CN113777920B - Fractional order chaotic synchronization control method based on RBF-NN and observer - Google Patents
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Abstract
The invention discloses a fractional order chaotic synchronization control method based on RBF-NN and an observer, which comprises the following steps of S1: taking the fractional order chaotic system as a driving system, and establishing an observer response system model based on RBF-NN; s2: defining the error between the driving system and the observer response system asS3: the controller u (t) is designed such that, when t → infinity,i.e. the drive system and the observer response system are synchronized. Aiming at the fractional order chaotic system, the invention approximates the uncertain item by utilizing the neural network technology and the self-adaptive technology under the condition that the system state is only output to be measurable, and simultaneously combines the observer technology, thereby realizing the synchronization of a driving system and a response system, and effectively avoiding the problem of unbounded control quantity caused by the increase of parameter estimation errors; through numerical simulation verification, the fractional order chaotic synchronization control method is effective and feasible.
Description
Technical Field
The invention relates to the technical field of chaotic systems, in particular to a fractional order chaotic synchronous control method based on RBF-NN and an observer.
Background
The discovery of relativity, quantum mechanics and chaos is a three-time revolution in the field of physics, and the discovery of chaos has the effect of turning over the sky and over the earth on the traditional world view of people, and explains and changes the life of people. In the research field of nonlinear dynamics, chaotic synchronization research has been attracting more and more attention. The chaos theory and chaos synchronization have made breakthrough progress in the 90 th century of the 20 th century, so that the engineering application prospect of chaos control is becoming clear. For example, secret communication based on chaotic synchronization plays an important role in the military field due to the outstanding security performance, and has important influence on various aspects of finance, commerce and daily life of people; the application of the chaos synchronization theory in human biomedical science leads the research and development of medical instruments to have a new direction; the application of chaotic synchronization in the fields of semiconductor lasers, plasmas, photoelectric ion beams and the like has revealed the head angle; the chaotic synchronous orbit technology and the chaotic inverse control technology are utilized, the design of the flexible system is carried out by utilizing chaos, when the system moves to an area close to a target orbit, the system is captured onto the target orbit by using small external force, and the collision between the ISEE-3I/C of the spaceship and comet is successfully realized at present. In recent years, research emphasis is not limited to weak chaotic systems, such as Lorenz systems, chen systems and the like, but is turned to hyper-chaotic and fractional-order chaotic systems with more complex dynamic behaviors and more development prospects.
However, in performing actual control, the information of the uncertain term boundaries of the system is unknown in most cases, and the difficulty of control is greater than in the case where the uncertain term upper boundaries are known. How to design a synchronous controller to suppress the influence of uncertainty factors without the upper bound of uncertainty terms is an urgent problem to be solved. On the other hand, due to practical engineering limitations, the system state quantity is not fully measurable, and how to design the controller when the system state quantity is not fully measurable, even when only the output is measurable, is also a problem to be considered in the design. Currently, most chaotic synchronization researches adopt lumped processing on uncertain terms and external interference, known information is not fully utilized, and the estimation of the uncertain terms is completely dependent on neural network technology.
Disclosure of Invention
Aiming at the problems, the invention aims to provide a fractional order chaotic synchronous control method based on RBF-NN and an observer, which enables a driving system and a response system to achieve synchronous control under the condition that the system state is only output to be measurable, thereby effectively avoiding the problem that the control quantity is unbounded caused by the increase of parameter estimation errors.
In order to achieve the above purpose, the technical scheme adopted by the invention is as follows:
the fractional order chaotic synchronization control method based on RBF-NN and observer is characterized by comprising the following steps,
s1: taking the fractional order chaotic system as a driving system, and establishing an observer response system model based on RBF-NN;
s2: defining the error between the driving system and the observer response system as
S3: the controller u (t) is designed such that, when t → infinity,i.e. the drive system and the observer response system are synchronized.
Further, the fractional order chaotic system is that in the formula ,representing the Riemann-Liouville fractional derivative, Γ (·) is a Gamma function; order α= (α) 1 ，α 2 ，…，α n ) T ，α i ∈(0，1)；x∈R n Is an n-dimensional state vector of a fractional order chaotic system, f (x) epsilon R n A epsilon R is a known smooth nonlinear function n×n Is a known parameter matrix of the system; y E R n As state variable, C T Is a matrix of known parameters of the system.
Further, the specific operation of step S1 includes the steps of,
s101: definition a=a 0 +ΔA，f(x)＝f 0 (x) +Δf (x), wherein A 0 and f0 The nominal parts of the matrices A and f are respectively, and delta A and delta f represent the uncertainty of the system, and then the fractional order chaotic system is converted into
S102: taking the converted fractional order chaotic system as a driving system, and establishing an observer response system model in the formula ,/>D E R as the estimated value of x n Is unknown and limited external disturbance and meets the requirement that d is less than or equal to theta 1 ，θ 1 Is a positive constant; u (t) is the control input to be designed;
s103: definition g (x) =Δax+Δf (x), approximating g (x) with a neural network, i.e., g (x) =w *T Phi (x) +epsilon (x), then the estimate of g (x) in the observer response system is
S104: based on the estimated value of g (x) obtained in step S103, the observer response system in step S102 is rewritten to in the formula ,/>Is designed as positive constant, W is less than or equal to theta 2 ，ε(y)＝[ε 1 (y)，ε 2 (y)，...，ε n (y)] T ∈R n Boundary condition epsilon (t) theta is less than or equal to theta 3 ，θ 2 and θ3 Is an unknown constant.
Further, the specific operation of step S3 includes the steps of,
s301: combining the fractional order chaotic system in step S101, i.e. the driving system, and the observer response system in step S104, it is possible to obtain
S302: definition matrix A c ＝A 0 -KC T Selecting proper gain K to make A c Satisfy the following requirementsWherein ε is less than or equal to 0,>
s303: obtaining an equivalent frequency distribution model of the pseudo-error variable according to the linear frequency distribution model of the fractional order systemWherein μ (ω) is a weight function, μ α (ω)＝sin(απ)/πω α ，z(ω，t)∈R n Is the actual error variable;
s304: definition v (ω, t) =z T (ω，t)Pz(ω，t)，Selecting Lyapunov function as +.>Deriving it to obtain
wherein ,
s305: because d is less than or equal to theta 1 ，||W * ||≤θ 2 ，||ε(y)||≤θ 3 ThenWherein θ is an unknown positive constant;
s306: the design observer u (t) is in the form of in the formula ,λmax (P) is the maximum feature root of the symmetric array P,>an estimated value of θ; selecting proper K to enable A c For a Hurwitz matrix, the symmetric matrix P satisfies the inequality +.>Establishment;
s307: select the adaptive law as in the formula ,/>Is positive array, Γ θ ，γ 1 Gamma, gamma 2 Is a positive number; when f 0 (x) When the nonlinear vector function meeting Lipschitz conditions is satisfied, RBF-NN weight and parameter Γ are adjusted according to the adaptive law θ 、Γ W γ 1 and γ2 The method comprises the steps of carrying out a first treatment on the surface of the Error between driving system and observer response system +.>Asymptotically converges to within a small neighborhood of the origin.
Further, in step S303, the linear frequency distribution model of the fractional order system is as followsWherein, the fractional order system D α y (t) =v (t), 0 < α < 1, y (t) ∈r, v (t) ∈r, weight function μ α (ω)sin(απ)/πωα，System state z (ω, t) ∈r.
The beneficial effects of the invention are as follows:
the fractional order chaotic synchronous control method based on RBF-NN and observer in the invention aims at uncertain factors including parameter perturbation, unknown function, external disturbance and the like existing in the fractional order chaotic system, approximates uncertain items by combining a neural network technology and a self-adaptive technology, and simultaneously realizes the synchronization of a driving system and a response system by combining the observer technology, thereby effectively avoiding the problem of unbounded control quantity caused by the increase of parameter estimation errors; through numerical simulation verification, the fractional order chaotic synchronization control method is effective and feasible.
Drawings
FIG. 1 is a graph of error between a drive system and an observer response system over time in a simulation test of the present invention.
FIG. 2 is a graph showing the control rate u over time in a simulation test according to the present invention.
Detailed Description
In order to enable those skilled in the art to better understand the technical solution of the present invention, the technical solution of the present invention is further described below with reference to the accompanying drawings and examples.
The fractional order chaotic synchronization control method based on RBF-NN and observer comprises the following steps,
s1: taking the fractional order chaotic system as a driving system, and establishing an observer response system model based on RBF-NN;
specifically, the fractional order chaotic system is in the formula ,representing the Riemann-Liouville fractional derivative, Γ (·) is a Gamma function; order α= (α) 1 ，α 2 ，…，α n ) T ，α i ∈(0，1)；x∈R n Is an n-dimensional state vector of a fractional order chaotic system, f (x) epsilon R n Is alreadyKnown smooth nonlinear function, A.epsilon.R n×n Is a known parameter matrix of the system; y E R n As state variable, C T Is a matrix of known parameters of the system.
The specific operation of building an observer response system model, based on RBF-NN, includes the steps of,
s101: definition a=a 0 +ΔA，f(x)＝f 0 (x) +Δf (x), wherein A 0 and f0 The nominal parts of the matrices A and f are respectively, and delta A and delta f represent the uncertainty of the system, and then the fractional order chaotic system is converted into
S102: taking the converted fractional order chaotic system as a driving system, and establishing an observer response system model in the formula ,/>D E R as the estimated value of x n Is unknown and limited external disturbance and meets the requirement that d is less than or equal to theta 1 ，θ 1 Is a positive constant; u (t) is the control input to be designed;
s103: definition g (x) =Δax+Δf (x), approximating g (x) with a neural network, i.e., g (x) =w *T Phi (x) +epsilon (x), then the estimate of g (x) in the observer response system is
S104: based on the estimated value of g (x) obtained in step S103, the observer response system in step S102 is rewritten to in the formula ,/>Is designed as positive constant, W is less than or equal to theta 2 ，ε(y)＝[ε 1 (y)，ε 2 (y)，...，ε n (y)] T ∈R n Boundary condition epsilon (y) theta less than or equal to theta 3 ，θ 2 and θ3 Is an unknown constant.
Further, step S2: defining the error between the driving system and the observer response system as
Further, step S3: the controller u (t) is designed such that, when t → infinity,i.e. the drive system and the observer response system are synchronized.
Specifically, S301: combining the fractional order chaotic system in step S101, i.e. the driving system, and the observer response system in step S104, it is possible to obtain
S302: definition matrix A c ＝A 0 -KC T Selecting proper gain K to make A c Satisfy the following requirementsWherein ε is less than or equal to 0,>
s303: obtaining an equivalent frequency distribution model of the pseudo-error variable according to the linear frequency distribution model of the fractional order systemWherein μ (ω) is a weight function, μ α (ω)＝sin(απ)/πω α ，z(ω，t)∈R n Is the actual error variable;
the linear frequency distribution model of the fractional order system is thatWherein, the fractional order system D α y (t) =v (t), 0 < α < 1, y (t) ∈r, v (t) ∈r, weight function μ α (ω)＝sin(απ)/πω α The system state z (ω, t) ∈r.
S304: definition v (ω, t) =z T (ω，t)Pz(ω，t)，Selecting Lyapunov function as +.>Deriving it to obtain
wherein ,
s305: because d is less than or equal to theta 1 ，||W * ||≤θ 2 ，||ε(y)||≤θ 3 ThenWherein θ is an unknown positive constant;
s306: the design observer u (t) is in the form of in the formula ,λmax (P) is the maximum feature root of the symmetric array P,>an estimated value of θ; selecting proper K to enable A c For a Hurwitz matrix, the symmetric matrix P satisfies the inequality +.>Establishment;
s307: select the adaptive law as in the formula ,/>Is positive array, Γ θ ，γ 1 Gamma, gamma 2 Is a positive number; when f 0 (x) When the nonlinear vector function meeting Lipschitz conditions is satisfied, RBF-NN weight and parameter Γ are adjusted according to the adaptive law θ 、Γ W γ 1 and γ2 The method comprises the steps of carrying out a first treatment on the surface of the Error between driving system and observer response system +.>Asymptotically converges to within a small neighborhood of the origin.
Embodiment one:
in this embodiment, the observer design process in step S3 is demonstrated.
Defining the estimation error asTheir frequency domain distribution models are respectively
Selecting Lyapunov function asDeriving V with respect to time t
Due to f 0 (x) To satisfy the nonlinear vector function of Lipschitz condition, then
Substituting the above three formulas into the formula of V deriving with respect to time t, and taking into account the adaptive law in step S307, it is possible to obtain
Selecting appropriate parameters to enableIs obtained by using the theorem of median value of definite integralWherein ρ is a positive number,is a finite constant.
Thus, the synchronization error and the parameter estimation error are both exponentially converged into a small field.
Simulation test:
the driving system (fractional order chaotic system) model is selected as an uncertain ultra Lorenz system
The controlled response system is
The initial choice of drive system is x (0) = (1, 1) T The initial value of the response system is selected as (0.1,0.1,0.1,0.1) T When αi=0.98, i= (1, 2, 3), the driving system and the response system are chaotic, in which the function uncertainty term and the external disturbance term are as follows
Adaptive law gain selection as Γ θ ＝0.01，Γ W ＝diag{0.01，0.01，0.01，0.01}，γ 1 ＝2，γ 2 =0.7, the response system order is selected to be α=0.98, c=diag (1, 0), δ=0.01.
The simulation result of the time-varying error (synchronous error) between the driving system and the observer response system is shown in the figure 1, the simulation result of the time-varying control rate u is shown in the figure 2, and as can be seen from the figure 1, when the error is not 0 initially, the error is finally 0 after the control rate adjustment, which indicates that the response system and the driving system are synchronous, and the error rapidly tends to 0 to achieve the synchronization; as can be seen from fig. 2, the control rate=0, and no control is applied anymore in accordance with the time when the system synchronization error tends to 0, that is, the control rate u effectively synchronizes the driving system and the response system, after a lapse of about 2 seconds.
The foregoing has shown and described the basic principles, principal features and advantages of the invention. It will be understood by those skilled in the art that the present invention is not limited to the embodiments described above, and that the above embodiments and descriptions are merely illustrative of the principles of the present invention, and various changes and modifications may be made without departing from the spirit and scope of the invention, which is defined in the appended claims. The scope of the invention is defined by the appended claims and equivalents thereof.
Claims (3)
1. The fractional order chaotic synchronization control method based on RBF-NN and observer is characterized by comprising the following steps,
s1: taking the fractional order chaotic system as a driving system, and establishing an observer response system model based on RBF-NN;
s2: defining the error between the driving system and the observer response system as
S3: the controller u (t) is designed such that, when t → infinity,that is, the driving system and the observer response system are synchronous;
the fractional order chaotic system is in the formula ,/>Representing the Riemann-Liouville fractional derivative, Γ (·) is a Gamma function; order α= (α) 1 ，α 2 ，…，α n ) T ，α i ∈(0，1)；x∈R n Is an n-dimensional state vector of a fractional order chaotic system, f (x) epsilon R n A epsilon R is a known smooth nonlinear function n×n Is a known parameter matrix of the system; y E R n As state variable, C T Is a known parameter matrix of the system;
the specific operation of step S1 includes the following steps,
s101: definition a=a 0 +ΔA，f(x)＝f 0 (x) +Δf (x), wherein A 0 and f0 The nominal parts of the matrices A and f are respectively, and delta A and delta f represent the uncertainty of the system, and then the fractional order chaotic system is converted into
S102: taking the converted fractional order chaotic system as a driving system, and establishing an observer response system model in the formula ,/>D E R as the estimated value of x n Is unknown and limited external disturbance and meets the requirement that d is less than or equal to theta 1 ，θ 1 Is a positive constant; u (t) is the control input to be designed;
s103: definition g (x) =Δax+Δf (x), approximating g (x) with a neural network, i.e., g (x) =w *T Phi (x) +epsilon (x), then the estimate of g (x) in the observer response system is
S104: based on the estimated value of g (x) obtained in step S103, the observer response system in step S102 is rewritten to in the formula ,/>Is designed as positive constant, ||W * ||≤θ 2 ，ε y (y)＝[ε 1 (y)，ε 2 (y)，...，ε n (y)] T ∈R n Boundary condition epsilon (y) theta less than or equal to theta 3 ，θ 2 and θ３ Is an unknown constant.
2. The method for controlling fractional order chaotic synchronization based on RBF-NN and observer according to claim 1, wherein the specific operation of step S3 comprises the steps of,
s301: combining the fractional order chaotic system in step S101, i.e. the driving system, and the observer response system in step S104, it is possible to obtain
S302: definition matrix A c ＝A 0 -KC T Selecting proper gain K to make A c Satisfy the following requirementsWherein ε is less than or equal to 0,>
s303: obtaining an equivalent frequency distribution model of the pseudo-error variable according to the linear frequency distribution model of the fractional order systemWherein μ (ω) is a weight function, μ α (ω)＝sin(απ)/πω α ，z(ω，t)∈R n Is the actual error variable;
s304: definition v (ω, t) =z T (ω，t)Pz(ω，t)，Selecting Lyapunov function as +.>Deriving it to obtain
wherein ,
s305: because d is less than or equal to theta 1 ，||W * ||≤θ 2 ，||ε(y))||≤θ 3 ThenWherein θ is an unknown positive constant;
s306: design observer ut) is in the form of in the formula ,λmax (P) is the maximum feature root of the symmetric array P,>an estimated value of θ; selecting proper K to enable A c For a Hurwitz matrix, the symmetric matrix P satisfies the inequality +.>Establishment;
s307: select the adaptive law as in the formula ,/>Is positive array, Γ θ ，γ 1 Gamma, gamma 2 Is a positive number; when f 0 (x) When the nonlinear vector function meeting Lipschitz conditions is satisfied, RBF-NN weight and parameter Γ are adjusted according to the adaptive law θ 、Γ W γ 1 and γ2 The method comprises the steps of carrying out a first treatment on the surface of the Error between driving system and observer response system +.>Asymptotically converges to within a small neighborhood of the origin.
3. The method for controlling fractional order chaotic synchronization based on RBF-NN and observer according to claim 2, wherein the linear frequency distribution model of the fractional order system in step S303 is as followsWherein, the fractional order system D α y (t) =v (t), 0 < α < 1, y (t) ∈r, v (t) ∈r, weight function μ α (ω)＝sin(απ)/πω α The system state z (ω, t) ∈r.
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