CN108931917B - Three-order strict feedback chaotic projection synchronization method - Google Patents

Three-order strict feedback chaotic projection synchronization method Download PDF

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CN108931917B
CN108931917B CN201811023769.XA CN201811023769A CN108931917B CN 108931917 B CN108931917 B CN 108931917B CN 201811023769 A CN201811023769 A CN 201811023769A CN 108931917 B CN108931917 B CN 108931917B
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赵海滨
刘冲
陆志国
于清文
颜世玉
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Northeastern University China
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Abstract

The invention provides a three-order strict feedback chaotic projection synchronization method, which comprises the following steps: establishing a driving system and a response system according to a state equation of a three-order strict feedback chaotic system, and establishing a projection synchronization error system according to the driving system and the response system; designing a nonlinear global sliding mode surface and a self-adaptive index approximation law; the global sliding mode controller adopting the nonlinear global sliding mode surface has robustness in the whole response process of the system, and the traditional sliding mode controller does not have robustness in the approach mode. The method combines a global sliding mode controller and a self-adaptive sliding mode controller, provides the self-adaptive global sliding mode controller, estimates modeling uncertainty and external interference signals through self-adaptive rate, has robustness in both an approach mode and a sliding mode, can realize three-order strict feedback chaotic projection synchronous control by only needing single control input, and overcomes the influence of the modeling uncertainty and the external interference signals.

Description

Three-order strict feedback chaotic projection synchronization method
Technical Field
The invention belongs to the technical field of automatic control, and particularly relates to a three-order strict feedback chaotic projection synchronization method.
Background
Chaos is a link connecting deterministic motion and stochastic motion, and widely exists in nature and in human society. Since the concept of projection synchronization proposed by Mainieri and Rehacek, the chaotic synchronization phenomena of different types are unified. The three-order strict feedback chaos can realize projection synchronization only by single input, and has wide application prospect in the aspect of secret communication. Due to the existence of modeling uncertainty and external interference signals, the projection synchronization of the three-order strict feedback chaotic system in different initial states is very difficult.
The sliding mode control has strong robustness for modeling uncertainty and external interference signals, has the advantages of high response speed, easiness in implementation and the like, and is widely applied to control of a nonlinear system. The global sliding mode controller adopting the nonlinear global sliding mode surface has global robustness, and has great advantages compared with the linear sliding mode surface. The adaptive sliding mode controller is capable of estimating the upper bound of modeling uncertainty and external interference signals by the adaptation rate. The nonlinear global sliding mode surface is combined with the adaptive sliding mode controller, and the design of the adaptive global sliding mode controller is very necessary for three-order strict feedback chaotic projection synchronization.
Disclosure of Invention
Based on the technical problems, the invention provides a three-order strict feedback chaotic projection synchronization method, which provides a nonlinear global sliding mode surface and an adaptive index approach law, adopts an adaptive rate to estimate the upper bound of modeling uncertainty and external interference signals, adopts a single adaptive global sliding mode controller, and ensures the isomorphic or heterogeneous three-order strict feedback chaotic projection synchronization in different initial states under the condition of modeling uncertainty and external interference signals.
The three-order strict feedback chaotic projection synchronization method comprises the following steps:
step 1: establishing a driving system and a response system according to a state equation of a three-order strict feedback chaotic system, and establishing a projection synchronization error system according to the driving system and the response system:
the driving system is a three-order strict feedback chaotic system, and the state equation is as follows:
Figure BDA0001788023210000011
wherein x is1，x2And x3Is the state variable of the system, x ═ x1,x2,x3]T，fx(x, t) is a continuous function and t is time. The formula (1) is used as a driving system;
the response system is a three-order strict feedback chaotic system, and the state equation is as follows:
Figure BDA0001788023210000021
wherein, y1，y2And y3Is the state variable of the system, y ═ y1,y2,y3]T，fy(y, t) is a continuous function, t is time;
a controlled response system with modeling uncertainty and external interference signals, the state equation is as follows:
Figure BDA0001788023210000022
where Δ f (y) is modeling uncertainty, d (t) is external interference signal, u is control input, and equation (3) is used as response system when f is uncertainx(x, t) and fy(y, t) when they have the same structure, the driving system and the response system are isomorphic chaotic; when f isx(x, t) and fy(y, t) when the structures are different, the driving system and the response system are heterogeneous chaos;
the modeling uncertainty Δ f (y) and the external interference signal d (t) are bounded, namely:
Figure BDA0001788023210000023
wherein d is1For modeling the upper bound of uncertainty Δ f (y), d2Is an upper bound of the external interference signal d (t), and d1≥0，d2≥0。d1And d2Estimating by adopting self-adaptive rate for unknown parameters;
the projection synchronization error of the driving system and the response system is ei＝yi-kxiWhere i ═ 1,2,3, k are proportionality constants, and k ≠ 0, from the drive system (1) and the response system (3), the projection synchronization error system is established as follows:
Figure BDA0001788023210000024
wherein e1，e2And e3Is a projection synchronization error system state variable;
step 2: designing a nonlinear global sliding mode surface and a self-adaptive index approximation law;
the nonlinear global sliding mode surface is as follows:
s＝e3+k1e2+k2e1-p(t) (6)
wherein k is1>0，k2>0, p (t) is a function designed for realizing global sliding mode control, and the function p (t) must satisfy the followingThree conditions of (c):
(1)p(0)＝e3(0)+k1e2(0)+k2e1(0)；
(2) when t → ∞, p (t) → 0;
(3) p (t) has a first derivative.
According to the above three conditions, the function p (t) is designed as:
p(t)＝p(0)e-βt (7)
wherein β is a constant, and β > 0. By taking the derivative of the function p (t), we can get:
Figure BDA00017880232100000310
the adaptive index approximation law is designed as follows:
Figure BDA0001788023210000031
wherein s is a nonlinear global sliding mode surface defined by equation (6),
Figure BDA0001788023210000032
λ0is constant, and λ0≥0。
Figure BDA0001788023210000033
And
Figure BDA0001788023210000034
respectively, are unknown parameters d1And d2The estimated value of the parameter lambda is obtained through the self-adaptive rate, the parameter lambda is self-adaptively adjusted according to the projection synchronization error, and the parameter lambda approaches to lambda along with the reduction of the projection synchronization error0；
And step 3: according to a projection synchronization error formula (5), a nonlinear global sliding mode surface formula (6) and a self-adaptive index approach law (9), a self-adaptive global sliding mode controller is designed, the single self-adaptive global sliding mode controller controls a projection synchronization error system to form a closed-loop system, the closed-loop system can realize projection synchronization of a driving system and a response system, and robustness is provided for modeling uncertainty and external interference signals;
according to the equations (5), (6) and (9), the adaptive global sliding mode controller is designed as follows:
Figure BDA0001788023210000035
the unknown parameter d1And d2The self-adaptive rate is as follows:
Figure BDA0001788023210000036
wherein, mu1And mu2Is constant and μ1>0，μ2>0。d10And d20Are respectively as
Figure BDA0001788023210000037
And
Figure BDA0001788023210000038
and d is an initial value of10>0，d20>0; s is a nonlinear global sliding mode surface defined by equation (6).
In the controller of equation (10) there is a sign function sgn(s),
Figure BDA0001788023210000039
the controller is discontinuous and generates a buffeting phenomenon, in order to weaken the influence of buffeting, a saturation function sat(s) is adopted to replace a sign function sgn(s), and finally the self-adaptive global sliding mode controller is as follows:
Figure BDA0001788023210000041
wherein the expression of the saturation function sat(s) is
Figure BDA0001788023210000042
Wherein δ is a constant, and δ>0；
The stability of a closed loop system is proved by a Lyapunov stability theory, wherein a Lyapunov function is
Figure BDA0001788023210000043
Wherein s is a nonlinear global sliding mode surface defined by equation (6), μ1And mu2Is constant and μ1>0，μ2>0，
Figure BDA0001788023210000044
And
Figure BDA0001788023210000045
respectively unknown parameters d obtained by the adaptation rate1And d2And (6) estimating the value.
Derivation of equation (13) and then substitution of equations (6), (5) and (11) to obtain
Figure BDA0001788023210000046
Then, the formula (10) is substituted and simplified to obtain
Figure BDA0001788023210000047
The Lyapunov stability theory proves that the closed-loop system consisting of the formula (5), the formula (10) and the formula (11) is stable, and the projection synchronization errors of the driving system and the response system gradually converge to zero. The single self-adaptive global sliding mode controller can realize the projection synchronization of the driving system and the response system in different initial states, and has good robustness on modeling uncertainty and external interference signals.
The beneficial technical effects are as follows:
the invention provides a three-order strict feedback chaotic projection synchronization method, a global sliding mode controller adopting a nonlinear global sliding mode surface has robustness in the whole response process of a system, and a traditional sliding mode controller does not have robustness in an approaching mode. The method combines a global sliding mode controller and a self-adaptive sliding mode controller, provides the self-adaptive global sliding mode controller, estimates modeling uncertainty and external interference signals through self-adaptive rate, has robustness in both an approach mode and a sliding mode, can realize three-order strict feedback chaotic projection synchronous control by only needing single control input, and overcomes the influence of the modeling uncertainty and the external interference signals.
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FIG. 1 is a schematic diagram of the overall structure of an embodiment of the present invention;
FIG. 2 is a response curve of a control input when a sign function is used in embodiment 1 of the present invention;
FIG. 3 is a response curve of a control input when a saturation function is used in embodiment 1 of the present invention;
FIG. 4 is a response curve of the projection synchronization error in embodiment 1 of the present invention;
FIG. 5 is a response curve of the control input when the sign function is adopted in embodiment 2 of the present invention;
FIG. 6 is a response curve of a control input when a saturation function is used in embodiment 2 of the present invention;
FIG. 7 is a response curve of the projection synchronization error in embodiment 2 of the present invention;
Detailed Description
The invention will be further described with reference to the accompanying drawings and specific examples:
as shown in fig. 1, according to a state equation of a three-order strict feedback chaotic system, a driving system and a response system are established, a projection synchronization error system is established, a nonlinear global sliding mode surface and a self-adaptive index approach law are designed, a self-adaptive rate and a self-adaptive global sliding mode controller are designed, the self-adaptive global sliding mode controller controls the projection synchronization error system to form a closed-loop control system, and the closed-loop control system realizes projection synchronization of the driving system and the response system.
In order to display more intuitivelyThe effectiveness of the three-order strict feedback chaotic projection synchronization method is provided, and MATLAB/Simulink software is adopted to carry out computer simulation experiments on the control scheme. In a simulation experiment, an ode45 algorithm and an ode45 algorithm, namely a fourth-fifth-order Runge-Kutta algorithm, are adopted, and are a numerical solution of a self-adaptive step-length ordinary differential equation, wherein the maximum step-length is 0.0001s, and the simulation time is 15 s. At the saturation function
Figure BDA0001788023210000051
In (2), δ is set to 0.001.
Specific example 1:
the driving system and the response system are isomorphic systems and are all Arneodo chaotic systems. The state equation of the Arneodo system is:
Figure BDA0001788023210000052
when parameter a1＝-1，a2＝-5.5，a3＝3.5，a4When the signal value is 1, the Arneodo system generates chaos. Equation (14) is a drive system. The initial state of the drive system is set to x1(0)＝2，x2(0)＝-2，x3(0)＝2.5。
The response system is also an Arneodo chaotic system, and the state equation is as follows:
Figure BDA0001788023210000061
the controlled response system with modeled uncertainty and external interference signals is represented as:
Figure BDA0001788023210000062
wherein the modeling uncertainty Δ f (y) is set to 1.5sin (2 y) where Δ f (y) is not determined2) The external interference signal d (t) is set to d (t) 1.5sin (3 t). Using as a response system a controlled response system (16) with modeled uncertainty and external interference signals. Initial state of response system is set to y1(0)＝1，y2(0)＝2，y2(0)＝0.5。
The projection synchronous error system is formula (5)
Figure BDA0001788023210000063
Where k is set to 0.5, i.e., the state variables of the drive system and the response system approach yi＝0.5xiWherein i is 1,2, 3.
The nonlinear global sliding mode surface adopts the formula (6):
s＝e3+k1e2+k2e1-p(t) (6)
wherein the parameter is set to k1＝2，k2＝2。
The function p (t) employs equation (7):
p(t)＝p(0)e-βt (7)
wherein, the parameter is set as beta-4.
The adaptive index approach law adopts the formula (9):
Figure BDA0001788023210000064
wherein,
Figure BDA0001788023210000065
parameter is set to lambda0＝0.2。
Figure BDA0001788023210000066
And
Figure BDA0001788023210000067
respectively, are unknown parameters d1And d2The estimated value of (2) is obtained by the adaptation rate.
Unknown parameter d1And d2The adaptive rate of (2) is represented by formula (11):
Figure BDA0001788023210000071
wherein the parameter is set to μ1＝50，μ2＝50，d10＝1.2，d20＝1.3。
The control parameters are set as before, and the system is simulated. Fig. 2 is a control input curve of the adaptive global sliding mode controller when using the sign function sgn(s). Fig. 3 is a control input curve of the adaptive global sliding mode controller when the saturation function sat(s) is used. In fig. 2, the control input exhibits a noticeable buffeting phenomenon. In fig. 3, the control input is relatively smooth without chattering. Fig. 4 is a response curve of the projection synchronization error. It can be intuitively observed from the simulation curve that the projection synchronization error basically converges to zero at 7s, and the projection synchronization speed is very high.
The projection synchronization error system formula (5) is controlled by the self-adaptive global sliding mode controller formula (12) and the self-adaptive rate formula (11) to form a closed-loop control system, and the closed-loop control system realizes the projection synchronization of the driving system and the response system. Under the condition of uncertain modeling and external interference signals, the driving system and the response system in different initial states realize projection synchronization, and have good robustness and high reliability.
Specific example 2:
the driving system and the response system are heterogeneous systems, the driving system is an Arneodo chaotic system, and the response system is a Genesio-Tesi chaotic system. The state equation for the Arneodo system takes equation (14):
Figure BDA0001788023210000072
when parameter a1＝-1，a2＝-5.5，a3＝3.5，a4When the signal value is 1, the Arneodo system generates chaos. Equation (14) is a drive system. The initial state of the drive system is set to x1(0)＝2.5，x2(0)＝-2.2，x3(0)＝2.6。
The response system is a Genesio-Tesi chaotic system, and the state equation is as follows:
Figure BDA0001788023210000073
wherein the parameters are a >0, b >0, c >0, and ab < c. When the parameters a is 1.2, b is 2.92 and c is 6, the Genesio-Tesi system generates chaos. The controlled response system with modeled uncertainty and external interference signals is represented as:
Figure BDA0001788023210000074
wherein the modeling uncertainty Δ f (y) is set to Δ f (y) 3sin (2 y)2)sin(y1) The external interference signal d (t) is set to d (t) 2sin (3 t). A controlled system (18) with modeled uncertainty and external disturbance signals is used as a response system. Initial state of response system is set to y1(0)＝1，y2(0)＝-2，y3(0)＝-1.5。
The projection synchronization error system is formula (5):
Figure BDA0001788023210000081
where the parameter is set to k-0.5, i.e., the state variables of the drive system and the response system approach yi＝-0.5xiWherein i is 1,2, 3.
The nonlinear global sliding mode surface adopts the formula (6):
s＝e3+k1e2+k2e1-p(t) (6)
wherein the parameter is set to k1＝2，k2＝2。
The function p (t) employs equation (7):
p(t)＝p(0)e-βt (7)
wherein, the parameter is set as beta-4.
The adaptive index approach law adopts the formula (9):
Figure BDA0001788023210000082
wherein,
Figure BDA0001788023210000083
parameter is set to lambda0＝0.2。
Figure BDA0001788023210000084
And
Figure BDA0001788023210000085
respectively, are unknown parameters d1And d2The estimated value of (2) is obtained by the adaptation rate.
Unknown parameter d1And d2The adaptive rate of (2) is expressed by the following formula (11):
Figure BDA0001788023210000086
wherein the parameter is set to μ1＝60，μ2＝60，d10＝2.6，d20＝1.8。
The control parameters are set as before, and the system is simulated. Fig. 5 is a control input curve for the adaptive global sliding mode controller when using the sign function sgn(s). Fig. 6 is a control input curve of the adaptive global sliding-mode controller when the saturation function sat(s) is used. In fig. 5, the control input exhibits a noticeable buffeting phenomenon. In fig. 6, the control input is relatively smooth without the occurrence of chattering. Fig. 7 is a response curve of the projection synchronization error. It can be intuitively observed from the simulation curve that the projection synchronization error basically converges to zero at 7s, and the projection synchronization speed is very high. The projection synchronization error system formula (5) is controlled by the self-adaptive global sliding mode controller formula (12) and the self-adaptive rate formula (11) to form a closed-loop control system, and the closed-loop control system realizes the projection synchronization of the driving system and the response system. Under the condition of uncertain modeling and external interference signals, the driving system and the response system in different initial states realize projection synchronization, and have good robustness and high reliability.

Claims (1)

1. A three-order strict feedback chaotic projection synchronization method is characterized by comprising the following steps:
step 1: establishing a driving system and a response system according to a state equation of a three-order strict feedback chaotic system, and establishing a projection synchronization error system according to the driving system and the response system:
the driving system is a three-order strict feedback chaotic system, and the state equation is as follows:
Figure FDA0002935440850000011
wherein x is1，x2And x3Is the state variable of the system, x ═ x1,x2,x3]T，fx(x, t) is a continuous function, t is time; the formula (1) is used as a driving system;
the response system is a three-order strict feedback chaotic system, and the state equation is as follows:
Figure FDA0002935440850000012
wherein, y1，y2And y3Is the state variable of the system, y ═ y1,y2,y3]T，fy(y, t) is a continuous function, t is time; a controlled response system with modeling uncertainty and external interference signals, the state equation is as follows:
Figure FDA0002935440850000013
wherein, Δ f (y) is uncertain, d (t) is external interference signal, u is control input, and formula (3) is used asResponse system, when fx(x, t) and fy(y, t) when they have the same structure, the driving system and the response system are isomorphic chaotic; when f isx(x, t) and fy(y, t) when the structures are different, the driving system and the response system are heterogeneous chaos;
the modeling uncertainty Δ f (y) and the external interference signal d (t) are bounded, i.e.:
Figure FDA0002935440850000014
wherein d is1For modeling the upper bound of uncertainty Δ f (y), d2Is an upper bound of the external interference signal d (t), and d1≥0，d2≥0，d1And d2Estimating by adopting self-adaptive rate for unknown parameters;
the projection synchronization error of the driving system and the response system is ei＝yi-kxiWhere i ═ 1,2,3, k are proportionality constants, and k ≠ 0, from the drive system (1) and the response system (3), the projection synchronization error system is established as follows:
Figure FDA0002935440850000021
wherein e1，e2And e3Is a projection synchronization error system state variable;
step 2: designing a nonlinear global sliding mode surface and a self-adaptive index approximation law;
the nonlinear global sliding mode surface is as follows:
s＝e3+k1e2+k2e1-p(t) (6)
wherein k is1＞0，k2The function p (t) is a function designed for realizing global sliding mode control, and the function p (t) must satisfy the following three conditions:
(1)p(0)＝e3(0)+k1e2(0)+k2e1(0)；
(2) when t approaches ∞, p (t) approaches 0;
(3) p (t) has a first derivative;
according to the above three conditions, the function p (t) is designed as:
p(t)＝p(0)e-βt (7)
where β is a constant and β >0, deriving the function p (t) yields:
Figure FDA0002935440850000022
the adaptive index approximation law is designed as follows:
Figure FDA0002935440850000023
wherein s is a nonlinear global sliding mode surface defined in equation (6),
Figure FDA0002935440850000024
λ0is constant, and λ0≥0，
Figure FDA0002935440850000025
And
Figure FDA0002935440850000026
respectively, are unknown parameters d1And d2An estimated value of (d);
and step 3: according to a projection synchronization error formula (5), a nonlinear global sliding mode surface formula (6) and a self-adaptive index approach law (9), a self-adaptive global sliding mode controller is designed, the single self-adaptive global sliding mode controller controls a projection synchronization error system to form a closed-loop system, the closed-loop system can realize projection synchronization of a driving system and a response system, and robustness is provided for modeling uncertainty and external interference signals;
according to the equations (5), (6) and (9), the adaptive global sliding mode controller is designed as follows:
Figure FDA0002935440850000027
the unknown parameter d1And d2The self-adaptive rate is as follows:
Figure FDA0002935440850000031
wherein, mu1And mu2Is constant and μ1＞0，μ2＞0，d10And d20Are respectively as
Figure FDA0002935440850000032
And
Figure FDA0002935440850000033
and d is an initial value of10＞0，d20Is greater than 0; s is a nonlinear global sliding mode surface defined in formula (6);
in the controller of equation (10) there is a sign function sgn(s),
Figure FDA0002935440850000034
the controller is discontinuous and generates a buffeting phenomenon, in order to weaken the influence of buffeting, a saturation function sat(s) is adopted to replace a sign function sgn(s), and finally the self-adaptive global sliding mode controller is as follows:
Figure FDA0002935440850000035
wherein the expression of the saturation function sat(s) is
Figure FDA0002935440850000036
Wherein, delta is a constant and is more than 0;
the stability of the closed-loop system is proved by the Lyapunov stability theory, wherein the Lyapunov function is as follows:
Figure FDA0002935440850000037
where s is the nonlinear global sliding mode surface defined in equation (6), μ1And mu2Is constant and μ1＞0，μ2＞0，
Figure FDA0002935440850000038
And
Figure FDA0002935440850000039
respectively unknown parameters d obtained by the adaptation rate1And d2And (6) estimating the value.
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