CN114721269B - 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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CN114721269B
CN114721269B CN202210375262.0A CN202210375262A CN114721269B CN 114721269 B CN114721269 B CN 114721269B CN 202210375262 A CN202210375262 A CN 202210375262A CN 114721269 B CN114721269 B CN 114721269B
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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 concept of a pulse time window is introduced into a system by using a discrete Lyapunov function method under the condition of considering a random pulse sequence by establishing a nonlinear multi-agent quasi-consistency model with disturbance, a controller is designed by fully combining various control methods such as pulse control, traction control and distributed control according to the influence of different pulse effects on the global index quasi-consistency of the system, and the like, and the method such as parameter variation and pulse comparison principle is utilized to give out sufficient quasi-consistency discrimination conditions under different pulse effects, so that the quasi-consistency convergence rate and error bound corresponding to the system are accurately calculated. In addition, numerical simulation is performed by using 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
Along with the continuous progress of the technology level, the multi-agent system is widely applied to the fields of wireless sensor networks, unmanned driving, robot formation and the like. In general, multiple agents are a combination of a plurality of independent agents, and the goal is to construct a large and complex system into a miniature system that is convenient for people to manage. Agents are mostly distributed with limited information processing and execution capabilities, while having limited bandwidth in sensing and communication capabilities. Therefore, how to enable the cooperation between agents is one of the main problems of interest. As a typical clustering phenomenon, consistency is widely found in nature, such as fish swarm foraging, bird swarm migration, welding robot cooperation, robot football game, and the like. The multi-agent consistency requires that each independent agent in the multi-agent system reach the same dynamic state, so that the maximization of production efficiency can be realized in the actual application scene.
In fact, only a small portion of multi-agent systems can achieve consistency through adjustment of their own parameters, and most of the multi-agent systems need to achieve consistency by externally applied controllers, i.e., adding control signals. Therefore, how to design a reasonable and efficient controller to achieve the consistency of multi-agent systems is a big focus of attention. The pulse signal is a typical transient and is characterized by discontinuities. By pulse control, the multi-agent system is controlled at each pulse time and simultaneously carries out information transmission, 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 consistency. However, most of the work today only focuses on the positive effect of the impulse effect on the system and ignores the negative effect of the impulse effect 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 actual production and life, there is always an error of unequal magnitude between each desired pulse input timing and actual pulse input timing for all pulses of one pulse sequence, so that minimum and maximum pulse intervals are generated. In short, pulses may be randomly generated over a time interval called a pulse time window, simply a pulse window. However, most of the existing works assume that the pulse sequence is fixed in advance, which undoubtedly ignores the randomness of the pulse sequence and thus increases the conservation of the study results. Therefore, the multi-agent consistency and the pulse window control work aiming at various pulse effects have certain theoretical and practical significance.
Disclosure of Invention
The invention aims to solve the technical problem of providing 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 pulse window-based disturbed nonlinear multi-agent quasi-consistency method, which comprises the following steps:
s1, constructing a mathematical model of a nonlinear multi-agent system subjected to external disturbance;
s2, constructing a mathematical model of an independent leading intelligent agent based on a mathematical model of an external disturbance-based nonlinear multi-intelligent agent system;
s3, constructing a controller by combining a mathematical model of the nonlinear multi-agent system based on external disturbance and a mathematical model of the independent leading agent with pulse control, containment control and a distributed control strategy;
s4, defining errors based on the controller and constructing a controlled error multi-intelligent system model;
s5, introducing a pulse time window by using a discrete Lyapunov function method so that pulses can be randomly generated in a period of time;
s6, respectively deriving the relation formula which is satisfied by the constructed Lyapunov function in the time interval and the pulse moment;
s7, obtaining a corresponding comparison system by utilizing a comparison principle according to a relational expression which is respectively derived and constructed and is met by the Lyapunov function in a time interval and a pulse moment;
s8, utilizing a comparison system and a pulse comparison principle to realize the global index quasi-consistency of the disturbed nonlinear multi-agent.
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 system i State vectors of (2); matrix array
Figure BDA0003590472520000033
A constant parameter matrix in the system; τ (t) is the time-varying time lag which is more than 0 and less than or equal to τ; nonlinear function->
Figure BDA0003590472520000034
Satisfy->
Figure BDA0003590472520000035
ξ i (t) is an external disturbance input, i=1, 2, …, N and
Figure BDA0003590472520000036
as a further improvement of the invention, the mathematical model of the independent lead agent is as follows:
Figure BDA0003590472520000037
wherein ,
Figure BDA0003590472520000038
is a state vector that independently leads agent η.
As a further improvement of the present invention, the controller is as follows:
Figure BDA0003590472520000039
wherein μ is the impulse effect; g is the control intensity; omega i More than or equal to 0 is feedback pinning gain, omega when the ith agent in the system is pinned i > 0; definition of the diagonal matrix Ω=diag { ω 1 ,ω 2 ,…,ω i -a }; distributed control matrix l= (L) ij ) N×N Meeting dissipation conditions, i.e.
Figure BDA00035904725200000310
For pulse sequences +.>
Figure BDA00035904725200000311
Assuming that it satisfies 0=t 0 ＜t 1 ＜t 2 ＜…＜t k < … and for->
Figure BDA00035904725200000312
There is->
Figure BDA00035904725200000313
Delta (t) is a dirac impulse 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:
e i (t)＝z i (t)-η(t)
and is also provided with
Figure BDA00035904725200000314
The controlled error multi-intelligent system model is constructed as follows:
Figure BDA00035904725200000315
wherein the nonlinear function
Figure BDA00035904725200000316
Suppose e i (t) at t=t k The moments are right consecutive, i.e. +.>
Figure BDA00035904725200000317
In addition, the initial values of the controlled error multi-intelligent system model satisfy:
e i (t)＝Λ i (t),i＝1,2,…,N
wherein, -tau < t is less than or equal to 0 and is a continuous function class
Figure BDA0003590472520000041
As a further improvement of the present invention, step S6 includes: the Dini derivatives of the constructed Lyapunov function are respectively derived at [ t ] k ,t k+1 )，
Figure BDA0003590472520000042
The following are satisfied:
Figure BDA0003590472520000043
at pulse time t k Satisfies the following between the left and right boundaries:
Figure BDA0003590472520000044
wherein delta is a parameter related to impulse effect, V (t) is a Lyapunov function, D + V (t) is the Dini derivative of the lyapunov function.
As a further development of the invention, the comparison system is as follows:
Figure BDA0003590472520000045
wherein ,
Figure BDA0003590472520000046
is a special solution of the pulse system, and epsilon > 0.
The invention also provides a computer readable storage medium comprising a stored program, wherein the program performs a pulse window based disturbed non-linear multi-agent quasi-compliance method as described in any of the above.
The invention also provides an electronic device, comprising: one or more processors, 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 pulse window based disturbed non-linear multi-agent quasi-compliance method as described 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 system comprises a mathematical model construction module of a nonlinear multi-agent system, a mathematical model generation module and a mathematical model generation module, wherein the mathematical model construction module is used for constructing a mathematical model of the nonlinear multi-agent system subjected to external disturbance;
the mathematical model of the independent leader intelligent agent is used for constructing the mathematical model of the independent leader intelligent agent based on the mathematical model of the nonlinear multi-intelligent agent system of external disturbance;
the controller construction module is used for constructing a controller based on a mathematical model of the nonlinear multi-agent system of external disturbance and a mathematical model of the independent leading agent by combining pulse control, containment control and a distributed control strategy;
the controlled error multi-intelligent system model is used for defining errors based on the controller and constructing the controlled error multi-intelligent system model;
the discrete Lyapunov function module is used for introducing a pulse time window by using a discrete Lyapunov function method so that pulses can be randomly generated in a period of time;
the relational expression deducing module is used for deducing relational expressions which are met by the constructed Lyapunov function in a time interval and a pulse moment respectively;
the comparison system acquisition module is used for obtaining a corresponding comparison system by utilizing a comparison principle according to the relation formula which is respectively deduced and established and is met by the Lyapunov function in a time interval and a 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 time-varying time-lag with external disturbance signals, and the like, 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 the concept of a pulse time window by utilizing a discrete Lyapunov function method, so that each pulse in a pulse sequence can be randomly generated within a certain time, the conservatism of the result is reduced, and the invention is convenient to realize in actual production.
The distributed pulse pinning controllers are designed by utilizing control strategies such as pulse control and pinning control. The system can counteract the possible negative effects of the pulse by using a feedback hold-down term in the controller, so that the smooth realization of the system consistency is ensured. In addition, because of the existence of the pulse, the control effect is only generated at each discrete pulse time, the control cost is reduced, and the control efficiency is improved.
The invention considers the negative effect of pulse on the realization process of system consistency, expands the discussion range of pulse effect, and gives consistency discrimination conditions under different pulse effects and convergence rate under different effects by using parameter variation method and pulse comparison principle, so that the research result is more practical.
The foregoing description is only an overview of the present invention, and is intended to be implemented in accordance with the teachings of the present invention, as well as the preferred embodiments thereof, together with the following detailed description of the invention, given by way of illustration only, together with the accompanying drawings.
Drawings
FIG. 1 is a flow chart of a disturbed non-linear multi-agent quasi-conformity method based on a pulse window in a preferred embodiment of the invention;
FIG. 2 is t in a preferred embodiment of the invention k A time-of-day pulse time window schematic;
FIG. 3 is a schematic diagram of control signals according to a preferred embodiment of the present invention;
FIG. 4 is a graph showing the evolution of systematic consistency errors for 0< delta < 1 in a preferred embodiment of the present invention;
FIG. 5 is a graph showing the first state evolution of each agent in the system when 0< delta < 1 in the preferred embodiment of the present invention;
FIG. 6 is a graph showing the error evolution of each agent in the system when 0< delta < 1 in the preferred embodiment of the present invention;
FIG. 7 is a graph showing the evolution of systematic consistency errors when delta > 1 in a preferred embodiment of the present invention;
FIG. 8 is a graph showing the first state evolution of each agent in the system when delta > 1 in the preferred embodiment of the present invention;
FIG. 9 is a graph showing the error evolution of each agent in the system when delta > 1 in the preferred embodiment of the present invention.
Detailed Description
The present invention will be further described with reference to the accompanying drawings and specific examples, which are not intended to be limiting, so that those skilled in the art will better understand the invention and practice it.
As shown in fig. 1, a pulse window-based disturbed nonlinear multi-agent quasi-consistency method in a preferred embodiment of the present invention comprises the following steps:
s1, constructing a mathematical model of a 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 system i State vectors of (2); matrix array
Figure BDA0003590472520000073
A constant parameter matrix in the system; t is time; τ (t) is the time-varying time lag which is more than 0 and less than or equal to τ; nonlinear function->
Figure BDA0003590472520000074
Satisfy->
Figure BDA0003590472520000075
ξ i (t) is an external disturbance input, i=1, 2, …, N and +.>
Figure BDA0003590472520000076
S2, constructing a mathematical model of an independent leading intelligent agent based on a mathematical model of an external disturbance nonlinear multi-intelligent agent system;
specifically, the mathematical model of the independent lead agent is as follows:
Figure BDA0003590472520000077
wherein ,
Figure BDA0003590472520000078
is a state vector that independently leads agent η. Furthermore, the solution η (t) of the system can be seen as a leader rich in rich information, while all agents in the nonlinear multi-agent system (1) can be seen as its followers to track its state. Thus, the problem of global index quasi-compliance between the nonlinear multi-agent system (1) and the target state (2) can be considered a lead-satellite problem.
S3, constructing a controller by combining a mathematical model of the nonlinear multi-agent system based on external disturbance and a mathematical model of the independent leading agent with pulse control, containment control and a distributed control strategy;
specifically, the controller is as follows:
Figure BDA0003590472520000079
wherein t is time; eta (t) is a state vector of independent leader agent eta; mu is the impulse effect; g is the control intensity; omega i More than or equal to 0 is feedback pinning gain, omega when the ith agent in the system is pinned i > 0; definition of the diagonal matrix Ω=diag { ω 1 ,ω 2 ,…,ω i -a }; distributed control matrix l= (L) ij ) N×N Meeting dissipation conditions, i.e.
Figure BDA00035904725200000710
For pulse sequences +.>
Figure BDA00035904725200000711
Assuming that it satisfies 0=t 0 ＜t 1 ＜t 2 ＜…＜t k < … and for->
Figure BDA00035904725200000712
There is->
Figure BDA00035904725200000713
Delta (t) is a dirac impulse function with respect to time t.
The controller is designed by combining control strategies such as pulse control, drag control and the like. The method can counteract the possible negative effects of the pulse by using feedback hold-down items in the self model, thereby ensuring the smooth realization of the system consistency. In addition, because of the existence of the pulse, the control effect is only generated at each discrete pulse time, the control cost is reduced, and the control efficiency is improved.
S4, defining errors based on the controller and constructing a controlled error multi-intelligent system model;
specifically, step S4 includes:
defining an error based on the controller as:
e i (t)＝z i (t)-η(t)
and is also provided with
Figure BDA0003590472520000081
The controlled error multi-intelligent system model is constructed as follows:
Figure BDA0003590472520000082
wherein the nonlinear function
Figure BDA0003590472520000083
Suppose e i (t) at t=t k The moments are right consecutive, i.e. +.>
Figure BDA0003590472520000084
In addition, the initial values of the controlled error multi-intelligent system model satisfy:
e i (t)＝Λ i (t),i＝1,2,...,N (5)
wherein,- τ < t.ltoreq.0 and continuous function class
Figure BDA0003590472520000085
S5, introducing a pulse time window by using a discrete Lyapunov function method so that pulses can be randomly generated in a period of time;
first the concept of a time window is introduced,
Figure BDA0003590472520000086
representing the time t of the pulse k Is referred to as a pulse time window (abbreviated as a pulse window). Indicating at t k The pulses generated at the moment can be +.>
Figure BDA0003590472520000087
Randomly generated during the time interval, and fig. 2 is a schematic diagram. Further considering all pulse moments, define +.>
Figure BDA0003590472520000088
and />
Figure BDA0003590472520000089
The minimum and maximum pulse intervals in the pulse train, respectively. All pulses will be in the pulse time window +.>
Figure BDA00035904725200000810
Internally randomly generated, i.e. for +.>
Figure BDA00035904725200000811
t k ∈[t k-1 +d min ,t k-1 +d max ]。
It can be seen that unlike conventional intermittent control strategies, the pulses are generated only once within a pulse time window, i.e. the control signal is generated only at a certain instant instead of continuously for 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 condition of industrial production.
A definition of global index quasi-consistency is given: based on an arbitrary initial value, if λ > 0, T 0 > 0 and β > 0, then global exponential quasi-agreement between the perturbed nonlinear multi-agent system (1) and the independent lead agent (2) will be achieved in the form of:
Figure BDA0003590472520000091
wherein ,
Figure BDA0003590472520000092
is an error bound.
The above definition plays a very important role in the final decision of 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, time interval [ t k ,t k+1 ) Is divided into [ t ] k ,t k +d min) and [tk +d min ,t k+1 ) Two major parts and
Figure BDA0003590472520000093
next, [ t ] k ,t k +d min ) This interval is equally divided into H smaller time intervals, each of which can be described as +.>
Figure BDA0003590472520000094
And the length between each cell is +.>
Figure BDA0003590472520000095
Furthermore, define
Figure BDA0003590472520000096
Subsequently, it is assumed that the continuous matrix function Θ (t) is +/in each interval>
Figure BDA0003590472520000097
All are piecewise continuous, defining Θ r ＝Θ(t k +χ r ) With linear interpolation, there may be the following conversions:
Figure BDA0003590472520000098
wherein
Figure BDA0003590472520000099
For interpolation coefficients, the continuous matrix function Θ (t) is in the interval t by the above conversion k ,t k +d min ) And->
Figure BDA00035904725200000910
The upper is related to the interpolation coefficient iota only; further, assume that the continuous matrix function Θ (t) is in the interval [ t ] k +d min ,t k+1 ) Above is a constant matrix Θ H The method comprises the steps of carrying out a first treatment on the surface of the Finally, based on the above analysis, the matrix function Θ (t) with discrete form can be expressed as
Figure BDA00035904725200000911
S6, respectively deducing the relation formula which is satisfied by the constructed Lyapunov function in the time interval and the pulse moment;
by using the discrete Lyapunov function method, we can well handle two different intervals [ t ] due to the introduction of a pulse window k ,t k +d min) and [tk +d min ,t k+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 conditions and methods for realizing global index quasi-consistency of the system are obtained, wherein the specific process is as follows:
the lyapunov function was constructed as follows:
Figure BDA0003590472520000102
for interval t k ,t k +d min )，
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 that:
Figure BDA0003590472520000106
next, an analysis was performed for each term:
Figure BDA0003590472520000107
Figure BDA0003590472520000108
Figure BDA0003590472520000109
from the above conclusions we can draw:
Figure BDA00035904725200001010
let phi be r (ι)＝A T Θ r (ι)+Θ r (ι)A+Θ r (ι)E+κ 1 B T Θ r (ι)Bκ 1 +2Θ r (ι)I+Ξ r ，Π r (ι)＝κ 2 D T Θ r (ι)Dκ 2 。
Using linear interpolation, it can be derived that:
Figure BDA0003590472520000111
if present
Figure BDA0003590472520000112
It can be derived that
Figure BDA0003590472520000113
From the above analysis, it is possible to obtain:
Figure BDA0003590472520000114
wherein t is [ t ] k ,t k +d min )，
Figure BDA0003590472520000115
Next consider the interval t k +d min ,t k+1 )，
Figure BDA0003590472520000116
Θ(t)＝Θ H . Similar to the above procedure we can derive:
Figure BDA0003590472520000117
if present
Figure BDA0003590472520000118
Then:
Figure BDA0003590472520000119
wherein t∈[tk +d min ,t k+1 )，
Figure BDA00035904725200001110
Next, according to formulas (7) and (8), it can be derived that:
Figure BDA00035904725200001111
wherein ,
Figure BDA00035904725200001112
consider at t k ，
Figure BDA00035904725200001113
Time controlled error multi-agent system. It can be rewritten into a Cronecker product
Figure BDA0003590472520000121
Based on formula (9) and if present:
Figure BDA0003590472520000122
it can be derived that:
Figure BDA0003590472520000123
s7, obtaining a corresponding comparison system by utilizing a comparison principle according to a relational expression which is respectively deduced and constructed and is met by the Lyapunov function in a time interval and a pulse moment;
in particular, using the comparison principle, the method
Figure BDA0003590472520000128
Set as a special solution for the pulse system as follows and ε > 0:
the comparison system was obtained as follows:
Figure BDA0003590472520000124
and S8, utilizing a comparison system and a pulse comparison principle to realize the global index quasi-consistency of the disturbed nonlinear multi-agent.
In particular, according to the pulse comparison principle, it is possible to derive
Figure BDA0003590472520000125
And t is more than or equal to 0.
According to the parameter variation method, the following integral formula can be obtained:
Figure BDA0003590472520000126
wherein Y (t, s) (t > s.gtoreq.0) is the Cauchy matrix of the following linear pulse system:
Figure BDA0003590472520000127
next, considering different impulse 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 that:
Figure BDA0003590472520000131
definition of the definition
Figure BDA0003590472520000132
and />
Figure BDA0003590472520000133
Then there are:
Figure BDA0003590472520000134
defining a parametric function as
Figure BDA0003590472520000135
If there is->
Figure BDA0003590472520000136
And F (≡) a) of > 0, can be derived from +.>
Figure BDA0003590472520000137
Furthermore, the->
Figure BDA0003590472520000138
Meaning that the parameter function is strictly monotonically increasing. Based on the above analysis, for equation +.>
Figure BDA0003590472520000139
There must be a special solution->
Figure BDA00035904725200001310
For- τ.ltoreq.t.ltoreq.0, it can be derived that:
Figure BDA00035904725200001311
next, we need to prove that at t > 0, the following formula is correct:
Figure BDA00035904725200001312
according to the countercheck method, if the formula (13) is correct, there must be
Figure BDA00035904725200001313
Such that:
Figure BDA00035904725200001314
and for formula (12), in
Figure BDA0003590472520000141
This is true, meaning:
Figure BDA0003590472520000142
next, according to
Figure BDA0003590472520000143
Equations (11) and (15) give the following detailed calculation procedure: />
Figure BDA0003590472520000144
It can be seen that the results obtained conflict with equation (14). Therefore, the previous assumption is not true, that is, equation (13) is correct.
Thus, when ε→0, it can be derived that:
Figure BDA0003590472520000151
that is to say:
Figure BDA0003590472520000152
i.e. after 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 that
Figure BDA0003590472520000153
The error bound is:
Figure BDA0003590472520000154
case 2: for delta > 1, it can be derived that:
Figure BDA0003590472520000155
similar to the procedure of case 1, it can be derived that:
Figure BDA0003590472520000156
i.e. after introduction of the pulse window, when delta > 1, a global index agreement between the disturbed non-linear multi-agent system (1) and the independent lead agent (2) is achieved by applying the designed controller (3), wherein the convergence rate is
Figure BDA0003590472520000157
The error bound is:
Figure BDA0003590472520000161
conclusion: through the analysis and the deduction, the disturbed nonlinear multi-agent quasi-consistency implementation method based on the pulse window is summarized as follows:
consider a pulse time window
Figure BDA0003590472520000162
The next random pulse sequence { t } 0 ,t 1 ,t 2 ,...,t k }，/>
Figure BDA0003590472520000163
We define +.>
Figure BDA0003590472520000164
If there is a parameter +.>
Figure BDA0003590472520000165
Delta and a series of positive definite symmetry matrices theta r R=1, 2,..h-1, such that:
case 1: for 0< delta.ltoreq.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->
Figure BDA00035904725200001612
Is a special solution of->
Figure BDA00035904725200001613
And:
Figure BDA00035904725200001614
Figure BDA00035904725200001615
referred to as error bound. I.e. after 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 delta > 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->
Figure BDA0003590472520000175
Is a special solution of->
Figure BDA0003590472520000176
And:
Figure BDA0003590472520000177
Figure BDA0003590472520000178
referred to as error bound. I.e. after introduction of the pulse window, when delta > 1, a global index agreement between the disturbed non-linear multi-agent system (1) and the independent lead agent (2) is achieved by applying the designed controller (3), wherein the convergence rate is +.>
Figure BDA0003590472520000179
Error bound is->
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 by using a zeiss circuit with a very wide application range, and each intelligent agent can be regarded 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 a 1 ＝9.02，a 2 ＝14.97，a 3 ＝0，a 4 ＝-1，a 5 ＝0，a 6 = -0.67, nonlinear function
Figure BDA00035904725200001712
In addition, in the case of the optical fiber,
Figure BDA00035904725200001713
h=3, the control intensity is set to g=0.1, Θ 0 ＝9×10 -7 I，Θ 1 ＝0.1I，Θ 2 ＝0.4I，Θ H ＝0.9I。
Step 2: taking different pulse effects into consideration, respectively carrying out numerical simulation on 0< delta less than or equal to 1 and delta > 1.
With Simulink, when 0< δ+.1:
after the pulse time window is considered, the pulse effect μ=1.3, Ω=diag {0.1,0,0,0,0.1,0},
Figure BDA0003590472520000181
Figure BDA0003590472520000182
definition of coherence error->
Figure BDA0003590472520000183
Delta=0.95 < 1 is the case at this time. By calculation, it can be derived that
Figure BDA0003590472520000184
In addition, the error bound is:
Figure BDA0003590472520000185
fig. 3 shows a schematic diagram of the control signal, it can be seen that the control signal is not uniform but randomly generated, because the introduction of a pulse time window results in random generation of pulses over a certain period of time. Fig. 4, 5, and 6 show a consistency error evolution curve, an evolution curve of the first state of each agent, and an error curve of each agent, respectively. It can be seen that over a period of time, the global index of errors is converged and kept within the allowable error limit of 0.228, that is, the quasi-consistency of the disturbed nonlinear multi-agent is realized when 0< delta is less than or equal to 1, and the reliability and rationality of the invention are verified.
With Simulink, when δ > 1:
after the pulse time window is considered, the pulse effect μ=3, Ω=diag {0.5,0.3,0.3,0,0.5,0.5},
Figure BDA0003590472520000186
Figure BDA0003590472520000187
definition of coherence error->
Figure BDA0003590472520000188
Delta=1.32 > 1 is the case at this time. By calculation to obtain
Figure BDA0003590472520000189
In addition, the error bound is:
Figure BDA00035904725200001810
fig. 7, 8, and 9 show a consistency error evolution curve, an evolution curve of the first state of each agent, and an error curve of each agent, respectively. It can be seen that over a period of time, the global index of errors is converged and kept within the allowable error bound of 0.108, i.e. the quasi-consistency of the disturbed nonlinear multi-agent is realized when delta > 1, and the reliability and rationality of the invention are verified.
Wherein delta is a parameter related to impulse effect, V (t) is a Lyapunov function, D + V (t) is the Dini derivative of the lyapunov function.
The invention establishes a novel multi-intelligent system model. The invention combines time-varying time-lag with external disturbance signals, and the like, 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 previous work it was assumed that the pulse instants were known and fixed, i.e. the randomness of the pulse sequence was ignored. The invention introduces the concept of a pulse time window by utilizing a discrete Lyapunov function method, so that each pulse in a pulse sequence can be randomly generated within a certain time, the conservatism of the result is reduced, and the invention is convenient to realize in actual production.
The invention designs a novel and efficient controller. Unlike other controller structures proposed by research, the present invention utilizes control strategies such as pulse control and pinning control to design a distributed pulse pinning controller. The system can counteract the possible negative effects of the pulse by using a feedback hold-down term in the controller, so that the smooth realization of the system consistency is ensured. In addition, because of the existence of the pulse, the control effect is only generated at each discrete pulse time, the control cost is reduced, and the control efficiency is improved.
The present invention considers the multiple pulse effect. Unlike the previous research work, the invention considers only the positive effect brought by the pulse, and simultaneously considers the negative effect possibly brought by the pulse to the realization process of the system consistency, expands the discussion range of the pulse effect, and gives consistency discrimination conditions under different pulse effects and convergence rates under different effects by using 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 pulse effect with positive effect, thereby promoting the pulse effect to all feasible ranges and greatly expanding the application range of pulse control in the industrial field. In addition, the pulse time window is introduced to enable the pulse to be randomly generated within a certain time, so that a pulse sequence at an accurate moment is not required to be determined, the implementation difficulty is reduced to a certain extent, and the method is more beneficial to industrial implementation.
The preferred embodiment of the invention also discloses a computer readable storage medium comprising a stored program, wherein the program executes the disturbed non-linear multi-agent quasi-consistency method based on the pulse window according to any of the above embodiments.
The preferred embodiment of the invention also discloses an electronic device, which comprises: the system comprises 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 pulse window based disturbed non-linear multi-agent quasi-compliance 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 system comprises a mathematical model construction module of a nonlinear multi-agent system, a mathematical model generation module and a mathematical model generation module, wherein the mathematical model construction module is used for constructing a mathematical model of the nonlinear multi-agent system subjected to external disturbance;
the mathematical model of the independent leader intelligent agent is used for constructing the mathematical model of the independent leader intelligent agent based on the mathematical model of the nonlinear multi-intelligent agent system of external disturbance;
the controller construction module is used for constructing a controller based on a mathematical model of the nonlinear multi-agent system of external disturbance and a mathematical model of the independent leading agent by combining pulse control, containment control and a distributed control strategy;
the controlled error multi-intelligent system model is used for defining errors based on the controller and constructing the controlled error multi-intelligent system model;
the discrete Lyapunov function module is used for introducing a pulse time window by using a discrete Lyapunov function method so that pulses can be randomly generated in a period of time;
the relational expression deducing module is used for deducing relational expressions which are met by the constructed Lyapunov function in a time interval and a pulse moment respectively;
the comparison system acquisition module is used for obtaining a corresponding comparison system by utilizing a comparison principle according to the relation formula which is respectively deduced and established and is met by the Lyapunov function in a time interval and a 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 pulse window-based disturbed nonlinear multi-agent quasi-consistency system of the embodiment of the invention is used for realizing the pulse window-based disturbed nonlinear multi-agent quasi-consistency method, so that the specific implementation of the system can be seen from the embodiment part of the pulse window-based disturbed nonlinear multi-agent quasi-consistency method in the foregoing, and therefore, the specific implementation of the system can be referred to the description of the corresponding embodiment of each part and will not be further described herein.
In addition, since the pulse window-based disturbed nonlinear multi-agent quasi-consistency system of the present embodiment is used for implementing the aforementioned pulse window-based disturbed nonlinear multi-agent quasi-consistency method, the actions thereof correspond to the actions of the above-mentioned method, and the details thereof are not repeated here.
The above embodiments are merely preferred embodiments for fully explaining the present invention, and the scope of the present invention is not limited thereto. Equivalent substitutions and modifications will occur to those skilled in the art based on the present invention, and are intended to be within the scope of the present invention. The protection scope of the invention is subject to the claims.

Claims (4)

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 a nonlinear multi-agent system subjected to external disturbance; the following are provided:
Figure FDA0003967880010000011
wherein ,
Figure FDA0003967880010000012
is the ith agent z in the system i State vectors of (2); matrix array
Figure FDA0003967880010000013
A constant parameter matrix in the system; t is time; τ (t) is the time-varying time lag which is more than 0 and less than or equal to τ; nonlinear function->
Figure FDA0003967880010000014
Satisfy->
Figure FDA0003967880010000015
ξ i (t) is an external disturbance input, i=1, 2,..n and +.>
Figure FDA0003967880010000016
S2, constructing a mathematical model of an independent leading intelligent agent based on a mathematical model of an external disturbance-based nonlinear multi-intelligent agent system; the following are provided:
Figure FDA0003967880010000017
wherein ,
Figure FDA0003967880010000018
a state vector that is an independent leader agent η;
s3, constructing a controller by combining a mathematical model of the nonlinear multi-agent system based on external disturbance and a mathematical model of the independent leading agent with pulse control, containment control and a distributed control strategy; the controller is as follows:
Figure FDA0003967880010000019
wherein t is time; eta (t) is a state vector of independent leader agent eta; mu is the impulse effect; g is the control intensity; omega i More than or equal to 0 is feedback pinning gain, omega when the ith agent in the system is pinned i > 0; definition of the diagonal matrix Ω=diag { ω 1 ,ω 2 ,...,ω i -a }; distributed control matrix l= (L) ij ) N×N Meeting dissipation conditions, i.e.
Figure FDA00039678800100000110
For pulse sequences
Figure FDA00039678800100000111
Assuming that it satisfies 0=t 0 ＜t 1 ＜t 2 ＜…＜t k < … and for->
Figure FDA00039678800100000112
There is->
Figure FDA00039678800100000113
Delta (t) is a dirac impulse function with respect to time t;
s4, defining errors based on the controller and constructing a controlled error multi-intelligent system model; comprising the following steps:
defining an error based on the controller as:
e i (t)＝z i (t)-η(t)
and is also provided with
Figure FDA0003967880010000021
The controlled error multi-intelligent system model is constructed as follows:
Figure FDA0003967880010000022
wherein the nonlinear function
Figure FDA0003967880010000023
Suppose e i (t) at t=t k The moments are right consecutive, i.e. +.>
Figure FDA0003967880010000024
In addition, the initial values of the controlled error multi-intelligent system model satisfy:
e i (t)＝Λ i (t),i＝1,2,...,N
wherein, -tau < t is less than or equal to 0 and is a continuous function class
Figure FDA0003967880010000025
S5, introducing a pulse window by using a discrete Lyapunov function method so that pulses can be randomly generated in a period of time; comprising the following steps:
first, time interval [ t k ,t k+1 ) Is divided into [ t ] k ,t k +d min) and [tk +d min ,t k+1 ) Two major parts and
Figure FDA0003967880010000026
next, [ t ] k ,t k +d min ) This interval is equally divided into H smaller time intervals, each of which can be described as
Figure FDA00039678800100000212
And the length between each cell is +.>
Figure FDA0003967880010000027
Furthermore, define
Figure FDA0003967880010000028
Subsequently, it is assumed that the continuous matrix function Θ (t) is +/in each interval>
Figure FDA00039678800100000213
All are piecewise continuous, defining Θ r ＝Θ(t k +χ r ) With linear interpolation, there may be the following conversions:
Figure FDA00039678800100000214
wherein
Figure FDA0003967880010000029
For interpolation coefficients, the continuous matrix function Θ (t) is in the interval t by the above conversion k ,t k +d min ) And->
Figure FDA00039678800100000210
The upper is related to the interpolation coefficient iota only; assume that the continuous matrix function Θ (t) is in interval t k +d min ,t k+1 ) Above is a constant matrix Θ H The method comprises the steps of carrying out a first treatment on the surface of the Finally, the matrix function Θ (t) with discrete form can be expressed as:
Figure FDA00039678800100000211
s6, respectively deriving the relation formula which is satisfied by the constructed discrete Lyapunov function in the time interval and the pulse moment; comprising the following steps: the Dini derivatives of the discrete Lyapunov functions constructed are derived at [ t k ,t k+1 )，
Figure FDA0003967880010000031
The following are satisfied:
Figure FDA0003967880010000032
at pulse time t k Satisfies the following between the left and right boundaries:
Figure FDA0003967880010000037
wherein delta is a parameter related to impulse effect, V (t) is a discrete Lyapunov function, D + V (t) is the Dini derivative of the discrete lyapunov function;
s7, obtaining a corresponding comparison system by utilizing a comparison principle according to a relation formula which is respectively deduced and constructed and is met by the discrete Lyapunov function in a time interval and a pulse moment; the comparison system is as follows:
Figure FDA0003967880010000033
wherein ,
Figure FDA0003967880010000034
is a special solution of a pulse system, and epsilon is more than 0;
s8, utilizing a comparison system and a pulse comparison principle to realize the global index quasi-consistency of the disturbed nonlinear multi-agent;
definition of global index quasi-consistency: based on an arbitrary initial value, if λ > 0, T 0 If > 0 and beta > 0, then global index quasi-agreement between the perturbed nonlinear multi-agent system and the independent lead agent is achieved in the form of:
Figure FDA0003967880010000036
/>
wherein ,
Figure FDA0003967880010000035
is an error bound.
2. A computer readable storage medium, characterized in that the storage medium comprises a stored program, wherein the program performs the pulse window based disturbed non-linear multi-agent quasi-conformity method as claimed in claim 1.
3. An electronic device, comprising: one or more processors, 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 pulse window based disturbed non-linear multi-agent quasi-compliance method of claim 1.
4. The disturbed nonlinear multi-agent quasi-consistency system based on the pulse window is characterized by comprising the following modules:
the system comprises a mathematical model construction module of a nonlinear multi-agent system, a mathematical model generation module and a mathematical model generation module, wherein the mathematical model construction module is used for constructing a mathematical model of the nonlinear multi-agent system subjected to external disturbance; the following are provided:
Figure FDA0003967880010000041
wherein ,
Figure FDA0003967880010000042
is the ith agent z in the system i State vectors of (2); matrix array
Figure FDA0003967880010000043
A constant parameter matrix in the system; t is time; τ (t) is the time-varying time lag which is more than 0 and less than or equal to τ; nonlinear function->
Figure FDA0003967880010000044
Satisfy->
Figure FDA0003967880010000045
ξ i (t) is an external disturbance input, i=1, 2,..n and +.>
Figure FDA0003967880010000046
The mathematical model of the independent leader intelligent agent is used for constructing the mathematical model of the independent leader intelligent agent based on the mathematical model of the nonlinear multi-intelligent agent system of external disturbance; the following are provided:
Figure FDA0003967880010000047
wherein ,
Figure FDA0003967880010000048
a state vector that is an independent leader agent η;
the controller construction module is used for constructing a controller based on a mathematical model of the nonlinear multi-agent system of external disturbance and a mathematical model of the independent leading agent by combining pulse control, containment control and a distributed control strategy; the controller is as follows:
Figure FDA0003967880010000049
wherein t is time; eta (t) is a state vector of independent leader agent eta; mu is the impulse effect; g is the control intensity; omega i More than or equal to 0 is feedback pinning gain, omega when the ith agent in the system is pinned i > 0; definition of the diagonal matrix Ω=diag { ω 1 ,ω 2 ,...,ω i -a }; distributed control matrix l= (L) ij ) N×N Meeting dissipation conditions, i.e.
Figure FDA00039678800100000410
For pulse sequences
Figure FDA00039678800100000411
Assuming that it satisfies 0=t 0 ＜t 1 ＜t 2 ＜…＜t k < … and for->
Figure FDA00039678800100000413
There is->
Figure FDA00039678800100000412
Delta (t) is a dirac impulse function with respect to time t;
the controlled error multi-intelligent system model is used for defining errors based on the controller and constructing the controlled error multi-intelligent system model; comprising the following steps:
defining an error based on the controller as:
e i (t)＝z i (t)-η(t)
and is also provided with
Figure FDA0003967880010000051
The controlled error multi-intelligent system model is constructed as follows:
Figure FDA0003967880010000052
wherein the nonlinear function
Figure FDA0003967880010000053
Suppose e i (t) at t=t k The moments are right consecutive, i.e. +.>
Figure FDA0003967880010000054
In addition, the initial values of the controlled error multi-intelligent system model satisfy:
e i (t)＝Λ i (t),i＝1,2,...,N
wherein, -tau < t is less than or equal to 0 and is a continuous function class
Figure FDA0003967880010000055
The discrete Lyapunov function module is used for introducing a pulse window by using a discrete Lyapunov function method so that pulses can be randomly generated in a period of time; comprising the following steps:
first, time interval [ t k ,t k+1 ) Is divided into [ t ] k ,t k +d min) and [tk +d min ,t k+1 ) Two major parts and
Figure FDA0003967880010000056
next, [ t ] k ,t k +d min ) This interval is equally divided into H smaller time intervals, each of which can be described as
Figure FDA00039678800100000512
And the length between each cell is +.>
Figure FDA0003967880010000057
Furthermore, define
Figure FDA0003967880010000058
Subsequently, it is assumed that the continuous matrix function Θ (t) is +/in each interval>
Figure FDA00039678800100000513
All are piecewise continuous, defining Θ r ＝Θ(t k +χ r ) With linear interpolation, there may be the following conversions:
Figure FDA00039678800100000514
wherein
Figure FDA0003967880010000059
For interpolation coefficients, the continuous matrix function Θ (t) is in the interval t by the above conversion k ,t k +d min ) And->
Figure FDA00039678800100000510
The upper is related to the interpolation coefficient iota only; assume that the continuous matrix function Θ (t) is in interval t k +d min ,t k+1 ) Above is a constant matrix Θ H The method comprises the steps of carrying out a first treatment on the surface of the Finally, the matrix function Θ (t) with discrete form can be expressed as:
Figure FDA00039678800100000511
the relational expression deducing module is used for deducing relational expressions which are met by the constructed discrete Lyapunov function in a time interval and a pulse moment respectively; comprising the following steps: the Dini derivatives of the discrete Lyapunov functions constructed are derived at [ t k ,t k+1 )，/>
Figure FDA0003967880010000061
The following are satisfied:
Figure FDA0003967880010000062
at pulse time t k Satisfies the following between the left and right boundaries:
Figure FDA0003967880010000063
wherein delta is a parameter related to impulse effect, V (t) is a discrete Lyapunov function, D + V (t) is the Dini derivative of the discrete lyapunov function;
the comparison system acquisition module is used for obtaining a corresponding comparison system by utilizing a comparison principle according to a relation formula which is respectively deduced and constructed and is met by the discrete Lyapunov function in a time interval and a pulse moment; the comparison system is as follows:
Figure FDA0003967880010000064
wherein ,
Figure FDA0003967880010000065
is a special solution of a pulse system, and epsilon is more than 0;
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;
definition of global index quasi-consistency: based on an arbitrary initial value, if λ > 0, T 0 If > 0 and beta > 0, then global index quasi-agreement between the perturbed nonlinear multi-agent system and the independent lead agent is achieved in the form of:
Figure FDA0003967880010000066
wherein ,
Figure FDA0003967880010000067
is an error bound. />
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