CN109932897A - A method of small-world network model bifurcation point is adjusted with PD control device - Google Patents
A method of small-world network model bifurcation point is adjusted with PD control device Download PDFInfo
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Abstract
The present invention provides a kind of method for adjusting small-world network model bifurcation point with PD control device, method includes the following steps: analysis obtains the unique equalization point of small-world network model of no control；Fractional order PD control device is applied for the small-world network model of no control, obtains controlled small-world network model；Controlled small-world network model is linearized at equalization point, obtains the correlated characteristic equation of controlled model；The Time Delay of Systems of small-world network is chosen as fork parameter, stability analysis is carried out to the controlled model after linearisation, obtains the stable state of the condition and system that generate fork；The scale parameter value and integral parameter value of controller are determined by the result of stability analysis, are effectively shifted to an earlier date or are lagged the time of origin of Hopf fork, small-world network is made to become controllable.The present invention solves the problems, such as that the bifurcation point that small-world network generates is difficult to quickly and accurately shift to an earlier date or lag.
Description
Technical field
The invention belongs to controller technology fields, and in particular to a kind of to adjust small-world network model fork with PD control device
The method of point.
Background technique
Small-world network model includes W-S small-world network model, N-W small-world network model and some other changes
Shape model.Wherein Watt s and Strogatz is initiative proposes small-world network and gives W-S small-world network model；
Then Newman and Watts again WS small-world network model is improved, propose N-W small-world network model, they with
Machine edged is instead of randomization reconnection, so as to avoid the possibility for generating isolated node.If by each element in system,
Each unit regards a node as, and the relationship between each element, unit regards side as, then these elements can be abstracted as one
Network, and information is inside propagated, such as the propagation of news, disease etc..
N-W small-world network model is the network that randomization plus a small amount of side generate on proximity rules network foundation.When
In random edged on proximity rules network, for fixed two o'clock, it will be able to " rich and varied " edged possibility is provided, from one
Point arrives another point, just have it is many preferably may, to be easy to obtain shortest path, and artificially fix edged distance not
The a variety of edgeds of energy offer are possible, so random edged is more advantageous to the diminution of average distance, " the letter being more conducive between node
Breath exchange ", to be more advantageous to synchronization.The artificially network of purposely fixed edged distance construction, can not show a candle to random edged construction
Network have worldlet property, the network that synchronizing capacity also can not show a candle to random edged comes well.
During small-world network carries out information interchange, may information exchange stagnate or run quickly and burst, it is existing that fork occurs
As being unfavorable for the operation of small-world network.At this moment need to be added controller just to ensure that network system within controlled range, allows
Its dynamic behavior towards it is intended that direction develop.In the prior art for the bifurcation point controlling party of small-world network model
Method is often with only integer rank controller, but integer rank controller is inaccurate to the control of bifurcation point and quick, nothing
Method meets the application needs of specific occasion.
Summary of the invention
Goal of the invention: aiming at the above problems existing in the prior art, a kind of PD control device adjusting small-word networks are provided
The method of network model bifurcation point, to solve the problems, such as that the bifurcation point that small-world network generates is difficult in advance or lags, it is ensured that net
The stability of network is more controllable.
Technical solution: the method for the invention for adjusting small-world network model bifurcation point with PD control device includes following step
It is rapid:
(1) analysis obtains the unique equalization point of small-world network model of no control；
(2) PD control device is applied for the small-world network model of no control, obtains controlled small-world network model；Its
In, the scale parameter of the PD control device is denoted as kp, integral parameter is denoted as kd；
(3) controlled small-world network model is linearized at equalization point, obtains the correlated characteristic equation of controlled model；
(4) Time Delay of Systems of small-world network is chosen as fork parameter, and the controlled model after linearisation is stablized
Property analysis, obtain generate fork condition and system stable state；
(5) the scale parameter k of controller is determined by the result of stability analysispValue and integral parameter kdValue, effectively
Ground shifts to an earlier date or the time of origin of lag Hopf fork, and small-world network is made to become controllable.
The utility model has the advantages that compared with the existing technology, the present invention has the following advantages: 1, have stronger generality, can be generalized to height
Tie up fractional order edible organic acid mixture Bifurcation System；2, convenient for operation；3, by the way that appropriate system parameter and order is arranged, it is effectively improved fork
The starting of point promotes small-world network to have better dynamic behavior.
Detailed description of the invention
Fig. 1 is the flow chart that the method for small-world network model bifurcation point is adjusted with PD control device；
Fig. 2 and Fig. 3 is controlled system in the case where no control (h=0.01, μ=1.5), obtains primal system
Bifurcation point τ00=0.6349, system is in equalization point V*Waveform diagram that is stable and that Hopf fork occur at=0.815；
Fig. 4 and Fig. 5 be respectively in the case that controlled system is in integer rank PD control device (n=1, h=0.01, μ=1.5,
kp=0.25, kd=0.25) the bifurcation point τ of controlled system, can be shifted to an earlier date01=0.4478, system is in equalization point V*At=0.815
Stablize and occur the waveform diagram of Hopf fork；
Fig. 6 and Fig. 7 be respectively in the case that controlled system is in integer rank PD control device (n=1, h=0.01, μ=1.5,
kp=-0.25, kdIt=- 0.25), can be with the bifurcation point τ of hysteresis controlled system01=0.849, system is in equalization point V*At=0.815
Stablize and occur the waveform diagram of Hopf fork；
Fig. 8 and Fig. 9 be respectively in the case that controlled system is in fractional order PD control device (n=2, h=0.01, μ=1.5,
kp=1, kd=-0.1) the bifurcation point τ of controlled system, can be shifted to an earlier date02=0.5454, system is in equalization point V*Stablize at=0.815
With the waveform diagram for Hopf fork occur；
Figure 10 and Figure 11 be respectively in the case that controlled system is in fractional order PD control device (n=2, h=0.01, μ=
1.5, kp=1, kdIt=- 0.7), can be with the bifurcation point τ of hysteresis controlled system02=0.8015, system is in equalization point V*At=0.815
Stablize and occur the waveform diagram of Hopf fork；
Figure 12 is to work as kdWhen=- 1, kpWith the relational graph of τ；
Figure 13 is to work as kpWhen=- 1, kdWith the relational graph of τ.
Specific embodiment
It is that the present invention is described in detail in conjunction with attached drawing below.
As shown in Figure 1, the present invention is a kind of method for adjusting small-world network model bifurcation point with PD control device, it is specific to wrap
Include following steps:
Step 1: unique equalization point of the small-world network model of no control is analyzed
Small-world network model without control is as follows:
Wherein, V (t) is the total influence amount transmitted in the random edged of t moment to information in small-world network；τ is worldlet
The Time Delay of Systems (τ >=0) of network；H is the system parameter (0≤h≤1) of small-world network, and what it determined small-world network is
System topological structure；μ is random edged probability, is a positive real number, the Positive balance point V of model is readily available by above-mentioned parameter*
Are as follows:
Step 2: PD control device is applied for the small-world network model of no control, obtains controlled small-world network model
Particularly, the PD control device that bifurcation point is adjusted is as follows:
Wherein, u (t) is the function of state of PD control device, kpAnd kdIt is the scale parameter and integral of controller, k respectivelyd≠
1。Fractional calculus is defined for Kai Putuo, and is had:
Wherein α is fractional order order,n∈Z+；Γ (x) is defined as gamma function, and:
PD control device is applied to can be characterized without small-world network model controlled obtained in control model by the following formula:
Step 3: there is the small-world network system of PD control device in equalization point V addition*Place's linearisation, obtains controlled system
The correlated characteristic equation of system；
Particularly, first there is the small-world network of PD control device in equalization point V addition*Place's linearisation, even
Wherein, n ∈ Z+,
Simultaneous above equation can obtain:
It is replaced by mathematics, converts corresponding same order subsystem for the small-world network control system of different orders, it may be assumed that
After addition has the small-world network system of PD control device to linearize at equalization point, obtain:
Therefore, it can be deduced that the correlated characteristic equation of controlled system are as follows:
By arrangement, can obtain:
Wherein s is characterized the characteristic value of equation.
Step 4: the Time Delay of Systems of small-world network is chosen as fork parameter, the controlled model after linearisation is carried out
Stability analysis obtains the stable state of the condition and system that generate fork
Small-world network system can stable condition be system features equation root have negative real part, it is therefore desirable to find
System is in the condition of neutrality, i.e. the case where pure imaginary root occurs in the characteristic equation of system.
1) when controller is integer rank and system is without time delay (i.e. n=1, τ=0), characteristic equation be can be rewritten as:
It obtains
It is apparent that available conclusion one:
Work as satisfactionAnd kdIn the case where < 1, the root of above formula has a negative real part, at this time whole
Number rank controllers control under without time lag system in equalization point V*Locate a wide range of asymptotically stability.
2) when controller is integer rank and system has time delay (i.e. n=1, τ > 0), characteristic equation be can be rewritten as:
S=i ω is enabled, wherein ω is angular speed, ω > 0, then e-sτ=cos (ω τ)-isin (ω τ).Original equation is substituted into, and
And separation real part imaginary part obtains,
It obtains,
WhenThenIt is available
Therefore, this characteristic equation has pure imaginary root.
Bifurcation point is system from a unstable critical point is stabilized to, then the root of corresponding characteristic equation will be from the point
Place pass through the imaginary axis reach the imaginary axis right half plane, therefore this point out characteristic root for diverge parameter derivative in τ01The real part at place
It is greater than zero, then characteristic root could traverse to right half plane from Left half-plane.
Characteristic equation both sides herein obtain τ derivation:
Due toAnd:
So
It is available to draw a conclusion two by above-mentioned analysis:
(i) work as satisfactionAnd kdIn the case where < 1, the integer rank controller when diverging parameter τ >=0
(n=1) there is time lag controlled system (τ > 0) a wide range of asymptotically stability at equalization point under controlling；
(ii) work as satisfactionAnd kdIn the case where < 1, in fork parameter τ ∈
[0,τ01) when integer rank controller (n=1) control under have time lag controlled system (τ > 0) progressive on a large scale at equalization point
Stablize, in fork parameter τ > τ01When integer rank controller (n=1) control under there is time lag controlled system (τ > 0) to become not
Stablize, and τ=τ01When at equalization point occur Hopf fork.
3) when controller is fractional order and system is without time delay (i.e. n > 1, τ=0), characteristic equation be can be rewritten as:
Assuming that above formula has a non-negative root, it is defined as s=Aeθi=A (cos θ+isin θ),A and θ points
Not Wei s the mould of complex plane is long and argument.
Above formula is brought into obtain,
Imaginary part obtains in fact for separation,
Work as kd0 He of <When, then Acos θ > 0, So whenWhen, real and imaginary part cannot be set up simultaneously from obtained formula, that is, be assumed
It is invalid.This means that all are all being put down again for being the characteristic equation that integer rank and system have time delay when controller
The Left half-plane in face.
It is available to draw a conclusion three by above-mentioned analysis:
Work as satisfactionAnd kdIn the case where < 0, fractional order controller (the n > in bifurcation point τ >=0
1) control under without time lag controlled system at equalization point a wide range of asymptotically stability.
4) when controller is fractional order and system has time delay (i.e. n > 1, τ > 0), characteristic equation are as follows:
As τ > 0, s=i ω is enabled, wherein ω > 0, substitutes into original equation, and separate real part imaginary part and obtain,
Above formula both sides simultaneously square and be added, obtain
It enablesSo above-mentioned formula can be write
It enables
Single order continuous derivative and Second Order Continuous derivative are asked to g (z), obtained
(1) whenAnd kdWhen < 0, obtain With
Therefore, g ' (z) > 0 is permanent on z > 0 sets up.And becauseSo g (z)=0
Without positive root, i.e., this characteristic equation is without pure imaginary root for any τ > 0.
(2) whenAnd kd< 0, obtainsWith
Therefore, g " (z) > 0 is permanent on z > 0 to be set up.Then release g ' (z) monotonic increase on z > 0.And becauseWithSo certainly exist zc(zc> 0) so that g ' (zc)=0.Therefore g (z) is in z ∈
(0,zc) on monotone decreasing, in z ∈ (zc,+∞) on monotonic increase.Due toWithSo there is unique zero point z in g (z)=00, i.e. this characteristic equation has pure imaginary root.
Therefore, shifting out equation onto has unique positive root, is defined as
Because of g ' (z0) > 0, soAnd becauseSo
The above results can be seen that in τ02Place, which meets, passes through condition, therefore, τ02It is the fork of the controlled system as n > 1
Point.It obtains to draw a conclusion four:
(i) work as satisfactionAnd kdIn the case where < 0, for τ >=0 at fractional order control device (n > 1)
There is time lag controlled system a wide range of asymptotically stability at equalization point under control.
(ii) work as satisfactionAnd kdIn the case where < 0, for τ ∈ [0,
τ02) fractional order control device (n > 1) control under have time lag controlled system a wide range of asymptotically stability at equalization point；For τ >
τ02Controlled system is unstable.
(iii) work as satisfactionkd< 0 and g ' (z0) in the case where > 0, when
τ=τ02When, there is time lag controlled system (τ >=0) that Hopf fork occurs in equalization point under fractional order control device (n > 1) control.
With example, the present invention is described further below, the correctness analyzed by Matlab come proof theory.
Step 1: compare for convenience, setting h=0.01, μ=1.5,
Obtain system balancing point V*=0.815.
Step 2: being obtained after applying fractional order PD control device to above-mentioned model
Step 3: carrying out a point case study to above-mentioned model, obtain in different situations, the different characteristics of controlled system.
Fig. 2 and Fig. 3 is controlled system in the case where no control (h=0.01, μ=1.5), obtains primal system
Bifurcation point τ00=0.6349, system is in equalization point V*Waveform diagram that is stable and that Hopf fork occur at=0.815.
Fig. 4 and Fig. 5 be respectively in the case that controlled system is in integer rank PD control device (n=1, h=0.01, μ=1.5,
kp=0.25, kd=0.25) the bifurcation point τ of controlled system, can be shifted to an earlier date01=0.4478, system is in equalization point V*At=0.815
Stablize and occur the waveform diagram of Hopf fork.
Fig. 6 and Fig. 7 be respectively in the case that controlled system is in integer rank PD control device (n=1, h=0.01, μ=1.5,
kp=-0.25, kdIt=- 0.25), can be with the bifurcation point τ of hysteresis controlled system01=0.849, system is in equalization point V*At=0.815
Stablize and occur the waveform diagram of Hopf fork.
Fig. 8 and Fig. 9 be respectively in the case that controlled system is in fractional order PD control device (n=2, h=0.01, μ=1.5,
kp=1, kd=-0.1) the bifurcation point τ of controlled system, can be shifted to an earlier date02=0.5454, system is in equalization point V*Stablize at=0.815
With the waveform diagram for Hopf fork occur.
Figure 10 and Figure 11 be respectively in the case that controlled system is in integer rank PD control device (n=2, h=0.01, μ=
1.5, kp=1, kdIt=- 0.7), can be with the bifurcation point τ of hysteresis controlled system02=0.8015, system is in equalization point V*At=0.815
Stablize and occur the waveform diagram of Hopf fork.
Figure 12 is to work as kdWhen=- 1, kpWith the relational graph of τ.Figure 13 is to work as kpWhen=- 1, kdWith the relational graph of τ.From this two figure
In, it can be realized that bifurcation point τ0With kp、kdIncrease and decline and its decrease speed is inversely proportional with order n.Meanwhile working as n
When=1, kdWith τ0There are linear relationships.Step 5: the scale parameter k of controller is determined by the result of BifurcationpValue and
Integral parameter kdValue, effectively in advance or lag Hopf fork time of origin, so that small-world network is become controllable.
Claims (6)
1. a kind of method for adjusting small-world network model bifurcation point with PD control device, which comprises the following steps:
(1) analysis obtains unique equalization point of the small-world network model of no control；
(2) PD control device is applied for the small-world network model of no control, obtains controlled small-world network model；Wherein, will
The scale parameter of the PD control device is denoted as kp, integral parameter is denoted as kd；
(3) controlled small-world network model is linearized at equalization point, obtains the correlated characteristic equation of controlled model；
(4) Time Delay of Systems of small-world network is chosen as fork parameter, and analysis of stability is carried out to the controlled model after linearisation
Analysis obtains the stable state of the condition and system that generate fork；
(5) the scale parameter k of PD control device is determined by the result of stability analysispValue and integral parameter kdValue, effectively
In advance or the time of origin that Hopf diverges is lagged, small-world network is made to become controllable.
2. the method according to claim 1 for adjusting small-world network model bifurcation point with PD control device, which is characterized in that
In step (1), the small-world network model of no control is characterized by the following formula:
Wherein, V (t) is the total influence amount transmitted in the random edged of t moment to information in small-world network；τ is small-world network
Time Delay of Systems, τ >=0；H is the parameter for determining the system topology of small-world network, 0≤h≤1；μ is that random edged is general
Rate is a positive real number；The Positive balance point V is obtained by the small-world network model of the no control*Are as follows:
3. the method according to claim 1 for adjusting small-world network model bifurcation point with PD control device, which is characterized in that
In step (2), PD control device is as follows:
Wherein, u (t) is the function of state of PD control device；kd≠ 1, n are positive integer.
4. the method according to claim 3 for adjusting small-world network model bifurcation point with PD control device, which is characterized in that
In step (2), the controlled small-world network model can be characterized by following formula:
5. the method according to claim 3 for adjusting small-world network model bifurcation point with PD control device, which is characterized in that
In step (3), the correlated characteristic equation of controlled model are as follows:
Wherein, s is characterized the characteristic value of equation.
6. the method according to claim 3 for adjusting small-world network model bifurcation point with PD control device, which is characterized in that
In step (5), the result of the stability analysis includes:
Work as satisfactionAnd kdIn the case where < 1, order n=1 PD control device control under without time lag
System is in equalization point V*Locate a wide range of asymptotically stability；
Work as satisfactionAnd kdIn the case where < 1, when diverging parameter τ >=0 and in the PD control of order n=1
There is time lag controlled system a wide range of asymptotically stability at equalization point under device control；
Work as satisfactionAnd kdIn the case where < 1, in fork parameter τ ∈ [0, τ01) when
And have time lag controlled system a wide range of asymptotically stability at equalization point under the control of the fractional order PD control device of order n=1,
Diverge parameter τ > τ01When and there is time lag controlled system to become unstable under the control of the PD control device of order n=1, and τ=
τ01When at equalization point occur Hopf fork；
Work as satisfactionAnd kdPD control in the case where < 0, when diverging parameter τ >=0 and in order n > 1
Device control under without time lag controlled system at equalization point a wide range of asymptotically stability；
Work as satisfactionAnd kdPD control in the case where < 0, when diverging parameter τ >=0 and in order n > 1
There is time lag controlled system a wide range of asymptotically stability at equalization point under device control；
Work as satisfactionAnd kdIn the case where < 0, in bifurcated parameter τ ∈ [0, τ02) when
And there is time lag controlled system a wide range of asymptotically stability at equalization point under the control of the PD control device of order n > 1；Join in bifurcated
Number τ > τ02When and the PD control device of order n > 1 control under have time lag controlled system unstable；
Work as satisfactionkd< 0 and g ' (z0) in the case where > 0, in bifurcated parameter τ
=τ02When and the PD control device of order n > 1 control under have time lag controlled system equalization point occur Hopf fork；
Wherein:
Wherein, z0For unique zero point of g (z)=0, and
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