CN108873690B - Trajectory tracking method of second-order strict feedback chaotic system - Google Patents

Trajectory tracking method of second-order strict feedback chaotic system Download PDF

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CN108873690B
CN108873690B CN201810996273.4A CN201810996273A CN108873690B CN 108873690 B CN108873690 B CN 108873690B CN 201810996273 A CN201810996273 A CN 201810996273A CN 108873690 B CN108873690 B CN 108873690B
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sliding mode
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赵海滨
陆志国
刘冲
于清文
颜世玉
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Northeastern University China
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Abstract

The invention provides a trajectory tracking method of a second-order strict feedback chaotic system, and relates to the technical field of automatic control. The invention comprises the following steps: step 1: establishing a track tracking error system through a controlled second-order strict feedback chaotic system with modeling uncertainty and external interference signals and an expected track; 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 track tracking control on the second-order strict feedback chaos to form a closed-loop system, the track tracking control of the second-order strict feedback chaos system is realized, and the stability of the closed-loop system is proved through the Lyapunov stability theory. Under the condition of uncertain modeling and external interference signals, the nonsingular fast terminal sliding mode controller realizes the track tracking control of a second order strict feedback chaotic system in different initial states, and has good robustness.

Description

Trajectory tracking method of second-order strict feedback chaotic system
Technical Field
The invention relates to the technical field of automatic control, in particular to a trajectory tracking method of a second-order strict feedback chaotic system.
Background
Chaos is a link connecting deterministic motion and stochastic motion, and widely exists in nature and in human society. The second-order strict feedback chaos can realize the track tracking control only by a 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 trajectory tracking control 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. The nonsingular fast terminal sliding mode controller is used for track tracking control of a second-order strict feedback chaotic system, and no relevant report is found yet. In the design of the nonsingular fast terminal sliding mode controller, an exponential approach law is usually adopted. In the exponential approach law, the parameters are fixed and do not have the adaptive adjustment function. 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 provides a track tracking method of a second-order strict feedback chaotic system, which aims to solve the technical problem of the prior art, provides a self-adaptive index approach law in the design of a nonsingular fast terminal sliding mode controller, can automatically adjust according to a track tracking error, and can accelerate the convergence speed. The method has good robustness for modeling uncertainty and external interference signals.
In order to solve the technical problems, the technical scheme adopted by the invention is as follows: a track tracking method of a second order strict feedback chaotic system; the method comprises the following steps:
step 1: establishing a track tracking error system according to a controlled second-order strict feedback chaotic system with modeling uncertainty and external interference signals and an expected track;
the state equation of the second-order strict feedback chaotic system is as follows
Figure BDA0001781948170000011
Wherein x is1And x2Is the state variable of the system, x ═ x1,x2]TF (x, t) is a continuous function, t is time; the controlled second-order strict feedback chaotic system with modeling uncertainty and external interference signals has the following state equation:
Figure BDA0001781948170000021
wherein, Δ f (x) is modeling uncertainty, d (t) is an external interference signal, and u is a control input; the modeling uncertainty Δ f (x) and the external interference signal d (t) are bounded, i.e.
|△f(x)|+|d(t)|≤d1 (3)
Wherein d is1To model an upper bound of uncertainty and external interference signals, and d1>0。d1For unknown parameters, the adaptive rate is used for estimation.
For the controlled second order strict feedback chaotic system, the state variable x1Is xdThe state variable x2Is the desired trajectory
Figure BDA0001781948170000022
Desired trajectory xdThe track tracking error of the second order strict feedback chaotic system and the expected track is e1＝x1-xd，
Figure BDA0001781948170000023
The system for establishing the tracking error of the track according to the formula (2) and the expected track is
Figure BDA0001781948170000024
Wherein e is1And e2Is a state variable of the trajectory tracking error system;
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
Figure BDA0001781948170000025
Wherein, alpha, beta, r1And r2Is a constant, α>0，β>0，1<r2<2，r1>r2；
The adaptive exponential approximation law is designed as
Figure BDA0001781948170000026
Wherein,
Figure BDA0001781948170000027
λ0is constant, and λ0≥0。
Figure BDA0001781948170000028
As an unknown parameter d1The estimated value of (2) is obtained by the adaptation rate. The parameter lambda is self-adaptively adjusted according to the size of the track tracking error, and the parameter lambda approaches to lambda along with the reduction of the track tracking error0；
And step 3: designing an unknown parameter d according to a trajectory tracking error formula (4), a nonsingular rapid terminal sliding mode surface formula (5) and a self-adaptive index approximation law formula (6)1The nonsingular fast terminal sliding mode controller controls a track tracking error system to form a closed-loop system, track tracking control of a second-order strict feedback chaotic system is achieved, and stability of the closed-loop system is proved through a Lyapunov stability theory.
In the step 3, according to the formula (4), the formula (5) and the formula (6), the nonsingular fast terminal sliding mode controller is designed as
Figure BDA0001781948170000031
The unknown parameter d1The self-adaptive rate is as follows:
Figure BDA0001781948170000032
wherein μ is a constant, and μ>0，
Figure BDA0001781948170000033
Namely, it is
Figure BDA0001781948170000034
Has an initial value of d0And d is0>0；
Replacing sgn(s) with saturation functions sat(s) to weaken buffeting in the controller of formula (7) 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:
Figure BDA0001781948170000035
wherein the expression of the saturation function sat(s) is
Figure BDA0001781948170000036
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
Figure BDA0001781948170000037
Wherein s is a nonsingular fast terminal sliding mode surface defined in formula (5), μ is a constant, and μ>0，
Figure BDA0001781948170000038
For unknown parameters d obtained by adaptive rate1Is estimated value of。
The derivation is performed on equation (10), and then equations (5) and (4) are substituted into the derivation
Figure BDA0001781948170000039
Then substituting the formula (7) and the formula (8) into the formula, and obtaining the product after simplification
Figure BDA0001781948170000041
The Lyapunov stability theory proves that the closed-loop system composed of the formula (4), the formula (7) and the formula (8) is stable, and the track tracking error of the second-order strict feedback chaotic system and the expected track gradually converges to zero. The nonsingular fast terminal sliding mode controller can realize the track tracking control of the second-order strict feedback chaotic system.
Adopt the produced beneficial effect of above-mentioned technical scheme to lie in: the invention provides a track tracking method of a second-order strict feedback chaotic system, which uses a nonsingular fast terminal sliding mode controller for track tracking control of the 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 a track tracking error, and the convergence speed can be accelerated. The method can realize the track tracking control of the second-order strict feedback chaotic system 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 shows a state provided by the first embodiment of the present inventionVariable x1And 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 a tracking 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 a tracking error according to a 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 controlled second-order strict feedback chaotic system with modeling uncertainty and an external interference signal and an expected track, a track tracking 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 track tracking error system to form a closed-loop control system, and the closed-loop control system realizes track tracking control of the second-order strict feedback chaotic system.
In order to more intuitively display the effectiveness of the trajectory tracking method of the second-order strict feedback chaotic system, MATLAB/Simulink software is adopted to carry out 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.
First embodiment
Step 1: establishing a track tracking error system according to a controlled second-order strict feedback chaotic system with modeling uncertainty and external interference signals and an expected track;
the second-order strict feedback chaos is a Duffing chaos system, and the state equation is as follows:
Figure BDA0001781948170000051
wherein x is1And x2Is the state variable of the system, x ═ x1,x2]T，a1A, b and ω are constants and t is time. When 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 controlled Duffing chaotic system with modeling uncertainty and external interference signals is represented as
Figure BDA0001781948170000052
Wherein the modeling uncertainty Δ f (x) is set to 1.2sin (x) at Δ f (x)1+x2) The external interference signal d (t) is set to d (t) 0.8sin (3 t). The initial state of the Duffing chaotic system is set as x1(0)＝1.2，x2(0)＝-0.8。
Duffing chaotic system state variable x1Is xdThe state variable x2Is the desired trajectory
Figure BDA0001781948170000054
Desired trajectory xdHas a second derivative of
Figure BDA0001781948170000053
The track tracking error of the Duffing chaotic system and the expected track is e1＝x1-xd，
Figure BDA0001781948170000055
The trajectory tracking error system adopts a formula (4)
Figure BDA0001781948170000069
Wherein e is1And e2Is a state variable of the trajectory tracking error system;
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 (5)
Figure BDA0001781948170000061
Wherein, the parameters are set as alpha-1, beta-1, r1＝1.8，r2＝1.4。
The adaptive index approach law adopts a formula (6)
Figure BDA0001781948170000062
Wherein,
Figure BDA0001781948170000063
parameter is set to lambda0＝0.5。
Figure BDA0001781948170000064
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 size of the track tracking error, and the parameter lambda approaches to lambda along with the reduction of the track tracking error0；
And step 3: according to a trajectory tracking error formula (4), a nonsingular rapid terminal sliding mode surface formula (5) and a self-adaptive exponential trendNear law equation (6), design unknown parameter d1The nonsingular fast terminal sliding mode controller controls a track tracking error system to form a closed-loop system, track tracking control of a second-order strict feedback chaotic system is achieved, and stability of the closed-loop system is proved through a Lyapunov stability theory.
Designing a nonsingular fast terminal sliding mode controller as follows:
Figure BDA0001781948170000065
unknown parameter d1The self-adaptation rate of (8)
Figure BDA0001781948170000066
Wherein,
Figure BDA0001781948170000067
the parameter is set as mu-1, d0＝2；
In the controller of equation (7), chattering occurs due to the presence of sgn(s) which would cause the controller to be discontinuous. To attenuate the effect of buffeting, the saturation function sat(s) is used instead of sgn(s). And finally, the nonsingular fast terminal sliding mode controller comprises the following steps:
Figure BDA0001781948170000068
wherein the expression of the saturation function sat(s) is
Figure BDA0001781948170000071
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 as follows:
Figure BDA0001781948170000072
wherein s is a nonsingular rapid terminal sliding mode surface defined in the formula (5), μ is a constant, and μ>0，
Figure BDA0001781948170000073
For unknown parameters d obtained by adaptive rate1An estimated value of (d);
the derivation is performed on equation (10), and then equations (5) and (4) are substituted to obtain:
Figure BDA0001781948170000074
then, substituting the formula (7) and the formula (8) to obtain:
Figure BDA0001781948170000075
the control parameters are set as before, and the system is simulated. Fig. 2 shows a control input curve of the nonsingular fast terminal sliding mode controller when the sign function sgn(s) is adopted, and in fig. 2, the control input has a significant buffeting phenomenon. Fig. 3 shows a control input curve of the nonsingular fast terminal sliding mode controller when the saturation function sat(s) is used, and in fig. 3, the control input is smooth without buffeting. Shown in FIG. 4 is a state variable x1And a desired trajectory xdIs the state variable x as shown in fig. 52And
Figure BDA0001781948170000076
the response curve of (c). Fig. 6 shows a response curve of the tracking error. The track tracking error can be intuitively observed from the simulation curve to be basically converged to zero in 3s, and the track tracking speed is very high; under modeling uncertainty and external interference signals, the second-order strict feedback chaotic system realizes track tracking control and has good robustness and high reliability.
Second embodiment:
step 1: establishing a track tracking error system according to a controlled second-order strict feedback chaotic system with modeling uncertainty and external interference signals and an expected track;
the second-order strict feedback chaos is a van der Pol chaos system, and the state equation is
Figure BDA0001781948170000081
Wherein x is1And x2Is the state variable of the system, x ═ x1,x2]TA, B and ω2Is constant and t is time. When the parameters are selected to be A-5, B-3, omega2When 1.788, the system represented by equation (1) is in a chaotic state. The controlled van der Pol chaotic system with modeling uncertainty and external interference signals is expressed as
Figure BDA0001781948170000082
Wherein the modeling uncertainty Δ f (x) is set to 0.8sin (2 x) at Δ f (x)1) The external interference signal d (t) is set to d (t) 1.2sin (4 t). The initial state of the van der Pol chaotic system is set as x1(0)＝-1.4，x2(0)＝3。
van der Pol chaotic system state variable x1Is xdThe state variable x2Is the desired trajectory
Figure BDA0001781948170000087
Desired trajectory xdHas a second derivative of
Figure BDA0001781948170000083
The track tracking error of the van der Pol chaotic system and the expected track is e1＝x1-xd，
Figure BDA0001781948170000084
The trajectory tracking error system adopts a formula (4)
Figure BDA0001781948170000085
Wherein e is1And e2Is a state variable of the trajectory tracking error system;
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 (5)
Figure BDA0001781948170000086
Wherein, the parameters are set as alpha-1, beta-1, r1＝1.9，r2＝1.4。
The adaptive index approach law adopts a formula (6)
Figure BDA0001781948170000091
Wherein,
Figure BDA0001781948170000092
parameter is set to lambda0＝1。
Figure BDA0001781948170000093
As an unknown parameter d1The estimated value of (2) is obtained by the adaptation rate.
And step 3: designing an unknown parameter d according to a trajectory tracking error formula (4), a nonsingular rapid terminal sliding mode surface formula (5) and a self-adaptive index approximation law formula (6)1The adaptive rate and nonsingular fast terminal sliding mode controller controls a track tracking error system to form a closed loop system, realizes track tracking control of a second order strict feedback chaotic system, and controls the track tracking error system through the nonsingular fast terminal sliding mode controllerThe Lyapunov stability theory demonstrates the stability of closed loop systems.
Unknown parameter d1The self-adaptation rate of (8)
Figure BDA0001781948170000094
Wherein,
Figure BDA0001781948170000095
the parameter is set as mu-1, d0＝1.9；
In the controller of equation (7), chattering occurs due to the presence of sgn(s) which would cause the controller to be discontinuous. To attenuate the effect of buffeting, the saturation function sat(s) is used instead of sgn(s). And finally, the nonsingular fast terminal sliding mode controller comprises the following steps:
Figure BDA0001781948170000096
wherein the expression of the saturation function sat(s) is
Figure BDA0001781948170000097
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 as follows:
Figure BDA0001781948170000098
wherein s is a nonsingular rapid terminal sliding mode surface defined in the formula (5), μ is a constant, and μ>0，
Figure BDA0001781948170000099
For unknown parameters d obtained by adaptive rate1An estimated value of (d);
the derivation is performed on equation (10), and then equations (5) and (4) are substituted to obtain:
Figure BDA00017819481700000910
then, substituting the formula (7) and the formula (8) to obtain:
Figure BDA0001781948170000101
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 used, 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 a desired trajectory xdIs the state variable x as shown in fig. 102And
Figure BDA0001781948170000102
the response curve of (c). Fig. 11 shows a response curve of the tracking error. The track tracking error can be intuitively observed from the simulation curve to be basically converged to zero in 3s, and the track tracking speed is very high; under the condition of uncertain modeling and external interference signals, the second-order strict feedback chaotic system realizes track tracking control and 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 trajectory tracking method of a second-order strict feedback chaotic system is characterized by comprising the following steps: the method comprises the following steps:
step 1: establishing a track tracking error system according to a controlled second-order strict feedback chaotic system with modeling uncertainty and external interference signals and an expected track;
the state equation of the second order strict feedback chaotic system is as follows:
Figure FDA0002810999060000011
wherein x is1And x2Is the state variable of the system, x ═ x1,x2]TF (x, t) is a continuous function, t is time; the controlled second-order strict feedback chaotic system with modeling uncertainty and external interference signals has the following state equation:
Figure FDA0002810999060000012
wherein, Δ f (x) is uncertain modeling, d (t) is external interference signal, and u is control input; the modeling uncertainty Δ f (x) and the external interference signal d (t) are bounded:
|Δf(x)|+|d(t)|≤d1 (3)
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;
for the controlled second order strict feedback chaotic system, the state variable x1Is xdThe state variable x2Is the desired trajectory
Figure FDA0002810999060000013
Desired trajectory xdThe track tracking error of the second order strict feedback chaotic system and the expected track is e1＝x1-xd，
Figure FDA0002810999060000014
The system for establishing a trajectory tracking error based on equation (2) and the desired trajectory is:
Figure FDA0002810999060000015
wherein e is1And e2Is a state variable of the trajectory tracking error system;
step 2: designing a nonsingular rapid terminal sliding mode surface and a self-adaptive index approximation law;
the nonsingular rapid terminal sliding mode surface is as follows:
Figure FDA0002810999060000016
wherein, alpha, beta, r1And r2Is constant, alpha is greater than 0, beta is greater than 0, 1 is greater than r2＜2，r1＞r2；
The adaptive index approximation law is designed as follows:
Figure FDA0002810999060000017
wherein,
Figure FDA0002810999060000021
λ0is constant, and λ0≥0，
Figure FDA0002810999060000022
As an unknown parameter d1The estimated value of (a) is obtained by the adaptive rate; the parameter lambda is self-adaptively adjusted according to the size of the track tracking error, and the parameter lambda approaches to lambda along with the reduction of the track tracking error0；
And step 3: according to the formula (4) of the track tracking error, the nonsingular fast terminal sliding mode surface is disclosedThe unknown parameter d is designed according to the formula (5) and the adaptive exponential approximation law formula (6)1The nonsingular fast terminal sliding mode controller controls a track tracking error system to form a closed-loop system, realizes track tracking control of a second-order strict feedback chaotic system, and proves the stability of the closed-loop system through a Lyapunov stability theory;
according to the formula (4), the formula (5) and the formula (6), the nonsingular fast terminal sliding mode controller is designed as follows:
Figure FDA0002810999060000023
the unknown parameter d1The self-adaptive rate is as follows:
Figure FDA0002810999060000024
wherein mu is a constant and mu > 0,
Figure FDA0002810999060000025
namely, it is
Figure FDA0002810999060000026
Has an initial value of d0And d is0＞0；
Replacing sgn(s) with saturation functions sat(s) to weaken buffeting in the controller of formula (7) 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:
Figure FDA0002810999060000027
wherein the expression of the saturation function sat(s) is
Figure FDA0002810999060000028
Where δ is a constant and δ > 0.
2. The trajectory tracking method of the second order strict feedback chaotic system 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 the Lyapunov function is
Figure FDA0002810999060000029
Wherein s is a nonsingular fast terminal sliding mode surface defined in formula (5), μ is a constant, and μ > 0,
Figure FDA00028109990600000210
for unknown parameters d obtained by adaptive rate1An estimate of (d).
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