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:
yi(t)=Cxi(t),
wherein i belongs to {1,2, L, N }; x is the number of
i(t)∈R
n,y
i(t)∈R
pAnd u
i(t)∈R
pRespectively the state, output and input of the ith agent at the time t; c is belonged to R
p×nIs an output matrix; alpha is alpha
0、A
0And B
0The normal parameters are corresponding to a nominal model of the system;
alpha defined for using Caputo differential
0The + 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:
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
Wherein
And
column vector, diag { σ { representing that the kth element is 1 and the other elements are all 0
11…σ
1n…σ
n1…σ
nnDenotes a diagonal matrix
Thus Δ a ═ D
AF
AE
AAnd Δ B ═ D
BF
BE
B。
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
The settling of the state zero point of (c);
b. designing distributed output feedback controllers
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
Using orthogonal transformation
Xi is an orthogonal matrix of appropriate dimensions, then the N-1 fractional order subsystems are:
wherein,
wherein the upper right subscript "T" denotes the transpose of the matrix or vector, λ
i(i-2, 3, …, N) is L
22+1
N-1·β
TCharacteristic value of (1), beta
T=[h
12,h
13,…,h
1N],
Symbol
Represents the kronecker product, 1
N-1∈R
N-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
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,
I
2a 2 x 2 identity matrix is represented,
represents 2n
2×2n
2The unit matrix of (a) is,
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:
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.
Drawings
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 of
i(t)∈R
n,y
i(t)∈R
pAnd u
i(t)∈R
pRespectively the state, output and input of the ith agent at the time t; c is belonged to R
p×nIs an output matrix, the singular value decomposition of C satisfies the condition that C ═ U [ S0%]V
TU 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 alpha
0、A
0And B
0The normal parameters are corresponding to a nominal model of the system;
alpha defined for using Caputo differential
0The + Δ α order derivative; Δ α is the uncertainty of the order; alpha is alpha
MZeta 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 ray
ijAnd beta
ijIs a positive scalar constant, σ
ijAnd η
ijIs in the interval [ -1,1 [)]A random number of (a), and
Mand B
MIs a known matrix with definite values, the symbol "o" represents the Hadamard product; symbol
Representing the kronecker product, the upper right subscript "T" representing the transpose of the matrix or vector;
and
column vector representing the kth element as 1 and the other elements as 0, 1
N-1∈R
N-1Representing a column vector, I, in which all elements are 1
nAn identity matrix representing n × n; n is a radical of
iA neighbor set for agent i; h is
ijIs 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 + M
T;diag{σ
1,σ
2Lσ
nDenotes a diagonal matrix
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:
where i ∈ {1,2, L, N }. x is the number of
i(t)∈R
n,y
i(t)∈R
pAnd u
i(t)∈R
pRespectively, the state, output and input of the ith agent at the time t. C is belonged to R
p×nIs an output matrix, alpha
0、A
0And B
0Is a constant parameter corresponding to a nominal model of the system (1).
Alpha defined for using Caputo differential
0The + 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
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
Wherein
And
column vector, diag { σ { representing that the kth element is 1 and the other elements are all 0
11…σ
1n…σ
n1…σ
nnDenotes a diagonal matrix
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
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
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)
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):
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
Definition of
The upper right subscript "T" denotes the transpose of the matrix or vector, then (6) becomes
Wherein beta is
T=[h
12,h
13,…,h
1N]And
symbol
Representing the kronecker product.
Applying orthogonal transformation to (7)
To obtain
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
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
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),
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
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,
I
2a 2 x 2 identity matrix is represented,
represents 2n
2×2n
2The unit matrix of (a) is,
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)
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
It is assumed that the topology of information interaction between fractional order agents is as shown in fig. 2. Its Laplacian matrix
Thus, it is possible to provide
And
singular value decomposition of C into
Initial state of fractional order agent is set to x
1(0)=[3,-4,-5]
T,x
2(0)=[-1,2,-7]
T,x
3(0)=[7,-3,6.5]
T,x
4(0)=[4,3,-0.5]
TAnd x
5(0)=[-7,-3,0.9]
T. Solving and using Matlab's LMI toolbox
And
corresponding linear matrix inequality (10), we get X-2.5370, Q
11=0.5589,
ε
1=ε
2=131.5959,ρ
1=ρ
2134.0178, 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.