Adaptive Neural Networks Synchronization of a Four-Dimensional Energy Resource Stochastic System

Abstract

An adaptive neural networks chaos synchronization control method is proposed for a four-dimensional energy resource demand-supply system with input constraints. Assuming the response system contains unknown uncertain nonlinearities and unknown stochastic disturbances, the neural networks and robust terms are used to identify the nonlinearities and overcome the stochastic disturbances, respectively. Based on stochastic Lyapunov stability and robust adaptive theories, an adaptive neural networks synchronization control method is developed. In the design process, an auxiliary design system is employed to address input constraints. Simulation results, which fully coincide with theoretical results, are presented to demonstrate the obtained results.

1. Introduction

Energy resource system is a kind of complex nonlinear system. Over the last two decades, much attention has been paid to the chaos synchronization in this class system. Reference [1] established a three-dimensional energy resource demand-supply system based on the real energy resources demand-supply system in the East and the West of China. By adding a new variable to consider the renewable resources, a four-dimensional energy resource system was proposed in [2]. The dynamics behaviors of the four-dimensional energy resource system have been analyzed by means of the Lyapunov exponents and bifurcation diagrams. Also the same as the above-mentioned power systems, this four-dimensional energy resource system is with rich chaos behaviors. The problem of chaotic control for the energy resource system was considered in [3]. Feedback control and adaptive control methods were used to suppress chaos to unstable equilibrium or unstable periodic orbits, where only three of the system’s parameters were supposed to be unknown. Reference [4] investigated the robust chaos synchronization problem for the four-dimensional energy resource systems based on the sliding mode control technique. The control of energy resource chaotic system was investigated by time-delayed feedback control method in [5]. Four linear control schemes are proposed to a four-dimensional energy resource system in [6]. Based on stability criterion of linear system and Lyapunov stability theory, respectively, the chaos synchronization problems for energy resource demand-supply system were discussed using two novel different control methods in [7].

In many practical dynamic systems (including the energy resource demand-supply system), physical input saturation on hardware dictates that the magnitude of the control signal is always constrained. Saturation is a potential problem for actuators of control systems. It often severely limits system performance, giving rise to undesirable inaccuracy or leading instability [8, 9]. The development of control schemes for systems with input saturation has been a task of major practical interest as well as theoretical significance. The proposed approaches in [1–7] assume that all the components of the considered energy resource demand-supply systems are in good operating conditions and do not consider the problem of saturation. Reference [10] proposed two different chaos synchronization methods for a class of energy resource demand-supply systems with input saturation, but the response system in [10] did not contain unknown uncertain nonlinearities and unknown stochastic disturbances. It is well known that stochastic disturbances also often exist in many practical systems. Their existence is a source of instability of the control systems; thus, the investigations on stochastic control systems have received considerable attention in recent years [11–22]. Since the emergence of the stochastic stabilization theory in the 1960s, the progress has been constructed by a fundamental technical Itô lemma, and the control design for stochastic systems is more difficult compared with deterministic systems.

Motivated by the above observations, an adaptive neural networks chaos synchronization method is proposed for a four-dimensional energy resource demand-supply system with input constraints. Assume that the response system contains unknown uncertain nonlinearities and unknown stochastic disturbances. In the design, the neural networks and robust terms are used to identify the nonlinearities and overcome the stochastic disturbances, respectively. Based on Lyapunov stability, an adaptive synchronization method is developed in order to make the states of two chaotic systems asymptotically synchronized. The new auxiliary design system is employed to address input constraints. Numerical simulations are provided to illustrate the effectiveness of the proposed approach.

Compared with the existing results, the main contributions of the proposed method are as follows: (i) the controlled response system of this paper contains unknown nonlinearities, and the proposed method can solve the unknown nonlinearity problem by neural networks, but the methods of [1–8, 10] cannot solve this problem; (ii) the controlled response system of this paper contains stochastic disturbances, and the proposed method can solve the stochastic disturbances problem based on Itô’s lemma and stochastic LaSalle’s theorem, but the methods of [1–8, 10] cannot solve this problem; (iii) an auxiliary design system is employed to address input constraints problem, and the methods of [1–8] can solve this problem.

2. Energy Resource Chaotic System

The four-dimensional energy resource system can be expressed as follows (see [2, 4, 6]): 𝑥̇=𝑎₁𝑥𝑥1−𝑀−𝑎₂𝑦+𝑧−𝑑₃̇𝑤,𝑦=−𝑏₁𝑦−𝑏₂𝑧+𝑏₃𝑥[(,̇𝑁−𝑥−𝑧)]𝑧=𝑐₁𝑧𝑐₂𝑥−𝑐₃,𝑤̇=𝑑₁𝑥−𝑑₂𝑤, (1) where 𝑥(𝑡) is the energy resource shortage in A region, 𝑦(𝑡) is the energy resource supply increment in B region, and 𝑧(𝑡) and 𝑤(𝑡) are energy resource import in A region and renewable energy resource in A region, respectively; 𝑀, 𝑁, 𝑎_𝑖, 𝑏_𝑗, 𝑐_𝑗, and 𝑑_𝑗(𝑖=1,2,𝑗=1,2,3) are parameters that are all positive real. The dynamics of this system has been extensively studied in [2, 4, 6].

When the system parameters are taken as the following values, this system exhibits chaotic behavior: 𝑀=1.8, 𝑁=1, 𝑎₁=0.1, 𝑎₂=0.15, 𝑏₁=0.06, 𝑏₂=0.082, 𝑏₃=0.07, 𝑐₁=0.2, 𝑐₂=0.5, 𝑐₃=0.4, 𝑑₁=0.1, 𝑑₂=0.06, and 𝑑₃=0.07. Without the particular statement, these values are adopted in this whole paper. Figures 1, 2, and 3 show the phase portraits with initial conditions of 𝑥(0)=0.82, 𝑦(0)=0.29, 𝑧(0)=0.48, and 𝑤(0)=0.1.

Figure 1 

Three-dimensional view 𝑥−𝑦−𝑧.

Figure 2 

Three-dimensional view 𝑦−𝑥−𝑤.

Figure 3 

Three-dimensional view 𝑤−𝑧−𝑥.

3. Synchronization of the Energy Resource System

In this section, a controller will be designed in order to make the response system track the drive system. The drive system with subscript 1 is written as 𝑥̇₁=𝑎₁𝑥₁𝑥1−¹𝑀−𝑎₂𝑦₁+𝑧₁−𝑑₃𝑤₁,𝑦̇₁=−𝑏₁𝑦₁−𝑏₂𝑧₁+𝑏₃𝑥₁𝑁−𝑥₁−𝑧₁,𝑧̇₁=𝑐₁𝑧₁𝑐₂𝑥₁−𝑐₃,𝑤̇₁=𝑑₁𝑥₁−𝑑₂𝑤₁. (2) Assume that the controlled response system with subscript 2 contained uncertain nonlinearities (unknown smooth nonlinear functions) and unknown external stochastic disturbance, and it can be expressed as the following dynamics: 𝑑𝑥₂=𝑎₁𝑥₂𝑥1−²𝑀−𝑎₂𝑦₂+𝑧₂−𝑑₃𝑤₂+𝑓₁𝑥₂+𝑢₁𝑣₁(𝑡)𝑑𝑡+𝑝₁𝑒₁𝑑𝑊,𝑑𝑦₂=−𝑏₁𝑦₂−𝑏₂𝑧₂+𝑏₃𝑥₂𝑁−𝑥₂−𝑧₂+𝑓₂𝑦₂+𝑢₂𝑣₂(𝑡)𝑑𝑡+𝑝₂𝑒₂𝑑𝑊,𝑑𝑧₂=𝑐₁𝑧₂𝑐₂𝑥₂−𝑐₃+𝑓₃𝑧₂+𝑢₃𝑣₃(𝑡)𝑑𝑡+𝑝₃𝑒₃𝑑𝑊,𝑑𝑤₂=𝑑₁𝑥₂−𝑑₂𝑤₂+𝑓₄𝑤₂+𝑢₄𝑣₄(𝑡)𝑑𝑡+𝑝₄𝑒₄𝑑𝑊, (3) where 𝑣_𝑖 is the actual controller to be designed and 𝑢_𝑖(𝑣_𝑖(𝑡))(𝑖=1,2,3,4) is the plant input subject to saturation type nonlinearly. 𝑝_𝑖(𝑒_𝑖), 𝑖=1,2,3,4, are uncertain functions, and 𝑊∈𝑅^𝑛 is an independent standard Brownian motion defined on a complete probability space, with the incremental covariance 𝐸{𝑑𝑊⋅𝑑𝑊^𝑇}=𝜎(𝑡)𝜎(𝑡)^𝑇𝑑𝑡.

Remark 1. If no input saturation, uncertain nonlinearities, and unknown external stochastic disturbance (i.e., 𝑢_𝑖(𝑣_𝑖(𝑡))=𝑣_𝑖(𝑡)), 𝑝_𝑖(𝑡)=0, and 𝑓_𝑖(⋅)=0(𝑖=1,2,3,4)) are included in (3), then (3) becomes the chaotic systems studied widely, see [8, 10], where 𝑢_𝑖(𝑣_𝑖(𝑡)) can be described as𝑢_𝑖𝑣_𝑖(𝑡)=sat𝑣_𝑖(𝑡)=sign𝑣_𝑖(𝑡)𝑢_𝑖𝑀,𝑣_𝑖(𝑡)≥𝑢_𝑖𝑀,𝑣_𝑖(𝑡),𝑣_𝑖(𝑡)<𝑢_𝑖𝑀, (4) where 𝑢_𝑖𝑀 is a known bound of 𝑢_𝑖(𝑣_𝑖(𝑡)).

To design an adaptive controller, the following basic assumption is made for the system (3).

Assumption 2. The disturbance covariance 𝑃^𝑇𝜎𝜎^𝑇𝑃≤∑⁴_𝑖=1𝑒_𝑖𝜎_𝑖(𝑒_𝑖), where 𝑃=𝑝₁(𝑒₁)𝑝₂(𝑒₂)𝑝₃(𝑒₃)𝑝₄(𝑒₄)^𝑇 and 𝜎_𝑖(𝑒_𝑖) is a known function.

To establish stochastic stability as a preliminary, consider a stochastic nonlinear system: 𝑑𝑥=𝑓(𝑥,𝑡)𝑑𝑡+𝑔(𝑥,𝑡)𝑑𝑊, (5) where 𝑥∈𝑅^𝑛, 𝑊 is an independent 𝑟-dimensional Wiener process, defined on the probability space (Ω,𝐹,𝑃), with the incremental covariance 𝐸{𝑑𝑊,𝑑𝑊^𝑇=𝜎(𝑡)𝜎(𝑡)^𝑇𝑑𝑒}, and 𝑓∶𝑅^𝑛×𝑅→𝑅^𝑛 and 𝑓∶𝑅^𝑛×𝑅→𝑅^𝑛×𝑟 are locally Lipschitz continuous in 𝑥, uniformly in 𝑡∈𝑅, with 𝑓(0,𝑡)=0 and 𝑔(0,𝑡)=0, for all 𝑡≥0.

Lemma 3 (see [16, 17] (stochastic LaSalle’s theorem)). Consider (5) and suppose that there exists a twice continuously differentiable function 𝑉(𝑥,𝑡), which is positive definite, decrescent, and radially unbounded, and another nonnegative continuous function 𝑄(𝑥)≥0 such that the infinitesimal generator 𝑉(𝑥,𝑡) along (5) satisfies ℓ𝑉∶=𝜕𝑉+𝜕𝑡𝜕𝑉𝑓(𝑥)+1𝜕𝑥2Tr𝜎^𝑇𝑔^𝑇𝜕²𝑉𝜕𝑥²𝑔𝜎≤−𝑄(𝑥),∀𝑥∈𝑅^𝑛,𝑡≥0, (6) where Tr denotes the matrix trace. Then, the equilibrium 𝑥=0 is globally stable in probability and 𝑃lim_𝑡→∞𝑄(𝑥(𝑡))=0=1,∀𝑥(0)∈𝑅^𝑛. (7)

In order to solve the unknown nonlinear 𝑓_𝑖(⋅)(𝑖=1,2,3,4), the following radial basis function neural networks (RBFNNs) [23] are used to identify them similar to fuzzy logic systems [24–27].

An RBFNN can approximate a continuous function ℎ(𝑋): 𝑅^𝑞→𝑅, ℎ_𝑛𝑛(𝑋)=𝑊^𝑇𝜑(𝑋), (8) where the input vector 𝑋∈Ω⊂𝑅^𝑞, weight vector 𝑊=[𝑊₁,…,𝑊_𝑚]^𝑇∈𝑅^𝑚, the NN node number 𝑚>1, and 𝜑(𝑋)=[𝜑₁(𝑋)⋯𝜑_𝑚(𝑋)]^𝑇, with 𝜑_𝑖(𝑋) being Gaussian functions, which have the form 𝜑_𝑖(𝑋)−=exp𝑋−𝜇_𝑖^𝑇𝑋−𝜇_𝑖𝜂²,𝑖=1,2,…,𝑚, (9) where 𝜇_𝑖=[𝜇_𝑖1,…,𝜇_𝑖𝑞]^𝑇 is the center of the receptive field and 𝜂 is the width of the Gaussian function.

According to the literatures [23], the neural network (8) can approximate any continuous function ℎ(𝑋) over a compact set 𝐷⊂𝑅^𝑞 to arbitrary any accuracy as ℎ(𝑋)=𝑊^∗𝑇𝜑(𝑋)+𝜀(𝑋),∀𝑋∈𝐷, (10) where 𝑊^∗ is an ideal constant weight, 𝜀(𝑋) is the bounded approximation error, and 𝑊^∗ is defined as 𝑊^∗=argmin_𝑊∈Ωsup_𝑋∈𝐷ℎ(𝑋)−𝑊^𝑇𝜑(𝑋). (11)

4. Adaptive Synchronization of the Energy Resource System

For different initial conditions of systems (2) and (3), the two coupled systems can achieve synchronization by designing an appropriate control input 𝑢_𝑖(𝑡). First, we define the synchronization error vector between systems (2) and (3) as 𝑒₁=𝑥₂−𝑥₁−ℎ₁,𝑒₂=𝑦₂−𝑦₁−ℎ₂,𝑒₃=𝑧₂−𝑧₁−ℎ₃,𝑒₄=𝑤₂−𝑤₁−ℎ₄, (12) where ℎ_𝑖(𝑖=1,2,3,4) is filter signal and will be given later.

From (2), (3), and (12), the error dynamical system can be written as 𝑑𝑒₁=−ℎ̇₁+𝑎₁𝑒₁−𝑎₂𝑒₂+𝑒₃−𝑎₁𝑥²₂𝑀+𝑎₁𝑥²₁𝑀−𝑑₃𝑒₄+𝑎₁ℎ₁−𝑎₂ℎ₂+ℎ₃−𝑑₃ℎ₄+𝑓₁𝑥₂+𝑢₁𝑣₁(𝑡)𝑑𝑡+𝑝₁𝑒₁𝑑𝑊,𝑑𝑒₂=−ℎ̇₂−𝑏₁𝑒₂−𝑏₂𝑒₃+𝑏₃𝑁𝑒₁−𝑏₃𝑥²₂+𝑏₃𝑥²₁+𝑏₃𝑥₂𝑧₂−𝑏₃𝑥₁𝑧₁−𝑏₁ℎ₂−𝑏₂ℎ₃+𝑏₃𝑁ℎ₁+𝑓₂𝑦₂+𝑢₂𝑣₂(𝑡)𝑑𝑡+𝑝₂𝑒₂𝑑𝑊,𝑑𝑒₃=−ℎ̇₃−𝑐₁𝑐₃𝑒₃+𝑐₁𝑐₂𝑥₂𝑧₂−𝑐₁𝑐₂𝑥₁𝑧₁−𝑐₁𝑐₃ℎ₃+𝑓₃𝑧₂+𝑢₃𝑣₃(𝑡)𝑑𝑡+𝑝₃𝑒₃𝑑𝑊,𝑑𝑒₄=−ℎ̇₄+𝑑₁𝑒₁−𝑑₂𝑒₄+𝑑₁ℎ₁−𝑑₂ℎ₄+𝑓₄𝑤₂+𝑢₄𝑣₄(𝑡)𝑑𝑡+𝑝₄𝑒₄𝑑𝑊. (13) RBFNNs are used to identify 𝑓_𝑖(⋅)(𝑖=1,2,3,4), and (13) can be rewritten as 𝑑𝑒₁=−ℎ̇₁+𝑎₁𝑒₁−𝑎₂𝑒₂+𝑒₃−𝑎₁𝑥²₂𝑀+𝑎₁𝑥²₁𝑀−𝑑₃𝑒₄+𝑎₁ℎ₁−𝑎₂ℎ₂+ℎ₃−𝑑₃ℎ₄+𝑊₁^∗𝑇𝜑₁𝑥₂+𝜀₁𝑥₂+𝑢₁𝑣₁(𝑡)𝑑𝑡+𝑝₁𝑒₁𝑑𝑊,𝑑𝑒₂=−ℎ̇₂−𝑏₁𝑒₂−𝑏₂𝑒₃+𝑏₃𝑁𝑒₁−𝑏₃𝑥²₂+𝑏₃𝑥²₁+𝑏₃𝑥₂𝑧₂−𝑏₃𝑥₁𝑧₁−𝑏₁ℎ₂−𝑏₂ℎ₃+𝑏₃𝑁ℎ₁+𝑊₂^∗𝑇𝜑₂𝑦₂+𝜀₂𝑦₂+𝑢₂𝑣₂(𝑡)𝑑𝑡+𝑝₂𝑒₂𝑑𝑊,𝑑𝑒₃=−ℎ̇₃−𝑐₁𝑐₃𝑒₃+𝑐₁𝑐₂𝑥₂𝑧₂−𝑐₁𝑐₂𝑥₁𝑧₁−𝑐₁𝑐₃ℎ₃+𝑊₃^∗𝑇𝜑₃𝑧₂+𝜀₃𝑧₂+𝑢₃𝑣₃(𝑡)𝑑𝑡+𝑝₃𝑒₃𝑑𝑊,𝑑𝑒₄=−ℎ̇₄+𝑑₁𝑒₁−𝑑₂𝑒₄+𝑑₁ℎ₁−𝑑₂ℎ₄+𝑊₄^∗𝑇𝜑₄𝑤₂+𝜀₄𝑤₂+𝑢₄𝑣₄(𝑡)𝑑𝑡+𝑝₄𝑒₄𝑑𝑊, (14) where |𝜀_𝑖|≤𝜀^∗_𝑖(𝑖=1,2,3,4) and 𝜀^∗_𝑖 is a positive constant.

In this section, we assume that all the parameters of the energy resource system are unknown. For convenience, similar to [7], we define 𝑎₃=𝑎₁/𝑀, 𝑏₄=𝑏₃𝑁, 𝑞₁=𝑐₁𝑐₂, and 𝑞₂=𝑐₁𝑐₃; the system (14) can be rewritten as 𝑑𝑒₁=−ℎ̇₁+𝑎₁𝑒₁−𝑎₂𝑒₂+𝑒₃−𝑎₃𝑥²₂+𝑎₃𝑥²₁−𝑑₃𝑒₄+𝑎₁ℎ₁−𝑎₂ℎ₂+ℎ₃−𝑑₃ℎ₄+𝑊₁^∗𝑇𝜑₁𝑥₂+𝜀₁𝑥₂+𝑢₁𝑣₁(𝑡)𝑑𝑡+𝑝₁𝑒₁𝑑𝑊,𝑑𝑒₂=−ℎ̇₂−𝑏₁𝑒₂−𝑏₂𝑒₃+𝑏₄𝑒₁−𝑏₃𝑥²₂+𝑏₃𝑥²₁+𝑏₃𝑥₂𝑧₂−𝑏₃𝑥₁𝑧₁−𝑏₁ℎ₂−𝑏₂ℎ₃+𝑏₄ℎ₁+𝑊₂^∗𝑇𝜑₂𝑦₂+𝜀₂𝑦₂+𝑢₂𝑣₂(𝑡)𝑑𝑡+𝑝₂𝑒₂𝑑𝑊,𝑑𝑒₃=−ℎ̇₃−𝑞₂𝑒₃+𝑞₁𝑥₂𝑧₂−𝑞₁𝑥₁𝑧₁−𝑞₂ℎ₃+𝑊₃^∗𝑇𝜑₃𝑧₂+𝜀₃𝑧₂+𝑢₃𝑣₃(𝑡)𝑑𝑡+𝑝₃𝑒₃𝑑𝑊,𝑑𝑒₄=−ℎ̇₄+𝑑₁𝑒₁−𝑑₂𝑒₄+𝑑₁ℎ₁−𝑑₂ℎ₄+𝑊₄^∗𝑇𝜑₄𝑤₂+𝜀₄𝑤₂+𝑢₄𝑣₄(𝑡)𝑑𝑡+𝑝₄𝑒₃𝑑𝑊. (15) Define the dynamic system as ℎ̇_𝑖=−ℎ_𝑖+𝑢_𝑖−𝑣_𝑖,𝑖=1,2,3,4. (16) Choose the following Lyapunov function candidate 𝑉 as 1𝑉=2𝑒²₁+𝑒²₂+𝑒²₃+𝑒²₄+𝑊^𝑇₁𝑊₁+𝑊^𝑇₂𝑊₂+𝑊^𝑇₃𝑊₃+𝑊^𝑇₄𝑊₄+𝑎²₁+𝑎²₂+𝑎²₃+𝑏²₁+𝑏²₂+𝑏²₃+𝑏²₄+𝑑²₁+𝑑²₂+𝑑²₃+𝑞²₁+𝑞²₂, (17) where 𝑎_𝑖=𝑎_𝑖−𝑎_𝑖, 𝑑_𝑖=𝑑_𝑖−𝑑_𝑖(𝑖=1,2,3), 𝑊_𝑗=𝑊^∗_𝑗−𝑊_𝑗, 𝑏_𝑗=𝑏_𝑗−𝑏_𝑗(𝑗=1,2,3,4), and 𝑞_𝑘=𝑞_𝑘−𝑞_𝑘(𝑘=1,2).

Similar to [16, 17], the ℓ infinitesimal generator of 𝑉 along with the solutions of (15) is ℓ𝑉=𝑒₁ℎ₁+𝑣₁+𝑎₁𝑒₁−𝑎₂𝑒₂+𝑒₃−𝑎₃𝑥²₂+𝑎₃𝑥²₁−𝑑₃𝑒₄+𝑎₁ℎ₁−𝑎₂ℎ₂+ℎ₃−𝑑₃ℎ₄+𝑝₁(𝑡)+𝑊^𝑇₁𝜑₁𝑥₂+𝜀₁𝑥₂+𝑒₂ℎ₂+𝑣₂−𝑏₁𝑒₂−𝑏₂𝑒₃+𝑏₄𝑒₁−𝑏₃𝑥²₂+𝑏₃𝑥²₁+𝑏₃𝑥₂𝑧₂−𝑏₃𝑥₁𝑧₁−𝑏₁ℎ₂−𝑏₂ℎ₃+𝑏₄ℎ₁+𝑝₂(𝑡)+𝑊^𝑇₂𝜑₂𝑦₂+𝜀₂𝑦₂+𝑒₃ℎ₃+𝑣₃−𝑞₂𝑒₃+𝑞₁𝑥₂𝑧₂−𝑞₁𝑥₁𝑧₁−𝑞₂ℎ₃+𝑝₃(𝑡)+𝑊^𝑇₃𝜑₃𝑧₂+𝜀₃𝑧₂+𝑒₄ℎ₄+𝑣₄+𝑑₁𝑒₁−𝑑₂𝑒₄+𝑑₁ℎ₁−𝑑₂ℎ₄+𝑝₄(𝑡)+𝑊^𝑇₄𝜑₄𝑤₂+𝜀₄𝑤₂+𝑊^𝑇₁𝑒₁𝜑₁𝑥₂−𝑊̇₁+𝑊^𝑇₂𝑒₂𝜑₂𝑦₂−𝑊̇₂+𝑊^𝑇₃𝑒₃𝜑₃𝑧₂−𝑊̇₃+𝑊^𝑇₄𝑒₄𝜑₄𝑤₂−𝑊̇₄+𝑎₁𝑒²₁+𝑒₁ℎ₁−𝑎̇₁+𝑎₂𝑒₁𝑒₂+𝑒₃−𝑒₁ℎ₂+ℎ₃−𝑎̇₂+𝑎₃−𝑒₁𝑥²₂+𝑒₁𝑥²₁−𝑎̇₃+𝑏₁−𝑒²₂−𝑒₂ℎ₂−̇̂𝑏₁+𝑏₂−𝑒₂𝑒₃−𝑒₂ℎ₃−̇̂𝑏₂+𝑏₃−𝑒₂𝑥²₂+𝑒₂𝑥²₁+𝑒₂𝑥₂𝑧₂−𝑒₂𝑥₁𝑧₁−̇̂𝑏₃+𝑏₄𝑒₂𝑒₁+𝑒₂ℎ₁−̇̂𝑏₄+𝑑₁𝑒₄𝑒₁+𝑒₄ℎ₄+𝑒₄ℎ₁−̇𝑑₁+𝑑₂−𝑒²₄−̇𝑑₂+𝑑₃−𝑒₁𝑒₄+ℎ₄−̇𝑑₃+𝑞₁𝑒₃𝑥₂𝑧₂−𝑒₃𝑥₁𝑧₁−𝑞̇₁+𝑞₂−𝑒²₃−𝑒₃ℎ₃−𝑞̇₂+⁴_𝑖=1𝑒_𝑖𝜎_𝑖𝑒_𝑖. (18) Design the actual controllers 𝑣_𝑗 and parameters update laws of 𝑊_𝑗, 𝑎_𝑖, 𝑑_𝑖(𝑖=1,2,3), 𝑏_𝑗(𝑗=1,2,3,4), and 𝑞_𝑘(𝑘=1,2) as follows: 𝑣₁=−𝑙₁𝑒₁−𝜎₁𝑒₁−ℎ₁−𝑎₁𝑒₁+𝑎₂𝑒₂+𝑒₃+𝑎₃𝑥²₂−𝑎₃𝑥²₁+𝑑₃𝑒₄−𝑎₁ℎ₁+𝑎₂ℎ₂+ℎ₃+𝑑₃ℎ₄−𝑊^𝑇₁𝜑₁𝑥₂−sgn𝑒₁𝜀^∗₁+𝛼₁,𝑣₂=−𝑙₂𝑒₂−𝜎₂𝑒₂−ℎ₂+𝑏₁𝑒₂+𝑏₂𝑒₃−𝑏₄𝑒₁+𝑏₃𝑥²₂−𝑏₃𝑥²₁−𝑏₃𝑥₂𝑧₂+𝑏₃𝑥₁𝑧₁+𝑏₁ℎ₂+𝑏₂ℎ₃−𝑏₄ℎ₁−𝑊^𝑇₂𝜑₂𝑦₂−sgn𝑒₂𝜀^∗₂+𝛼₂,𝑣₃=−𝑙₃𝑒₃−𝜎₃𝑒₃−ℎ₃+𝑞₂𝑒₃−𝑞₁𝑥₂𝑧₂+𝑞₁𝑥₁𝑧₁+𝑞₂ℎ₃−𝑊^𝑇₃𝜑₃𝑧₂−sgn𝑒₃𝜀^∗₃+𝛼₃,𝑣₄=−𝑙₄𝑒₄−𝜎₄𝑒₄−ℎ₄−𝑑₁𝑒₁+𝑑₂𝑒₄−𝑑₁ℎ₁+𝑑₂ℎ₄−𝑊^𝑇₄𝜑₄𝑤₂−sgn𝑒₄𝜀^∗₄+𝛼₄, (19)(20)(21)(22) where 𝑙_𝑖(𝑖=1,2,3,4) are positive design parameters. Consider the following: 𝑊̇₁=𝑒₁𝜑₁𝑥₂,𝑊̇₂=𝑒₂𝜑₂𝑦₂,𝑊̇₃=𝑒₃𝜑₃𝑧₂,𝑊̇₄=𝑒₄𝜑₄𝑤₂,𝑎̇₁=𝑒²₁+𝑒₁ℎ₁,𝑎̇₂=𝑒₁𝑒₂+𝑒₃−𝑒₁ℎ₂+ℎ₃,𝑎̇₃=−𝑒₁𝑥²₂+𝑒₁𝑥²₁,̇̂𝑏₁=−𝑒²₂−𝑒₂ℎ₂,̇̂𝑏₂=−𝑒₂𝑒₃−𝑒₂ℎ₃,̇̂𝑏₃=−𝑒₂𝑥²₂+𝑒₂𝑥²₁+𝑒₂𝑥₂𝑧₂−𝑒₂𝑥₁𝑧₁,̇̂𝑏₄=𝑒₂𝑒₁+𝑒₂ℎ₁,̇𝑑₁=𝑒₄𝑒₁+𝑒₄ℎ₄+𝑒₄ℎ₁,̇𝑑₂=−𝑒²₄,̇𝑑₃=−𝑒₁𝑒₄+ℎ₄,𝑞̇₁=𝑒₃𝑥₂𝑧₂−𝑒₃𝑥₁𝑧₁,𝑞̇₂=−𝑒²₃−𝑒₃ℎ₃. (23)(24)(25)(26)(27)(28)(29)(30)(31)(32)(33)(34)(35)(36)(37)(38) Substituting (19)–(38) into (18) results in ℓ𝑉≤−𝑙₁𝑒²₁−𝑙₂𝑒²₂−𝑙₃𝑒²₃−𝑙₄𝑒²₄. (39)

From (39) and Lemma 3, we can conclude that the states 𝑥₂, 𝑦₂, 𝑧₂, and 𝑤₂ of response system (2) and the states 𝑥₁, 𝑦₁, 𝑧₁, and 𝑤₁ of drive system (3) are ultimately synchronized asymptotically in probability.

5. Simulation Results

In this section, external perturbations 𝑝_𝑖(𝑒_𝑖)=𝑒_𝑖; uncertain nonlinear 𝑓₁(𝑥₂)=0.1𝑥²₂, 𝑓₂(𝑦₂)=0.1𝑦²₂, 𝑓₃(𝑧₂)=𝑧₂, and 𝑓₄(𝑤₂)=𝑤₂. Consider 𝛼₁=𝛼₂=𝛼₂=𝛼₂=0.1 and 𝜀^∗₁=𝜀^∗₂=𝜀^∗₃=𝜀^∗₄=0.1. RBFNNs, 𝑊^𝑇_𝑖𝜑_𝑖(⋅), contain 25 nodes, with centers evenly spaced in [−4,4] and width is 2. 𝑊̇(𝑡) is assumed to be Gaussian white noise with zero mean and variance 1.0.

The initial values are chosen as 𝑥₁(0)=0.1, 𝑦₁(0)=−0.8, 𝑧₁(0)=0.2, 𝑤₁(0)=0.1, 𝑥₂(0)=0.4, 𝑦₂(0)=0.1, 𝑧₂(0)=0.6, and 𝑤₂(0)=−0.3, and the other initial values are chosen as zeros. The saturation values are 𝑢_2𝑀=5, 𝑢_3𝑀=2, and 𝑢_4𝑀=2. Design parameters in controllers are 𝑙₁=20, 𝑙₂=20, 𝑙₃=20, and 𝑙₄=20. The simulation results are shown in Figures 4, 5, 6, 7, 8, 9, 10, and 11.

Figure 4 

The trajectory of 𝑒1.

Figure 5 

The trajectory of 𝑒2.

Figure 6 

The trajectory of 𝑒3.

Figure 7 

The trajectory of 𝑒4.

Figure 8 

The trajectories of 𝑣1 (solid line) and 𝑢1 (dash-dotted line).

Figure 9 

The trajectories of 𝑣2 (solid line) and 𝑢2 (dash-dotted line).

Figure 10 

The trajectories of 𝑣3 (solid line) and 𝑢3 (dash-dotted line).

Figure 11 

The trajectories of 𝑣4 (solid line) and 𝑢4 (dash-dotted line).

Remark 4. It is worth pointing out that the method of [10] cannot be used to control the systems of this paper. There exist three reasons: (i) the system of this paper is four dimensional and the system in [10] is three dimensional; (ii) the system of this paper contains stochastic disturbances, and the system in [10] does not contain them; (iii) the controlled response system of this paper contains unknown nonlinearities, and [10] does not contain them.

6. Conclusions

This paper has solved the synchronization problems of a class of unknown parameters four-dimensional energy resource system. The main features of the proposed algorithm are that (i) the problems of the input constraint have been solved by employing a new auxiliary system; (ii) the unknown nonlinearities and stochastic disturbances that existed in the response system have been overcome by the neural networks and some special robust terms, respectively; (iii) the stability of the energy resource demand-supply system has been guaranteed based on stochastic Lyapunov theory.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (no. 51308275) and the Foundation of Liaoning Educational Committee (no. L2012225).