CN108267957B - Control method for robustness output consistency of multi-agent system in fractional order interval - Google Patents

Control method for robustness output consistency of multi-agent system in fractional order interval Download PDF

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CN108267957B
CN108267957B CN201810065562.2A CN201810065562A CN108267957B CN 108267957 B CN108267957 B CN 108267957B CN 201810065562 A CN201810065562 A CN 201810065562A CN 108267957 B CN108267957 B CN 108267957B
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王立明
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Dragon Totem Technology Hefei Co ltd
Shenzhen Dragon Totem Technology Achievement Transformation Co ltd
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Abstract

A control method for robustness output consistency of a fractional order interval multi-agent system comprises the following steps: a. converting the control problem of the robustness output consistency of the multi-agent system in the fractional order interval into the stabilization problem of the state zero point of the multi-agent system in the fractional order interval; b. designing a distributed output feedback controller; c. converting the stability problem of the state zero point of the closed-loop fractional order interval multi-agent system into the stability analysis problem of the state zero points of N-1 fractional order subsystems; d. giving a condition capable of ensuring that the state zero points of the N-1 fractional order subsystems are stable at the same time; e. and solving the undetermined feedback matrix in the output feedback controller. The output feedback controller has simple design and convenient solution, can resist the interference caused by the interval uncertainty of the order and other model parameters, has good control effect, and well solves the problem of robust output consistency control of a fractional order interval multi-agent system.

Description

Control method for robustness output consistency of multi-agent system in fractional order interval
Technical Field
The invention relates to a distributed output feedback control method with an uncertain capability of resisting intervals for a fractional order interval multi-agent system, belonging to the technical field of control.
Background
The problem of consistency in multi-agent systems has received increasing attention over the last few years. This is mainly due to the fact that multiple agents are widely used in aircraft formation, attitude adjustment, mobile robots, sensor networks, and the like. The goal of the consistency problem is to design a suitable protocol for a group of multi-agents to agree on a certain physical quantity by interacting with neighbors. In recent years, researchers have conducted extensive research into the problem of consistency of multi-agent systems with single-integral dynamics or dual-integral dynamics.
However, real physical systems are not always described in terms of integer order dynamics, and the integer order multi-agent systems that have been extensively studied are only special cases of fractional order multi-agent systems. Research shows that the existing research result of the consistency of the integer-order multi-agent system cannot be directly applied to the consistency problem of the fractional-order multi-agent system. More importantly, recent studies have found that many real physical systems, including vehicles moving in viscoelastic materials and aircraft operating at high speeds in a particulate environment, are better suited to be described by fractional order differential dynamics.
In recent years, a great deal of research and development have been carried out on the problem of controlling the consistency of the multi-level and multi-agent by numerous scholars at home and abroad, but in the existing literature, for the convenience of discussion, it is generally assumed that the dynamic model of the multi-level and multi-agent is completely determined and known. In practical engineering applications, however, most controlled objects are not ideal linear steady-state systems, but rather have model uncertainties to some degree. It is therefore very realistic and necessary to consider the consistency problem of a fractional order multi-agent system with model uncertainty, especially with order uncertainty. If these problems are not solved, the true application and generalization of the fractional order multi-agent system theory cannot be realized.
In addition, for a traditional single controlled fractional order interval system, when uncertainty exists in the order and other model parameters at the same time, a mature robust control theory can be utilized to design a controller, so that the corresponding closed-loop fractional order interval system realizes robust stability. In view of this, applying the conventional robust control theory and method to the output consistency control of the fractional order interval multi-agent system would be a feasible solution. However, considering the complexity of the multi-agent system in the fractional order interval, the coupling of the fractional order dynamics of the intelligent individuals and the network topology, the particularity of the problem of robustness consistency and the like, how to apply the existing robust control theory and method becomes the key for solving the problem of consistency control of the multi-agent system in the fractional order interval.
Disclosure of Invention
The invention aims to provide a control method for consistency of robust output of a multi-agent system in a fractional order interval aiming at the defects of the prior art, and thoroughly solves the problem of consistency control of robust output of the system.
The problems of the invention are solved by the following technical scheme:
a method for controlling robustness output consistency of a fractional order interval multi-agent system, the method comprising the steps of:
a. the control problem of the robustness output consistency of the fractional order interval multi-agent system is converted into the stabilization problem of the state zero point of the fractional order interval multi-agent system:
assuming that an undirected topological fractional order interval multi-agent system is composed of N fractional order agents with interval uncertainty, the dynamic model of the ith agent is as follows:
Figure BDA0001556576440000021
yi(t)＝Cxi(t),
wherein i belongs to {1,2, L, N }; x is the number ofi(t)∈Rn，yi(t)∈RpAnd ui(t)∈RpRespectively the state, output and input of the ith agent at the time t; c is belonged to Rp×nIs an output matrix; alpha is alpha0、A0And B0The normal parameters are corresponding to a nominal model of the system;
Figure RE-GDA0001617998700000022
alpha defined for using Caputo differential0The + Δ α order derivative; the uncertainty Δ α of the order is defined as:
Δα＝αMζ
wherein alpha isMIs the maximum perturbation range of the order and satisfies alpha0+αM< 1 and alpha0-αM> 0, zeta is in the interval [ -1,1 ]]A random number of (c);
the uncertain parts delta A and delta B of the system matrix respectively satisfy:
Figure BDA0001556576440000023
and
Figure BDA0001556576440000024
wherein gamma isijAnd betaijIs a positive scalar constant, σijAnd ηijIs in the interval [ -1,1 [)]A random number of (a), andMand BMIs a known matrix with definite values, the symbol "o" represents the Hadamard product;
to facilitate handling of uncertainties Δ A and Δ B, variables are introduced
Figure BDA0001556576440000031
Wherein
Figure BDA0001556576440000032
And
Figure BDA0001556576440000033
column vector, diag { σ { representing that the kth element is 1 and the other elements are all 011…σ1n…σn1…σnnDenotes a diagonal matrix
Figure BDA0001556576440000034
Thus Δ a ═ DAFAEAAnd Δ B ═ DBFBEB。
Introduction of a new variable deltai(t)＝x1(t)-xi(t) converting the control problem of the robustness output consistency of the system into a fractional order interval multi-intelligent system
Figure BDA0001556576440000035
The settling of the state zero point of (c);
b. designing distributed output feedback controllers
Figure BDA0001556576440000036
Wherein N isiAs neighbors of agent iGathering; h isijThe weight value of the edge in the information interaction topology G is obtained; h if agent i is able to receive the output information of agent j ij1 is ═ 1; otherwise, hijF is the pending feedback matrix; the Laplacian matrix of the undirected graph G is marked as L;
c. the stabilization problem of the state zero point of the closed-loop fractional order interval multi-agent system is converted into the stability analysis problem of the state zero points of N-1 fractional order subsystems:
definition of
Figure BDA0001556576440000037
Using orthogonal transformation
Figure BDA0001556576440000038
Xi is an orthogonal matrix of appropriate dimensions, then the N-1 fractional order subsystems are:
Figure BDA0001556576440000039
wherein the content of the first and second substances,
Figure BDA00015565764400000310
wherein the upper right subscript "T" denotes the transpose of the matrix or vector, λi(i-2, 3, …, N) is L22+1N-1·βTCharacteristic value of (1), betaT＝[h12,h13,…,h1N]，
Figure BDA0001556576440000041
Symbol
Figure BDA0001556576440000042
Represents the kronecker product, 1N-1∈RN-1A column vector representing all elements as 1;
d. and giving a condition capable of ensuring that the state zero points of the N-1 fractional order subsystems are stable at the same time:
assuming that the singular value decomposition of the output matrix C satisfies the condition that C ═ U [ S0 ]]VTU and V are both unitary matrices of appropriate dimensions, S is a diagonal matrix, whichThe elements on the main diagonal are the singular values of C in descending order. If there is a matrix X ∈ Rp×pTwo symmetric positive definite matrices Q11∈Rp×p,Q22∈R(n-p)×(n-p)And 4 real constants εj＞0,ρj> 0(j ═ 1,2) make the following 4 inequalities
Figure BDA0001556576440000043
And
Figure BDA0001556576440000044
and wherein sym (M) represents M + MTThen N-1 subsystems are stable at the same time, i.e. a fractional order interval multi-agent system can achieve robust output consistency under the action of a distributed output feedback controller, wherein,
Figure BDA0001556576440000045
Figure BDA0001556576440000046
Figure BDA0001556576440000047
I2a 2 x 2 identity matrix is represented,
Figure BDA0001556576440000048
Figure BDA00015565764400000412
represents 2n2×2n2The unit matrix of (a) is,
Figure BDA0001556576440000049
Figure BDA00015565764400000410
Figure BDA00015565764400000411
e. solving an undetermined feedback matrix in the output feedback controller:
the calculation method of the feedback matrix F in the controller comprises the following steps:
Figure BDA0001556576440000051
wherein
Figure BDA0001556576440000052
The output feedback controller provided by the invention is simple in design and convenient to solve, can resist interference caused by interval uncertainty of orders and other model parameters, has a good control effect and strong practicability, and well solves the problem of robust output consistency control of a multi-agent system in fractional order intervals.
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The present invention will be described in further detail with reference to the accompanying drawings.
FIG. 1 is a schematic diagram of the design process of the output feedback controller of the fractional order interval multi-agent system of the present invention;
FIG. 2 is a topological diagram of information interaction among the agents of the present invention;
FIG. 3(a) is the position in the complex plane of randomly generated 500 characteristic values of a fractional order multi-agent system that satisfy a given interval without any control;
FIG. 3(b) is a trace of the output error of a fractional order multi-agent system randomly generated and satisfying a given interval without any control;
FIG. 4(a) is the position in the complex plane of the randomly generated 500 eigenvalues of the fractional order multi-agent system that meet a given interval under the action of the output feedback controller;
fig. 4(b) is a trace of the output error of the fractional order multi-agent system randomly generated under the action of the output feedback controller and satisfying a given interval.
The individual symbols herein are: x is the number ofi(t)∈Rn，yi(t)∈RpAnd ui(t)∈RpRespectively the state, output and input of the ith agent at the time t; c is belonged to Rp×nIs an output matrix, the singular value decomposition of C satisfies the condition that C ═ U [ S0%]VTU and V are unitary matrixes with proper dimensions, S is a diagonal matrix, and elements on a main diagonal of the matrix are singular values of C arranged in a descending order; alpha is alpha0、A0And B0The normal parameters are corresponding to a nominal model of the system;
Figure RE-GDA0001617998700000052
alpha defined for using Caputo differential0The + Δ α order derivative; Δ α is the uncertainty of the order; alpha is alphaMZeta is in the interval [ -1,1 ] for the maximum perturbation range of the order]A random number of (c); delta A and delta B are uncertain parts of a system matrix; gamma rayijAnd betaijIs a positive scalar constant, σijAnd ηijIs in the interval [ -1,1 [)]A random number of (a), andMand BMIs a known matrix with definite values, the symbol "o" represents the Hadamard product; symbol
Figure RE-GDA0001617998700000053
Representing the kronecker product, the upper right subscript "T" representing the transpose of the matrix or vector;
Figure RE-GDA0001617998700000054
and
Figure RE-GDA0001617998700000055
column vector representing the kth element as 1 and the other elements as 0, 1N-1∈RN-1Representing a column vector, I, in which all elements are 1nAn identity matrix representing n × n; n is a radical ofiA neighbor set for agent i; h isijIs a letterThe weight of the edge in the interactive topology G is determined; f is the pending feedback matrix; l is a Laplacian matrix of the undirected graph G; xi is an orthogonal matrix of suitable dimensions; sym (M) represents M + MT；diag{σ1,σ2LσnDenotes a diagonal matrix
Figure RE-GDA0001617998700000061
If A is a vector, | A | | | represents the Euclidean norm of vector A, and if A is a matrix, | A | | | represents the induced 2 norm of matrix A.
Detailed Description
The invention aims to provide an output feedback control method based on local output information for a fractional order multi-agent system with interval uncertainty, so that the fractional order interval multi-agent system can realize robust output consistency.
As shown in fig. 1, the technical solution of the present invention is implemented as follows:
1. converting the control problem of the robustness output consistency of the multi-agent system in the fractional order interval into the stabilization problem of the state zero point of the multi-agent system in the fractional order interval;
2. designing a distributed output feedback controller;
3. converting the stability problem of the state zero point of the closed-loop fractional order interval multi-agent system into the stability analysis problem of the state zero points of N-1 fractional order subsystems;
4. giving a condition capable of ensuring that the state zero points of the N-1 fractional order subsystems are stable at the same time;
5. and solving the undetermined feedback matrix in the output feedback controller.
The invention has the following technical characteristics:
(1) in the step 1, an intermediate variable is introduced, so that the control problem of robustness consistency of the fractional order interval multi-agent system is converted into the stabilization problem of the state zero point of the fractional order interval multi-agent system.
(2) Designed in step 2 is an output feedback controller based on the local mutual information between the fractional order intelligent agents, and a feedback matrix of the output feedback controller is to be determined.
(3) And 3, converting the stabilization problem of the state zero point of the closed-loop fractional order interval multi-agent system into the stability analysis problem of the state zero point of the N-1 fractional order subsystems by utilizing orthogonal transformation.
(4) And 4, analyzing the N-1 fractional order subsystems by applying the existing robust control theory to obtain a condition which can ensure that the N-1 fractional order subsystems are stable at the same time, namely a condition for realizing the consistency of robust output.
(5) In step 5, the solving conditions and the calculation formula of the feedback matrix in the output feedback controller are given in the form of a linear matrix inequality, so that the matrix can be conveniently solved by using an LMI tool box of Matlab.
It is known that: the undirected topological fractional order interval multi-agent system consists of N fractional order agents with interval uncertainty, and the dynamic model of the ith agent is as follows:
Figure BDA0001556576440000071
where i ∈ {1,2, L, N }. x is the number ofi(t)∈Rn，yi(t)∈RpAnd ui(t)∈RpRespectively, the state, output and input of the ith agent at the time t. C is belonged to Rp×nIs an output matrix, alpha0、A0And B0Is a constant parameter corresponding to a nominal model of the system (1).
Figure RE-GDA0001617998700000072
Alpha defined for using Caputo differential0The + Δ α order derivative. The uncertainty Δ α of the order is defined as
Δα＝αMζ (2)
Wherein alpha isMIs the maximum perturbation range of the order and satisfies alpha0+αM< 1 and alpha0-αM> 0, zeta is in the interval [ -1,1 ]]A random number of (2).
The uncertain parts of the system matrix delta A and delta B respectively satisfy
Figure BDA0001556576440000073
And
Figure BDA0001556576440000074
wherein gamma isijAnd betaijIs a positive scalar constant, σijAnd ηijIs in the interval [ -1,1 [)]A random number of (a), andMand BMIs a known matrix with certain values, the symbol "o" denotes the Hadamard product.
To facilitate handling of uncertainties Δ A and Δ B, variables are introduced
Figure BDA0001556576440000075
Wherein
Figure BDA0001556576440000076
And
Figure BDA0001556576440000077
column vector, diag { σ { representing that the kth element is 1 and the other elements are all 011…σ1n…σn1…σnnDenotes a diagonal matrix
Figure BDA0001556576440000078
Thus, it is possible to provide
ΔA＝DAFAEAAnd Δ B ═ DBFBEB (3)
The objects of the invention are: for a fractional order interval multi-agent system (1), a distributed output feedback controller with the capability of resisting interval uncertainty is designed, so that the closed-loop fractional order interval multi-agent system can realize output consistency
Figure BDA0001556576440000079
Wherein yj(t)-yi(t) | | denotes the vector yj(t)-yi(t) Euclidean norm.
Referring to fig. 1, the specific implementation process of the present invention is as follows:
step 1: robust output consistency control problem of fractional order interval multi-agent system is converted into stabilization problem of state zero point of fractional order interval multi-agent system
Since yj(t)-yi(t)||＝||C(xj(t)-xi(t))||≤||C||||xj(t)-xi(t) | |, consistency of output
Figure BDA0001556576440000081
Is equivalent to
Figure BDA0001556576440000082
Introduction of a new variable deltai(t)＝x1(t)-xi(t) converting the control problem of the robustness output consistency of the system (1) into a fractional order interval multi-agent system by using (2) and (3)
Figure BDA0001556576440000083
The state zero point of (2).
Step 2: output feedback controller design
Designing a distributed output feedback controller aiming at the stabilization problem of the state zero point of (4):
Figure BDA0001556576440000084
wherein N isiA neighbor set for agent i; h isijAnd the weight value of the edge in the information interaction topology G. H if agent i is able to receive the output information of agent j ij1 is ═ 1; otherwise, h ij0. F is the pending feedback matrix. Laplacian moment of undirected graph GThe array is denoted L.
And step 3: converting the stabilization problem of the state zero point of a closed-loop fractional order interval multi-agent system into the stability analysis problem of the state zero point of N-1 fractional order sub-systems
Under the action of the output feedback controller (5), (4) can be rewritten into
Figure BDA0001556576440000085
Definition of
Figure BDA0001556576440000086
The upper right subscript "T" denotes the transpose of the matrix or vector, then (6) becomes
Figure BDA0001556576440000087
Wherein beta isT＝[h12,h13,…,h1N]And
Figure BDA0001556576440000088
symbol
Figure BDA0001556576440000089
Representing the kronecker product.
Applying orthogonal transformation to (7)
Figure BDA00015565764400000810
To obtain
Figure BDA0001556576440000091
Wherein xi is an orthogonal matrix of a suitable dimension, Λ @ xiT(L22+1N-1·βT)Ξ＝diag{λ2,λ3,L,λN}。
Due to the fact that
Figure BDA0001556576440000093
Is block diagonalized, the problem of stabilization of the state zero of a fractional order interval multi-agent system (8) is equivalent to N-1 subsystems
Figure BDA0001556576440000094
Stability analysis of state zero of (1). Wherein λ isi(i-2, 3, …, N) is L22+1N-1·βTIs determined by the characteristic value of (a),
Figure BDA0001556576440000095
and 4, step 4: analysis of robust output consistency conditions
The key to giving a robust output consistency condition is to determine the conditions that will enable N-1 subsystems in (9) to be stable simultaneously. (9) The conditions for simultaneous stabilization of the N-1 subsystems are as follows:
assuming that the singular value decomposition of the output matrix C satisfies the condition that C ═ U [ S0 ]]VTU and V are unitary matrices of appropriate dimensions, and S is a diagonal matrix whose elements on the main diagonal are singular values of C in descending order. If there is a matrix X ∈ Rp×pTwo symmetric positive definite matrices Q11∈Rp×p,Q22∈R(n-p)×(n-p)And 4 real constants εj＞0,ρj> 0(j ═ 1,2) make the following 4 inequalities
Figure BDA0001556576440000096
And
Figure BDA0001556576440000097
and meanwhile, the N-1 subsystems in the (9) are stable at the same time, namely the multi-agent system (1) with the fractional order interval can realize the robustness output consistency under the action of the distributed output feedback controller (5).
Wherein the content of the first and second substances,
Figure BDA0001556576440000098
Figure BDA0001556576440000099
Figure BDA00015565764400000910
I2a 2 x 2 identity matrix is represented,
Figure BDA00015565764400000911
Figure BDA00015565764400000912
represents 2n2×2n2The unit matrix of (a) is,
Figure BDA00015565764400000913
Figure BDA0001556576440000101
Figure BDA0001556576440000102
and 5: solving of feedback matrix
Based on the robust output consistency condition in step 4, the calculation method for providing the feedback matrix F in the controller is as follows:
F＝XUSQ11 -1S-1U-1 (11)
wherein
Figure BDA0001556576440000103
The effects of the present invention can be further illustrated by the following simulations:
simulation content: consider a fractional order multi-agent system (1) consisting of four fractional order agents whose parameters satisfy
Figure BDA0001556576440000104
It is assumed that the topology of information interaction between fractional order agents is as shown in fig. 2. Its Laplacian matrix
Figure BDA0001556576440000105
Thus, it is possible to provide
Figure BDA0001556576440000106
And
Figure BDA0001556576440000107
singular value decomposition of C into
Figure BDA0001556576440000108
Initial state of fractional order agent is set to x1(0)＝[3,-4,-5]T，x2(0)＝[-1,2,-7]T，x3(0)＝[7,-3,6.5]T，x4(0)＝[4,3,-0.5]TAnd x5(0)＝[-7,-3,0.9]T. Solving and using Matlab's LMI toolbox
Figure BDA0001556576440000109
And
Figure BDA00015565764400001010
corresponding linear matrix inequality (10), we get X-2.5370, Q11＝0.5589,
Figure BDA00015565764400001011
ε1＝ε2＝131.5959,ρ1＝ρ2134.0178, F XUSQ can be obtained by (11)11 -1S-1U-1＝-4.5396。
Fig. 3(a) depicts the positions in the complex plane of the randomly generated 500 fractional order multi-agent systems (9) satisfying (12) without any control action, and it can be seen from fig. 3(a) that some of the characteristic values are distributed in the unstable region. The random trial results in fig. 3(a) therefore show that the output of the fractional order multi-agent system (9) is not robust stable without any control effort. Fig. 3(b) shows the output error trajectory of the fractional order multi-agent system (9) randomly generated and satisfying (12) without any control action, and it can be seen from fig. 3(b) that the output error trajectory of the fractional order multi-agent system (9) does not converge without any control action, which further confirms the conclusion from fig. 3 (a). Fig. 4(a) shows the positions in the complex plane of the randomly generated 500 characteristic values of the fractional order multi-agent system (9) satisfying (12) under the action of the output feedback controller, and it can be seen from fig. 4(a) that all the characteristic values are distributed in the stable region. The random trial results in fig. 4(a) therefore show that the output of the fractional order multi-agent system (9) is robust and stable under the action of the output feedback controller. Fig. 4(b) shows the output error trajectory of the fractional order multi-agent system (9) randomly generated and satisfying (12) under the action of the output feedback controller, and it can be seen from fig. 4(b) that the output error trajectory of the fractional order multi-agent system (9) under the action of the output feedback controller converges progressively, which further confirms the conclusion obtained from fig. 4 (a). Thus, as can be seen from fig. 3 and 4, the distributed output feedback control method of the present invention is efficient and robust.

Claims (1)

1. A control method for robustness output consistency of a fractional order interval multi-agent system is characterized by comprising the following steps:
a. the control problem of the robustness output consistency of the fractional order interval multi-agent system is converted into the stabilization problem of the state zero point of the fractional order interval multi-agent system:
assuming that an undirected topological fractional order interval multi-agent system is composed of N fractional order agents with interval uncertainty, the dynamic model of the ith agent is as follows:
Figure FDA0002718225500000011
yi(t)＝Cxi(t)，
where i ∈ {1,2, …, N }; x is the number ofi(t)∈Rn，yi(t)∈RpAnd ui(t)∈RpRespectively the state, output and input of the ith agent at the time t; c is belonged to Rp×nIs an output matrix; alpha is alpha0、A0And B0The normal parameters are corresponding to a nominal model of the system;
Figure FDA0002718225500000012
alpha defined for using Caputo differential0The + Δ α order derivative; the uncertainty Δ α of the order is defined as:
Δα＝αMζ
wherein alpha isMIs the maximum perturbation range of the order and satisfies alpha0+αM< 1 and alpha0-αM> 0, zeta is in the interval [ -1,1 ]]A random number of (c);
the uncertain parts delta A and delta B of the system matrix respectively satisfy:
ΔA＝AMo[σij]n×n＝[γij]n×no[σij]n×n＝[γijσij]n×nand Δ B ═ BMo[ηij]n×n＝[βij]n×no[ηij]n×n＝[βijηij]n×n，
Wherein gamma isijAnd betaijIs a positive scalar constant, σijAnd ηijIs in the interval [ -1,1 [)]A random number of (a), andMand BMIs a known matrix with definite values, the symbol "o" represents the Hadamard product;
to facilitate handling of uncertainties Δ A and Δ B, variables are introduced
Figure FDA0002718225500000021
Wherein
Figure FDA0002718225500000022
And
Figure FDA0002718225500000023
column vector, diag { σ { representing that the kth element is 1 and the other elements are all 011Lσ1nLσn1LσnnDenotes a diagonal matrix
Figure FDA0002718225500000024
Thus Δ a ═ DAFAEAAnd Δ B ═ DBFBEB；
Introduction of a new variable deltai(t)＝x1(t)-xi(t) converting the control problem of the robustness output consistency of the system into a fractional order interval multi-agent system
Figure FDA0002718225500000025
The settling of the state zero point of (c);
b. designing distributed output feedback controllers
Figure FDA0002718225500000026
Wherein N isiA neighbor set for agent i; h isijThe weight value of the edge in the information interaction topology G is obtained; h if agent i is able to receive the output information of agent jij1 is ═ 1; otherwise, hijF is the pending feedback matrix; the Laplacian matrix of the undirected graph G is marked as L;
c. the stabilization problem of the state zero point of the closed-loop fractional order interval multi-agent system is converted into the stability analysis problem of the state zero points of N-1 fractional order subsystems:
definition of
Figure FDA0002718225500000027
Using orthogonal transformation
Figure FDA0002718225500000028
Xi is an orthogonal matrix of appropriate dimensions, then the N-1 fractional order subsystems are:
Figure FDA0002718225500000029
wherein the content of the first and second substances,
Figure FDA00027182255000000210
wherein the upper right subscript "T" represents the transpose of the matrix or vector; lambda [ alpha ]i(i ═ 2,3, L, N) is L22+1N-1·βTCharacteristic value of (1), betaT＝[h12,h13,L,h1N]；
Figure FDA0002718225500000031
1N-1∈RN -1Column vector, symbol, representing all elements being 1
Figure FDA0002718225500000032
Represents the kronecker product;
d. and giving a condition capable of ensuring that the state zero points of the N-1 fractional order subsystems are stable at the same time:
assuming that the singular value decomposition of the output matrix C satisfies the condition that C ═ U [ S0 ]]VTU and V are unitary matrixes with proper dimensions, S is a diagonal matrix, and elements on a main diagonal of the diagonal matrix are singular values of C arranged in a descending order; if there is a matrix X ∈ Rp×pTwo symmetric positive definite matrices Q11∈Rp×p,Q22∈R(n-p)×(n-p)And 4 real constants εj＞0,ρj> 0(j ═ 1,2) make the following 4 inequalities
Figure FDA0002718225500000033
And
Figure FDA0002718225500000034
and at the same time, the N-1 subsystems are stable at the same time, namely the fractional order interval multi-agent system can realize robust output consistency under the action of the distributed output feedback controller, wherein,
Figure FDA0002718225500000035
Figure FDA0002718225500000036
Figure FDA0002718225500000037
I2a 2 x 2 identity matrix is represented,
Figure FDA0002718225500000038
Figure FDA0002718225500000039
represents 2n2×2n2The unit matrix of (a) is,
Figure FDA00027182255000000310
Figure FDA00027182255000000311
Figure FDA00027182255000000312
e. solving an undetermined feedback matrix in the output feedback controller:
the calculation method of the feedback matrix F in the controller comprises the following steps:
F＝XUSQ11 -1S-1U-1
wherein
Figure FDA0002718225500000041
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