CN108845494B - Second-order strict feedback chaotic projection synchronization method - Google Patents
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Abstract
The invention provides a second-order strict feedback chaotic projection synchronization method, and relates to the technical field of automatic control. The invention comprises the following steps of 1: establishing a driving system and a controlled response system according to a state equation of a second-order strict feedback chaotic system, and establishing a projection synchronization error system; step 2: designing a nonsingular rapid terminal sliding mode surface and a self-adaptive index approximation law; and step 3: the adaptive rate is designed to estimate the upper bound of modeling uncertainty and external interference signals, the nonsingular fast terminal sliding mode controller is designed to perform projection synchronization control on the second-order strict feedback chaos to form a closed-loop system, the projection synchronization of a driving system and a controlled response system is realized, and the stability of the closed-loop system is proved through the Lyapunov stability theory. The invention provides a self-adaptive index approximation law in the design of a nonsingular fast terminal sliding mode controller, performs automatic adjustment according to a projection synchronization error, can accelerate the convergence speed, and has good robustness on modeling uncertainty and external interference signals.
Description
Technical Field
The invention relates to the technical field of automatic control, in particular to a second-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 second-order strict feedback chaos can realize projection synchronization only by single control 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 second 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. In order to realize the characteristic of limited time convergence, a terminal sliding mode controller is provided. Since the convergence speed of the terminal sliding mode controller is relatively slow when the terminal sliding mode controller is close to a balanced state, some scholars propose a fast terminal sliding mode controller. The fast terminal sliding mode controller has a faster convergence speed than the terminal sliding mode controller, but has a singular problem. In order to overcome the singularity problem, a nonsingular fast terminal sliding mode controller is provided. The nonsingular fast terminal sliding mode controller has the advantages of being high in convergence speed, strong in robustness, capable of converging in limited time and the like. In the design of the nonsingular fast terminal sliding mode controller, an exponential approach law is usually adopted. In the exponential approximation law, the parameters are fixed and cannot be adaptively adjusted. The upper bound of modeling uncertainty and external interference signals in the chaotic system is unknown, and the design of the controller is very difficult.
Disclosure of Invention
The invention aims to solve the technical problem of providing a second-order strict feedback chaotic projection synchronization method aiming at the defects of the prior art, the method can be used for carrying out projection synchronization on a driving system and a controlled response system, the projection synchronization speed is high, and the method has good robustness on modeling uncertainty and external interference signals.
In order to solve the technical problems, the technical scheme adopted by the invention is as follows: a second-order strict feedback chaotic projection synchronization method comprises the following steps:
step 1: establishing a driving system and a controlled response system according to a state equation of a second-order strict feedback chaotic system, and establishing a projection synchronization error system;
the driving system is a second-order strict feedback chaotic system, and the state equation is as follows
Wherein x is1And x2Is the state of the systemVariable, x ═ x1,x2]T，fx(x, t) is a continuous function, t is time, and the formula (1) is taken as a driving system;
the response system is a second-order strict feedback chaotic system, and the state equation is as follows:
wherein, y1And y2Is the state variable of the system, y ═ y1,y2]T，fy(y, t) is a continuous function, t is time, a controlled response system with modeling uncertainty and external interference signals, and the state equation is as follows
Wherein, Deltaf (y) is uncertain modeling, d (t) is external interference signal, u is control input, and the formula (3) is used as controlled response system, when f is uncertain, d (t) is uncertain modeling, d (t) is external interference signal, u is control input, and f is uncertain modeling output, d (t) is output, and u is outputx(x, t) and fy(y, t) have the same structure, the driving system and the controlled response system are isomorphic chaos, and when f isx(x, t) and fy(y, t) when the structures are different, the driving system and the controlled response system are heterogeneous chaos;
the modeling uncertainty Δ f (y) and the external interference signal d (t) are bounded, i.e.
|△f(y)|+|d(t)|≤d1 (4)
Wherein d is1To model an upper bound of uncertainty and external interference signals, and d1≥0，d1Estimating by adopting self-adaptive rate for unknown parameters;
the projection synchronization error of the driving system and the controlled response system is ei＝yi-kxiWhere i is 1,2, k is a proportionality constant, and k is not equal to 0, a projection synchronization error system is established according to the driving system equation (1) and the controlled response system equation (3) as follows
Wherein g (x, y, t) ═ fy(y,t)-kfx(x,t)，e1And e2Is a projection synchronization error system state variable;
step 2: designing a nonsingular rapid terminal sliding mode surface and a self-adaptive index approximation law;
the nonsingular rapid terminal sliding form surface is
Wherein, alpha, beta, r1And r2Is constant and α>0，β>0，1<r2<2，r1>r2。
The adaptive exponential approximation law is designed as
Wherein, the lambda is a parameter,
λ0is constant, and λ0≥0，
As an unknown parameter d1The 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: and designing a self-adaptive rate and a nonsingular rapid terminal sliding mode controller according to a projection synchronization error formula (5), a nonsingular rapid terminal sliding mode surface formula (6) and a self-adaptive index approach law formula (7), wherein the nonsingular rapid terminal sliding mode controller controls a projection synchronization error system to form a closed-loop system, so that projection synchronization of a driving system and a controlled response system is realized, and the stability of the closed-loop system is proved through a Lyapunov stability theory.
In the step 3, the nonsingular fast terminal sliding mode controller is designed according to the formula (5), the formula (6) and the formula (7) as follows:
the unknown parameter d1Has an adaptive rate of
Wherein μ is a constant, and μ>0，
I.e. d0Is composed of
And d is an initial value of0>0；
Replacing sgn(s) with saturation functions sat(s) to weaken buffeting in the controller of formula (8) caused by the fact that the controller is discontinuous due to the existence of sgn(s); and finally, the nonsingular fast terminal sliding mode controller comprises the following steps:
wherein the expression of the saturation function sat(s) is
Wherein δ is a constant, and δ>0。
The stability of the closed loop system is proved by a Lyapunov stability theory in the step 3, wherein the Lyapunov function is
Wherein s is a nonsingular fast terminal sliding mode surface defined in formula (6), μ is a constant, and μ>0，
For unknown parameters d obtained by adaptive rate1An estimate of (d).
Adopt the produced beneficial effect of above-mentioned technical scheme to lie in: the invention provides a second-order strict feedback chaotic projection synchronization method, which uses a nonsingular fast terminal sliding mode controller for projection synchronization of a second-order strict feedback chaotic system and provides estimation of an upper bound of modeling uncertainty and an external interference signal through a self-adaptive rate. In the design of the nonsingular fast terminal sliding mode controller, a self-adaptive index approach law is provided, automatic adjustment can be performed according to projection synchronization errors, and the convergence speed can be accelerated. The method can perform projection synchronization of the driving system and the controlled response system, has high projection synchronization speed, and has good robustness on modeling uncertainty and external interference signals.
Drawings
FIG. 1 is a general schematic diagram provided by an embodiment of the present invention;
FIG. 2 is a response curve of a control input using a sign function according to a first embodiment of the present invention;
FIG. 3 is a response curve of a control input using a saturation function according to a first embodiment of the present invention;
FIG. 4 is a diagram of a state variable x according to a first embodiment of the present invention1And y1The response curve of (a);
FIG. 5 is a diagram of a state variable x according to a first embodiment of the present invention2And y2The response curve of (a);
FIG. 6 is a response curve of the projective synchronization error provided by the first embodiment of the present invention;
FIG. 7 is a response curve of a control input using a sign function according to a second embodiment of the present invention;
FIG. 8 is a response curve of a control input using a saturation function according to a second embodiment of the present invention;
FIG. 9 is a diagram of a state variable x according to a second embodiment of the present invention1And y1The response curve of (a);
FIG. 10 is a diagram of a state variable x according to a second embodiment of the present invention2And y2The response curve of (a);
fig. 11 is a response curve of the projective synchronization error according to the second embodiment of the present invention.
Detailed Description
The following detailed description of embodiments of the present invention is provided in connection with the accompanying drawings and examples. The following examples are intended to illustrate the invention but are not intended to limit the scope of the invention.
As shown in fig. 1, according to a state equation of a second order strict feedback chaotic system, a driving system and a controlled response system are established, a projection synchronization error system is established, a nonsingular fast terminal sliding mode surface and a self-adaptive index approach law are designed, a self-adaptive rate and a nonsingular fast terminal sliding mode controller are designed, the nonsingular fast terminal 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 controlled response system.
In order to more intuitively display the effectiveness of the second-order strict feedback chaotic projection synchronization method, 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 8 s. In the saturation function sat(s), the parameter δ is set to 0.001.
The first embodiment:
step 1: establishing a driving system and a controlled response system according to a state equation of a second-order strict feedback chaotic system, and establishing a projection synchronization error system;
the driving system and the controlled response system are isomorphic systems and are both Duffing chaotic systems. The Duffing chaos is a second-order strict feedback chaos system, and the state equation is
The parameter is selected as a1When a is-1, a is 0.25, b is 0.3, and ω is 1.0, the system represented by formula (1) is in a chaotic state. The formula (1) is used as a driving system. The initial state of the drive system is set to x1(0)＝-0.5，x2(0)＝0.5。
The response system is also a Duffing chaotic system, and the state equation is
A controlled response system with modeled uncertainty and external interference signals is shown as
Wherein the modeling uncertainty Δ f (y) is set to 1.5y1The external interference signal d (t) is set to d (t) 1.5sin (5 t). Taking the formula (3) as a controlled response system, the initial state is set to y1(0)＝2.5，y2(0)＝1.2。
And (3) establishing a projection synchronization error system according to the driving system of the formula (1) and the controlled response system of the formula (3). Projection synchronous error system adopts formula (5)
Wherein g (x, y, t) ═ fy(y,t)-kfx(x, t) and the parameter is set to k-2, namely the state variables of the driving system and the controlled response system approach to yi＝-2xiWherein i is 1, 2.
Step 2: designing a nonsingular rapid terminal sliding mode surface and a self-adaptive index approximation law;
nonsingular rapid terminal sliding mode surface adopting formula (6)
Wherein, the parameters are set as alpha-1, beta-1, r1＝1.8，r2＝1.4。
The adaptive index approach law adopts a formula (7)
Wherein the content of the first and second substances,
parameter is set to lambda0＝1。
As an unknown parameter d1The estimated value of (2) is obtained by the adaptation rate.
And step 3: and designing a self-adaptive rate and a nonsingular rapid terminal sliding mode controller according to a projection synchronization error formula (5), a nonsingular rapid terminal sliding mode surface formula (6) and a self-adaptive index approach law (7), wherein the nonsingular rapid terminal sliding mode controller controls a projection synchronization error system to form a closed-loop system, so that projection synchronization of a driving system and a controlled response system is realized, and the stability of the closed-loop system is proved through a Lyapunov stability theory.
Designing a nonsingular fast terminal sliding mode controller according to a formula (5), a formula (6) and a formula (7) as follows:
unknown parameter d1The self-adaptive rate of the method adopts a formula (9)
Wherein the content of the first and second substances,
the parameter is set as mu 2, d0＝2；
In order to weaken the influence of buffeting, a saturation function sat(s) is adopted to replace the sign function sgn(s), and finally the nonsingular fast terminal sliding mode controller is as follows:
wherein the expression of the saturation function sat(s) is
Wherein δ is a constant, and δ>0；
The stability of the closed-loop system is proved by the Lyapunov stability theory; wherein the Lyapunov function is
Wherein s is a nonsingular fast terminal sliding mode surface defined in formula (6), μ is a constant, and μ>0，
For unknown parameters d obtained by adaptive rate1An estimate of (d).
Derivation of equation (11) and then substitution of equations (6) and (5) to obtain
Then, the formula (8) and the formula (9) are substituted into the formula, and the formula can be obtained after simplification
The control parameters are set as before, and the system is simulated. As shown in fig. 2, which is a control input curve of the nonsingular fast terminal sliding mode controller when the sign function sgn(s) is adopted, in fig. 2, the control input has a significant buffeting phenomenon; fig. 3 is a control input curve of the nonsingular fast terminal sliding mode controller when the saturation function sat(s) is adopted, and in fig. 3, the control input is smooth without buffeting; shown in FIG. 4 is a state variable x1And y1The response curve of (a); shown in FIG. 5 is a state variable x2And y2The response curve of (a); FIG. 6 shows a response curve of the projection synchronization error; from the simulation curve, it can be visually observed that the projection synchronization error basically converges to zero in 3s, and the projection synchronization speed is very high.
The projection synchronization error system formula (5) is controlled by a nonsingular fast terminal sliding mode controller formula (8) and a self-adaptive rate formula (9) to form a closed-loop control system, and the closed-loop control system realizes the projection synchronization of a driving system and a controlled response system. Under modeling uncertainty and external interference signals, the projection synchronization is realized by the driving system and the controlled response system in different initial states, and the method has good robustness and high reliability.
Second embodiment:
step 1: establishing a driving system and a controlled response system according to a state equation of a second-order strict feedback chaotic system, and establishing a projection synchronization error system;
the driving system and the controlled response system are heterogeneous systems, the driving system is a Duffing chaotic system, and the controlled response system is a van der Pol chaotic system. The equation of state of the Duffing chaotic system adopts a formula (1)
The parameter is selected as a1When a is-1, a is 0.25, b is 0.3, and ω is 1.0, the system represented by formula (1) is inA chaotic state. Formula (1) as a driving system, with an initial state set to x1(0)＝0.8，x2(0)＝-0.5。
The response system is a van der Pol chaotic system, and the state equation is
The parameters are selected from the group consisting of A5, B3, and omega2When 1.788, the system represented by equation (2) is in a chaotic state. A controlled response system with modeled uncertainty and external interference signals is shown as
Wherein the modeling uncertainty Δ f (y) is set to 1.3y2The external interference signal d (t) is set to d (t) 1.2sin (4 t). Taking the formula (3) as a controlled response system, the initial state is set to y1(0)＝-2，y2(0)＝1.5。
And (3) establishing a projection synchronization error system according to the driving system of the formula (1) and the controlled response system of the formula (3). Projection synchronous error system adopts formula (5)
Wherein g (x, y, t) ═ fy(y,t)-kfx(x, t) and the parameter is set to k equal to 1, namely the state variables of the driving system and the controlled response system approach to yi＝xiWherein i is 1, 2.
Step 2: designing a nonsingular rapid terminal sliding mode surface and a self-adaptive index approximation law;
nonsingular rapid terminal sliding mode surface adopting formula (6)
Wherein, the parameters are set as alpha-1, beta-1, r1＝1.6，r2＝1.4。
The adaptive index approach law adopts a formula (7)
Wherein the content of the first and second substances,
parameter is set to lambda0＝1。
As an unknown parameter d1The estimated value of (2) is obtained by the adaptation rate.
And step 3: and designing a self-adaptive rate and a nonsingular rapid terminal sliding mode controller according to a projection synchronization error formula (5), a nonsingular rapid terminal sliding mode surface formula (6) and a self-adaptive index approach law (7), wherein the nonsingular rapid terminal sliding mode controller controls a projection synchronization error system to form a closed-loop system, so that projection synchronization of a driving system and a controlled response system is realized, and the stability of the closed-loop system is proved through a Lyapunov stability theory.
Designing a nonsingular fast terminal sliding mode controller according to a formula (5), a formula (6) and a formula (7) as follows:
unknown parameter d1The self-adaptive rate of the method adopts a formula (9)
Wherein the content of the first and second substances,
the parameter is set as mu-5, d0＝2；
In order to weaken the influence of buffeting, a saturation function sat(s) is adopted to replace the sign function sgn(s), and finally the nonsingular fast terminal sliding mode controller is as follows:
wherein the expression of the saturation function sat(s) is
Wherein δ is a constant, and δ>0；
The stability of the closed-loop system is proved by the Lyapunov stability theory; wherein the Lyapunov function is
Wherein s is a nonsingular fast terminal sliding mode surface defined in formula (6), μ is a constant, and μ>0，
For unknown parameters d obtained by adaptive rate1An estimate of (d).
Derivation of equation (11) and then substitution of equations (6) and (5) to obtain
Then, the formula (8) and the formula (9) are substituted into the formula, and the formula can be obtained after simplification
The control parameters are set as before, and the system is simulated.Fig. 7 shows a control input curve of the nonsingular fast terminal sliding mode controller when the sign function sgn(s) is adopted, and in fig. 7, the control input has a significant buffeting phenomenon; fig. 8 shows a control input curve of the nonsingular fast terminal sliding mode controller when the saturation function sat(s) is used, and in fig. 8, the control input is smooth without buffeting; shown in FIG. 9 is a state variable x1And y1The response curve of (a); shown in FIG. 10 is a state variable x2And y2The response curve of (a); fig. 11 shows a response curve of the projection synchronization error. From the simulation curve, it can be visually observed that the projection synchronization error basically converges to zero in 3s, and the projection synchronization speed is very high.
The projection synchronization error system formula (5) is controlled by a nonsingular fast terminal sliding mode controller formula (8) and a self-adaptive rate formula (9) to form a closed-loop control system, and the closed-loop control system realizes the projection synchronization of a driving system and a controlled response system. Under modeling uncertainty and external interference signals, the projection synchronization is realized by the driving system and the controlled response system in different initial states, and the method has good robustness and high reliability.
Finally, it should be noted that: the above embodiments are only used to illustrate the technical solution of the present invention, and not to limit the same; while the invention has been described in detail and with reference to the foregoing embodiments, it will be understood by those skilled in the art that: the technical solutions described in the foregoing embodiments may still be modified, or some or all of the technical features may be equivalently replaced; such modifications and substitutions do not depart from the spirit of the corresponding technical solutions and scope of the present invention as defined in the appended claims.
Claims (2)
1. A second-order strict feedback chaotic projection synchronization method is characterized in that: the method comprises the following steps:
step 1: establishing a driving system and a controlled response system according to a state equation of a second-order strict feedback chaotic system, and establishing a projection synchronization error system;
the driving system is a second-order strict feedback chaotic system, and the state equation is as follows
Wherein x is1And x2Is the state variable of the system, x ═ x1,x2]T，fx(x, t) is a continuous function, t is time, and the formula (1) is taken as a driving system;
the response system is a second-order strict feedback chaotic system, and the state equation is as follows:
wherein, y1And y2Is the state variable of the system, y ═ y1,y2]T，fy(y, t) is a continuous function, t is time, a controlled response system with modeling uncertainty and external interference signals, and the state equation is as follows
Wherein, Δ f (y) is uncertain modeling, d (t) is external interference signal, u is control input, formula (3) is used as controlled response system, when f is uncertain, d (t) is uncertain modeling, d (t) is external interference signal, u is uncertain modeling, u is external interference signal, u is uncertain modeling input, and f is uncertain modeling inputx(x, t) and fy(y, t) have the same structure, the driving system and the controlled response system are isomorphic chaos, and when f isx(x, t) and fy(y, t) when the structures are different, the driving system and the controlled response system are heterogeneous chaos;
the modeling uncertainty Δ f (y) and the external interference signal d (t) are bounded, i.e.
|Δf(y)|+|d(t)|≤d1 (4)
Wherein d is1To model an upper bound of uncertainty and external interference signals, and d1≥0，d1Estimating by adopting self-adaptive rate for unknown parameters;
the projection synchronization error of the driving system and the controlled response system is ei＝yi-kxiWhere i is 1,2, k is a proportionality constant, and k is not equal to 0, a projection synchronization error system is established according to the driving system equation (1) and the controlled response system equation (3) as follows
Wherein g (x, y, t) ═ fy(y,t)-kfx(x,t)，e1And e2Is a projection synchronization error system state variable;
step 2: designing a nonsingular rapid terminal sliding mode surface and a self-adaptive index approximation law;
the nonsingular rapid terminal sliding form surface is
Wherein, alpha, beta, r1And r2Is constant, and alpha is greater than 0, beta is greater than 0, 1 is greater than r2＜2，r1＞r2；
The adaptive exponential approximation law is designed as
Wherein, the lambda is a parameter,
λ0is constant, and λ0≥0，
As an unknown parameter d1The 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 the projection synchronization error along with the reduction of the projection synchronization errorλ0；
And step 3: designing a self-adaptive rate and a nonsingular rapid terminal sliding mode controller according to a projection synchronous error formula (5), a nonsingular rapid terminal sliding mode surface formula (6) and a self-adaptive index approach law formula (7), wherein the nonsingular rapid terminal sliding mode controller controls a projection synchronous error system to form a closed-loop system, so that projection synchronization of a driving system and a controlled response system is realized, and the stability of the closed-loop system is proved through a Lyapunov stability theory;
the non-singular fast terminal sliding mode controller designed according to the formula (5), the formula (6) and the formula (7) is as follows:
the unknown parameter d1Has an adaptive rate of
Wherein mu is a constant and mu > 0,
d0is composed of
And d is an initial value of0＞0；
Replacing sgn(s) with saturation functions sat(s) to weaken buffeting in the controller of formula (8) caused by the fact that the controller is discontinuous due to the existence of sgn(s); and finally, the nonsingular fast terminal sliding mode controller comprises the following steps:
wherein the expression of the saturation function sat(s) is
Wherein δ is a constant and δ > 0.
2. The second-order strict feedback chaotic projection synchronization method according to claim 1, characterized in that: the stability of the closed-loop system is proved by a Lyapunov stability theory in the step 3, wherein a Lyapunov function is as follows:
wherein s is a nonsingular fast terminal sliding mode surface defined in formula (6), μ is a constant, and μ > 0,
for unknown parameters d obtained by adaptive rate1An estimate of (d).
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| CN109240093B (en) * | 2018-12-04 | 2021-08-10 | 东北大学 | Three-order strict feedback chaotic track tracking method based on global integral sliding mode |
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| CN110007596A (en) * | 2019-03-29 | 2019-07-12 | 东北大学 | A kind of second order chaos ratio projective synchronization method controlling input-bound |
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Citations (9)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| US7630948B2 (en) * | 2006-09-07 | 2009-12-08 | International Business Machines Corporation | System and method for managing a chaotic event |
| CN102024180A (en) * | 2010-12-23 | 2011-04-20 | 浙江大学 | Support vector machine-based parameter-adaptive motion prediction method |
| CN103528776A (en) * | 2013-09-27 | 2014-01-22 | 东北大学 | High-order rotating blade dynamic similarity test experiment table and test method |
| CN104238572A (en) * | 2014-07-23 | 2014-12-24 | 南京理工大学 | Motor servo system jitter-free sliding mode position control method based on disturbance compensation |
| CN104303115A (en) * | 2011-11-03 | 2015-01-21 | 牛津布鲁克斯大学 | A method of controlling a dynamic physical system that exhibits a chaotic behaviour |
| CN105306193A (en) * | 2015-11-16 | 2016-02-03 | 郑州轻工业学院 | Finite time synchronization sliding-mode control method of complex chaotic system with unknown parameters |
| US9286877B1 (en) * | 2010-07-27 | 2016-03-15 | Diana Dabby | Method and apparatus for computer-aided variation of music and other sequences, including variation by chaotic mapping |
| CN105785763A (en) * | 2016-03-24 | 2016-07-20 | 郑州轻工业学院 | Finite time combination synchronization sliding mode control method for composite chaotic systems with uncertain parameters |
| CN106647277A (en) * | 2017-01-06 | 2017-05-10 | 淮阴工学院 | Self-adaptive dynamic surface control method of arc-shaped mini-sized electromechanical chaotic system |
Family Cites Families (1)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| US6763271B2 (en) * | 2001-04-12 | 2004-07-13 | The United States Of America As Represented By The Secretary Of The Navy | Tracking sustained chaos |
-
2018
- 2018-08-29 CN CN201810996295.0A patent/CN108845494B/en not_active Expired - Fee Related
Patent Citations (9)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| US7630948B2 (en) * | 2006-09-07 | 2009-12-08 | International Business Machines Corporation | System and method for managing a chaotic event |
| US9286877B1 (en) * | 2010-07-27 | 2016-03-15 | Diana Dabby | Method and apparatus for computer-aided variation of music and other sequences, including variation by chaotic mapping |
| CN102024180A (en) * | 2010-12-23 | 2011-04-20 | 浙江大学 | Support vector machine-based parameter-adaptive motion prediction method |
| CN104303115A (en) * | 2011-11-03 | 2015-01-21 | 牛津布鲁克斯大学 | A method of controlling a dynamic physical system that exhibits a chaotic behaviour |
| CN103528776A (en) * | 2013-09-27 | 2014-01-22 | 东北大学 | High-order rotating blade dynamic similarity test experiment table and test method |
| CN104238572A (en) * | 2014-07-23 | 2014-12-24 | 南京理工大学 | Motor servo system jitter-free sliding mode position control method based on disturbance compensation |
| CN105306193A (en) * | 2015-11-16 | 2016-02-03 | 郑州轻工业学院 | Finite time synchronization sliding-mode control method of complex chaotic system with unknown parameters |
| CN105785763A (en) * | 2016-03-24 | 2016-07-20 | 郑州轻工业学院 | Finite time combination synchronization sliding mode control method for composite chaotic systems with uncertain parameters |
| CN106647277A (en) * | 2017-01-06 | 2017-05-10 | 淮阴工学院 | Self-adaptive dynamic surface control method of arc-shaped mini-sized electromechanical chaotic system |
Non-Patent Citations (2)
| Title |
|---|
| Dynamics, circuit implementation and synchronization of a new three-dimensional fractional-order chaotic system;xu zhang;《AEU - International Journal of Electronics and Communications》;20171230;PP 435-445 * |
| 分数阶混沌系统的终端滑模同步控制;邵克勇;《化工自动化及仪表》;20180430;第298-303页 * |
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