CN113703317B - Bifurcation delay controller design method based on predation model - Google Patents
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Abstract
The invention discloses a bifurcation delay controller design method based on a predation model, which comprises the following steps of improving the traditional predation model, and establishing a partial differential predation model containing fear time lag and diffusion to obtain balance point information; applying a bifurcation delay controller to the uncontrolled partial differential predation model containing fear time lag and diffusion, and adding the bifurcation delay controller at a balance point to obtain a predation model added with the bifurcation delay controller; linearizing the predation model at a balance point to obtain a characteristic equation of a linearized controlled network; selecting time lag, and selecting proper controller parameters by carrying out stability analysis and bifurcation analysis on the characteristic equation of the linearized controlled network so that the network is locally stable at the balance point accessory; the method solves the problem of low fitting degree between the traditional predation model and the actual population quantity change, and improves the accuracy of the model.
Description
Technical Field
The invention relates to the technical field of controllers, in particular to a bifurcation delay controller design method based on a predation model.
Background
Leslie first proposed a predator model in 1958, which opened the hot tide for studying ecological models. In a practical ecological environment, different location factors and the fear effect of predators have a significant impact on predation behavior. Thus, the fear factors and the spatial effects of diffusion are essential to build predation predated models. Time delays often lead to abrupt changes in the system dynamics, making the system unstable. Therefore, the effect of time lags must be considered when exploring the performance of the kinetic model. Predator models with diffusion and fear time lags can better fit population quantity changes in the actual ecological environment. The patent introduces fear time lag and reaction diffusion on the basis of a classical model, and can better accord with the diffraction situation of the population in the actual environment. The existing predation model has the problems of low fitting degree with the actual natural reproduction law and inaccurate reaction law. Compared with the traditional predatory model, the improved predatory model provided by the invention is more close to the reproduction rule of actual natural population, better reflects the population quantity change in the natural world, and has better fitting degree. The improved novel predation model solves the problem of low fitting degree between the traditional predation model and the actual population quantity change, and improves the accuracy of model description.
Disclosure of Invention
The invention aims to: in order to overcome the defects of the prior art, the invention provides a bifurcation delay controller design method based on a predator model, which solves the problem of integral simulation precision in the combination of a controller and the predator model in the prior art. By applying the controller design method provided by the invention, the effect of expanding the effective stable domain can be realized by setting two parameters, and the simulation effect is better.
The technical scheme is as follows: the invention discloses a bifurcation delay controller design method based on a predation model, which comprises the following steps:
the method comprises the steps of improving a traditional predation model, and establishing a partial differential predation model containing fear time lags and diffusion to obtain balance point information;
applying a bifurcation delay controller to the uncontrolled partial differential predation model containing fear time lag and diffusion, and adding the bifurcation delay controller at a balance point to obtain a predation model added with the bifurcation delay controller;
linearizing a predation model acted by the bifurcation delay controller at a balance point to obtain a characteristic equation of a linearized controlled network;
and selecting time lag, and selecting proper controller parameters by carrying out stability analysis and bifurcation analysis on the characteristic equation of the linearized controlled network, so that the network is locally stable at the balance point.
Alternatively, the partial differential predation containing fear time lags and diffusion is represented by a predation model:
wherein u (t, x) and v (t, x) represent population densities of predators and time t and location x of predators, respectively; d, d 1 And d 2 Is the diffusion coefficient of predators and predators, delta is Laplace operator, and the characteristic root of delta is-k 2 K is N; both populations follow a logistic growth, r/1+cv (t- τ, x) is the modified natural growth rate of the predator, r is the intrinsic growth rate of the prey, C is the fear parameter, τ is the fear time lag, K is the ambient load rate of the predator, s is the natural growth rate of the predator, u (t, x)/h is the modified ambient load rate of the predator, representing the ambient load capacity of the predator in direct proportion to the predator density, h is the ratio of ambient load rate to u (t, x); mu (t, x) v (t, x)/[ u (t, x) +Av (t, x)]Is a Holling-II functional response function, m and A are parameters of the Holling-II function;
balance point E * (u * ,v * ) Expressed as:
u * ＝hv *
u * and v * Respectively represent the system at the balance point E * Where predators and predators take values.
Optionally, the adding of the bifurcation delay controller at the balance point is expressed as follows:
wherein α is an adjustment parameter, u is a u component in the balance point, and β is a state feedback parameter.
Alternatively, the predation model incorporating the bifurcation delay controller is as follows:
alternatively, the neuron model to be acted upon by the bifurcation delay controller is at equilibrium point E * (u * ,v * ) Linearization at the position to obtain:
the characteristic equation of the linearized controlled network is expressed as:
namely:
λ 2 +(d 1 k 2 +d 2 k 2 -a 11 -a 22 )λ+(d 1 k 2 -a 11 )(d 2 k 2 -a 22 )-a 21 (a 12 +a 13 e -λτ )＝0 (7)
wherein:
optionally, the model is at equilibrium point E * (u * ,v * ) The condition of nearby local asymptotic stability is that the root of the characteristic equation of the model has a negative real part, so that a critical stable condition, namely the condition that the characteristic equation has a pure virtual root, is found.
Optionally, the case of making the root of the characteristic equation have a negative real part specifically includes:
when the system has no time lag τ=0, the characteristic equation is:
λ 2 +(d 1 k 2 +d 2 k 2 -a 11 -a 22 )λ+(d 1 k 2 -a 11 )(d 2 k 2 -a 22 )-a 21 (a 12 +a 13 )＝0 (9)
the root of the above equation has a negative real part, and the following Laws-Hurwitz criterion is satisfied:
d 1 k 2 +d 2 k 2 -a 11 -a 22 ＞0 (10)
(d 1 k 2 -a 11 )(d 2 k 2 -a 22 )-a 21 (a 12 +a 13 )＞0 (11)
thus, when the controller parameters satisfy the two inequalities described above, the model without time lags is stable;
when the system is time-lag (τ > 0), taking λ=iω into the characteristic equation, separating the real and imaginary parts can obtain:
wherein the method comprises the steps of
At this time, let h (ω) =ω 4 +[p 1 2 (k 2 )-2q 1 (k 2 )+2a 21 a 12 ]ω 2 +[q 1 (k 2 )-a 21 a 12 ] 2 -a 21 2 a 13 2 When [ q ] 1 (k 2 )-a 21 a 12 ] 2 -a 21 2 a 13 2 < 0, the above equation has at least one positive root ω 0 The corresponding time lag at this time can be solved:
the bifurcation time lag is a critical threshold for the system to stabilize to unsteady, then the root of the corresponding feature equation is going to traverse the imaginary axis from the left half-plane of the imaginary axis to the right half-plane, so the traversing condition at this point: the derivative of the feature root with respect to the bifurcation parameter τ is at τ 0 The real part at the point is larger than zero, and the characteristic root can pass through from the left half plane to the right half plane of the virtual axis, thereby obtaining the following components:
the time delay selection satisfies tau epsilon [0, tau ] 0 ) The controlled model is at equilibrium point E * (u * ,v * ) Local asymptotic stabilization;
time lag satisfies τ=τ 0 At the point of equilibrium E * (u * ,v * ) Where Hopf branches are generated when tau crosses tau 0 When the system generates a set of periodic solutions.
The beneficial effects are that: the improved predation model fully considers the space effect influence of time delay, fear factors and diffusion on the model, and better fits population quantity change in the actual ecological environment.
Drawings
FIG. 1 is a flow chart of a method according to the present invention;
fig. 2 is a waveform diagram of predator stabilization when τ=10 for uncontrolled model (19);
fig. 3 is a waveform of predator stabilization when τ=10 for uncontrolled model (19);
fig. 4 is a waveform diagram of the instability of predators in the case where τ=24 of the uncontrolled model (19);
fig. 5 is a waveform diagram of predator instability with τ=24 for the uncontrolled model (19);
fig. 6 is a waveform diagram of the controlled model (20) being re-stabilized by the predator with controller parameters α=1.4, β= -0.03 and τ=24;
fig. 7 is a waveform diagram of the controlled model (20) with controller parameters α=1.4, β= -0.03 and τ=24 for predators to resume steady;
fig. 8 is a waveform diagram of the controlled model (20) for predator instability with controller parameters α=1.4, β= -0.03 and τ=33;
fig. 9 is a waveform diagram of the controlled model (20) with the controller parameters α=1.4, β= -0.03 and τ=33 for predator instability.
Detailed Description
The invention is further described below with reference to the accompanying drawings. The following examples are only for more clearly illustrating the technical aspects of the present invention, and are not intended to limit the scope of the present invention.
The invention relates to a bifurcation delay controller design method based on a predation model, which comprises the following steps:
and improving the traditional predation model, and establishing a partial differential predation model containing fear time lags and diffusion to obtain balance point information.
Classical predation is represented by the predation model as:
where u and v represent population densities of predators and predators, respectively. ru (1-u/K) represents the predator following a logistic growth, the inherent growth rate of the prey is r, and the environmental load bearing capacity is K. muv/u+Av is a classical Holling-II coupling function, which represents predator predation behavior for predators, m and A are parameters of the Holling-II function. sv (1-hv/u) represents that predators follow a logistic growth, the natural growth rate is s, the environmental bearing capacity is u/h, and h represents the ratio of the environmental bearing capacity to u.
And introducing fear time lags and reaction diffusion on the basis of the classical predation model to obtain a diffusion predation model with uncontrolled fear time lags.
The uncontrolled fear time-lapse diffuse predation is expressed by the predation model as:
where u (t, x) and v (t, x) represent population densities of predators and time t and location x of predators, respectively. d, d 1 And d 2 Is the diffusion coefficient of predators and predators, delta is Laplace operator, and the characteristic root of delta is-k 2 K is N. Both populations follow a logistic growth, r/1+cv (t- τ, x) is the modified natural growth rate of predators, r is the intrinsic growth rate of the prey, C is the fear parameter, τ is the fear time lag, K is the ambient load rate of the predators, s is the natural growth rate of the predators, u (t, x)/h is the modified ambient load rate of the predators, representing the ambient load capacity of the predators in direct proportion to the density of the predators, h is the ratio of the ambient load rate to u (t, x). mu (t, x) v (t, x)/[ u (t, x) +Av (t, x)]Is a Holling-II functional response function, the Holling-II coupling function represents predator predation behavior of predator, and m and A are parameters of the Holling-II function;
the Neumann boundary conditions for the predation predated model are:
wherein n is inThe upper outer unit normal vector, Ω= (0, pi) is a smooth boundary +.>A bounded region thereon. Neumann boundary condition means that no population crosses the bounded region +.>
At this time, the only positive balance point of the model is E * (u * ,v * )，
And applying a bifurcation delay controller to the uncontrolled partial differential predation model with fear time lag and diffusion, and adding the bifurcation delay controller at the balance point to obtain the predation model added with the bifurcation delay controller.
At equilibrium point E * (u * ,v * ) The expression of the add-in bifurcation delay controller is as follows:
wherein α is an adjustment parameter, u is a component of u in the balance point, and β is a state feedback parameter.
The mathematical expression of the predation model incorporating the bifurcation delay controller is as follows:
and linearizing the predation model acted by the bifurcation delay controller at a balance point to obtain a characteristic equation of a linearized controlled network.
Linearizing the predation model acted by the bifurcation delay controller at a balance point to obtain:
wherein the method comprises the steps of
The characteristic equation of the linearized controlled network is expressed as:
namely:
λ 2 +(d 1 k 2 +d 2 k 2 -a 11 -a 22 )λ+(d 1 k 2 -a 11 )(d 2 k 2 -a 22 )-a 21 (a 12 +a 13 e -λτ )＝0 (10)
and selecting time lag, and selecting proper controller parameters by carrying out stability analysis and bifurcation analysis on the characteristic equation of the linearized controlled network, so that the network is locally stable at the balance point.
The controlled model is at balance point E * (u * ,v * ) The condition of nearby local asymptotic stability is that the root of the characteristic equation of the model has a negative real part, so that a critical stable condition, namely the condition that the characteristic equation has a pure virtual root, is found.
The case of having the root of the characteristic equation with a negative real part specifically includes:
(1) When the system has no time lag τ=0, the characteristic equation (10) becomes:
λ 2 +(d 1 k 2 +d 2 k 2 -a 11 -a 22 )λ+(d 1 k 2 -a 11 )(d 2 k 2 -a 22 )-a 21 (a 12 +a 13 )＝0 (11)
the filling condition for the root of the characteristic equation (11) having a negative real part is that the following Lawset-Herwitz Routh-Hurwitz criterion is satisfied:
wherein,
thus, when the controller parameters are such that the controlled system satisfies the above inequality (12), the controlled model without time lags is at equilibrium point E * (u * ,v * ) The position is locally asymptotically stable;
(2) When the system is time-lag (τ > 0), assuming that the characteristic equation of the controlled system has a pair of pure virtual roots±iω, and bringing λ=iω into the characteristic equation, the separation of the real and imaginary parts can be obtained:
wherein the method comprises the steps of
Squaring the trigonometric function (13) to obtain an equation
ω 4 +[p 1 2 (k 2 )-2q 1 (k 2 )+2a 21 a 12 ]ω 2 +[q 1 (k 2 )-a 21 a 12 ] 2 -a 21 2 a 13 2 ＝0 (14)
At this time, let
h(ω)＝ω 4 +[p 1 2 (k 2 )-2q 1 (k 2 )+2a 21 a 12 ]ω 2 +[q 1 (k 2 )-a 21 a 12 ] 2 -a 21 2 a 13 2 When [ q ] 1 (k 2 )-a 21 a 12 ] 2 -a 21 2 a 13 2 < 0, the above equation has at least one positive root ω 0 Corresponds to omega 0 Can solve the time tau at this time 0 ：
The bifurcation time lag is a critical threshold for the system to stabilize to unsteady, then the root of the corresponding feature equation is going to traverse the imaginary axis from the left half-plane of the imaginary axis to the right half-plane, so the traversing condition at this point: the derivative of the feature root with respect to the bifurcation parameter τ is at τ 0 The real part is larger than zero, and the characteristic root can pass from the left half plane to the right half plane of the virtual axis.
The τ is derived from two sides of the characteristic equation and the reciprocal is obtained:
the real part is taken from the pair (16)
It is obvious that the process is not limited to,
as can be derived from the crossing condition (18), the root of the feature equation (10) crosses the imaginary axis from the left half plane to the right half plane, at which point Hopf bifurcation occurs.
The above results can be seen at τ 0 Where the crossing condition is satisfied, therefore τ 0 Is the bifurcation point of the original controlled system. We can conclude that:
A. the time delay selection satisfies tau epsilon [0, tau ] 0 ) The controlled model is at equilibrium point E * (u * ,v * ) Local asymptotic stabilization;
B. time lag satisfies τ=τ 0 At the point of equilibrium E * (u * ,v * ) Where Hopf branches are generated when tau crosses tau 0 When the system generates a set of periodic solutions.
The invention is further illustrated by the following examples. The invention uses Matlab simulation examples for verification.
The first step: selecting uncontrolled predation models containing fear time lags and diffusion:
the bifurcation time lag of the uncontrolled system is as follows, which is obtained by calculating the Hopf bifurcation
As shown in fig. 2 and 3, the time lag is selectedWhen the u (t, x) sum of the uncontrolled system (1)v (t, x) at equilibrium point E * (u * ,v * ) The local part is asymptotically stable.
As shown in fig. 4 and 5, the time lag is selectedAt the equilibrium point E, u (t, x) and v (t, x) of the uncontrolled system (1) * (u * ,v * ) The position is out of stability and is in an oscillation state, and the uncontrolled system (1) is at the moment E * (u * ,v * ) Hopf bifurcation occurs.
And a second step of: the predation model with fear time lag and diffusion is added to the bifurcation delay controller, and the controller parameters are alpha=1.4 and beta= -0.03. The mathematical expression of the controlled system is as follows:
the bifurcation time lag of the controlled system is as follows, which is obtained by calculating the Hopf bifurcation
As shown in fig. 6 and 7, the time lag is selectedWhen the controlled system (3) is under the action of the bifurcation delay controller (2), at the balance point E * (u * ,v * ) The position returns to be stable.
As shown in fig. 8 and 9, the time lag is selectedAt the point of equilibrium E, u (t, x) and v (t, x) of the controlled system (3) * (u * ,v * ) Is out of stability in an oscillating state, the controlled system (3) is at this time at E * (u * ,v * ) Hopf bifurcation occurs.
For system/device embodiments, the description is relatively simple as it is substantially similar to method embodiments, with reference to the description of method embodiments in part.
It should be noted that in this document relational terms such as first and second, and the like are used solely to distinguish one entity or action from another entity or action without necessarily requiring or implying any actual such relationship or order between such entities or actions.
It will be appreciated by those skilled in the art that embodiments of the present application may be provided as a method, system, or computer program product. Accordingly, the present application may take the form of an entirely hardware embodiment, an entirely application embodiment, or an embodiment combining application and hardware aspects. Furthermore, the present application may take the form of a computer program product embodied on one or more computer-usable storage media (including, but not limited to, disk storage, CD-ROM, optical storage, and the like) having computer-usable program code embodied therein.
The present invention is described with reference to flowchart illustrations and/or block diagrams of methods, apparatus (systems) and computer program products according to embodiments of the invention. It will be understood that each flow and/or block of the flowchart illustrations and/or block diagrams, and combinations of flows and/or blocks in the flowchart illustrations and/or block diagrams, can be implemented by computer program instructions. These computer program instructions may be provided to a processor of a general purpose computer, special purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, create means for implementing the functions specified in the flowchart flow or flows and/or block diagram block or blocks.
These computer program instructions may also be stored in a computer-readable memory that can direct a computer or other programmable data processing apparatus to function in a particular manner, such that the instructions stored in the computer-readable memory produce an article of manufacture including instruction means which implement the function specified in the flowchart flow or flows and/or block diagram block or blocks.
These computer program instructions may also be loaded onto a computer or other programmable data processing apparatus to cause a series of operational steps to be performed on the computer or other programmable apparatus to produce a computer implemented process such that the instructions which execute on the computer or other programmable apparatus provide steps for implementing the functions specified in the flowchart flow or flows and/or block diagram block or blocks.
While preferred embodiments of the present invention have been described, additional variations and modifications in those embodiments may occur to those skilled in the art once they learn of the basic inventive concepts. It is therefore intended that the following claims be interpreted as including the preferred embodiments and all such alterations and modifications as fall within the scope of the invention.
It will be apparent to those skilled in the art that various modifications and variations can be made to the present invention without departing from the spirit or scope of the invention. Thus, it is intended that the present invention also include such modifications and alterations insofar as they come within the scope of the appended claims or the equivalents thereof.
Claims (3)
1. A method for designing a bifurcation delay controller based on a predated model, the method comprising:
the method comprises the steps of improving a traditional predation model, and establishing a partial differential predation model containing fear time lags and diffusion to obtain balance point information;
applying a bifurcation delay controller to the uncontrolled partial differential predation model containing fear time lag and diffusion, and adding the bifurcation delay controller at a balance point to obtain a predation model added with the bifurcation delay controller;
linearizing a predation model acted by the bifurcation delay controller at a balance point to obtain a characteristic equation of a linearized controlled network;
selecting time lag, and selecting controller parameters by carrying out stability analysis and bifurcation analysis on the characteristic equation of the linearized controlled network so that the network is locally stable at the balance point;
the partial differential predation containing fear time lags and diffusion is represented by the predation model as:
wherein u (t, x) and v (t, x) represent population densities of predators and time t and location x of predators, respectively; d, d 1 And d 2 Is the diffusion coefficient of predators and predators, delta is Laplace operator, and the characteristic root of delta is-k 2 K is N; both populations follow a logistic growth, r/1+cv (t- τ, x) is the modified natural growth rate of the predator, r is the intrinsic growth rate of the prey, C is the fear parameter, τ is the fear time lag, K is the ambient load rate of the predator, s is the natural growth rate of the predator, u (t, x)/h is the modified ambient load rate of the predator, representing the ambient load capacity of the predator in direct proportion to the predator density, h is the ratio of ambient load rate to u (t, x); mu (t, x) v (t, x)/[ u (t, x) +Av (t, x)]Is a coupling function of Holling-II type, m and A are parameters of the Holling-II type function;
balance point E * (u * ,v * ) Expressed as:
u * ＝hv *
u * and v * Respectively represent the system at the balance point E * The value of the predators and predators;
predation models incorporating the bifurcation delay controller are as follows:
wherein alpha is an adjusting parameter, u is a u component in the calculated balance point, and beta is a state feedback parameter;
predation model to be acted on by bifurcation delay controller is at equilibrium point E * (u * ,v * ) Linearization at the position to obtain:
the characteristic equation of the linearized controlled model is expressed as:
namely:
λ 2 +(d 1 k 2 +d 2 k 2 -a 11 -a 22 )λ+(d 1 k 2 -a 11 )(d 2 k 2 -a 22 )-a 21 (a 12 +a 13 e -λτ )＝0(7)
wherein:
2. the method of designing a bifurcation delay controller based on a predated model as defined in claim 1 wherein said model is at equilibrium point E * (u * ,v * ) The condition of nearby local asymptotic stability is that the root of the characteristic equation of the model has a negative real part, so that a critical stable condition, namely the condition that the characteristic equation has a pure virtual root, is found.
3. The method for designing a bifurcation delay controller based on predation model according to claim 2 wherein the case of having the root of the characteristic equation with a negative real part specifically comprises:
when the system has no time lag τ=0, the characteristic equation is:
λ 2 +(d 1 k 2 +d 2 k 2 -a 11 -a 22 )λ+(d 1 k 2 -a 11 )(d 2 k 2 -a 22 )-a 21 (a 12 +a 13 )＝0(9)
the root of the above equation has a negative real part, and the following Laws-Hurwitz criterion is satisfied:
d 1 k 2 +d 2 k 2 -a 11 -a 22 ＞0(10)
(d 1 k 2 -a 11 )(d 2 k 2 -a 22 )-a 21 (a 12 +a 13 )＞0(11)
thus, when the controller parameters satisfy the two inequalities described above, the model without time lags is stable;
when the system time lag τ > 0, λ=iω is brought into the characteristic equation, and the separation of the real part and the imaginary part can be obtained:
wherein the method comprises the steps of
At this time, let h (ω) =ω 4 +[p 1 2 (k 2 )-2q 1 (k 2 )+2a 21 a 12 ]ω 2 +[q 1 (k 2 )-a 21 a 12 ] 2 -a 21 2 a 13 2 When [ q ] 1 (k 2 )-a 21 a 12 ] 2 -a 21 2 a 13 2 < 0, the above equation has at least one positive root ω 0 The corresponding time lag at this time can be solved:
the bifurcation time lag is a critical threshold for the system to stabilize to unsteady, then the root of the corresponding feature equation is going to traverse the imaginary axis from the left half-plane of the imaginary axis to the right half-plane, so the traversing condition at this point: the derivative of the feature root with respect to the bifurcation parameter τ is at τ 0 The real part at the point is larger than zero, and the characteristic root can pass through from the left half plane to the right half plane of the virtual axis, thereby obtaining the following components:
the time delay selection satisfies tau epsilon [0, tau ] 0 ) The controlled model is at the equilibrium pointE * (u * ,v * ) Local asymptotic stabilization;
time lag satisfies τ=τ 0 At the point of equilibrium E * (u * ,v * ) Where Hopf branches are generated when tau crosses tau 0 When the system generates a set of periodic solutions.
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