CN110018636B - Three-order strict feedback chaotic rail proportional projection synchronization method under saturation constraint - Google Patents
Three-order strict feedback chaotic rail proportional projection synchronization method under saturation constraint Download PDFInfo
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Abstract
The invention belongs to the technical field of automatic control, and particularly relates to a three-order strict feedback chaotic proportion projection synchronization method under saturation constraint, which comprises the following steps: s1, establishing a proportional projection synchronous error system equation according to the driving system equation and the response system equation; s2, establishing a global sliding mode controller equation by adopting an improved global sliding mode surface equation and a combination approach law equation, and performing balance control on a proportional projection synchronous error system equation by adopting the global sliding mode controller equation under saturation constraint; the combined approximation law equation is established by adopting a double-power approximation law equation and an isokinetic approximation law equation. According to the proportional projection synchronization method provided by the invention, an improved global sliding mode surface and a combination approach law are adopted to establish a global sliding mode controller equation, the global sliding mode controller equation performs balance control on a proportional projection synchronization error system equation under saturation constraint, and the proportional projection synchronization control of a driving system and a response system is realized.
Description
Technical Field
The invention belongs to the technical field of automatic control, and particularly relates to a three-order strict feedback chaotic proportion projection synchronization method under saturation constraint.
Background
Chaos is a link connecting deterministic motion and stochastic motion, and widely exists in nature and in human society. The synchronous control of the chaotic system is one of the research hotspots in the nonlinear field. 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 proportional projection synchronous control only by single control input, and has wide application prospect in the aspect of secret communication. 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 global sliding mode surface has robustness in both the approach mode and the sliding mode, and has better robustness than a common sliding mode controller. When the initial state deviation of the chaotic system is large, the initial value of the global sliding mode controller is large, so that the actuator is in a saturated state, even the actuator is damaged, and the synchronization is difficult to realize.
Disclosure of Invention
Technical problem to be solved
Aiming at the existing technical problems, the invention provides a three-order strict feedback chaotic proportional projection synchronization method under saturation constraint, which adopts an improved global sliding mode surface and a combination approach law to establish a global sliding mode controller equation, and the global sliding mode controller equation under saturation constraint carries out balance control on a proportional projection synchronization error system equation so as to realize proportional projection synchronization control of a driving system and a response system.
(II) technical scheme
In order to achieve the purpose, the invention adopts the main technical scheme that:
a three-order strict feedback chaotic proportion projection synchronization method in a saturation state comprises the following steps:
s1, establishing a proportional projection synchronous error system equation according to the driving system equation and the response system equation;
s2, establishing a global sliding mode controller equation by adopting an improved global sliding mode surface equation and a combination approach law equation, and performing balance control on a proportional projection synchronous error system equation by adopting the global sliding mode controller equation under saturation constraint;
the combined approximation law equation is established by adopting a double-power approximation law equation and an isokinetic approximation law equation.
Preferably, the step S1 further includes the following sub-steps:
s101, respectively establishing a driving system equation and a response system equation;
s102, establishing a proportional projection synchronous error system equation by means of the driving system equation and the response system equation obtained in the step S101.
Preferably, the step S2 further includes the following sub-steps:
s201, establishing an improved global sliding mode surface equation and a combination approach law equation;
s202, establishing a global sliding mode controller equation by adopting an improved global sliding mode surface equation and a combination approach law equation;
and S203, substituting the global sliding mode controller equation under saturation constraint into the proportional projection synchronous error system equation to obtain a three-order strict feedback chaotic proportional projection synchronous error system equation with control input.
Preferably, the driving system in step S1 is a three-order strict feedback chaotic system, and the state equation is as follows:
wherein x is1，x2And x3Is the state variable of the system, x ═ x1,x2,x3]T，f1(x) Is a continuous function.
Preferably, the response system in step S101 is a third-order strict feedback chaotic system with modeling uncertainty and external interference signals, and the state equation is as follows:
wherein, y1，y2And y3Is the state variable of the system, y ═ y1,y2,y3]T，g1(y) is a continuous function,. DELTA.g1(y) is modeling uncertainty, d (t) is external interference signal, and t is time;
modeling uncertainty Δ g in response systems1(y) and the external interference signal d (t) are bounded and are represented as:
|Δg1(y)|+|d(t)|≤d1；
wherein d is1Is constant, and d1＞0。
Preferably, the step S102 further includes:
the proportional projection synchronization error of the driving system equation and the response system equation is defined as:
wherein k is a proportionality constant, and k is not equal to 0, e1，e2And e3Proportional projection synchronization error;
and (3) carrying out derivation on the proportional projection synchronous error, and establishing a proportional projection synchronous error system equation as follows:
preferably, the global sliding-mode surface equation modified in step S201 is:
wherein k is1Is constant, and k1The p (t) is a function for realizing the establishment of the global sliding mode control;
the function p (t) is:
wherein n is an even number, t0Is constant, and t0> 0, t is time.
Preferably, the step S201 further includes:
the double power approach law equation is:
wherein k is2And k3Is constant, and k2＞0，k3Alpha and beta are constants, alpha is more than 0 and less than 1, and beta is more than 1;
the constant velocity approximation law equation is:
wherein k is4Is constant, and k4≥d1Very small positive numbers, and > 0;
combining the double power approximation law equation with the constant velocity approximation law equation to obtain a combined approximation law equation as follows:
preferably, the global sliding-mode controller equation in step 202 is:
preferably, the three-order strict feedback chaotic proportion projection synchronization error system equation with control input is as follows:
the global sliding mode controller equation is subject to saturation constraints as follows:
wherein u ismaxIs the maximum control input value, and umax＞0，uminIs a minimum control input value, and uminIf the value is less than 0, u is a global sliding mode controller equation under saturation constraint;
the saturation constraint is approximately expressed by adopting a hyperbolic tangent function, and a global sliding mode controller equation under the saturation constraint is finally as follows:
(III) advantageous effects
The invention has the beneficial effects that: the invention provides a three-order strict feedback chaotic proportional projection synchronization method under saturation constraint, which is characterized in that a driving system is a three-order strict feedback chaotic system equation, a response system is a three-order strict feedback chaotic system equation with modeling uncertainty and external interference signals, a proportional projection synchronous error system equation is established according to the driving system equation and the response system equation, an improved global sliding mode surface equation and a combined approximation law equation are adopted to establish a global sliding mode controller equation, and finally the global sliding mode controller equation under saturation constraint is adopted to carry out proportional projection synchronous error system equation for balance control to form a closed loop system, so that the proportional projection synchronous control of the driving system equation and the response system equation can be realized, and the method has good robustness on the modeling uncertainty and the external interference signals.
Drawings
FIG. 1 is a general schematic diagram of a three-order strict feedback chaotic proportion projection synchronization method under saturation constraint provided by the present invention;
fig. 2 is a schematic diagram of a response curve of a global sliding mode controller under saturation constraint in the specific embodiment 1 of the three-order strict feedback chaotic proportion projection synchronization method under saturation constraint provided by the present invention;
FIG. 3 is a schematic diagram of a response curve of a proportional projection synchronization error in the specific embodiment 1 of the three-order strict feedback chaotic proportional projection synchronization method under saturation constraint according to the present invention;
fig. 4 is a schematic diagram of a response curve of a global sliding mode controller under saturation and constraint in a specific embodiment 2 of a three-order strict feedback chaotic proportion projection synchronization method under saturation constraint provided by the present invention;
fig. 5 is a schematic diagram of a response curve of a proportional projection synchronization error in a specific embodiment 2 of a three-order strict feedback chaotic proportional projection synchronization method under saturation constraint provided by the present invention.
Detailed Description
For the purpose of better explaining the present invention and to facilitate understanding, the present invention will be described in detail by way of specific embodiments with reference to the accompanying drawings.
The embodiment shown in fig. 1 discloses a three-order strict feedback chaotic proportion projection synchronization method in a saturation state, which comprises the following steps:
s1, establishing a proportional projection synchronous error system equation according to the driving system equation and the response system equation;
s2, establishing a global sliding mode controller equation by adopting an improved global sliding mode surface equation and a combination approach law equation, and performing balance control on a proportional projection synchronous error system equation by adopting the global sliding mode controller equation under saturation constraint;
the combined approximation law equation is established by adopting a double-power approximation law equation and an isokinetic approximation law equation.
It should be noted that: the step S1 in this embodiment further includes the following sub-steps:
s101, respectively establishing a driving system equation and a response system equation;
s102, establishing a proportional projection synchronous error system equation by means of the driving system equation and the response system equation obtained in the step S101.
Next, the step S2 in this embodiment further includes the following sub-steps:
s201, establishing an improved global sliding mode surface equation and a combination approach law equation;
s202, establishing a global sliding mode controller equation by adopting an improved global sliding mode surface equation and a combination approach law equation;
and S203, substituting the global sliding mode controller equation under saturation constraint into the proportional projection synchronous error system equation to obtain a three-order strict feedback chaotic proportional projection synchronous error system equation with control input.
Next, it should be explained that: here, the driving system in step S1 is a three-order strict feedback chaotic system, and the state equation is as follows:
wherein x is1，x2And x3Is the state variable of the system, x ═ x1,x2,x3]T，f1(x) Is a continuous function.
In detail, in this embodiment, the response system in step S101 is a three-order strict feedback chaotic system with modeling uncertainty and external interference signals, and the state equation is as follows:
wherein, y1，y2And y3Is the state variable of the system, y ═ y1,y2,y3]T，g1(y) is a continuous function,. DELTA.g1(y) is modeling uncertainty, d (t) is external interference signal, and t is time;
modeling uncertainty Δ g in response systems1(y) and the external interference signal d (t) are bounded and are represented as:
|Δg1(y)|+|d(t)|≤d1；
wherein d is1Is constant, and d1＞0。
In this embodiment, the step S102 further includes:
the proportional projection synchronization error of the driving system equation and the response system equation is defined as:
wherein k is a proportionality constant, and k is not equal to 0, e1，e2And e3Proportional projection synchronization error;
and (3) carrying out derivation on the proportional projection synchronous error, and establishing a proportional projection synchronous error system equation as follows:
in this embodiment, the improved global sliding-mode surface equation in step S201 is:
wherein k is1Is constant, and k1The p (t) is a function for realizing the establishment of the global sliding mode control;
the function p (t) is:
wherein n is an even number, t0Is constant, and t0> 0, t is time.
In this embodiment, the step S201 further includes:
the double power approach law equation is:
wherein k is2And k3Is constant, and k2＞0，k3Alpha and beta are constants, alpha is more than 0 and less than 1, and beta is more than 1;
the constant velocity approximation law equation is:
wherein k is4Is constant, and k4≥d1Very small positive numbers, and > 0;
combining the double power approximation law equation with the constant velocity approximation law equation to obtain a combined approximation law equation as follows:
it should be noted here that the global sliding-mode controller equation in step 202 is:
finally, it should be noted that: the three-order strict feedback chaotic proportion projection synchronization error system equation with control input is as follows:
the global sliding mode controller equation is subject to saturation constraints as follows:
wherein u ismaxIs the maximum control input value, and umax＞0，uminIs a minimum control input value, and uminIf the value is less than 0, u is a global sliding mode controller equation under saturation constraint;
the saturation constraint is approximately expressed by adopting a hyperbolic tangent function, and a global sliding mode controller equation under the saturation constraint is finally as follows:
as shown in fig. 1, the driving system is a three-order strict feedback chaotic system, the response system is a three-order strict feedback chaotic system with modeling uncertainty and external interference signals, a proportional projection synchronous error system is established according to the driving system and the response system, a global sliding mode controller is established by adopting an improved global sliding mode surface and a combination approach law, and finally the global sliding mode controller under saturation constraint is adopted to perform balance control on the proportional projection synchronous error system to form a closed-loop system, so that the proportional projection synchronous control of the driving system and the response system can be realized, and the three-order strict feedback chaotic system has good robustness on the modeling uncertainty and the external interference signals.
In order to more intuitively display the effectiveness of the three-order strict feedback chaotic proportional projection synchronization method under saturation constraint, 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 4 s.
Specific example 1:
the driving system is a Genesio-Tesi chaotic system, and the response system is an Arneodo chaotic system. The state equation of the driving system is as follows:
the response system with modeling uncertainty and external interference signals is:
wherein the modeling is uncertain by Δ g1(y) is set to Δ g1(y)＝0.6sin(y1+2y2) The external interference signal d (t) is set to d (t) 0.5cos (7 t). Due to | Δ g1(y)|+|d(t)|≤d1Then d is1＝1.1。
The proportional projection synchronous error equation of the driving system and the response system adopts the following formula:
wherein, the parameter is set as k 1.5.
The improved global sliding-mode surface equation uses the following formula:
wherein the parameter is set to k1＝5。
The function p (t) takes the following formula:
wherein the parameter is set to t0＝0.8，n＝4。
The combined approximation law equation takes the following formula:
wherein the parameter is set to k2＝1，k3＝1，k4＝1.2，α＝0.6，β＝1.4，＝0.001。
The global sliding-mode controller equation takes the following equation:
the saturation constraint is approximately expressed by adopting a hyperbolic tangent function, and a global sliding mode controller equation under the final saturation constraint adopts the following formula:
wherein the parameter is set to umax＝40，umin＝-60。
The initial state of the drive system is set to x1(0)＝2，x2(0)＝-2，x3(0) 1.2. Initial state of response system is set to y1(0)＝1，y2(0)＝2，y2(0) 0.5. The control parameters are set as before, and the system is simulated. FIG. 2 is a response curve of a global sliding-mode controller equation under saturation constraints. FIG. 3 is a response curve of the proportional projection synchronization error, which substantially converges to zero at 1.5s, and the proportional projection synchronization is very fast and robust to modeling uncertainty and external interference signals.
Under saturation constraint, the global sliding mode controller equation performs balance control on the proportional projection synchronous error system to form a closed-loop control system, proportional projection synchronous control of a driving system and a response system in different initial states is achieved, the proportional projection synchronous speed is very high, and the method has good robustness and high reliability.
Specific example 2:
the driving system and the response system are all Arneodo chaotic systems.
The state equation of the driving system is as follows:
the response system equation with modeling uncertainty and external interference signal is:
wherein the modeling is uncertain by Δ g1(y) is set to Δ g1(y)＝0.5sin(y1+y2) The external interference signal d (t) is set to d (t) 0.5sin (6 t). Due to | Δ g1(y)|+|d(t)|≤d1Then d is1＝1。
The proportional projection synchronous error equation of the driving system and the response system adopts the following formula:
wherein, the parameter is set as k-0.9.
The improved global sliding-mode surface equation uses the following formula:
wherein the parameter is set to k1＝5。
The function p (t) takes the following formula:
wherein the parameter is set to t0＝0.9，n＝4。
The combined approximation law equation takes the following formula:
wherein the parameter is set to k2＝1，k3＝1，k4＝1.3，α＝0.6，β＝1.4，＝0.001。
The global sliding-mode controller equation takes the following equation:
the saturation constraint is approximately expressed by adopting a hyperbolic tangent function, and a global sliding mode controller equation under the final saturation constraint adopts the following formula:
wherein the parameter is set to umax＝16，umin＝-18。
The initial state of the drive system is set to x1(0)＝1，x2(0)＝2，x3(0) 1. Initial state of response system is set to y1(0)＝-2，y2(0)＝-2，y2(0) 3. The control parameters are set as before, and the system is simulated. FIG. 4 is a response curve of a global sliding-mode controller equation under saturation constraints. FIG. 5 is a response curve of the proportional projection synchronization error, which substantially converges to zero at 1.9s, and the proportional projection synchronization is very fast and robust to modeling uncertainty and external interference signals.
Under saturation constraint, the global sliding mode controller equation performs balance control on the proportional projection synchronous error system equation to form a closed-loop control system, proportional projection synchronous control of a driving system and a response system in different initial states is achieved, the proportional projection synchronous speed is very high, and the method has good robustness and high reliability.
The technical principles of the present invention have been described above in connection with specific embodiments, which are intended to explain the principles of the present invention and should not be construed as limiting the scope of the present invention in any way. Based on the explanations herein, those skilled in the art will be able to conceive of other embodiments of the present invention without inventive efforts, which shall fall within the scope of the present invention.
Claims (4)
1. A three-order strict feedback chaotic proportion projection synchronization method in a saturated state is characterized by comprising the following steps:
s1, establishing a proportional projection synchronous error system equation according to the driving system equation and the response system equation;
s2, establishing a global sliding mode controller equation by adopting an improved global sliding mode surface equation and a combination approach law equation, and performing balance control on a proportional projection synchronous error system equation by adopting the global sliding mode controller equation under saturation constraint;
the combined approximation law equation is established by adopting a double-power approximation law equation and an isokinetic approximation law equation;
the step S1 further includes the following sub-steps:
s101, respectively establishing a driving system equation and a response system equation;
s102, establishing a proportional projection synchronous error system equation by means of the driving system equation and the response system equation obtained in the step S101;
the step S2 further includes the following sub-steps:
s201, establishing an improved global sliding mode surface equation and a combination approach law equation;
s202, establishing a global sliding mode controller equation by adopting an improved global sliding mode surface equation and a combination approach law equation;
s203, substituting the global sliding mode controller equation under saturation constraint into a proportional projection synchronous error system equation to obtain a three-order strict feedback chaotic proportional projection synchronous error system equation with control input;
the improved global sliding-mode surface equation in step S201 is:
wherein k is1Is constant, and k1The p (t) is a function for realizing the establishment of the global sliding mode control;
the function p (t) is:
wherein n is an even number, t0Is constant, and t0T is time when the temperature is more than 0;
the step S201 further includes:
the double power approach law equation is:
wherein k is2And k3Is constant, and k2＞0，k3Alpha and beta are constants, alpha is more than 0 and less than 1, and beta is more than 1;
the constant velocity approximation law equation is:
wherein k is4Is constant, and k4≥d1Very small positive numbers, and > 0;
combining the double power approximation law equation with the constant velocity approximation law equation to obtain a combined approximation law equation as follows:
the global sliding mode controller equation in step 202 is:
the three-order strict feedback chaotic proportion projection synchronization error system equation with control input is as follows:
the global sliding mode controller equation is subject to saturation constraints as follows:
wherein u ismaxIs the maximum control input value, and umax＞0，uminIs a minimum control input value, and uminIf the value is less than 0, u is a global sliding mode controller equation under saturation constraint;
the saturation constraint is approximately expressed by adopting a hyperbolic tangent function, and a global sliding mode controller equation under the saturation constraint is finally as follows:
2. the method according to claim 1, wherein the driving system in step S1 is a three-order strict feedback chaotic system, and the state equation is as follows
Wherein x is1，x2And x3Is the state variable of the system, x ═ x1,x2,x3]T，f1(x) Is a continuous function.
3. The method according to claim 2, wherein the response system in step S101 is a third order strict feedback chaotic system with modeling uncertainty and external interference signal, and the state equation is as follows:
wherein, y1，y2And y3Is the state variable of the system, y ═ y1,y2,y3]T，g1(y) is a continuous function,. DELTA.g1(y) is modeling uncertainty, d (t) is external interference signal, t is time;
modeling uncertainty Δ g in response systems1(y) and the external interference signal d (t) are bounded and are represented as:
|Δg1(y)|+|d(t)|≤d1；
wherein d is1Is constant, and d1＞0。
4. The method according to claim 3, wherein the step S102 further comprises: the proportional projection synchronization error of the driving system equation and the response system equation is defined as:
wherein k is a proportionality constant, and k is not equal to 0, e1，e2And e3Proportional projection synchronization error;
and (3) carrying out derivation on the proportional projection synchronous error, and establishing a proportional projection synchronous error system equation as follows:
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