CN111339489A - Controller design method of multi-agent system under limited domain condition - Google Patents

Controller design method of multi-agent system under limited domain condition Download PDF

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CN111339489A
CN111339489A CN202010092880.5A CN202010092880A CN111339489A CN 111339489 A CN111339489 A CN 111339489A CN 202010092880 A CN202010092880 A CN 202010092880A CN 111339489 A CN111339489 A CN 111339489A
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张博闻
黄佳卉
马立丰
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Nanjing University of Science and Technology
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Abstract

The invention discloses a method for designing a controller of a multi-agent system under a limited domain condition, which comprises the following steps: establishing a nonlinear discrete time-varying random multi-agent system mathematical model under the condition of a limited domain; establishing a controller protocol and a closed-loop system equation according to a nonlinear discrete time-varying random multi-agent system mathematical model under the condition of a limited domain; establishing a mean square H infinity consistent performance index according to a closed loop system equation; determining a design objective of controller parameters for enabling the closed loop system to meet a mean square H-infinity consistent performance index; determining a linear matrix inequality for solving the controller parameters according to the determined design objective of the controller parameters which enables the closed-loop system to meet the mean square H-infinity consistent performance index; and solving the linear matrix inequality to obtain corresponding controller parameters, and finishing the design of the controller. The invention comprehensively considers the influences of nonlinearity, time-varying parameters and random effects, and has strong practicability.

Description

Controller design method of multi-agent system under limited domain condition
Technical Field
The invention belongs to a controller design technology, in particular to a controller design method of a multi-agent system under the condition of a limited domain.
Background
The distributed cooperative control of the multi-agent system is an important research field in the aspect of control theory. The problem of controlling the consistency of a multi-agent system is more a current research hotspot, and the consistency means that the states of all individuals in a complex system tend to be the same value over time, and describes the information exchange process between the individual and the adjacent individuals.
Currently, many achievements have been made in the research of consistency control, output regulation, tracking control and synchronization of a general multi-agent system. However, some multi-agent systems with complex topology, nonlinearity and randomness are not concerned, and the H-infinity consistency mean square consistency performance is not researched.
Disclosure of Invention
The invention aims to provide a method for designing a controller of a multi-agent system under a limited domain situation.
The technical solution for realizing the purpose of the invention is as follows: a method for designing a controller of a multi-agent system under a limited domain condition comprises the following specific steps:
step 1, establishing a nonlinear discrete time-varying random multi-agent system mathematical model under a finite field condition;
step 2, establishing a controller protocol and a closed-loop system equation according to a nonlinear discrete time-varying random multi-agent system mathematical model under the condition of a finite field;
step 3, establishing a mean square H infinity consistent performance index according to a closed loop system equation;
step 4, determining a design target of the controller parameters which enable the closed-loop system to meet the mean square H-infinity consistent performance index;
step 5, determining a linear matrix inequality for solving the controller parameters according to the determined design target of the controller parameters which enables the closed-loop system to meet the mean square H infinity consistent performance index;
and 6, solving the linear matrix inequality to obtain corresponding controller parameters, and finishing the design of the controller.
Preferably, the established mathematical model of the nonlinear discrete time-varying random multi-agent system under the condition of the finite field is specifically as follows:
Figure BDA0002384301560000021
wherein k is more than or equal to 0 and less than or equal to T is a finite field, T is a given normal number,
Figure BDA0002384301560000022
is the state vector for time k, and,
Figure BDA0002384301560000023
is the state vector at time k +1,
Figure BDA0002384301560000024
in order to control the input of the electronic device,
Figure BDA0002384301560000025
is the measured output of the agent i,
Figure BDA0002384301560000026
is the controlled output of the agent and,
Figure BDA0002384301560000027
represents the covariance Wi,kN uncorrelated zero mean gaussian white noise sequences > 0,
Figure BDA0002384301560000028
is and Wi,kUncorrelated zero mean white Gaussian noise sequences, Ak,Bk,Ck,Dk,Ek,FkAnd LkIs a time-varying matrix of different dimensions, fi,kIs a non-linear random function.
Preferably, the non-linear random function
Figure BDA0002384301560000029
And satisfies the following constraints:
Figure BDA00023843015600000210
Figure BDA00023843015600000211
Figure BDA00023843015600000212
Figure BDA00023843015600000213
wherein the content of the first and second substances,
Figure BDA00023843015600000214
and
Figure BDA00023843015600000215
is a matrix of the corresponding dimension.
Preferably, the controller protocol established in step 2 is:
Figure BDA00023843015600000216
wherein the content of the first and second substances,
Figure BDA00023843015600000217
is the feedback gain matrix to be designed,
Figure BDA00023843015600000218
represents the output of the jth agent, hi,jAre non-negative real numbers.
Preferably, the specific method for establishing the closed-loop system equation according to the nonlinear discrete time-varying random multi-agent system mathematical model under the finite field condition is as follows:
according to a controller protocol and a mathematical model of a nonlinear discrete time-varying random multi-agent system, obtaining:
Figure BDA0002384301560000031
by extending the system state xi,kAnd obtaining a closed loop system equation:
Figure BDA0002384301560000032
wherein:
Figure BDA0002384301560000033
Figure BDA0002384301560000034
Figure BDA0002384301560000035
Figure BDA0002384301560000036
Figure BDA0002384301560000037
Figure BDA0002384301560000038
Figure BDA0002384301560000039
Figure BDA00023843015600000310
Figure BDA00023843015600000311
Figure BDA00023843015600000312
Figure BDA00023843015600000313
Figure BDA00023843015600000314
the given incidence matrix embodies the topological structure among the multiple intelligent agents;
Figure BDA00023843015600000315
the operation symbols represent that the Crohn's products are made among the matrixes; i isNIs an N-dimensional identity matrix, U, V are both block matrices, InNIs one of the identity matrices;
1Nrepresenting an n-dimensional column vector with all elements 1,
Figure BDA00023843015600000316
is its transpose, taking into account the average state values of all agents in the system:
Figure BDA00023843015600000317
will be provided with
Figure BDA00023843015600000318
Substituting, we can get:
Figure BDA0002384301560000041
order to
Figure BDA0002384301560000042
Obtaining:
Figure BDA0002384301560000043
and from the average state values of all agents:
Figure BDA0002384301560000044
wherein:
Figure BDA0002384301560000045
Figure BDA0002384301560000046
order to
Figure BDA0002384301560000047
Obtaining a system equation:
Figure BDA0002384301560000048
Figure BDA0002384301560000049
preferably, establishing the mean square H ∞ consistent performance indexes according to the closed-loop system equation is as follows:
at time k, the H ∞ match performance index is as follows:
Figure BDA00023843015600000410
wherein
Figure BDA00023843015600000411
Φ > 0 is a given weighting matrix;
at time k, the mean square consistent performance index of agent i (i ═ 1, 2.., N) is as follows:
Figure BDA0002384301560000051
each agent i, its initial state xi,0(i ═ 1,2,. N) is known and satisfies the following condition:
Figure BDA0002384301560000052
by combining the two points, the mean square H infinity consistent performance index is established as follows:
Figure BDA0002384301560000053
wherein γ is a predetermined noise level, { XI-k}0≤k≤TIs a predetermined positive definite matrix sequence, which represents the upper bound of the mean square consistent performance.
Preferably, the design objective of determining the output feedback controller parameters that cause the system to satisfy the mean square H ∞ consistent performance index is specifically: controller output feedback gain to be designed Kk}0≤k≤TThe following three sets of recursive matrix inequality conditions should be satisfied, so that the mean square H ∞ consistent performance index of the system is satisfied:
first, in the following inequalities, the following relationships are present:
Figure BDA0002384301560000054
Figure BDA0002384301560000055
Figure BDA0002384301560000056
Figure BDA0002384301560000057
Figure BDA0002384301560000058
Figure BDA0002384301560000059
Figure BDA00023843015600000510
Figure BDA00023843015600000511
Figure BDA00023843015600000512
Figure BDA00023843015600000513
Figure BDA00023843015600000514
(1) for closed loop systems), gives a undirected communication graph G and a sequence of output feedback gains Kk}0≤k≤TThe noise level γ > 0 and the positive definite matrix Φ > 0 are known, if the initial conditions exist
Figure BDA0002384301560000061
Positive definite matrix of { Qk}0≤k≤T+1When the following recursion matrix inequality is satisfied, the output feedback gain { K } can be knownk}0≤k≤TSo that the closed loop system satisfies the condition J[0,T]＜γ2：
Figure BDA0002384301560000062
Wherein
Figure BDA0002384301560000063
Figure BDA0002384301560000064
Figure BDA0002384301560000065
Figure BDA0002384301560000066
(2) For a closed loop system, a undirected communication graph G and a sequence of output feedback gains { K }are givenk}0≤k≤TIf an initial condition P exists0＝∑kPositive definite matrix sequence { Pk}0≤k≤T+1Satisfies the following recursion matrix inequality, then k is less than or equal to T, P for all 0 ≦ kk＞∑kIf true, the output feedback gain { K }is knownk}0≤k≤TSo that the closed loop system satisfies the conditions
Figure BDA0002384301560000067
Figure BDA0002384301560000068
Without loss of generality, matrices
Figure BDA0002384301560000069
The following decomposition is carried out:
Figure BDA00023843015600000610
Figure BDA00023843015600000611
wherein
Figure BDA00023843015600000612
Is a column vector, further having the following equality relationship:
Figure BDA0002384301560000071
Figure BDA0002384301560000072
Figure BDA0002384301560000073
Figure BDA0002384301560000074
Figure BDA0002384301560000075
Figure BDA0002384301560000076
Figure BDA0002384301560000077
(3) for a closed loop system, given a triplet (G, γ, xi)k) Positive definite matrix phi and output feedback gain sequence Kk}0≤k≤TNormal norm ∈ > 0 for a closed loop system), if there are two sequences of positive scalars
Figure BDA0002384301560000078
Initial values are respectively
Figure BDA0002384301560000079
P0＝∑0Two positive definite matrices of { Qk}0≤k≤T+1,{Pk}0≤k≤T+1The following recursion matrix inequality is satisfied, then the output feedback gain { K } is knownk}0≤k≤TThe closed loop system meets the uniform performance index of mean square H infinity:
Figure BDA00023843015600000710
Figure BDA00023843015600000711
Figure BDA00023843015600000712
Figure BDA00023843015600000713
Figure BDA00023843015600000714
preferably, step 5 determines the linear matrix inequality for solving the controller parameters according to the design objective of the determined controller parameters that enables the closed-loop system to satisfy the mean square H ∞ consistent performance index as follows:
for a closed loop system, given a triplet (G, γ, xi)k) Positive definite matrix phi, normal norm ∈ > 0 if there are two sequences of positive scalars
Figure BDA0002384301560000081
Output feedback controller sequence Kk}0≤k≤TPositive definite matrix sequence
Figure BDA0002384301560000082
The following recursion linear matrix inequality is satisfied, and the output feedback controller parameter { K ] can be obtainedk}0≤k≤T：
Figure BDA0002384301560000083
Figure BDA0002384301560000084
Figure BDA0002384301560000085
Figure BDA0002384301560000086
Figure BDA0002384301560000087
Wherein
Figure BDA0002384301560000088
Parameter { X in formulak}1≤k≤T+1,{Yk}1≤k≤T+1Iterate through the following equation:
Figure BDA0002384301560000089
initial value X0,Y0Satisfy the requirement of
Figure BDA00023843015600000810
Compared with the prior art, the invention has the following remarkable advantages:
1) the multi-agent system based on the invention comprehensively considers the influences of nonlinearity, time-varying parameters and random effect, better conforms to the physical model in the actual engineering and has strong practicability;
2) the performance index provided by the invention establishes a unified framework for the H-infinity consistent performance and the mean square consistent performance index, can reflect the consistent dynamic state of the system under the constraints of a given topological structure, a noise attenuation level and a mean square level, and has more comprehensive performance.
The present invention is described in further detail below with reference to the attached drawings.
Drawings
FIG. 1 is a schematic diagram of the H ∞ consistency error of agents 1,2,3 ( agents 1,2,3 in the figure) in the system under the action of a controller.
FIG. 2 is a schematic diagram of the mean square deviation of the first term of the state vectors of agents 1,2,3 ( agents 1,2,3 in the figure) in the system under the action of the controller.
FIG. 3 is a diagram of the mean square deviation of the second term of the state vector of agents 1,2,3 ( agents 1,2,3 in the figure) in the system under the action of the controller.
Detailed Description
In a multi-agent system, N agents communicate according to a topology described by an undirected graph G (V, Θ, H), where V {1, 2.., N } is a set of vertices,
Figure BDA0002384301560000091
is set as edge, H ═ Hij]Is a symmetrically weighted adjacency matrix, i.e. hijIf (i, j) ∈ Θ, we call j the neighbor node of i, where self-edge is forbidden, i.e. for any i ∈ V,
Figure BDA0002384301560000092
the neighborhood of agent i is defined as
Figure BDA0002384301560000093
The degree of input is defined as
Figure BDA0002384301560000094
A method for designing a controller of a multi-agent system under a limited domain condition comprises the following specific steps:
step 1, establishing a mathematical model of a nonlinear discrete time-varying random multi-agent system under a finite field condition, specifically:
Figure BDA0002384301560000095
wherein k is more than or equal to 0 and less than or equal to T is a finite field, T is a given normal number,
Figure BDA0002384301560000096
is the state vector for time k, and,
Figure BDA0002384301560000101
is the state vector at time k +1,
Figure BDA0002384301560000102
in order to control the input of the electronic device,
Figure BDA0002384301560000103
is the measured output of the agent i,
Figure BDA0002384301560000104
is the controlled output of the agent and,
Figure BDA0002384301560000105
represents the covariance Wi,kN uncorrelated zero mean Gaussian white noise sequences > 0.
Figure BDA0002384301560000106
Is and Wi,kAn uncorrelated zero-mean gaussian white noise sequence. A. thek,Bk,Ck,Dk,Ek,FkAnd LkAre time-varying matrices of different dimensions. Non-linear random function
Figure BDA0002384301560000107
And satisfies the following constraint conditions
Figure BDA0002384301560000108
Figure BDA0002384301560000109
Wherein the content of the first and second substances,
Figure BDA00023843015600001010
and
Figure BDA00023843015600001011
is a matrix of the corresponding dimension.
Step 2, establishing a controller protocol and a closed-loop system equation according to a nonlinear discrete time-varying random multi-agent system mathematical model under the condition of a limited domain, wherein the controller protocol is as follows:
Figure BDA00023843015600001012
wherein the content of the first and second substances,
Figure BDA00023843015600001013
is the feedback gain matrix to be designed,
Figure BDA00023843015600001014
represents the output of the jth agent, hi,jAre non-negative real numbers.
According to a controller protocol and a mathematical model of a nonlinear discrete time-varying random multi-agent system, obtaining:
Figure BDA00023843015600001015
by extending the system state xi,kThe following closed-loop system equation is obtained:
Figure BDA00023843015600001016
wherein:
Figure BDA0002384301560000111
Figure BDA0002384301560000112
is a given incidence matrix which embodies the topology among the multiple agents;
Figure BDA0002384301560000113
the operation symbols represent that the Crohn's products are made among the matrixes; i isNIs an N-dimensional identity matrix, U, V are both block matrices, InNIs one of the identity matrices.
1NRepresenting an n-dimensional column vector with all elements 1,
Figure BDA0002384301560000114
is the transpose thereof.
The average state values of all agents in the system are considered simultaneously:
Figure BDA0002384301560000115
then, will
Figure BDA0002384301560000116
Substituting, we can get:
Figure BDA0002384301560000117
further, let
Figure BDA0002384301560000118
Obtaining:
Figure BDA0002384301560000121
and can be obtained from (8):
Figure BDA0002384301560000122
wherein
Figure BDA0002384301560000123
Figure BDA0002384301560000124
Figure BDA0002384301560000125
Further, let
Figure BDA0002384301560000126
The following enhancement system equation is obtained:
Figure BDA0002384301560000127
Figure BDA0002384301560000128
step 3, establishing a mean square H infinity consistent performance index according to a closed loop system equation, which specifically comprises the following steps:
for the closed loop system (6), at time k, the H ∞ match performance index is as follows:
Figure BDA0002384301560000129
wherein
Figure BDA00023843015600001210
Φ > 0 is the given weighting matrix.
For a closed-loop system (6), at time k, the mean-square agreement performance index of agent i (i ═ 1, 2.., N) is as follows:
Figure BDA0002384301560000131
each agent i, its initial state xi,0(i ═ 1,2,. N) is known and satisfies the following condition:
Figure BDA0002384301560000132
by combining the two points, for the closed-loop system equation (6), the mean square H ∞ consistent performance index is established as follows:
Figure BDA0002384301560000133
wherein γ is a predetermined noise level, { XI-k}0≤k≤TIs a predetermined positive definite matrix sequence, representing mean squareUpper bound of consistent performance.
Step 4, determining the design objective of the output feedback controller parameters which enable the system to meet the mean square H infinity consistent performance index specifically comprises the following steps: controller output feedback gain to be designed Kk}0≤k≤TThe following three sets of recursive matrix inequality conditions should be satisfied so that the mean square H ∞ uniform performance index (16) of the system is satisfied:
first, in the following inequalities, the following relationships are present:
Figure BDA0002384301560000134
Figure BDA0002384301560000135
Figure BDA0002384301560000136
Figure BDA0002384301560000137
Figure BDA0002384301560000138
Figure BDA0002384301560000139
Figure BDA00023843015600001310
Figure BDA00023843015600001311
Figure BDA00023843015600001312
Figure BDA00023843015600001313
Figure BDA00023843015600001314
(1) for the closed loop system (6), a undirected communication graph G and a sequence of output feedback gains { K }are givenk}0≤k≤TThe noise level γ > 0 and the positive definite matrix Φ > 0 are known, if the initial conditions exist
Figure BDA0002384301560000141
Positive definite matrix of { Qk}0≤k≤T+1When the following recursion matrix inequality is satisfied, the output feedback gain { K } can be knownk}0≤k≤TThe closed loop system (7) can be made to satisfy the condition J[0,T]＜γ2
Figure BDA0002384301560000142
Wherein
Figure BDA0002384301560000143
The proof that the controller causes the system to satisfy this condition is as follows:
generally speaking
Figure BDA0002384301560000144
From system equation (12), the following is obtained:
Figure BDA0002384301560000145
by using the properties of the matrix trace, it can be derived
Figure BDA0002384301560000146
After a series of mathematical operations, obtaining
Figure BDA0002384301560000151
It is obvious that the H ∞ consistency performance index in (13) is equivalently expressed as:
Figure BDA0002384301560000152
wherein
Figure BDA0002384301560000153
Then, in order to
Figure BDA0002384301560000154
In which zero term is added
Figure BDA0002384301560000155
To obtain
Figure BDA0002384301560000156
For (24), k sums the two sides separately from [0, T ], yielding:
Figure BDA0002384301560000157
thus, it is possible to provide
Figure BDA0002384301560000158
Due to Ψk＜0，QT+1> 0, according to the initial conditions
Figure BDA0002384301560000159
Can obtain the product
Figure BDA00023843015600001510
Satisfies J[0,T]＜γ2And finishing the verification.
Since the mean square consistency performance indicator in (11) can be equivalently expressed as follows:
Figure BDA00023843015600001511
definition of
Figure BDA0002384301560000161
The above formula can be further written as
Figure BDA0002384301560000162
(2) For the closed loop system (6), a undirected communication graph G and a sequence of output feedback gains { K }are givenk}0≤k≤TIf an initial condition P exists0＝∑kPositive definite matrix sequence { Pk}0≤k≤T+1Satisfies the following recursion matrix inequality, then k is less than or equal to T, P for all 0 ≦ kk＞∑kThis is true. Then the output feedback gain K is knownk}0≤k≤TCan make the closed loop system (6) satisfy the condition
Figure BDA0002384301560000163
Figure BDA0002384301560000164
Without loss of generality, matrices
Figure BDA0002384301560000165
The following decomposition can be performed:
Figure BDA0002384301560000166
wherein
Figure BDA0002384301560000167
Is a vector of the columns of the image,further has the following equation relation:
Figure BDA0002384301560000168
the proof that the controller causes the system to satisfy this condition is as follows:
first, ∑ is derivedkIn a finite field [0, T]Considering the enhancement system (12), ∑ can be calculatedk+1The following were used:
Figure BDA0002384301560000171
note the non-linear random function
Figure BDA0002384301560000172
The constraint conditions are satisfied by
Figure BDA0002384301560000173
Obtained by the two formulas:
Figure BDA0002384301560000174
the induction method is used below. Obviously, when k is 0, P0＞∑0If true; assuming that P is when k > 0k＞∑kIf it is, then P needs to be provedk+1＞∑k+1The same is true. In fact, taking into account the characteristics of the matrix traces, the combination (30) and Pk＞∑k
Figure BDA0002384301560000175
The certification is completed according to the induction method.
(3) For a closed loop system (6), the triplets (G, γ, xi) are givenk) Positive definite matrix phi and output feedback gain sequence Kk}0≤k≤TNormal ∈ > 0 for seriesSystem (7), if there are two positive scalar sequences
Figure BDA0002384301560000176
Initial values are respectively
Figure BDA0002384301560000177
P0＝∑0Two positive definite matrices of { Qk}0≤k≤T+1,{Pk}0≤k≤T+1The following recursion matrix inequality is satisfied, then the output feedback gain { K } is knownk}0≤k≤TThe closed loop system (7) can be made to meet the mean square H ∞ consistent performance index.
Figure BDA0002384301560000181
Figure BDA0002384301560000182
Figure BDA0002384301560000183
Figure BDA0002384301560000184
Figure BDA0002384301560000185
The proof that the controller causes the system to satisfy this condition is as follows:
first of all, two arguments are introduced,
theorem 1, (Schur complement theorem) given constant matrix
Figure BDA0002384301560000186
Wherein
Figure BDA0002384301560000187
Then
Figure BDA0002384301560000188
If and only if
Figure BDA0002384301560000189
Theorem 2, for any real vector a, b, P > 0 is a matrix of appropriate dimensions, then
aTPb+bTPa≤∈aTPa+∈-1bTPb (32)
Where ∈ > 0 is a given constant.
According to Schur's complementary theory, (37) true and only true
Figure BDA00023843015600001810
According to the nature of the matrix tracks, i.e.
Figure BDA0002384301560000191
Similarly, (38) holds and only holds
Figure BDA0002384301560000192
Wherein
Figure BDA0002384301560000193
Figure BDA0002384301560000194
According to Lesion 2, for any ∈ > 0, the following holds:
Figure BDA0002384301560000195
(46) and (47) both satisfy the requirement of the inequality (17), that is, the H ∞ consistency performance is satisfied.
Similarly, the formula (39) is obtained by applying Schur supplement theory
Figure BDA0002384301560000196
Is equivalent to:
Figure BDA0002384301560000197
according to (29) and (41)
Figure BDA0002384301560000198
Namely, the mean square consistency performance is satisfied and proved.
And 5, according to the design target of the controller parameters which are determined in the step 4 and enable the closed-loop system to meet the mean square H infinity consistent performance index, determining a linear matrix inequality for solving the controller parameters as follows:
for a closed loop system (6), the triplets (G, γ, xi) are givenk) Positive definite matrix phi, normal norm ∈ > 0 if there are two sequences of positive scalars
Figure BDA0002384301560000199
Output feedback controller sequence Kk}0≤k≤TPositive definite matrix sequence
Figure BDA0002384301560000201
The following recursion linear matrix inequality is satisfied, and the output feedback controller parameter { K ] can be obtainedk}0≤k≤T：
Figure BDA0002384301560000202
Figure BDA0002384301560000203
Figure BDA0002384301560000204
Figure BDA0002384301560000205
Figure BDA0002384301560000206
Wherein
Figure BDA0002384301560000207
Parameter { X in formulak}1≤k≤T+1,{Yk}1≤k≤T+1Iterate through the following equation:
Figure BDA0002384301560000208
initial value X0,Y0Satisfy the requirement of
Figure BDA0002384301560000209
And 6, solving the linear matrix inequality to obtain corresponding controller parameters, and finishing the design of the controller.
The effectiveness of the designed controller is verified according to a set of numerical examples, specifically:
by providing a numerical simulation example, a Matlab/LMI tool box is utilized to solve the designed controller parameters, and the effectiveness of the controller on the H-infinity consistency control problem of the nonlinear discrete time-varying random multi-agent system under the condition of a finite field is verified.
Consider the following nonlinear discrete time-varying random multi-agent:
Figure BDA0002384301560000211
Figure BDA0002384301560000212
Ck＝[0.15 0.24]
Figure BDA0002384301560000213
Ek＝0.20
Figure BDA0002384301560000214
Lk＝[0.20 0.30]
suppose three agents linked by a directed communication graph G, their correlation matrix
Figure BDA00023843015600002114
The following were used:
Figure BDA0002384301560000215
let the random non-linearity in the system be of the form:
Figure BDA0002384301560000216
wherein the content of the first and second substances,
Figure BDA0002384301560000217
ρi,k，θi,k( i 1,2,3) are uncorrelated zero-mean white gaussian noise sequences with the same covariance,
Figure BDA0002384301560000218
and
Figure BDA0002384301560000219
are the first and second terms of the system state. It can be easily derived that:
Figure BDA00023843015600002110
Figure BDA00023843015600002111
the initial state is as follows:
Figure BDA00023843015600002112
X0＝2I6,Y0＝2I12.
let T equal to 100, gamma equal to 2, phi equal to I2，Ξk＝7I2It can be seen that (12) and (60) are satisfied.
The results of the verification are shown in FIGS. 1-3, where FIG. 1 shows the H ∞ consistency error of the system at a predetermined noise attenuation level
Figure BDA00023843015600002113
The H infinity consistency error of any agent fluctuates around the 0 value, and the requirement of the performance index is met; to observe the mean square consistency performance index, define
Figure BDA0002384301560000221
To characterize the deviation of the state of the agent from the mean value, fig. 2 and 3 show
Figure BDA0002384301560000222
And
Figure BDA0002384301560000223
i.e., the deviation of the values of the first and second terms of the state vector of the agent from the average value at a certain time, it can be clearly seen that the state deviation of any agent is less than a predetermined upper bound at each time. Simulation results show that the mean square H infinity consistency controller of the system is very effective.
In summary, the invention provides a design method of an H infinity consistency controller of a nonlinear discrete time-varying random multi-agent system under a finite field condition. The method comprises the steps of establishing a mean square H infinity consistent performance index according to a closed loop system equation, determining a design target of controller parameters enabling the closed loop system to meet the mean square H infinity consistent performance index, determining a linear matrix inequality for solving the controller parameters meeting the mean square H infinity consistent performance condition based on the target, and then solving the linear matrix inequality to obtain corresponding controller parameters.

Claims (8)

1. A method for designing a controller of a multi-agent system under a limited domain condition is characterized by comprising the following specific steps:
step 1, establishing a nonlinear discrete time-varying random multi-agent system mathematical model under a finite field condition;
step 2, establishing a controller protocol and a closed-loop system equation according to a nonlinear discrete time-varying random multi-agent system mathematical model under the condition of a finite field;
step 3, establishing a mean square H infinity consistent performance index according to a closed loop system equation;
step 4, determining a design target of the controller parameters which enable the closed-loop system to meet the mean square H-infinity consistent performance index;
step 5, determining a linear matrix inequality for solving the controller parameters according to the determined design target of the controller parameters which enables the closed-loop system to meet the mean square H infinity consistent performance index;
and 6, solving the linear matrix inequality to obtain corresponding controller parameters, and finishing the design of the controller.
2. The method as claimed in claim 1, wherein the mathematical model of the non-linear discrete time-varying stochastic multi-agent system under the finite field condition is specifically:
Figure FDA0002384301550000011
wherein k is more than or equal to 0 and less than or equal to T is a finite field, T is a given normal number,
Figure FDA0002384301550000012
is the state vector for time k, and,
Figure FDA0002384301550000013
is the state vector at time k +1,
Figure FDA0002384301550000014
in order to control the input of the electronic device,
Figure FDA0002384301550000015
is the measured output of the agent i,
Figure FDA0002384301550000016
is the controlled output of the agent and,
Figure FDA0002384301550000017
represents the covariance Wi,kN uncorrelated zero mean gaussian white noise sequences > 0,
Figure FDA0002384301550000018
is and Wi,kUncorrelated zero mean white Gaussian noise sequences, Ak,Bk,Ck,Dk,Ek,FkAnd LkIs a time-varying matrix of different dimensions, fi,kIs a non-linear random function.
3. The method of claim 2, wherein the non-linear random function is a design of a controller for a multi-agent system in a limited domain scenario
Figure FDA0002384301550000019
And satisfies the following constraints:
Figure FDA0002384301550000021
Figure FDA0002384301550000022
Figure FDA0002384301550000023
Figure FDA0002384301550000024
wherein the content of the first and second substances,
Figure FDA0002384301550000025
and
Figure FDA0002384301550000026
is a matrix of the corresponding dimension.
4. The method for designing a controller of a multi-agent system under a limited domain situation as claimed in claim 1, wherein the controller protocol established in step 2 is:
Figure FDA0002384301550000027
wherein the content of the first and second substances,
Figure FDA0002384301550000028
is the feedback gain matrix to be designed,
Figure FDA0002384301550000029
represents the output of the jth agent, hi,jAre non-negative real numbers.
5. The method for designing a controller of a multi-agent system under a finite field condition as claimed in claim 1, wherein the specific method for establishing the closed-loop system equation according to the mathematical model of the nonlinear discrete time-varying random multi-agent system under the finite field condition is as follows:
according to a controller protocol and a mathematical model of a nonlinear discrete time-varying random multi-agent system, obtaining:
Figure FDA00023843015500000210
by extending the system state xi,kAnd obtaining a closed loop system equation:
Figure FDA00023843015500000211
wherein:
Figure FDA0002384301550000031
Figure FDA0002384301550000032
Figure FDA0002384301550000033
Figure FDA0002384301550000034
Figure FDA0002384301550000035
Figure FDA0002384301550000036
Figure FDA0002384301550000037
Figure FDA0002384301550000038
Figure FDA0002384301550000039
Figure FDA00023843015500000310
Figure FDA00023843015500000311
Figure FDA00023843015500000312
the given incidence matrix embodies the topological structure among the multiple intelligent agents;
Figure FDA00023843015500000313
the operation symbols represent that the Crohn's products are made among the matrixes; i isNIs an N-dimensional identity matrix, U, V are both block matrices, InNIs one of the identity matrices;
1Nrepresenting an n-dimensional column vector with all elements 1,
Figure FDA00023843015500000314
is its transpose, taking into account the average state values of all agents in the system:
Figure FDA00023843015500000315
will be provided with
Figure FDA00023843015500000316
Substituting, we can get:
Figure FDA00023843015500000317
order to
Figure FDA00023843015500000318
Obtaining:
Figure FDA0002384301550000041
and from the average state values of all agents:
Figure FDA0002384301550000042
wherein:
Figure FDA0002384301550000043
Figure FDA0002384301550000044
order to
Figure FDA0002384301550000045
Obtaining a system equation:
Figure FDA0002384301550000046
Figure FDA0002384301550000047
6. the method of controller design for multi-agent system in finite field situation as claimed in claim 1, wherein the mean square H ∞ uniform performance index is established according to the closed loop system equation as follows:
at time k, the H ∞ match performance index is as follows:
Figure FDA0002384301550000048
wherein
Figure FDA0002384301550000049
Phi > 0 is givenA fixed weighting matrix;
at time k, the mean square consistent performance index of agent i (i ═ 1, 2.., N) is as follows:
Figure FDA00023843015500000410
each agent i, its initial state xi,0(i ═ 1,2,. N) is known and satisfies the following condition:
Figure FDA0002384301550000051
by combining the two points, the mean square H infinity consistent performance index is established as follows:
Figure FDA0002384301550000052
wherein γ is a predetermined noise level, { XI-k}0≤k≤TIs a predetermined positive definite matrix sequence, which represents the upper bound of the mean square consistent performance.
7. The method for controller design of multi-agent system in finite field situation as claimed in claim 1, wherein the design objective of determining the output feedback controller parameters that make the system meet the mean square H ∞ consistency performance index is specified as: controller output feedback gain to be designed Kk}0≤k≤TThe following three sets of recursive matrix inequality conditions should be satisfied, so that the mean square H ∞ consistent performance index of the system is satisfied:
first, in the following inequalities, the following relationships are present:
Figure FDA0002384301550000053
Figure FDA0002384301550000054
Figure FDA0002384301550000055
Figure FDA0002384301550000056
Figure FDA0002384301550000057
Figure FDA0002384301550000058
Figure FDA0002384301550000059
Figure FDA00023843015500000510
Figure FDA00023843015500000511
Figure FDA00023843015500000512
Figure FDA00023843015500000513
(1) for a closed loop system, a undirected communication graph G and a sequence of output feedback gains { K }are givenk}0≤k≤TThe noise level γ > 0 and the positive definite matrix Φ > 0 are known, if the initial conditions exist
Figure FDA00023843015500000514
Positive definite matrix of { Qk}0≤k≤T+1If the following recursion matrix inequality is satisfied, the output can be knownOut feedback gain Kk}0≤k≤TSo that the closed loop system satisfies the condition J[0,T]＜γ2：
Figure FDA0002384301550000061
Wherein
Figure FDA0002384301550000062
Figure FDA0002384301550000063
Figure FDA0002384301550000064
Figure FDA0002384301550000065
(2) For a closed loop system, a undirected communication graph G and a sequence of output feedback gains { K }are givenk}0≤k≤TIf an initial condition P exists0＝∑kPositive definite matrix sequence { Pk}0≤k≤T+1Satisfies the following recursion matrix inequality, then k is less than or equal to T, P for all 0 ≦ kk＞∑kIf true, the output feedback gain { K }is knownk}0≤k≤TSo that the closed loop system satisfies the conditions
Figure FDA0002384301550000066
Figure FDA0002384301550000067
Without loss of generality, matrices
Figure FDA0002384301550000068
The following decomposition is carried out:
Figure FDA0002384301550000069
Figure FDA00023843015500000610
wherein
Figure FDA00023843015500000611
Is a column vector, further having the following equality relationship:
Figure FDA0002384301550000071
Figure FDA0002384301550000072
Figure FDA0002384301550000073
Figure FDA0002384301550000074
Figure FDA0002384301550000075
Figure FDA0002384301550000076
Figure FDA0002384301550000077
(3) for a closed loop system, given a triplet (G, γ, xi)k) Positive definite matrix phi and output feedback gain sequence Kk}0≤k≤TNormal norm ∈ > 0 for a closed loop system), if there are two sequences of positive scalars
Figure FDA0002384301550000078
Initial values are respectively
Figure FDA0002384301550000079
P0＝∑0Two positive definite matrices of { Qk}0≤k≤T+1,{Pk}0≤k≤T+1The following recursion matrix inequality is satisfied, then the output feedback gain { K } is knownk}0≤k≤TThe closed loop system meets the uniform performance index of mean square H infinity:
Figure FDA00023843015500000710
Figure FDA00023843015500000711
Figure FDA00023843015500000712
Figure FDA00023843015500000713
Figure FDA00023843015500000714
8. the method for designing a controller for a multi-agent system in a finite field situation as claimed in claim 1, wherein the step 5 determines the linear matrix inequality for solving the controller parameters according to the design objective of the controller parameters determined to make the closed loop system satisfy the mean square H ∞ consistency performance index as follows:
for closed loop systems, a triplet is given(G，γ，Ξk) Positive definite matrix phi, normal norm ∈ > 0 if there are two sequences of positive scalars
Figure FDA0002384301550000081
Output feedback controller sequence Kk}0≤k≤TPositive definite matrix sequence
Figure FDA0002384301550000082
The following recursion linear matrix inequality is satisfied, and the output feedback controller parameter { K ] can be obtainedk}0≤k≤T：
Figure FDA0002384301550000083
Figure FDA0002384301550000084
Figure FDA0002384301550000085
Figure FDA0002384301550000086
Figure FDA0002384301550000087
Wherein
Figure FDA0002384301550000088
Parameter { X in formulak}1≤k≤T+1,{Yk}1≤k≤T+1Iterate through the following equation:
Figure FDA0002384301550000089
initial value X0,Y0Satisfy the requirement of
Figure FDA00023843015500000810
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