CN114721269A - Disturbed nonlinear multi-agent quasi-consistency method and system based on pulse window - Google Patents

Disturbed nonlinear multi-agent quasi-consistency method and system based on pulse window Download PDF

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CN114721269A
CN114721269A CN202210375262.0A CN202210375262A CN114721269A CN 114721269 A CN114721269 A CN 114721269A CN 202210375262 A CN202210375262 A CN 202210375262A CN 114721269 A CN114721269 A CN 114721269A
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汤泽
王鲲鹏
王艳
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Jiangnan University
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Abstract

The invention discloses a disturbed nonlinear multi-agent quasi-consistency method based on a pulse window, which is characterized in that a disturbed nonlinear multi-agent system model is established, under the condition of considering a random pulse sequence, a discrete Lyapunov function method is utilized to introduce a pulse time window concept into a system, a controller is designed by fully combining various control methods such as pulse control, containment control and distributed control according to the influence of different pulse effects on the system global index quasi-consistency, and the method of parameter variation, pulse comparison principle and the like is utilized to provide sufficient quasi-consistency distinguishing conditions under different pulse effects, so that the quasi-consistency convergence rate and the error bound corresponding to the system are accurately calculated. In addition, numerical simulation is carried out by utilizing a Chua's circuit so as to verify the effectiveness of the quasi-consistency method provided by the invention.

Description

Disturbed nonlinear multi-agent quasi-consistency method and system based on pulse window
Technical Field
The invention relates to the technical field of information, in particular to a disturbed nonlinear multi-agent quasi-consistency method and system based on a pulse window.
Background
With the continuous progress of the technology level, the multi-agent system is widely applied in the fields of wireless sensor networks, unmanned driving, robot formation and the like. In general, a multi-agent is an interconnection of a plurality of independent agents, and the aim of the multi-agent is to construct a large and complex system into a miniature system which is convenient for people to manage. Most agents are distributed, with limited information processing and execution capabilities, and limited bandwidth in sensing and communication capabilities. Therefore, how to enable cooperative work among agents is one of the main issues that people pay attention to. As a typical clustering phenomenon, consistency exists widely in nature, such as collective foraging of fish groups, migration of bird groups, cooperative work of welding robots, robot football games, and the like. The multi-agent consistency requires that each independent agent in the multi-agent system reaches the same dynamic state, and further, the production efficiency can be maximized in an actual application scene.
In fact, only a few multi-agent systems can achieve consistency through adjustment of their parameters, and most multi-agent systems need to achieve consistency through an externally applied controller, i.e., adding control signals. Therefore, how to design a reasonable and efficient controller to achieve the consistency of multi-agent system is a big focus of attention. The pulse signal is a typical transient phenomenon and has a discontinuous characteristic. Through pulse control, the multi-agent system is controlled to transmit information at the same time only at every pulse moment, so that the control cost can be well saved, and the control efficiency is improved. Thus, pulse control has an irreplaceable role in the field of multi-agent coherence. However, most work today is only concerned about the positive effects of impulse effects on the system and ignores the negative effects of impulse effects on the system. For a pulse signal, when the pulse signal is input into the system, the pulse signal not only can promote the consistency of the multi-agent system, but also can generate certain disturbance action on the system and the controller so as to destroy the stability of the whole multi-agent system and prevent the consistency from being realized. Furthermore, in real production life, there is always an error of unequal magnitude between each desired pulse input timing and the actual pulse input timing for all pulses of a pulse train, resulting in minimum and maximum pulse intervals. In short, the pulses may be randomly generated within a time interval referred to as a pulse time window, or pulse window for short. However, most of the existing work assumes that the pulse sequence is fixed in advance, which undoubtedly ignores the randomness of the pulse sequence and increases the conservatism of the research results. Therefore, the multi-agent consistency and pulse window control work aiming at various pulse effects has certain theoretical and practical significance.
Disclosure of Invention
The technical problem to be solved by the invention is to provide a disturbed nonlinear multi-agent quasi-consistency method based on a pulse window, which has high precision, high control efficiency and low control cost.
In order to solve the above problems, the present invention provides a pulsed window based perturbed nonlinear multi-agent quasi-consistency method, which comprises the following steps:
s1, constructing a mathematical model of the nonlinear multi-agent system subjected to external disturbance;
s2, constructing a mathematical model of the independent leading agent based on the mathematical model of the nonlinear multi-agent system of the external disturbance;
s3, constructing a controller by combining a pulse control strategy, a containment control strategy and a distributed control strategy based on a mathematical model of the nonlinear multi-agent system based on external disturbance and a mathematical model of an independent leader agent;
s4, defining errors based on the controller and constructing a controlled error multi-agent system model;
s5, introducing a pulse time window by using a discrete Lyapunov function method so that pulses can be randomly generated within a period of time;
s6, respectively deducing a relational expression which is satisfied by the constructed Lyapunov function in a time interval and a pulse moment;
s7, obtaining a corresponding comparison system by utilizing a comparison principle according to the relational expression which is satisfied by the respectively deduced and constructed Lyapunov function in the time interval and the pulse moment;
and S8, realizing disturbed nonlinear multi-agent global index quasi-consistency by using a comparison system and a pulse comparison principle.
As a further improvement of the present invention, the mathematical model of the externally perturbed nonlinear multi-agent system is as follows:
Figure BDA0003590472520000031
wherein ,
Figure BDA0003590472520000032
is the ith agent z in the systemiA state vector of (a); matrix array
Figure BDA0003590472520000033
Is a constant parameter matrix in the system; tau (t) is time-varying time lag and meets the condition that tau (t) is more than 0 and less than or equal to tau; non-linear function
Figure BDA0003590472520000034
Satisfy the requirement of
Figure BDA0003590472520000035
ξi(t) is the external disturbance input, i is 1,2, …, N and
Figure BDA0003590472520000036
as a further refinement of the present invention, the mathematical model of the independent leader agent is as follows:
Figure BDA0003590472520000037
wherein ,
Figure BDA0003590472520000038
is the state vector of the independent leader agent eta.
As a further improvement of the present invention, the controller is as follows:
Figure BDA0003590472520000039
wherein μ is the pulseImpact effect; g is control strength; omegaiThe feedback containment gain is more than or equal to 0, and when the ith intelligent agent in the system is contained, omega isiIs greater than 0; define diagonal matrix Ω ═ diag { ω1,ω2,…,ωi}; distributed control matrix L ═ Lij)N×NSatisfy the dissipation condition, i.e.
Figure BDA00035904725200000310
For pulse sequences
Figure BDA00035904725200000311
Suppose it satisfies 0 ═ t0＜t1＜t2＜…＜tk< … and for
Figure BDA00035904725200000312
Is provided with
Figure BDA00035904725200000313
δ (t) is the dirac pulse function with respect to time t.
As a further improvement of the present invention, step S4 includes:
defining an error based on the controller as:
ei(t)＝zi(t)-η(t)
and is
Figure BDA00035904725200000314
The controlled error multi-agent system model is constructed as follows:
Figure BDA00035904725200000315
wherein the non-linear function
Figure BDA00035904725200000316
Let ei(t) at t ═ tkThe moments being successive to the right, i.e.
Figure BDA00035904725200000317
In addition, the initial values of the controlled-error multi-agent system model satisfy:
ei(t)＝Λi(t),i＝1,2,…,N
where- τ < t ≦ 0 and continuous function class
Figure BDA0003590472520000041
As a further improvement of the present invention, step S6 includes: respectively deducing the Dini derivatives of the constructed Lyapunov functions at [ t ]k,tk+1)，
Figure BDA0003590472520000042
The following requirements are met:
Figure BDA0003590472520000043
and at the pulse time tkSatisfies the following conditions between the left and right boundaries:
Figure BDA0003590472520000044
where δ is a parameter related to the impulse effect, V (t) is the Lyapunov function, D+V (t) is the Dini derivative of the Lyapunov function.
As a further improvement of the present invention, the comparison system is as follows:
Figure BDA0003590472520000045
wherein ,
Figure BDA0003590472520000046
is a special solution of a pulse system, and epsilon is more than 0.
The present invention also provides a computer readable storage medium comprising a stored program, wherein the program performs the method of pulsed window based perturbed nonlinear multi-agent quasi-coherence according to any of the above.
The present invention also provides an electronic device, comprising: one or more processors, a memory, and one or more programs, wherein the one or more programs are stored in the memory and configured to be executed by the one or more processors, the one or more programs including instructions for performing the pulsed window based perturbed nonlinear multi-agent quasi-coherence method as recited in any of the above.
The invention also provides a disturbed nonlinear multi-agent quasi-consistency system based on the pulse window, which comprises the following modules:
the mathematical model construction module of the nonlinear multi-agent system is used for constructing a mathematical model of the nonlinear multi-agent system subjected to external disturbance;
the mathematical model of the independent leading intelligent agent is used for constructing the mathematical model of the independent leading intelligent agent based on the mathematical model of the externally disturbed nonlinear multi-intelligent-agent system;
the controller building module is used for building a controller based on a mathematical model of an external disturbance nonlinear multi-agent system and a mathematical model of an independent leading agent and combining pulse control, containment control and distributed control strategies;
a controlled error multi-agent system model for defining an error based on the controller and constructing a controlled error multi-agent system model;
a discrete lyapunov function module for introducing a pulse time window using a discrete lyapunov function method such that pulses can be randomly generated over a period of time;
the relational expression derivation module is used for respectively deriving the relational expressions which are satisfied by the constructed Lyapunov function in the time interval and the pulse moment;
the comparison system acquisition module is used for obtaining a corresponding comparison system by utilizing a comparison principle according to the relational expression which is satisfied by the respectively deduced and constructed Lyapunov function in the time interval and the pulse moment;
and the global index quasi-consistency module is used for realizing the global index quasi-consistency of the disturbed nonlinear multi-agent by utilizing a comparison system and a pulse comparison principle.
The invention has the beneficial effects that:
the invention combines the time-varying time lag and the external disturbance signal, etc., establishes the disturbed time-varying time-lag nonlinear multi-agent model, greatly improves the adaptability of the model, and is more in line with the practice.
The invention introduces the concept of pulse time window by utilizing the discrete Lyapunov function method, can ensure that each pulse in the pulse sequence is randomly generated within a certain time, reduces the conservative property of the result and is convenient to realize in the actual production.
The distributed pulse containment controller is designed by using control strategies such as pulse control, containment control and the like. The feedback control item in the controller can be used for counteracting the possible negative effect brought by the pulse, so that the smooth realization of the system consistency is ensured. In addition, due to the existence of the pulse, the control effect is only generated at each discrete pulse moment, the control cost is reduced, and the control efficiency is improved.
The invention simultaneously considers the negative effect which the pulse may bring to the realization process of the system consistency, expands the discussion range of the pulse effect, and provides consistency discrimination conditions under different pulse effects and convergence rates under different effects by utilizing a parameter variation method, a pulse comparison principle and the like, so that the research result is more practical.
The foregoing description is only an overview of the technical solutions of the present invention, and in order to make the technical means of the present invention more clearly understood, the present invention may be implemented in accordance with the content of the description, and in order to make the above and other objects, features, and advantages of the present invention more clearly understood, the following preferred embodiments are described in detail with reference to the accompanying drawings.
Drawings
FIG. 1 is a flow chart of a method for perturbed nonlinear multi-agent quasi-coherence based on a pulse window in a preferred embodiment of the present invention;
FIG. 2 shows a preferred embodiment of the present invention, tkTime of dayA schematic diagram of a pulse time window;
FIG. 3 is a schematic diagram of control signals in a preferred embodiment of the present invention;
FIG. 4 is a system consistency error evolution curve when δ is 0< δ ≦ 1 in the preferred embodiment of the present invention;
FIG. 5 is a first state evolution curve of each agent of the system when δ is 0< δ ≦ 1 in the preferred embodiment of the present invention;
FIG. 6 is an evolution curve of each agent error of the system when δ is 0< δ ≦ 1 in the preferred embodiment of the present invention;
FIG. 7 is a system consistency error evolution curve when δ > 1 in the preferred embodiment of the present invention;
FIG. 8 is a first state evolution curve for each agent of the system when δ > 1 in the preferred embodiment of the present invention;
FIG. 9 is an evolution curve of error of each agent of the system when δ > 1 in the preferred embodiment of the present invention.
Detailed Description
The present invention is further described below in conjunction with the following figures and specific examples so that those skilled in the art may better understand the present invention and practice it, but the examples are not intended to limit the present invention.
As shown in fig. 1, the method for the quasi-consistency of the disturbed non-linear multi-agent based on the pulse window in the preferred embodiment of the present invention comprises the following steps:
step S1, constructing a mathematical model of the nonlinear multi-agent system subjected to external disturbance;
specifically, the mathematical model of the externally perturbed nonlinear multi-agent system is as follows:
Figure BDA0003590472520000071
wherein ,
Figure BDA0003590472520000072
is the ith agent z in the systemiA state vector of (a); matrix array
Figure BDA0003590472520000073
Is a constant parameter matrix in the system; t is time; tau (t) is time-varying time lag and meets the condition that tau (t) is more than 0 and less than or equal to tau; non-linear function
Figure BDA0003590472520000074
Satisfy the requirement of
Figure BDA0003590472520000075
ξi(t) is the external disturbance input, i is 1,2, …, N and
Figure BDA0003590472520000076
step S2, building a mathematical model of the independent leader agent on the basis of the mathematical model of the nonlinear multi-agent system with external disturbance;
specifically, the mathematical model of the independent leader agent is as follows:
Figure BDA0003590472520000077
wherein ,
Figure BDA0003590472520000078
is the state vector of the independent leader agent eta. Furthermore, the solution η (t) of the system can be seen as a rich information leader, while all agents in the non-linear multi-agent system (1) can be seen as its followers to track its state. Thus, the problem of global index quasi-consistency between the non-linear multi-agent system (1) and the target state (2) can be seen as a leader-follower problem.
S3, constructing a controller based on the mathematical model of the nonlinear multi-agent system of the external disturbance and the mathematical model of the independent leading agent by combining pulse control, containment control and distributed control strategies;
specifically, the controller is as follows:
Figure BDA0003590472520000079
wherein t is time; η (t) is the state vector of independent leader agent η; mu is a pulse effect; g is control strength; omegaiThe feedback containment gain is more than or equal to 0, and when the ith intelligent agent in the system is contained, omega isiIs greater than 0; define diagonal matrix Ω ═ diag { ω1,ω2,…,ωi}; distributed control matrix L ═ (L)ij)N×NSatisfy the dissipation condition, i.e.
Figure BDA00035904725200000710
For pulse sequences
Figure BDA00035904725200000711
Suppose it satisfies 0 ═ t0＜t1＜t2＜…＜tk< … and for
Figure BDA00035904725200000712
Is provided with
Figure BDA00035904725200000713
δ (t) is the dirac pulse function with respect to time t.
The controller is designed by combining control strategies such as pulse control, drag control and the like. The feedback containment term in the self model can be used for counteracting the possible negative effect brought by the pulse, so that the smooth realization of the system consistency is ensured. In addition, due to the existence of the pulse, the control effect is only generated at each discrete pulse moment, the control cost is reduced, and the control efficiency is improved.
Step S4, defining errors based on the controller and constructing a controlled error multi-agent system model;
specifically, step S4 includes:
defining an error based on the controller as:
ei(t)＝zi(t)-η(t)
and is
Figure BDA0003590472520000081
The controlled error multi-agent system model is constructed as follows:
Figure BDA0003590472520000082
wherein the non-linear function
Figure BDA0003590472520000083
Let ei(t) at t ═ tkThe moments being successive to the right, i.e.
Figure BDA0003590472520000084
In addition, the initial values of the controlled-error multi-agent system model satisfy:
ei(t)＝Λi(t),i＝1,2,...,N (5)
where- τ < t ≦ 0 and continuous function class
Figure BDA0003590472520000085
Step S5, introducing a pulse time window by using a discrete Lyapunov function method so that pulses can be randomly generated within a period of time;
the concept of a time window is first introduced,
Figure BDA0003590472520000086
indicating belonging to pulse time tkPulse time window (pulse window for short). Indicates at tkThe pulses generated at the time may be at
Figure BDA0003590472520000087
Randomly generated during this time interval, and figure 2 is a schematic diagram. Further taking into account all pulse instants
Figure BDA0003590472520000088
And
Figure BDA0003590472520000089
respectively, the minimum and maximum pulse intervals in the pulse train. All pulses will be in the pulse time window
Figure BDA00035904725200000810
Internal random generation, that is to say for
Figure BDA00035904725200000811
tk∈[tk-1+dmin,tk-1+dmax]。
It can be seen that, unlike the conventional intermittent control strategy, the pulse is generated only once within a pulse time window, i.e. the control signal is generated only at a certain instant instead of continuously within a period of time, which undoubtedly reduces the control time to a great extent and thus can effectively reduce the control cost. Meanwhile, the randomness of the pulse sequence is considered in the invention process, so that the research result is more in line with the actual situation of industrial production.
Given the definition of global index quasi-consistency: based on an arbitrary initial value, T if λ > 0 exists0> 0 and β > 0, global index quasi-consistency between the disturbed non-linear multi-agent system (1) and the independent leading agent (2) will be achieved in the form:
Figure BDA0003590472520000091
wherein ,
Figure BDA0003590472520000092
is the error bound.
The above definition plays a very important role in the process of finally judging whether the system achieves quasi-consistency.
Next, a pulse time window is introduced using the discrete lyapunov function method. It is necessary to introduce the discrete lyapunov function method: first, a time interval [ t ]k,tk+1) Is divided into [ tk,tk+dmin) and [tk+dmin,tk+1) Two major parts are
Figure BDA0003590472520000093
Second, [ t ]k,tk+dmin) This interval is equally divided into H smaller time intervals, each of which can be described as
Figure BDA0003590472520000094
And the length of each cell is
Figure BDA0003590472520000095
In addition, define
Figure BDA0003590472520000096
Subsequently, it is assumed that the continuous matrix function Θ (t) is in each interval
Figure BDA0003590472520000097
All the above are piecewise continuous, and define thetar＝Θ(tk+χr) With linear interpolation, the following transformations can be made:
Figure BDA0003590472520000098
wherein
Figure BDA0003590472520000099
For interpolating coefficients, the continuous matrix function Θ (t) is in the interval [ t ] by the above conversionk,tk+dmin) And is
Figure BDA00035904725200000910
Above only relates to the interpolation coefficient iota; further, assume that the continuous matrix function Θ (t) is in the interval [ t ]k+dmin,tk+1) Above is a constant matrix thetaH(ii) a Finally, according to the above analysis, the matrix function Θ (t) with discrete form can beIs shown as
Figure BDA00035904725200000911
Step S6, respectively deducing a relational expression which is satisfied by the constructed Lyapunov function in a time interval and a pulse moment;
using discrete Lyapunov function method, we can well deal with two different intervals [ t ] caused by introducing a pulse windowk,tk+dmin) and [tk+dmin,tk+1)，
Figure BDA0003590472520000101
According to the discrete Lyapunov function method, a corresponding discrete Lyapunov function can be constructed, theoretical derivation and numerical simulation are carried out, and then the condition and the method for realizing global index quasi-consistency of the system are obtained, and the specific process is as follows:
the lyapunov function was constructed as follows:
Figure BDA0003590472520000102
for the interval [ tk,tk+dmin)，
Figure BDA0003590472520000103
It can be derived that:
Figure BDA0003590472520000104
furthermore, we have:
Figure BDA0003590472520000105
from the definition of the error and the controlled error system, it can be derived:
Figure BDA0003590472520000106
next, for each item, an analysis is performed:
Figure BDA0003590472520000107
Figure BDA0003590472520000108
Figure BDA0003590472520000109
from the above conclusions, we can conclude that:
Figure BDA00035904725200001010
let phir(ι)＝ATΘr(ι)+Θr(ι)A+Θr(ι)E+κ1BTΘr(ι)Bκ1+2Θr(ι)I+Ξr，Πr(ι)＝κ2DTΘr(ι)Dκ2。
Using linear interpolation, one can derive:
Figure BDA0003590472520000111
if present
Figure BDA0003590472520000112
Then can derive
Figure BDA0003590472520000113
From the above analysis, one can obtain:
Figure BDA0003590472520000114
wherein t ∈ [ t ]k,tk+dmin)，
Figure BDA0003590472520000115
Next consider the interval tk+dmin,tk+1)，
Figure BDA0003590472520000116
Θ(t)＝ΘH. Similar to the above steps, we can derive:
Figure BDA0003590472520000117
if present
Figure BDA0003590472520000118
Then:
Figure BDA0003590472520000119
wherein t∈[tk+dmin,tk+1)，
Figure BDA00035904725200001110
Next, according to equations (7) and (8), it can be found that:
Figure BDA00035904725200001111
wherein ,
Figure BDA00035904725200001112
consider being at tk，
Figure BDA00035904725200001113
A time controlled error multi-agent system. By using the kronecker product, it can be rewritten into
Figure BDA0003590472520000121
Based on equation (9) and if present:
Figure BDA0003590472520000122
then it can be derived:
Figure BDA0003590472520000123
step S7, obtaining a corresponding comparison system by utilizing a comparison principle according to the relational expression which is satisfied by the respectively deduced and constructed Lyapunov function in the time interval and the pulse moment;
specifically, using the principle of comparison, will
Figure BDA0003590472520000128
Set to the special solution of the following pulse system and ε > 0:
the comparative system was obtained as follows:
Figure BDA0003590472520000124
and step S8, realizing disturbed nonlinear multi-agent global index quasi-consistency by using a comparison system and a pulse comparison principle.
In particular, according to the pulse comparison principle, it can be derived
Figure BDA0003590472520000125
And t is more than or equal to 0.
According to the parametric variational method, the following integral expression can be obtained:
Figure BDA0003590472520000126
wherein Y (t, s) (t > s ≧ 0) is the Cauchy matrix of the following linear pulse system:
Figure BDA0003590472520000127
next, considering various pulse effects, the range of δ is divided into 0< δ ≦ 1 and δ > 1, and then discussed separately.
Case 1: for 0< δ ≦ 1, it can be derived:
Figure BDA0003590472520000131
definition of
Figure BDA0003590472520000132
And
Figure BDA0003590472520000133
then there are:
Figure BDA0003590472520000134
defining a parameter function as
Figure BDA0003590472520000135
If so, then
Figure BDA0003590472520000136
And F (∞) > 0, can be obtained
Figure BDA0003590472520000137
In addition to this, the present invention is,
Figure BDA0003590472520000138
meaning that the parameter function is strictly monotonically increasing. Based on the above analysis, for the equation
Figure BDA0003590472520000139
Must have a special solution
Figure BDA00035904725200001310
For- τ ≦ t ≦ 0, it may be found that:
Figure BDA00035904725200001311
next, we need to demonstrate that at t > 0, the following formula is correct:
Figure BDA00035904725200001312
according to the counter-syndrome method, if the formula (13) is correct, then it must be present
Figure BDA00035904725200001313
Such that:
Figure BDA00035904725200001314
and for formula (12), are
Figure BDA0003590472520000141
This is still true, which means:
Figure BDA0003590472520000142
then, according to
Figure BDA0003590472520000143
Equations (11) and (15), the following detailed calculation procedure is given:
Figure BDA0003590472520000144
it can be seen that the results obtained conflict with equation (14). Therefore, the previous assumption is not valid, that is, equation (13) is correct.
Thus, when ε → 0, it can be found that:
Figure BDA0003590472520000151
that is to say:
Figure BDA0003590472520000152
i.e. after the introduction of the pulse window, when 0<When delta is less than or equal to 1, the controller (3) is applied to realize the global index quasi-consistency between the disturbed nonlinear multi-agent system (1) and the independent leading agent (2), wherein the convergence rate is
Figure BDA0003590472520000153
The error bound is:
Figure BDA0003590472520000154
case 2: for δ > 1, it can be derived:
Figure BDA0003590472520000155
similar to the procedure of case 1, it can be derived:
Figure BDA0003590472520000156
i.e. after introduction of the pulse window, when delta > 1, a global index quasi-consistency between the disturbed non-linear multi-agent system (1) and the independent leading agent (2) is achieved by applying the designed controller (3), wherein the convergence rate is
Figure BDA0003590472520000157
The error bound is:
Figure BDA0003590472520000161
and (4) conclusion: through the analysis and the derivation, the method for realizing the disturbed nonlinear multi-agent quasi-consistency based on the pulse window is summarized as follows:
taking into account the pulse time window
Figure BDA0003590472520000162
Next random pulse sequence t0,t1,t2,...,tk}，
Figure BDA0003590472520000163
We define
Figure BDA0003590472520000164
If parameters exist
Figure BDA0003590472520000165
Delta and a series of positive definite symmetry matrices theta r1,2, H-1, such that:
case 1: for 0< δ ≦ 1, if present:
Figure BDA0003590472520000166
Figure BDA0003590472520000167
Figure BDA0003590472520000168
Figure BDA0003590472520000169
the controlled-error multi-agent system (4) will converge to a tight set in the form of a global index
Figure BDA00035904725200001610
Wherein the convergence rate is:
Figure BDA00035904725200001611
is composed of
Figure BDA00035904725200001612
The special solution of (1) is that,
Figure BDA00035904725200001613
and:
Figure BDA00035904725200001614
Figure BDA00035904725200001615
referred to as the error bound. I.e. after the introduction of the pulse window, when 0<When delta is less than or equal to 1, the controller (3) is applied to realize the global index quasi-consistency between the disturbed nonlinear multi-agent system (1) and the independent leading agent (2), wherein the convergence rate is
Figure BDA00035904725200001616
Error bound is
Figure BDA00035904725200001617
Case 2: for δ > 1, if present:
Figure BDA00035904725200001618
Figure BDA00035904725200001619
Figure BDA0003590472520000171
Figure BDA0003590472520000172
the controlled-error multi-agent system (4) will converge to a tight set in the form of a global index
Figure BDA0003590472520000173
Wherein the convergence rate is:
Figure BDA0003590472520000174
is composed of
Figure BDA0003590472520000175
The special solution of (a) is that,
Figure BDA0003590472520000176
and:
Figure BDA0003590472520000177
Figure BDA0003590472520000178
referred to as the error bound. I.e. after introduction of the pulse window, when delta > 1, a global index quasi-consistency between the disturbed non-linear multi-agent system (1) and the independent leading agent (2) is achieved by applying the designed controller (3), wherein the convergence rate is
Figure BDA0003590472520000179
Error bound of
Figure BDA00035904725200001710
To further verify the effectiveness of the present invention, the following steps were taken:
step 1: and constructing a Chua's circuit model and selecting parameters.
The disturbed nonlinear multi-agent system model and the independent leading agent model are respectively selected as (1) and (2), wherein the disturbed nonlinear multi-agent system comprises six agents. Next, numerical simulation verification is performed using a very wide range of applications of the zeiss circuit, and each agent can be seen as an independent zeiss circuit. The Chua's circuit model with time varying skew is as follows:
Figure BDA00035904725200001711
wherein, the parameter value in the Chua's circuit model is a1＝9.02，a2＝14.97，a3＝0，a4＝-1，a5＝0，a6-0.67, nonlinear function
Figure BDA00035904725200001712
In addition to this, the present invention is,
Figure BDA00035904725200001713
h-3, control strength set to g-0.1, Θ0＝9×10-7I，Θ1＝0.1I，Θ2＝0.4I，ΘH＝0.9I。
Step 2: and respectively carrying out numerical simulation on 0< delta less than or equal to 1 and delta greater than 1 by considering different pulse effects.
With Simulink, when 0< δ ≦ 1:
after considering the pulse time window, the pulse effect μ is set to 1.3, Ω is set to diag {0.1,0,0,0,0.1,0},
Figure BDA0003590472520000181
Figure BDA0003590472520000182
defining consistency errors
Figure BDA0003590472520000183
When delta is equal to 0.95 < 1, the situation is satisfied. Can be obtained by calculation
Figure BDA0003590472520000184
In addition, the error bound is:
Figure BDA0003590472520000185
fig. 3 shows a schematic diagram of the control signal, and it can be seen that the control signal is not uniform but randomly generated because the introduction of the pulse time window results in the random generation of pulses within a certain time. Figures 4, 5 and 6 show the evolution curves of the consistent error, the evolution curve of the first state of each agent and the error curve of each agent, respectively. It can be seen that after a period of time, the error is globally and exponentially converged and kept within the allowable error range of 0.228, so that the quasi-consistency of the disturbed nonlinear multi-agent is realized when the delta is more than 0 and less than or equal to 1, and the reliability and the rationality of the invention are verified.
With Simulink, when δ > 1:
after considering the pulse time window, the pulse effect μ is set to 3, Ω is set to diag {0.5,0.3,0.3,0,0.5,0.5},
Figure BDA0003590472520000186
Figure BDA0003590472520000187
defining consistency errors
Figure BDA0003590472520000188
In this case, δ 1.32 > 1 is the same. By calculation
Figure BDA0003590472520000189
In addition, the error bound is:
Figure BDA00035904725200001810
figures 7, 8 and 9 show the evolution curves of the consistent error, the evolution curve of the first state of each agent and the error curve of each agent, respectively. It can be seen that, after a period of time, the error average global index converges and is kept within the allowable error range of 0.108, namely, the quasi-consistency of the disturbed nonlinear multi-agent is realized when delta is larger than 1, and the reliability and the rationality of the invention are verified.
Where δ is a parameter related to the impulse effect, V (t) is the Lyapunov function, D+V (t) is the Dini derivative of the Lyapunov function.
The invention establishes a novel multi-agent system model. The invention combines the time-varying time lag and external disturbance signals, etc., establishes a disturbed time-varying time-lag nonlinear multi-agent model, greatly improves the adaptability of the model, and is more practical.
The invention introduces a pulse time window, and considers the randomness of the pulse sequence. In most of the past, it was assumed that the pulse time was known and fixed, i.e. the randomness of the pulse sequence was ignored. The invention introduces the concept of pulse time window by utilizing the discrete Lyapunov function method, can ensure that each pulse in the pulse sequence is randomly generated within a certain time, reduces the conservative property of the result and is convenient to realize in the actual production.
The invention designs a novel high-efficiency controller. Different from controller structures proposed by other researches, the distributed pulse containment controller is designed by using control strategies such as pulse control and containment control. The feedback control item in the controller can be used for counteracting the possible negative effect brought by the pulse, so that the smooth realization of the system consistency is ensured. In addition, due to the existence of the pulse, the control effect is only generated at each discrete pulse moment, the control cost is reduced, and the control efficiency is improved.
The present invention takes into account the effect of multiple pulses. Different from the previous research work which only considers the positive effect brought by the pulse, the invention also considers the negative effect which is possibly brought to the realization process of the system consistency by the pulse, expands the discussion range of the pulse effect, and provides consistency judging conditions under different pulse effects and convergence rates under different effects by utilizing a parameter variation method, a pulse comparison principle and the like, so that the research result is more practical.
The invention breaks through the limitation that the traditional multi-agent consistency analysis only considers the pulse effect with positive effect, thereby popularizing the pulse effect to all feasible ranges and greatly expanding the application range of pulse control in the industrial field. In addition, due to the introduction of the pulse time window, pulses can be randomly generated within a certain time, a pulse sequence at an accurate moment does not need to be determined, the realization difficulty is reduced to a certain extent, and the industrial realization is facilitated.
The preferred embodiment of the present invention also discloses a computer readable storage medium, which comprises a stored program, wherein the program performs the pulsed window based perturbed nonlinear multi-agent quasi-consistency method according to any of the above embodiments.
The preferred embodiment of the present invention also discloses an electronic device, comprising: one or more processors, a memory, and one or more programs, wherein the one or more programs are stored in the memory and configured to be executed by the one or more processors, the one or more programs comprising instructions for performing the pulsed window based perturbed nonlinear multi-agent quasi-coherence method of any of the embodiments described above.
The preferred embodiment of the invention also discloses a disturbed nonlinear multi-agent quasi-consistency system based on the pulse window, which comprises the following modules:
the mathematical model construction module of the nonlinear multi-agent system is used for constructing a mathematical model of the nonlinear multi-agent system subjected to external disturbance;
the mathematical model of the independent leading agent is used for constructing the mathematical model of the independent leading agent based on the mathematical model of the externally disturbed nonlinear multi-agent system;
the controller building module is used for building a controller based on a mathematical model of an external disturbance nonlinear multi-agent system and a mathematical model of an independent leading agent and combining pulse control, containment control and distributed control strategies;
a controlled error multi-agent system model for defining an error based on the controller and constructing a controlled error multi-agent system model;
a discrete lyapunov function module for introducing a pulse time window using a discrete lyapunov function method such that pulses can be randomly generated over a period of time;
the relational expression derivation module is used for respectively deriving the relational expressions which are satisfied by the constructed Lyapunov function in the time interval and the pulse moment;
the comparison system acquisition module is used for obtaining a corresponding comparison system by utilizing a comparison principle according to the relational expression which is satisfied by the respectively deduced and constructed Lyapunov function in the time interval and the pulse moment;
and the global index quasi-consistency module is used for realizing the global index quasi-consistency of the disturbed nonlinear multi-agent by utilizing a comparison system and a pulse comparison principle.
The disturbed nonlinear multi-agent quasi-consistency system based on the pulse window according to the embodiment of the present invention is used to implement the foregoing disturbed nonlinear multi-agent quasi-consistency method based on the pulse window, and therefore, the detailed implementation of the system can be found in the foregoing part of the embodiment of the disturbed nonlinear multi-agent quasi-consistency method based on the pulse window, and therefore, the detailed implementation thereof can refer to the description of the corresponding part of the embodiment, and will not be further described herein.
In addition, since the perturbed nonlinear multi-agent quasi-consistency system based on the pulse window of this embodiment is used to implement the aforementioned perturbed nonlinear multi-agent quasi-consistency method based on the pulse window, its role corresponds to that of the above method, and is not described herein again.
The above embodiments are merely preferred embodiments for fully illustrating the present invention, and the scope of the present invention is not limited thereto. The equivalent substitution or change made by the technical personnel in the technical field on the basis of the invention is all within the protection scope of the invention. The protection scope of the invention is subject to the claims.

Claims (10)

1. The disturbed nonlinear multi-agent quasi-consistency method based on the pulse window is characterized by comprising the following steps of:
s1, constructing a mathematical model of the nonlinear multi-agent system subjected to external disturbance;
s2, constructing a mathematical model of the independent leading agent based on the mathematical model of the nonlinear multi-agent system of the external disturbance;
s3, constructing a controller by combining a pulse control strategy, a containment control strategy and a distributed control strategy based on a mathematical model of the nonlinear multi-agent system based on external disturbance and a mathematical model of an independent leader agent;
s4, defining errors based on the controller and constructing a controlled error multi-agent system model;
s5, introducing a pulse time window by using a discrete Lyapunov function method so that pulses can be randomly generated within a period of time;
s6, respectively deducing a relational expression which is satisfied by the constructed Lyapunov function in a time interval and a pulse moment;
s7, obtaining a corresponding comparison system by utilizing a comparison principle according to the relational expression which is satisfied by the respectively deduced and constructed Lyapunov function in the time interval and the pulse moment;
and S8, realizing disturbed nonlinear multi-agent global index quasi-consistency by using a comparison system and a pulse comparison principle.
2. The pulsed window based perturbed nonlinear multi-agent quasi-consistency method according to claim 1, wherein the mathematical model of the externally perturbed nonlinear multi-agent system is as follows:
Figure FDA0003590472510000011
wherein ,
Figure FDA0003590472510000012
is the ith agent z in the systemiA state vector of (a); matrix array
Figure FDA0003590472510000013
Is a constant parameter matrix in the system; t is time; tau (t) is time-varying time lag which is more than 0 and less than or equal to tau (t); non-linear function
Figure FDA0003590472510000014
Satisfy the requirement of
Figure FDA0003590472510000015
ξi(t) is an external disturbance input, i 1,2, N and
Figure FDA0003590472510000016
3. the pulsed window-based perturbed nonlinear multi-agent quasi-conformance method of claim 2 wherein the mathematical model of the independent lead agent is as follows:
Figure FDA0003590472510000021
wherein ,
Figure FDA0003590472510000022
a state vector for independent leader agent eta.
4. The pulsed window based perturbed nonlinear multi-agent quasi-conformance method of claim 1 wherein the controller is as follows:
Figure FDA0003590472510000023
wherein t is time; η (t) is the state vector of independent leader agent η; mu is a pulse effect; g is control strength; omegaiThe feedback containment gain is more than or equal to 0, and when the ith intelligent agent in the system is contained, omega isiIs greater than 0; define diagonal matrix Ω ═ diag { ω1,ω2,…,ωi}; distributed control matrix L ═ (L)ij)N×NSatisfy the dissipation condition, i.e.
Figure FDA0003590472510000024
For pulse sequences
Figure FDA0003590472510000025
Provided that it satisfies 0 ═ t0＜t1＜t2＜…＜tk< … and for
Figure FDA0003590472510000026
Is provided with
Figure FDA0003590472510000027
δ (t) is the dirac impulse function with respect to time t.
5. The perturbed nonlinear multi-agent quasi-consistency method based on pulse windows as recited in claim 4, wherein the step S4 comprises:
defining an error based on the controller as:
ei(t)＝zi(t)-η(t)
and is
Figure FDA0003590472510000028
The controlled error multi-agent system model is constructed as follows:
Figure FDA0003590472510000029
wherein the non-linear function
Figure FDA00035904725100000210
Let ei(t) at t ═ tkThe moments being successive to the right, i.e.
Figure FDA00035904725100000212
In addition, the initial values of the controlled-error multi-agent system model satisfy:
ei(t)＝Λi(t),i＝1,2,…,N
where- τ < t ≦ 0 and the continuous function class
Figure FDA00035904725100000211
6. The pulsed window-based perturbed nonlinear multi-agent quasi-coherent method according to claim 5, wherein step S6 comprises: respectively deducing the Dini derivatives of the constructed Lyapunov functions at [ t ]k,tk+1)，
Figure FDA0003590472510000031
The above requirements are:
Figure FDA0003590472510000032
and at the pulse time tkSatisfies the following conditions between the left and right boundaries:
Figure FDA0003590472510000033
where δ is a parameter related to the pulse effect, V (t) is the Lyapunov function, D+V (t) is the Dini derivative of the Lyapunov function.
7. The pulsed window-based perturbed nonlinear multi-agent quasi-consistency method according to claim 6, wherein the comparison system is as follows:
Figure FDA0003590472510000034
wherein ,
Figure FDA0003590472510000035
is a special solution of a pulse system, and epsilon is more than 0.
8. Computer readable storage medium, characterized in that the storage medium comprises a stored program, wherein the program performs the pulse window based perturbed nonlinear multi-agent quasi-coherence method according to any of claims 1-7.
9. An electronic device, comprising: one or more processors, a memory, and one or more programs, wherein the one or more programs are stored in the memory and configured to be executed by the one or more processors, the one or more programs comprising instructions for performing the pulsed window based perturbed nonlinear multi-agent quasi-consistency method of any of claims 1-7.
10. The disturbed nonlinear multi-agent quasi-consistency system based on the pulse window is characterized by comprising the following modules:
the mathematical model construction module of the nonlinear multi-agent system is used for constructing a mathematical model of the nonlinear multi-agent system subjected to external disturbance;
the mathematical model of the independent leading agent is used for constructing the mathematical model of the independent leading agent based on the mathematical model of the externally disturbed nonlinear multi-agent system;
the controller building module is used for building a controller based on a mathematical model of an external disturbance nonlinear multi-agent system and a mathematical model of an independent leading agent and combining pulse control, containment control and distributed control strategies;
a controlled error multi-agent system model for defining an error based on the controller and constructing a controlled error multi-agent system model;
a discrete lyapunov function module for introducing a pulse time window using a discrete lyapunov function method such that pulses can be randomly generated over a period of time;
the relational expression derivation module is used for respectively deriving the relational expressions which are satisfied by the constructed Lyapunov function in the time interval and the pulse moment;
the comparison system acquisition module is used for obtaining a corresponding comparison system by utilizing a comparison principle according to the relational expression which is satisfied by the respectively deduced and constructed Lyapunov function in the time interval and the pulse moment;
and the global index quasi-consistency module is used for realizing the global index quasi-consistency of the disturbed nonlinear multi-agent by utilizing a comparison system and a pulse comparison principle.
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