{T} (x) Q i (x) - β \land i, i = 2, 3

(11)

\land \cdot d ¯ i = γ i [| x i | - d ¯ i \land]

(12)

Using the inequality $x i d i (t) \leq | x i | d ¯ i$ , (11) and (12), (10) becomes

L \cdot \leq - \sum_{i = 1}^{3} λ i x_{i}^{2} - \sum_{i = 2}^{3} \frac{1}{h i} β ~ i β \land i + \sum_{i = 2}^{3} \frac{1}{2} (m i + l i {ϕ ¯}_{i}^{2}) - \sum_{i = 2}^{3} d ¯ i ~ d ¯ i \land

(13)

Since

α \in [0, 1], λ 1 = (25 α + 10) \geq 10 .

In addition, for the term

β \land i β ~ i

, it follows

- β \land i β ~ i = - β ~ i (β i + β ~ i) = - {β ~}_{i}^{2} + β i β ~ i \leq - \frac{1}{2} {β ~}_{i}^{2} + \frac{1}{2} β_{i}^{2}

(14)

- d ¯ i ~ d ¯ i \land = - d ¯ i ~ (d ¯ i ~ + d ¯ i) = - d ¯ i ~ 2 + d ¯ i ~ d ¯ i \leq - \frac{1}{2} d ¯ i ~ 2 + \frac{1}{2} d ¯ i 2

(15)

Thus, in light of (14) and (15), (13) can verify

L \cdot \leq - \sum_{i = 1}^{3} λ i x_{i}^{2} - \sum_{i = 2}^{3} \frac{1}{2 h i} {β ~}_{i}^{2} - \sum_{i = 2}^{3} \frac{1}{2} d ¯ i ~ 2 + \sum_{i = 2}^{3} \frac{1}{2} (m i + l i ɛ 2 + \frac{1}{h i} β_{i}^{2} + d ¯ i 2) = - \sum_{i = 1}^{3} \frac{2 λ i}{2} x_{i}^{2} - \sum_{i = 2}^{3} \frac{1}{2 h i} {β ~}_{i}^{2} - \sum_{i = 2}^{3} \frac{1}{2} d ¯ i ~ 2 + \sum_{i = 2}^{3} \frac{1}{2} (m i + l i {ϕ ¯}_{i}^{2} + \frac{1}{h i} β_{i}^{2} + d ¯ i 2)

(16)

Now, define

a 0 = min {1, 2 λ 1, 2 λ i, γ i | i = 2, 3}

and

d 0 = \sum_{i = 2}^{3} \frac{1}{2} (m i + l i {ϕ ¯}_{i}^{2} + \frac{1}{h i} β_{i}^{2} + d ¯ i 2)

Define $k = a 0 / b 0$ . Then, (16) can be represented as

L \cdot \leq - a 0 L + d 0

which implies

L \cdot \leq (L (0) - k) e - a 0 t + k

for

t \geq 0 .

According to the definition of L and the inequality above, the following inequality becomes true.

\sum_{i = 1}^{3} x_{i}^{2} \leq (L (0) - k) e - a 0 t + 2 k

Note that $h i, m i, l i,$ and $λ i$ are all design parameters. Thus for any given $k \leq 1 / 2 μ 2$ can be obtained by appropriately choosing these design parameters. Furthermore, it is found that $lim_{t \to \infty} x_{i}^{2} \leq μ 2$ . Similarly, it can be proven that the control laws and the adaptive laws converge to a neighborhood of the origin, and the radius of this neighborhood can be made as small as possible by tuning appropriately the design parameters.

4. Simulation study

In this section, to illustrate the effectiveness of the proposed adaptive control scheme, a simulation example is given.

For the unified chaotic system (1), it is assumed that $α = 0.8$ , $n r 1 = 0.2, n r 2 = 0.5, d r 1 = 0.3, d r 2 = 0.4$ and $n l 1 = 0.3$ , $n l 2 = 0.2$ , $d l 1 = 0.5, d l 2 = 0.6$ .

Based on the above design method, the adaptive neural networks are constructed as follows

w i = \frac{1}{n i (t)} [- (λ i + \frac{1}{2 l i}) x i - sign (x i) d ¯ i \land - \frac{1}{2 m i} x i β \land i Q_{i}^{T} (x) Q i (x)]

(17)

and the adaptive laws are given as

\land \cdot β i = \frac{h i}{2 m i} x_{i}^{2} Q_{i}^{T} (x) Q i (x) - β \land i, i = 2, 3

(18)

\land \cdot d ¯ i = γ i [| x i | - d ¯ i \land]

(19)

The designed parameters of the proposed control approach are chosen as $λ 2 = 0.05, λ 3 = 2, l 1 = 0.05, l 2 = 0.05$ , $γ 1 = 3, γ 2 = 3,$ $h 1 = 2, h 2 = 0.5, m 1 = 0.2, m 2 = 0.2$ . The initial conditions for the states are chosen as $x 1 (0) = - 1,$ $x 2 (0) = 0.3$ , $x 3 (0) = - 1.5$ , $β \land 1 (0) = 0$ , $β \land 2 (0) = 0.1$ , and $¯ \land d 1 (0) = 0.5, ¯ \land d 2 (0) = 0$ .

The neural networks contain 25 nodes with the centers $π l$ evenly spaced in $[- 1.5, 1.5] \times [- 2, 2] \times [- 2.5, 2.5]$ , and widths $μ = 2$ where the input variable of the neural networks is given as $x = [x 1, x 2, x 3] T$ .

The simulation results are obtained in Figures 1 –3. From Figure 1, the response curves of x₁, x₂ and x₃ are shown. They are bounded to converge to the neighborhood of zero. Figures 2 and 3 illustrate the trajectory of the control inputs and adaptation laws. A good control performance is obtained. Hence, it can be seen that the proposed controllers guarantee the closed-loop systems stable.

Figure 1.

The curves of the state x₁ (solid line) x₂ (dashed line) and x₃ (dot-and-dash line).

Figure 2.

(a) u₂ and (b) u₃.

Figure 3.

(a) $β \land 1$ (solid line) and $β \land 2$ (dashed line); and (b) $¯ \land d 1$ (solid line) and $¯ \land d 2$ (dashed line).

5. Conclusion

In this article, an adaptive control algorithm is investigated for unified chaotic systems. The dead-zone input is considered in the chaotic systems. The unknown functions of the chaotic systems are approximated by using neural networks. The adaptive compensative terms are used to compensate for the parameters in the dead-zone. The main advantage of the approach is to reduce the computation burden compared with the approach for the chaotic systems with the dead-zone. The performance of the proposed control scheme was validated by an example. The future works will extend this approach to the discrete-time systems.

Footnotes

Funding

This work was supported by The Foundation of Educational Department of Liaoning Province (grant number L2013243), the National Natural Science Foundation of China (grant number 61104017), and the Program for Liaoning Excellent Talents in University (grant number LJQ2011064).

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