CN108873690A - A kind of trace tracking method of the tight feedback chaos system of second order - Google Patents
A kind of trace tracking method of the tight feedback chaos system of second order Download PDFInfo
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Abstract
The present invention provides a kind of trace tracking method of tight feedback chaos system of second order, is related to automatic control technology field.The present invention includes the following steps：Step 1：By establishing track following error system with the uncertain tight feedback chaos system of controlled second order and desired trajectory with external interference signals of modeling；Step 2：Design nonsingular fast terminal sliding-mode surface and adaptive exponentially approaching rule；Step 3：Design adaptive rate does not know modeling and the upper bound of external interference signals is estimated, it designs nonsingular fast terminal sliding mode controller and Trajectory Tracking Control is carried out to the tight feedback chaos of second order, form closed-loop system, the Trajectory Tracking Control for realizing the tight feedback chaos system of second order, proves closed-loop system stability by Lyapunov Theory of Stability.For the present invention in the case where modeling is not known with external interference signals, nonsingular fast terminal sliding mode controller realizes the Trajectory Tracking Control of the different tight feedback chaos systems of original state second order, and has good robustness.
Description
Technical field
The present invention relates to automatic control technology field more particularly to a kind of track following sides of the tight feedback chaos system of second order
Method.
Background technique
Chaos is the tie for connecting regular motion and random motion, is widely present in nature and human society.
The tight feedback chaos of second order only need single control input to can be achieved with Trajectory Tracking Control, have in terms of secret communication extensive
Application prospect.Uncertain and external interference signals are modeled due to existing, for the different tight feedback chaos systems of original state second order
Trajectory Tracking Control it is extremely difficult.
Sliding formwork control has very strong robustness for modeling uncertain and external interference signals, and has fast response time
And the advantages that easy to accomplish, it is widely used in the control of nonlinear system.The characteristics of in order to realize finite time convergence control, proposes end
Hold sliding mode controller.Since TSM control device is when close to equilibrium state, convergence rate is slow, some scholars propose
Fast terminal sliding mode controller.Fast terminal sliding mode controller has convergence rate more faster than TSM control device, still
There are singular problems.In order to overcome singular problem, nonsingular fast terminal sliding mode controller is had also been proposed.Nonsingular fast terminal
Sliding mode controller has fast convergence rate, and stronger robustness can be the Finite-time convergence the advantages that.It will be nonsingular quick
TSM control device is used for the Trajectory Tracking Control of the tight feedback chaos system of second order, yet there are no relevant report.Nonsingular fast
In the design of fast TSM control device, exponentially approaching rule is generallyd use.In exponentially approaching rule, parameter is fixed and invariable,
Without adaptive adjustment function.It is unknown, controller that uncertain and external interference signals the upper bounds are modeled in chaos system
Design it is extremely difficult.
Summary of the invention
The technical problem to be solved by the present invention is in view of the above shortcomings of the prior art, provide a kind of tight feedback chaos of second order
The trace tracking method of system proposes adaptive exponentially approaching rule in the design of nonsingular fast terminal sliding mode controller,
It can be automatically adjusted according to track following error, convergent speed can be accelerated.And external disturbance letter uncertain to modeling
Number have good robustness.
In order to solve the above technical problems, the technical solution adopted by the present invention is that：A kind of tight feedback chaos system of second order
Trace tracking method；Include the following steps：
Step 1：According to the uncertain tight feedback chaos system of controlled second order and expectation with external interference signals of modeling
Track following error system is established in track；
The tight feedback chaos system of second order, state equation are as follows
Wherein, x1And x2For the state variable of system, x=[x1,x2]T, f (x, t) is continuous function, and t is the time；With building
The uncertain tight feedback chaos system of controlled second order with external interference signals of mould, state equation are as follows：
Wherein, △ f (x) is that modeling is uncertain, and d (t) is external interference signals, and u is control input；Model uncertain △ f
(x) and the equal bounded of external interference signals d (t), i.e.,
|△f(x)|+|d(t)|≤d1 (3)
Wherein, d1To model the uncertain upper bound with external interference signals, and d1>0。d1For unknown parameter, using adaptive
Rate is estimated.
Feedback chaos system tight for controlled second order, state variable x1Desired trajectory be xd, state variable x2Expectation rail
Mark isDesired trajectory xdThere is a second dervative, the track following error of the tight feedback chaos system of second order and desired trajectory is e1=
x1-xd,Establishing track following error system according to formula (2) and desired trajectory is
Wherein, e1And e2For track following error system state variable；
Step 2：Design nonsingular fast terminal sliding-mode surface and adaptive exponentially approaching rule；
The nonsingular fast terminal sliding-mode surface is
Wherein, α, β, r1And r2For constant, α>0, β>0,1<r2<2, r1>r2；
The adaptive exponentially approaching rule is designed as
Wherein,λ0For constant, and λ0≥0。For unknown parameter d1Estimated value, pass through adaptive rate
It obtains.Parameter lambda is adaptively adjusted according to the size of track following error, and with the reduction of track following error, parameter lambda becomes
It is bordering on λ0；
Step 3：According to track following error formula (4), nonsingular fast terminal sliding-mode surface formula (5) and adaptive index
Reaching Law formula (6) designs unknown parameter d1Adaptive rate and nonsingular fast terminal sliding mode controller, nonsingular quick end
End sliding mode controller controls track following error system, forms closed-loop system, realizes the tight feedback chaos system of second order
Trajectory Tracking Control, and proved by stability of the Lyapunov Theory of Stability to closed-loop system.
According to formula (4), formula (5) and formula (6) in the step 3, nonsingular fast terminal sliding mode controller is designed
For
The unknown parameter d1Adaptive rate be：
Wherein, μ is constant, and μ>0,I.e.Initial value be d0, and d0>0；
Sgn (s) is replaced using saturation function sat (s), is weakened in the controller of formula (7) since there are sgn (s) to make
The chattering phenomenon that controller is discontinuous and occurs；Finally the nonsingular fast terminal sliding mode controller is：
Wherein, the expression formula of saturation function sat (s) isWherein δ is constant, and δ>0.
It is proved in the step 3 by stability of the Lyapunov Theory of Stability to closed-loop system, wherein
Lyapunov function is
Wherein, s is nonsingular fast terminal sliding-mode surface defined in formula (5), and μ is constant, and μ>0,For by certainly
The unknown parameter d that adaptation rate obtains1Estimated value.
Derivation is carried out to formula (10), is then brought formula (5) and formula (4) into available
Then formula (7) and formula (8) are brought into, is obtained after abbreviation
The closed-loop system being made of formula (4), formula (7) and formula (8) is demonstrated by Lyapunov Theory of Stability
It is stable, the track following error asymptotic convergence of the tight feedback chaos system of second order and desired trajectory to zero.Nonsingular quick end
End sliding mode controller can be realized the Trajectory Tracking Control of the tight feedback chaos system of second order.
Generated beneficial effect is by adopting the above technical scheme：The tight feedback chaos system of a kind of second order provided by the invention
Nonsingular fast terminal sliding mode controller is used for the track of the tight feedback chaos system of second order by the trace tracking method of system, this method
Tracing control proposes not knowing modeling by adaptive rate and the upper bound of external interference signals is estimated.Nonsingular fast
In the design of fast TSM control device, adaptive exponentially approaching rule is proposed, can be carried out according to track following error automatic
Adjustment, can accelerate convergent speed.This method can be realized the Trajectory Tracking Control of the tight feedback chaos system of second order, to modeling
Uncertain and external interference signals have good robustness.
Detailed description of the invention
Fig. 1 is general principles figure provided in an embodiment of the present invention；
The response curve of control input when the symbolization function that Fig. 2 provides for first embodiment of the invention；
Fig. 3 is the response curve using control input when saturation function that first embodiment of the invention provides；
Fig. 4 is the state variable x that first embodiment of the invention provides1And y1Response curve；
Fig. 5 is the state variable x that first embodiment of the invention provides2And y2Response curve；
Fig. 6 is the response curve for the track following error that first embodiment of the invention provides；
The response curve of control input when the symbolization function that Fig. 7 provides for second embodiment of the invention；
Fig. 8 is the response curve using control input when saturation function that second embodiment of the invention provides；
Fig. 9 is the state variable x that second embodiment of the invention provides1And y1Response curve；
Figure 10 is the state variable x that second embodiment of the invention provides2And y2Response curve；
Figure 11 is the response curve for the track following error that second embodiment of the invention provides.
Specific embodiment
With reference to the accompanying drawings and examples, specific embodiments of the present invention will be described in further detail.Implement below
Example is not intended to limit the scope of the invention for illustrating the present invention.
As shown in Figure 1, according to modeling is uncertain and the tight feedback chaos system of controlled second order of external interference signals and
Desired trajectory establishes track following error system, designs nonsingular fast terminal sliding-mode surface and adaptive exponentially approaching rule, design
Adaptive rate and nonsingular fast terminal sliding mode controller, the nonsingular fast terminal sliding mode controller is to track following error system
System is controlled, and closed-loop control system is formed, which realizes the track following control of the tight feedback chaos system of second order
System.
For a kind of more intuitive display trace tracking method of the tight feedback chaos system of second order proposed by the present invention
Validity carries out emulation experiment to this control program using MATLAB/Simulink software.In emulation experiment, using ode45
Algorithm ,-five rank Runge-Kutta algorithm of ode45 algorithm, that is, quadravalence, is a kind of numerical solution of ordinary differential equations of adaptive step
Method, maximum step-length 0.0001s, simulation time 8s.Parameter setting is δ=0.001 in saturation function sat (s).
First embodiment
Step 1：According to the uncertain tight feedback chaos system of controlled second order and expectation with external interference signals of modeling
Track following error system is established in track；
The tight feedback chaos of second order are Duffing chaos system, and state equation is：
Wherein, x1And x2For the state variable of system, x=[x1,x2]T, a1, a, b and ω be constant, t is the time.Work as parameter
It is selected as a1When=- 1, a=0.25, b=0.3, ω=1.0, the system that formula (1) indicates is in chaos state.With modeling
Uncertain and external interference signals controlled Duffing chaos systems are expressed as
Wherein, it models uncertain △ f (x) and is set as △ f (x)=1.2sin (x1+x2), external interference signals d (t) setting
For d (t)=0.8sin (3t).The original state of Duffing chaos system is set as x1(0)=1.2, x2(0)=- 0.8.
Duffing state of chaotic system variable x1Desired trajectory be xd, state variable x2Desired trajectory beIt is expected that
Track xdWith second dervative, it is set as
The track following error of Duffing chaos system and desired trajectory is e1=x1-xd,Track following
Error system uses formula (4)
Wherein, e1And e2For track following error system state variable；
Step 2：Design nonsingular fast terminal sliding-mode surface and adaptive exponentially approaching rule；
Nonsingular fast terminal sliding-mode surface uses formula (5)
Wherein, parameter setting is α=1, β=1, r1=1.8, r2=1.4.
Adaptive exponentially approaching rule uses formula (6)
Wherein,Parameter setting is λ0=0.5.For unknown parameter d1Estimated value, by adaptive
Rate obtains, and parameter lambda is adaptively adjusted according to the size of track following error, with the reduction of track following error, parameter lambda
Level off to λ0；
Step 3：According to track following error formula (4), nonsingular fast terminal sliding-mode surface formula (5) and adaptive index
Reaching Law formula (6) designs unknown parameter d1Adaptive rate and nonsingular fast terminal sliding mode controller, nonsingular quick end
End sliding mode controller controls track following error system, forms closed-loop system, realizes the tight feedback chaos system of second order
Trajectory Tracking Control, and proved by stability of the Lyapunov Theory of Stability to closed-loop system.
Designing nonsingular fast terminal sliding mode controller is：
Unknown parameter d1Adaptive rate use formula (8)
Wherein,Parameter setting is μ=1, d0=2；
In the controller of formula (7), since there are sgn (s), and controller can be made discontinuous, there is chattering phenomenon.In order to
Weaken the influence buffeted, sgn (s) is replaced using saturation function sat (s).The final nonsingular fast terminal sliding mode controller
For：
Wherein, the expression formula of saturation function sat (s) isWherein δ is constant, and δ>0；
It is proved by stability of the Lyapunov Theory of Stability to closed-loop system, wherein Lyapunov function is：
Wherein, s is nonsingular fast terminal sliding-mode surface defined in formula (5), and μ is constant, and μ>0,For by adaptive
It should the obtained unknown parameter d of rate1Estimated value；
Derivation is carried out to formula (10), then brings into obtain by formula (5) and formula (4)：
Then formula (7) and formula (8) are brought into, is obtained after abbreviation：
Control parameter is for example preceding set, carries out the emulation of system.When being symbolization function sgn (s) as shown in Figure 2, non-surprise
The control input curve of different fast terminal sliding mode controller, in Fig. 2, there is apparent chattering phenomenon in control input.Such as Fig. 3
Shown is the control input curve of nonsingular fast terminal sliding mode controller when using saturation function sat (s), in Fig. 3, control
There is not chattering phenomenon in system input, smoother.It is state variable x as shown in Figure 41With desired trajectory xdResponse curve, such as
It is state variable x shown in Fig. 52WithResponse curve.It is the response curve of track following error as shown in Figure 6.It is bent from emulation
Line can intuitively observe that track following error converges to zero in 3s substantially, and the speed of track following is very fast；It is modeling
Under indeterminate and external interference signals, the tight feedback chaos system of second order realizes Trajectory Tracking Control, has good robust
Property and very high reliability.
Second embodiment：
Step 1：According to the uncertain tight feedback chaos system of controlled second order and expectation with external interference signals of modeling
Track following error system is established in track；
The tight feedback chaos of second order are van der Pol chaos system, and state equation is
Wherein, x1And x2For the state variable of system, x=[x1,x2]T, A, B and ω2For constant, t is the time.When parameter is selected
It is selected as A=5, B=3, ω2When=1.788, the system that formula (1) indicates is in chaos state.It is uncertain and external with modeling
The controlled van der Pol chaos system of interference signal is expressed as
Wherein, it models uncertain △ f (x) and is set as △ f (x)=0.8sin (2x1), external interference signals d (t) is set as
D (t)=1.2sin (4t).The original state of van der Pol chaos system is set as x1(0)=- 1.4, x2(0)=3.
Van der Pol state of chaotic system variable x1Desired trajectory be xd, state variable x2Desired trajectory be
Desired trajectory xdWith second dervative, it is set as
The track following error of van der Pol chaos system and desired trajectory is e1=x1-xd,Track
Tracking error system uses formula (4)
Wherein, e1And e2For track following error system state variable；
Step 2：Design nonsingular fast terminal sliding-mode surface and adaptive exponentially approaching rule；
Nonsingular fast terminal sliding-mode surface uses formula (5)
Wherein, parameter setting is α=1, β=1, r1=1.9, r2=1.4.
Adaptive exponentially approaching rule uses formula (6)
Wherein,Parameter setting is λ0=1.For unknown parameter d1Estimated value, pass through adaptive rate
It obtains.
Step 3：According to track following error formula (4), nonsingular fast terminal sliding-mode surface formula (5) and adaptive index
Reaching Law formula (6) designs unknown parameter d1Adaptive rate and nonsingular fast terminal sliding mode controller, nonsingular quick end
End sliding mode controller controls track following error system, forms closed-loop system, realizes the tight feedback chaos system of second order
Trajectory Tracking Control, and proved by stability of the Lyapunov Theory of Stability to closed-loop system.
Unknown parameter d1Adaptive rate use formula (8)
Wherein,Parameter setting is μ=1, d0=1.9；
In the controller of formula (7), since there are sgn (s), and controller can be made discontinuous, there is chattering phenomenon.In order to
Weaken the influence buffeted, sgn (s) is replaced using saturation function sat (s).The final nonsingular fast terminal sliding mode controller
For：
Wherein, the expression formula of saturation function sat (s) isWherein δ is constant, and δ>0；
It is proved by stability of the Lyapunov Theory of Stability to closed-loop system, wherein Lyapunov function is：
Wherein, s is nonsingular fast terminal sliding-mode surface defined in formula (5), and μ is constant, and μ>0,For by adaptive
It should the obtained unknown parameter d of rate1Estimated value；
Derivation is carried out to formula (10), then brings into obtain by formula (5) and formula (4)：
Then formula (7) and formula (8) are brought into, is obtained after abbreviation：
Control parameter is for example preceding set, carries out the emulation of system.When being symbolization function sgn (s) as shown in Figure 7, non-surprise
The control input curve of different fast terminal sliding mode controller, in Fig. 7, there is apparent chattering phenomenon in control input.Such as Fig. 8
Shown is the control input curve of nonsingular fast terminal sliding mode controller when using saturation function sat (s), in fig. 8, control
There is not chattering phenomenon in system input, smoother.It is state variable x as shown in Figure 91With desired trajectory xdResponse curve, such as
It is state variable x shown in Figure 102WithResponse curve.It is the response curve of track following error as shown in figure 11.From emulation
Curve can intuitively observe that track following error converges to zero in 3s substantially, and the speed of track following is very fast；It is building
Mould is uncertain, and the tight feedback chaos system of second order realizes Trajectory Tracking Control under external interference signals, has good robust
Property and very high reliability.
Finally it should be noted that：The above embodiments are only used to illustrate the technical solution of the present invention., rather than its limitations；To the greatest extent
Present invention has been described in detail with reference to the aforementioned embodiments for pipe, those skilled in the art should understand that：Its according to
So be possible to modify the technical solutions described in the foregoing embodiments, or to some or all of the technical features into
Row equivalent replacement；And these are modified or replaceed, it does not separate the essence of the corresponding technical solution, and the claims in the present invention are limited
Fixed range.
Claims (3)
1. a kind of trace tracking method of the tight feedback chaos system of second order, it is characterised in that：Include the following steps：
Step 1：The uncertain tight feedback chaos system of controlled second order and desired trajectory with external interference signals is modeled according to having,
Establish track following error system；
The tight feedback chaos system of second order, state equation are as follows：
Wherein, x1And x2For the state variable of system, x=[x1,x2]T, f (x, t) is continuous function, and t is the time；Not with modeling
Determining and external interference signals the tight feedback chaos systems of controlled second order, state equation are as follows：
Wherein, △ f (x) is that modeling is uncertain, and d (t) is external interference signals, and u is control input；Model uncertain △ f (x) and
The equal bounded of external interference signals d (t)：
|△f(x)|+|d(t)|≤d1 (3)
Wherein, d1To model the uncertain upper bound with external interference signals, and d1>0, d1For unknown parameter, using adaptive rate into
Row estimation；
Feedback chaos system tight for controlled second order, state variable x1Desired trajectory be xd, state variable x2Desired trajectory beDesired trajectory xdThere is a second dervative, the track following error of the tight feedback chaos system of second order and desired trajectory is e1=x1-xd,Establishing track following error system according to formula (2) and desired trajectory is：
Wherein, e1And e2For track following error system state variable；
Step 2：Design nonsingular fast terminal sliding-mode surface and adaptive exponentially approaching rule；
The nonsingular fast terminal sliding-mode surface is：
Wherein, α, β, r1And r2For constant, α>0, β>0,1<r2<2, r1>r2；
The adaptive exponentially approaching rule is designed as：
Wherein,λ0For constant, and λ0>=0,For unknown parameter d1Estimated value, obtained by adaptive rate；
Parameter lambda is adaptively adjusted according to the size of track following error, and with the reduction of track following error, parameter lambda levels off to
λ0；
Step 3：According to track following error formula (4), nonsingular fast terminal sliding-mode surface formula (5) and adaptive exponential approach
It restrains formula (6), designs unknown parameter d1Adaptive rate and nonsingular fast terminal sliding mode controller, nonsingular fast terminal is sliding
Mould controller controls track following error system, forms closed-loop system, realizes the track of the tight feedback chaos system of second order
Tracing control, and proved by stability of the Lyapunov Theory of Stability to closed-loop system.
2. a kind of trace tracking method of the tight feedback chaos system of second order according to claim 1, it is characterised in that：It is described
According to formula (4), formula (5) and formula (6) in step 3, designing nonsingular fast terminal sliding mode controller is：
The unknown parameter d1Adaptive rate be：
Wherein, μ is constant, and μ>0,I.e.Initial value be d0, and d0>0；
Sgn (s) is replaced using saturation function sat (s), is weakened in the controller of formula (7) due to making to control there are sgn (s)
The chattering phenomenon that device is discontinuous and occurs；Finally the nonsingular fast terminal sliding mode controller is：
Wherein, the expression formula of saturation function sat (s) isWherein δ is constant, and δ>0.
3. a kind of trace tracking method of the tight feedback chaos system of second order according to claim 1, it is characterised in that：It is described
It is proved in step 3 by stability of the Lyapunov Theory of Stability to closed-loop system, wherein Lyapunov function is
Wherein, s is nonsingular fast terminal sliding-mode surface defined in formula (5), and μ is constant, and μ>0,To pass through adaptive rate
Obtained unknown parameter d1Estimated value.
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