Robust Adaptive Synchronization of the Energy Resource System with Constraint

Abstract

Two different chaos synchronization methods are proposed for a class of energy resource demand supply-system with input constraint. Firstly, chaotic synchronization is achieved for a class of energy resource demand supply system with known system parameters based on the Lyapunov theory. Secondly, an adaptive control approach is investigated for a class of energy resource demand supply system with input constraint, and the parameters of the system are unknown based on the Lyapunov stability and robust adaptive theory. To address the input constraint, new auxiliary signals and design systems are employed. Numerical simulations are provided to illustrate the effectiveness of the proposed approach.

1. Introduction

Energy resource system is a kind of complex nonlinear system. Energy resource including coal, petroleum, natural gas, water and electricity, and nuclear power can be classified as renewable energy and nonrenewable energy according to the capability of sustainable utilization. The issue of energy supply and demand has been valued worldwide with increasing development of economy. One of the most noticeable problems in the field of energy resource is how to study energy resource system deeply through nonlinear dynamics, which is currently a rapid developing method [1].

Reference [2] established a three-dimensional energy resource demand-supply system based on the real energy resources demand-supply in the East and the West of China. Furthermore, by adding a new variable to consider the renewable resources, a four-dimensional energy resource system was proposed in [3]. 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 [1]. 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]. 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 [6].

Although the adaptive synchronization control has achieved a great progress, the aforementioned control approaches assume that all the components of the considered energy resource demand-supply systems are in good operating conditions. As we know, many control systems have constraints on their inputs in the forms of input saturation or dead zone [7–17]. In practice, input saturation constraint is one of the most important input constraints which usually appear in many industries control systems. There are two main motivations for the saturation studies. One is that saturation is a potential problem for actuators of control systems. It often severely degrades the system performance, gives rise to undesirable inaccuracy, or even affects system stability. The other is that the control actions are usually limited in energy or magnitude; the saturation of the control output is necessary in practice.

Motivated by the above observations, two different chaos synchronization control methods are proposed for a class of energy resource demand-supply system with input constraint. Based on Lyapunov stability and robust adaptive theory, on the assumption that all the parameters of the system are known and unknown, nonadaptive and adaptive control approaches are proposed to make the states of two chaotic systems asymptotic synchronization. The main contributions of the proposed algorithm are that (i) the problems of the input constraint are solved by employing a new auxiliary system; (ii) the stability of the energy resource demand-supply system is guaranteed based on the Lyapunov theory.

2. Energy Resource Chaotic System

In the paper, we consider the following energy resource system (see [2, 6]): 𝑥̇=𝑎₁𝑥𝑥1−𝑀−𝑎₂,̇𝑦+𝑧𝑦=−𝑏₁𝑦−𝑏₂𝑧+𝑏₃𝑥[(,̇𝑁−𝑥−𝑧)]𝑧=𝑐₁𝑧𝑐₂𝑥−𝑐₃, (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) is the energy resource import in A region; 𝑀, 𝑁, 𝑎_𝑖, 𝑏_𝑗 (𝑖=1,2, 𝑗=1,2,3) are parameters that are all positive real.

Similar to [6], when the system parameters are taken as the following values, this system exhibits chaotic behavior: 𝑀=1.8, 𝑁=1, 𝑎₁=0.09, 𝑎₂=0.15, 𝑏₁=0.06, 𝑏₂=0.082, 𝑏₃=0.07, 𝑐₁=0.2, 𝑐₂=0.5, and 𝑐₃=0.4. Without the particular statement, these values are adopted in this whole paper. Figure 1 shows the phase portrait with initial conditions of 𝑥(0)=0.82, 𝑦(0)=0.29, and 𝑧(0)=0.48.

Figure 1 

Energy resource attractor.

3. Synchronization of the Energy Resource System

In this section, we will design a controller in order to make the response system trace the drive system. In order to obtain synchronization of the energy resource system (1), the drive system with subscript 1 is written as 𝑥̇₁=𝑎₁𝑥₁𝑥1−¹𝑀−𝑎₂𝑦₁+𝑧₁,𝑦̇₁=−𝑏₁𝑦₁−𝑏₂𝑧₁+𝑏₃𝑥₁𝑁−𝑥₁−𝑧₁,𝑧̇₁=𝑐₁𝑧₁𝑐₂𝑥₁−𝑐₃. (2) The controlled response system with subscript 2 can be expressed as 𝑥̇₂=𝑎₁𝑥₂𝑥1−²𝑀−𝑎₂𝑦₂+𝑧₂+𝑢₁𝑣₁(𝑡),𝑦̇₂=−𝑏₁𝑦₂−𝑏₂𝑧₂+𝑏₃𝑥₂𝑁−𝑥₂−𝑧₂+𝑢₂𝑣₂(𝑡),𝑧̇₂=𝑐₁𝑧₂𝑐₂𝑥₂−𝑐₃+𝑢₃𝑣₃(𝑡), (3) where 𝑣_𝑖 (𝑖=1,2,3) is the controller inputs to be designed, 𝑢_𝑖(𝑣_𝑖(𝑡)) (𝑖=1,2,3) denotes the plant input subject to saturation type nonlinearly.

Remark 1. If no input saturation (i.e., 𝑢_𝑖(𝑣_𝑖(𝑡))=𝑣_𝑖(𝑡)) (𝑖=1,2,3) is included in (3), then (3) becomes the chaotic systems studied widely; see [2, 6].
𝑢(𝑣(𝑡)) is described by 𝑢_𝑖𝑣_𝑖(𝑡)=sat𝑣_𝑖(𝑡)=sign𝑣_𝑖(𝑡)𝑢_𝑀,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)). Clearly, the relationship between the applied control 𝑢_𝑖(𝑣_𝑖(𝑡)) and the control input 𝑣_𝑖(𝑡) has a sharp corner when |𝑣_𝑖(𝑡)|=𝑢_𝑖𝑀. Similar to [15], define 𝑒₁=𝑥₂−𝑥₁−ℎ₁,𝑒₂=𝑦₂−𝑦₁−ℎ₂,𝑒₃=𝑧₂−𝑧₁−ℎ₃, (5) where ℎ_𝑖 (𝑖=1,2,3) are filter signals and will be given later.
By using (2), (3), and (5), the error dynamical system can be written as 𝑒₁̇=−ℎ̇₁+𝑎₁𝑒₁−𝑎₂𝑒₂+𝑒₃−𝑎₁𝑥²₂𝑀+𝑎₁𝑥²₁𝑀+𝑎₁ℎ₁+𝑎₂ℎ₂+ℎ₃+𝑢₁𝑣₁(𝑡),𝑒₂̇=−ℎ̇₂−𝑏₁𝑒₂−𝑏₂𝑒₃+𝑏₃𝑁𝑒₁−𝑏₃𝑥²₂+𝑏₃𝑥²₁+𝑏₃𝑥₂𝑧₂−𝑏₃𝑥₁𝑧₁+𝑏₁ℎ₂+𝑏₂ℎ₃−𝑏₃𝑁ℎ₁+𝑢₂𝑣₂(𝑡),𝑒₃̇=−ℎ̇₃−𝑐₁𝑐₃𝑒₃+𝑐₁𝑐₂𝑥₂𝑧₂−𝑐₁𝑐₂𝑥₁𝑧₁+𝑐₁𝑐₃ℎ₃+𝑢₃𝑣₃(𝑡). (6) Similar to [18, 19], choose Lyapunov function candidate 𝑉 as 1𝑉=2𝑒²₁+12𝑒²₂+12𝑒²₃. (7) The time derivative of 𝑉 is 𝑉̇=𝑒₁𝑒₁̇+𝑒₂𝑒₂̇+𝑒₃𝑒₃̇. (8) Define the dynamic system as ℎ̇_𝑖=−ℎ_𝑖+𝑢_𝑖−𝑣_𝑖,𝑖=1,2,3. (9) Substituting (6) and (9) into (8) results in 𝑉̇=𝑒₁ℎ₁+𝑣₁+𝑎₁𝑒₁−𝑎₂𝑒₂+𝑒₃−𝑎₁𝑥²₂𝑀+𝑎₁𝑥²₁𝑀+𝑎₁ℎ₁+𝑎₂ℎ₂+ℎ₃+𝑒₂ℎ₂+𝑣₂−𝑏₁𝑒₂−𝑏₂𝑒₃+𝑏₃𝑁𝑒₁−𝑏₃𝑥²₂+𝑏₃𝑥²₁+𝑏₃𝑥₂𝑧₂−𝑏₃𝑥₁𝑧₁+𝑏₁ℎ₂+𝑏₂ℎ₃−𝑏₃𝑁ℎ₁+𝑢₂+𝑒₃ℎ₃+𝑣₃−𝑐₁𝑐₃𝑒₃+𝑐₁𝑐₂𝑥₂𝑧₂−𝑐₁𝑐₂𝑥₁𝑧₁+𝑐₁𝑐₃ℎ₃. (10) By using Young’s inequality, we have −𝑎₂𝑒₁𝑒₂+𝑒₃≤𝑎²₂𝑒²₁+12𝑒²₂+12𝑒²₃,𝑒₂−𝑏₂𝑒₃+𝑏₃𝑁𝑒₁≤𝑏²₂+𝑏²₃𝑁²2𝑒²₂+12𝑒²₃+12𝑒²₁. (11) Substituting (11) into (10) results in 𝑉̇≤𝑎₁+𝑎²₂+12𝑒²₁+𝑏₁+12+𝑏²₂+𝑏²₃𝑁²2𝑒²₂+1−𝑐₁𝑐₃𝑒²₃+𝑒₁ℎ₁+𝑣₁−𝑎₁𝑥²₂𝑀+𝑎₁𝑥²₁𝑀+𝑎₁ℎ₁+𝑎₂ℎ₂+ℎ₃+𝑒₂ℎ₂+𝑣₂−𝑏₃𝑥²₂+𝑏₃𝑥²₁+𝑏₃𝑥₂𝑧₂−𝑏₃𝑥₁𝑧₁+𝑏₁ℎ₂+𝑏₂ℎ₃−𝑏₃𝑁ℎ₁+𝑒₃ℎ₃+𝑣₃+𝑐₁𝑐₂𝑥₂𝑧₂−𝑐₁𝑐₂𝑥₁𝑧₁+𝑐₁𝑐₃ℎ₃. (12) Choose the actual controllers 𝑣_𝑖𝑣₁=−𝑙₁𝑒₁−ℎ₁+𝑎₁𝑥²₂𝑀−𝑎₁𝑥²₁𝑀−𝑎₁ℎ₁−𝑎₂ℎ₂+ℎ₃,𝑣₂=−𝑙₂𝑒₂−ℎ₂+𝑏₃𝑥²₂−𝑏₃𝑥²₁−𝑏₃𝑥₂𝑧₂+𝑏₃𝑥₁𝑧₁−𝑏₁ℎ₂−𝑏₂ℎ₃+𝑏₃𝑁ℎ₁,𝑣₃=−𝑙₃𝑒₃−ℎ₃−𝑐₁𝑐₂𝑥₂𝑧₂+𝑐₁𝑐₂𝑥₁𝑧₁−𝑐₁𝑐₃ℎ₃, (13) where 𝑙_𝑖 (𝑖=1,2,3) are positive design parameters. Substituting (13) into (12) results in 𝑉̇≤−𝑙₁−𝑎₁−𝑎²₂−12𝑒²₁−𝑙₂−𝑏₁−12−𝑏²₂+𝑏²₃𝑁²2𝑒²₂−𝑙₃+𝑐₁𝑐₃−1𝑒²₃. (14) Let 𝑙_𝑖 satisfy 𝑙₁>𝑎₁+𝑎²₂+1/2,  𝑙₂>𝑏₁+1/2+(𝑏²₂+𝑏²₃𝑁²)/2, and 𝑙₃+𝑐₁𝑐₃>1. We can obtain 𝑉̇≤0, and it is concluded that 𝑒₁, 𝑒₂, and 𝑒₃ converge to zero as time 𝑡 tends to infinity. Therefore, the synchronization of response systems (3) and the drive system (2) is finally achieved.

Remark 2. It is noted that the controller in [6] was designed by using the stability criterion of linear system, not based on Lyapunov stability theory. However, this paper designed the control laws based on Lyapunov method. This paper can use a constructive way to obtain the control laws for this class of energy resource demand-supply system. In addition, the control laws’ design parameters of this paper have fewer restrictions compared to [6].

4. Simulation Results 1

In this section, the initial values of are chosen as 𝑥₁(0)=0.82, 𝑦₁(0)=0.29, 𝑧₁(0)=0.48, 𝑥₂(0)=0.69, 𝑦₂(0)=−0.03, 𝑧₂(0)=1.25. The saturation values are 𝑢_1𝑀=0.5, 𝑢_2𝑀=2, and 𝑢_3𝑀=2. The design parameters in controllers are chosen as 𝑙₁=20, 𝑙₂=10, and 𝑙₃=10. The simulation results are shown in Figures 2, 3, 4, and 5.

Figure 2 

The trajectories of 𝑒1 (solid line), 𝑒2 (star line), and 𝑒3 (dash-dotted line).

Figure 3 

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

Figure 4 

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

Figure 5 

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

5. Adaptive Synchronization of the Energy Resource System

In this section, we assume that all the parameters of the energy resource system are unknown. For convenience, we define 𝑎₃=𝑎₁/𝑀, 𝑏₄=𝑏₃𝑁, 𝑑₁=𝑐₁𝑐₂, and 𝑑₂=𝑐₁𝑐₃. The energy resource system (1) can be rewritten as: 𝑥̇=𝑎₁𝑥−𝑎₂𝑦+𝑧−𝑎₃𝑥²,𝑦̇=−𝑏₁𝑦−𝑏₂𝑧−𝑏₃𝑥(𝑥−𝑧)+𝑏₄̇𝑥,𝑧=𝑑₁𝑥𝑧−𝑑₂𝑧 (15) and the drive system can be also rewritten as 𝑥̇₁=𝑎₁𝑥₁−𝑎₂𝑦₁+𝑧₁−𝑎₃𝑥²₁,𝑦̇₁=−𝑏₁𝑦₁−𝑏₂𝑧₁−𝑏₃𝑥₁𝑥₁−𝑧₁+𝑏₄𝑥₁,𝑧̇₁=𝑑₁𝑥₁𝑧₁−𝑑₂𝑧₁. (16) The response system can be expressed as 𝑥̇₂=𝑎₁𝑥₂−𝑎₂𝑦₂+𝑧₂−𝑎₃𝑥²₂+𝑢₁𝑣₁(𝑡,𝑦̇₂)=−𝑏₁𝑦₂−𝑏₂𝑧₂−𝑏₃𝑥₂𝑥₂−𝑧₂+𝑏₄𝑥₂+𝑢₂𝑣₂(𝑡),𝑧̇₂=𝑑₁𝑥₂𝑧₂−𝑑₂𝑧₂+𝑢₃𝑣₃(𝑡). (17) Choose the same errors 𝑒_𝑖 as (5) and the same filter signals ℎ_𝑖 as (9), and we have 𝑒₁̇=−ℎ̇₁+𝑎₁𝑒₁−𝑎₂𝑒₂+𝑒₃−𝑎₃𝑥²₂+𝑎₃𝑥²₁+𝑎₁ℎ₁+𝑎₂ℎ₂+ℎ₃+𝑢₁𝑣₁(𝑡),𝑒₂̇=−ℎ̇₂−𝑏₁𝑒₂−𝑏₂𝑒₃+𝑏₄𝑒₁−𝑏₃𝑥²₂+𝑏₃𝑥²₁+𝑏₃𝑥₂𝑧₂−𝑏₃𝑥₁𝑧₁+𝑏₁ℎ₂+𝑏₂ℎ₃−𝑏₄ℎ₁+𝑢₂𝑣₂(𝑡),𝑒₃̇=−ℎ̇₃−𝑑₂𝑒₃+𝑑₁𝑥₂𝑧₂−𝑑₂𝑥₁𝑧₁+𝑑₁ℎ₃+𝑢₃𝑣₃(𝑡). (18) Choose the Lyapunov function candidate 𝑉 as 1𝑉=2𝑒²₁+𝑒²₂+𝑒²₃+𝑎²₁+𝑎²₂+𝑎²₃+𝑏̃²₁+𝑏̃²₂+𝑏̃²₃+𝑏̃²₄+𝑑²₁+𝑑²₂, (19) where 𝑎_𝑖=𝑎_𝑖−𝑎_𝑖 (𝑖=1,2,3), 𝑏̃_𝑗=𝑏_𝑗−𝑏̂_𝑗 (𝑗=1,2,3,4), and 𝑑_𝑘=𝑑_𝑘−𝑑_𝑘 (𝑘=1,2).

The time derivative of 𝑉 along with the solution of (18) is 𝑉̇‌=𝑒₁ℎ₁+𝑣₁+𝑎₁𝑒₁−𝑎₂𝑒₂+𝑒₃−𝑎₃𝑥²₂+𝑎₃𝑥²₁+𝑎₁ℎ₁+𝑎₂ℎ₂+ℎ₃+𝑒₂ℎ₂+𝑣₂−𝑏̂₁𝑒₂−𝑏̂₂𝑒₃+𝑏̂₄𝑒₁−𝑏̂₃𝑥²₂+𝑏̂₃𝑥²₁+𝑏̂₃𝑥₂𝑧₂−𝑏̂₃𝑥₁𝑧₁+𝑏̂₁ℎ₂+𝑏̂₂ℎ₃−𝑏̂₄ℎ₁+𝑒₃ℎ₃+𝑣₃−𝑑₂𝑒₃+𝑑₁𝑥₂𝑧₂−𝑑₂𝑥₁𝑧₁+𝑑₁ℎ₃+𝑎₁𝑒₁ℎ₁+𝑒²₁−𝑎̇₁+𝑎₂−𝑒₂+𝑒₃𝑒₁+ℎ₂+ℎ₃𝑒₁−𝑎̇₂+𝑎₃−𝑒₁𝑥²₂+𝑒₁𝑥²₁+𝑒₁ℎ₁+𝑒₁ℎ₂+ℎ₃−𝑎̇₃+𝑏̃₁−𝑒²₂+𝑒₂ℎ₂−̇̂𝑏₁+𝑏̃₂−𝑒₂𝑒₃+𝑒₂ℎ₃−̇̂𝑏₂+𝑏̃₃−𝑒₂𝑥²₂+𝑒₂𝑥²₁+𝑒₂𝑥₂𝑧₂−𝑒₂𝑥₁𝑧₁−̇̂𝑏₃+𝑏̃₄𝑒₁𝑒₂−𝑒₂ℎ₁−̇̂𝑏₄+𝑑₁𝑒₃𝑥₂𝑧₂+𝑒₃ℎ₃−̇𝑑₁+𝑑₂−𝑒²₃−𝑒₃𝑥₁𝑧₁−̇𝑑₂. (20) Choose the actual controllers 𝑣_𝑖 and update the laws of 𝑎_𝑖 (𝑖=1,2,3), 𝑏̂_𝑗 (𝑗=1,2,3,4), and 𝑑_𝑘 (𝑘=1,2) as follows: 𝑣₁=−𝑙₁𝑒₁−ℎ₁−𝑎₁𝑒₁+𝑎₂𝑒₂+𝑒₃+𝑎₃𝑥²₂−𝑎₃𝑥²₁−𝑎₁ℎ₁−𝑎₂ℎ₂+ℎ₃,𝑣₂=−𝑙₂𝑒₂−ℎ₂+𝑏̂₁𝑒₂+𝑏̂₂𝑒₃−𝑏̂₄𝑒₁+𝑏̂₃𝑥²₂−𝑏̂₃𝑥²₁−𝑏̂₃𝑥₂𝑧₂+𝑏̂₃𝑥₁𝑧₁−𝑏̂₁ℎ₂−𝑏̂₂ℎ₃+𝑏̂₄ℎ₁,𝑣₃=−𝑙₃𝑒₃−ℎ₃+𝑑₂𝑒₃−𝑑₁𝑥₂𝑧₂+𝑑₂𝑥₁𝑧₁−𝑑₁ℎ₃, (21) where 𝑙_𝑖 (𝑖=1,2,3) are positive design parameters: 𝑎̇₁=𝑒₁ℎ₁+𝑒²₁,𝑎̇₂=−𝑒₂+𝑒₃𝑒₁+ℎ₂+ℎ₃𝑒₁,𝑎̇₃=−𝑒₁𝑥²₂+𝑒₁𝑥²₁+𝑒₁ℎ₁+𝑒₁ℎ₂+ℎ₃,̇̂𝑏₁=−𝑒²₂+𝑒₂ℎ₂,̇̂𝑏₂=−𝑒₂𝑒₃+𝑒₂ℎ₃,̇̂𝑏₃=−𝑒₂𝑥²₂+𝑒₂𝑥²₁+𝑒₂𝑥₂𝑧₂−𝑒₂𝑥₁𝑧₁,̇̂𝑏₄=𝑒₁𝑒₂−𝑒₂ℎ₁,̇𝑑₁=𝑒₃𝑥₂𝑧₂+𝑒₃ℎ₃,̇𝑑₂=−𝑒²₃−𝑒₃𝑥₁𝑧₁. (22) Substituting (21) and (22) into (20) results in 𝑉̇‌=−𝑙₁𝑒²₁−𝑙₂𝑒²₂−𝑙₃𝑒²₃. (23)

From (23), we can conclude that the states 𝑥₂, 𝑦₂, and 𝑧₂ of response system (17) and the states 𝑥₁, 𝑦₁, and 𝑧₁ of drive system (16) are ultimately synchronized asymptotically.

6. Simulation Results 2

In this section, the initial values are chosen as 𝑥₁(0)=0.82, 𝑦₁(0)=0.29, 𝑧₁(0)=0.48, 𝑥₂(0)=0.69, 𝑦₂(0)=−0.03, and 𝑧₂(0)=1.25, and the other initial values are chosen as zeros. The saturation values are 𝑢_1𝑀=0.5, 𝑢_2𝑀=0.5, and 𝑢_3𝑀=2. The design parameters in controllers are 𝑙₁=5, 𝑙₂=4, and 𝑙₃=4. The simulation results are shown in Figures 6, 7, 8, and 9.

Figure 6 

The trajectories of 𝑒1 (solid line), 𝑒2 (star line), and 𝑒3 (dash-dotted line).

Figure 7 

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

Figure 8 

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

Figure 9 

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

7. Conclusions

For a class of known and unknown parameters for energy resource demand-supply system with input constraints, the chaos synchronization problems have been discussed using two different control methods, respectively. 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 stability of the energy resource demand-supply system has been guaranteed based on Lyapunov theory. The results all have demonstrated the validity and feasibility of the proposed approaches.

Acknowledgment

This work was supported by the Foundation of Liaoning Educational Committee (no. L2012225).