CN110568759A - robust synchronization control method of fractional order chaotic system - Google Patents

Abstract

the invention provides a robust synchronization control method of a fractional order chaotic system, which comprises the steps of establishing a dynamic equation of a fractional order chaotic system model; designing an integral slip-form surface, the slip-form surface designed to: according to the continuous change of the current state of the fractional order chaotic system, the system moves according to the state track of the sliding mode preset by the sliding mode surface, and the synchronization is achieved; and designing a sliding mode controller according to the sliding mode surface, wherein the sliding mode controller enables the fractional order chaotic system to reach the sliding mode surface. The invention considers disturbance and uncertainty and has stronger practicability.

Description

Robust synchronization control method of fractional order chaotic system

Technical Field

the invention belongs to a fractional order chaotic system control technology, and particularly relates to a robust synchronization control method of a fractional order chaotic system.

Background

Research shows that most physical systems in real application are fractional order systems, and fractional calculus can describe the dynamic characteristics of the systems more exactly compared with the traditional integer order systems. In addition, in the past decades, chaos and the engineering technical field present closer relation, so that research of the fractional order chaotic system gradually becomes a hot topic in the field range of research science. A plurality of working cases for researching synchronous control related modeling, simulation and analysis of the fractional order chaotic system appear at home and abroad. Document 1(Peng GJ, Jiang YL, Chen F. generalized projected synchronization of fractional order systems. Phys A Stat Mech Appl 2008; 387: 3738-46.) for a class of fractional order systems, a generalized projection method is used to synchronize the systems. Document 2(Wang Q, Dong-Lian Q. synchronization for fractional order systematic systems with fractional order parameters. int J Control Autom Syst 2016; 14:211.) designs a synchronous controller based on fractional order Lyapunov stability theory for an uncertain fractional order chaotic system. Document 3 (kuntaneadredas. adaptive control of fractional-order coherent systems using adaptive control. nonlinear Dyn 2016; 84: 2505-15.) proposes an adaptive synchronization method for fractional order chaotic systems based on a passive control method. Document 4(Lin TC, Lee TY. Chaossingsynchronization of uncertain fractional-order systems with time delay based on adaptive Fuzzy scaling mode control. IEEE Trans Fuzzy Syst 2011; 19: 623-35.) designs an adaptive Fuzzy controller for a fractional order chaotic system with time delay.

From the relevant published documents at present, most of the object models researched by the documents are ideal fractional order chaotic systems, the disturbance and the uncertainty are not considered, or the uncertainty is considered in only one input of a multi-state system, while most of the real systems present the uncertainty and have the disturbance, and the disturbance and the uncertainty are not considered and are difficult to be applied in the actual systems.

disclosure of Invention

the invention aims to provide a robust synchronization control method of a fractional order chaotic system.

The technical solution for realizing the purpose of the invention is as follows: a robust synchronization control method of a fractional order chaotic system comprises the following steps:

Step 1, establishing a dynamic equation of a fractional order chaotic system model;

Step 2, designing an integral sliding mode surface, wherein the sliding mode surface is designed as follows:

According to the continuous change of the current state of the fractional order chaotic system, the system moves according to the state track of the sliding mode preset by the sliding mode surface, and the synchronization is achieved;

and 3, designing a sliding mode controller according to the sliding mode surface, wherein the sliding mode controller enables the fractional order chaotic system to reach the sliding mode surface.

Preferably, the dynamic equation of the fractional order chaotic system model established in step 1 includes:

Main system dynamic equation:

wherein X ═ X₁,x₂,...,x_n]^T∈RⁿIs the state of the master system, f_i(X, t) { i ═ 1,2, 3.., n } is a differentiable, nonlinear function of the primary system states, α ∈ (0,1) is the order of differentiation;

From the system dynamics equation:

wherein Y ═ Y₁,y₂,...,y_n]^T∈Rⁿas a slave system state, f_i(Y, t) is a differentiable nonlinear function from the system state, Δ f_i(Y, t) is the unknown bounded uncertainty, d_i(t) is an external disturbance, U (t) ═ U₁(t),u₂(t),...,u_n(t)]^T∈RⁿIs a control input;

error system dynamic equation:

wherein D is_i(Y,t)＝Δf_i(Y,t)+d_i(t) unknown uncertainty, e_i＝y_i-x_i1,2,3, n, y is an error_iTo slave system state, x_iis the main system state.

preferably, the integral sliding mode surface designed in step 2 is specifically:

s_i(t)＝D^α-1e_i+∫(λ_ie_i+λ_isgn(e_i))dτ,i＝1,2,...,n

wherein e is_iis the error state, λ_i> 0, the sgn (. cndot.) function is defined as follows:

preferably, the sliding mode controller designed according to the sliding mode surface in the step 3 is as follows:

|Δf_i(Y,t)+d_i(t)|≤γ_i

Wherein, γ_iis an unknown normal number that is not known,representing uncertainty y_iIs determined by the estimated value of (c),representing the error between the estimated value and the actual value, mu_iIs a designed normal number, becauseIs unknown, so a fractional order adaptation rate is used:

Where ξ is a normal number.

Compared with the prior art, the invention has the following remarkable advantages: the invention fully considers two factors of uncertainty and disturbance, and improves the reliability of control and synchronization of the fractional order chaotic system.

Drawings

FIG. 1 shows x in 2-dimensional FO-Duffing Holmes chaotic system₁,y₁And (5) synchronizing the effect graphs.

FIG. 2 is a diagram of x in a 2-dimensional FO-Duffing Holmes chaotic system₂,y₂and (5) synchronizing the effect graphs.

FIG. 3 is a schematic diagram of the error system of the 2-dimensional FO-Duffing Holmes chaotic system converging in a finite time.

Detailed Description

a robust synchronization control method of a fractional order chaotic system comprises the following specific steps:

Step 1, establishing a dynamic equation of a fractional order chaotic system model, wherein the dynamic equation comprises a dynamic equation of a master system, a dynamic equation of a slave system and a dynamic equation of a synchronous error system, and the dynamic equations respectively comprise the following concrete steps:

Dynamic equations of the master system:

Wherein X ═ X₁,x₂,...,x_n]^T∈RⁿIs the state of the master system, f_i(X, t) { i ═ 1,2, 3.. times, n } is a differentiable, nonlinear function of the primary system states, and α ∈ (0,1) is the order of differentiation.

Dynamic equations of the slave system:

Wherein Y is [ Y ═ Y₁,y₂,...,y_n]^T∈RⁿAs a slave system state, f_i(Y, t) is a differentiable nonlinear function of the slave system state, given above, subject to an unknown bounded uncertainty Δ f from the chaotic system_i(Y, t) and external disturbance d_i(t) interference, U (t) ═ U₁(t),u₂(t),...,u_n(t)]^T∈RⁿIs a control input.

Dynamic equation of error system:

wherein D_i(Y,t)＝Δf_i(Y,t)+d_i(t) represents the unknown uncertainty, e_i＝y_i-x_i1,2,3,.. n } is an error.

Step 2, designing an integral sliding mode surface, wherein the sliding mode surface is designed as follows:

According to the continuous change of the current state of the fractional order chaotic system, the system moves according to the state track of the sliding mode preset by the sliding mode surface, and the synchronization is achieved;

s_i(t)＝D^α-1e_i+∫(λ_ie_i+λ_isgn(e_i))dτ,i＝1,2,...,n

Wherein e_iIs the error state, λ_i> 0, the sgn (. cndot.) function is defined as follows:

From s_i(t) ═ 0 is derived:

D^α-1e_i＝-∫(λ_ie_i+λ_isgn(e_i))

D^αe_i＝-(λ_ie_i+λ_isgn(e_i))

the system is proved to be stable under the action of the sliding mode surface. The process is as follows:

Leading:

Wherein eta_iIs a normal number.

Selecting a Lyapunov function as V_1i＝e_i ²and, the two sides of the equal equation are integrated:

It is obvious that

Derived from the theory of quotation

D^αV_1i≤e_iD^αe_i+η_i|e_i|

≤-e_iλ_i(e_i+sgn(e_i))+η_i|e_i|

≤-λ_i(e_i ²+|e_i|)+η_i|e_i|

≤-(λ_i-η_i)|e_i|

Wherein is selected from_i＞η_ithe derivative of the Lyapunov function is determined negatively, and the system is stable.

the system is proved to be converged in a limited time, and the process is as follows:

to D^αV_1i≤-(λ_i-η_i)|e_ii both sides from arrival time T_irTo the end time T_isIntegrating to obtain:

because of V_1i(T_is) Is equal to 0, to obtain

the resulting system converges in a limited time. Parameter M_i＝D^-α|e_i|。

and 3, designing a sliding mode controller according to the sliding mode surface, wherein the sliding mode controller enables the fractional order chaotic system to reach the sliding mode surface. The sliding mode controller is as follows:

Suppose that: | Δ f_i(Y,t)+d_i(t)|≤γ_i

Wherein, γ_iis an unknown normal number that is not known,Representative uncertainty y_iis determined by the estimated value of (c),representing the error between the estimated value and the actual value. Mu.s_iIs a designed normal number, becauseis unknown, so a fractional order adaptation rate is used:

Where ξ is a normal number.

proving that the derivative of the lyapunov function is negative:

Choosing Lyapunov function asIntegrating two sides to obtain:

the method is simplified to obtain:

it is obvious that

The derivative of the Lyapunov function is negative, and the system can reach the sliding mode surface.

Example 1

A robust synchronization control method of a fractional order chaotic system comprises the following specific steps:

step 1, selecting a 2-dimensional FO-Duffing Holmes chaotic system as an example, and establishing a system model:

A main system:

And the slave system:

the parameters are selected from a-0.25, b-0.3 and α -0.98. The uncertainty vector is:

step 2, establishing a slip form surface:

s_i(t)＝D^α-1e_i+∫(λ_ie_i+λ_isgn(e_i))dτ,i＝1,2,...,n

Wherein e_i＝y_i-x_iparameter λ_i＝0.2。

Step 3, establishing a self-adaptive sliding mode controller, adding the self-adaptive sliding mode controller into an error system state equation to enable the system to reach a sliding mode surface:

wherein the parameter mu_i＝2，ξ＝0.5。

Under the control of the designed adaptive sliding mode controller, as shown in FIG. 1, the synchronization effect of x1 and y1 is good, and as shown in FIG. 2, the synchronization effect of x2 and y2 is good. And it is seen from fig. 3 that the synchronization error converges to 0 within a finite time.

Claims (4)

1. A robust synchronization control method of a fractional order chaotic system is characterized by comprising the following steps:

Step 1, establishing a dynamic equation of a fractional order chaotic system model;

step 2, designing an integral sliding mode surface, wherein the sliding mode surface is designed as follows:

According to the continuous change of the current state of the fractional order chaotic system, the system moves according to the state track of the sliding mode preset by the sliding mode surface, and the synchronization is achieved;

And 3, designing a sliding mode controller according to the sliding mode surface, wherein the sliding mode controller enables the fractional order chaotic system to reach the sliding mode surface.

2. The robust synchronous control method of the fractional order chaotic system according to claim 1, wherein the dynamic equation of the fractional order chaotic system model established in the step 1 comprises:

Main system dynamic equation:

wherein X ═ X₁,x₂,...,x_n]^T∈RⁿIs the state of the master system, f_i(X, t) { i ═ 1,2, 3.., n } is a differentiable, nonlinear function of the primary system states, α ∈ (0,1) is the order of differentiation;

from the system dynamics equation:

wherein Y ═ Y₁,y₂,...,y_n]^T∈RⁿAs a slave system state, f_i(Y, t) is a differentiable nonlinear function from the system state, Δ f_i(Y, t) is the unknown bounded uncertainty, d_i(t) is an external disturbance, U (t) ═ U₁(t),u₂(t),...,u_n(t)]^T∈RⁿIs a control input;

error system dynamic equation:

Wherein D is_i(Y,t)＝Δf_i(Y,t)+d_i(t) unknown uncertainty, e_i＝y_i-x_i1,2,3, n, y is an error_iTo slave system state, x_iIs the main system state.

3. the robust synchronization control method of the fractional order chaotic system according to claim 1, wherein the integral sliding mode surface designed in the step 2 is specifically:

s_i(t)＝D^α-1e_i+∫(λ_ie_i+λ_isgn(e_i))dτ,i＝1,2,...,n

wherein e is_iIs the error state, λ_i> 0, the sgn (. cndot.) function is defined as follows:

4. The robust synchronization control method of the fractional order chaotic system according to claim 1, wherein a sliding mode controller designed according to a sliding mode surface in step 3 is:

|Δf_i(Y,t)+d_i(t)|≤γ_i

Wherein, γ_iis an unknown normal number that is not known,Representing uncertainty y_iIs determined by the estimated value of (c),Representing the error between the estimated value and the actual value, mu_iis a designed normal number, becauseIs unknown, so a fractional order adaptation rate is used:

Where ξ is a normal number.