CN108267957A - A kind of control method of fractional order section multi-agent system robust output consistency - Google Patents
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Abstract
A kind of control method of fractional order section multi-agent system robust output consistency, includes the following steps：A., the control problem of the robust output consistency of fractional order section multi-agent system is converted into the Stabilization of the state zero of fractional order section multi-agent system；B. distributed output feedback controller is designed；C., the Stabilization of the state zero of closed loop fractional order section multi-agent system is converted into the stability analysis problem of the state zero of 1 fractional order subsystem of N；D. the condition for the state zero Simultaneous Stabilization that can ensure 1 fractional order subsystem of N is provided；E. the feedback matrix undetermined in output feedback controller is solved.The output feedback controller design of the present invention is simple, it is convenient to solve, it can resist and be interfered caused by exponent number and the bounded-but-unknown uncertainty of other model parameters, possess good control effect, well solved the robust output consistency control problem of fractional order section multi-agent system.
Description
Technical field
The present invention relates to a kind of bounded-but-unknown uncertainty ability is resisted for having for fractional order section multi-agent system
Distributed output feedback ontrol method, belongs to control technology field.
Background technology
In the past few years, the consistency problem of multi-agent system has obtained more and more concerns.This is mainly
Since multiple agent is widely used in aircraft formation, pose adjustment, mobile robot and sensor network etc..Consistency
The purpose of problem is that the suitable agreement of design makes one group of multiple agent pass through with neighbours' progress information exchange so that they are directed to certain
A physical quantity is reached an agreement.In recent years, scholars have been directed to dynamic (dynamical) mostly intelligent with simple integral dynamics or double integrator
The consistency problem of system system conducts extensive research.
However, true physical system not always uses integer rank kinetic description, the integer rank furtherd investigate
Multi-agent system is only the special case of fractional order multi-agent system.Research shows that existing integer rank multiple agent system
The result of study of system consistency can not be directly applied in the consistency problem of fractional order multi-agent system.It is prior
It is that research recently finds many true physical systems, is included in the vehicle that is moved in viscoelastic material and in microparticle environment
Aircraft of high speed operation etc., is all more suitable for fractional order differential kinetic description.
In recent years, lot of domestic and foreign scholar has carried out numerous studies simultaneously to the control problem of fractional order multiple agent consistency
It makes some progress, but in having document, it is convenient to discuss, generally assume that the dynamic analog of fractional order multiple agent
Type is completely determining and known.However in practical engineering application, most of controlled devices are not ideal Linear Time Invariant systems
System, but there are model uncertainties to a certain extent.Accordingly, it is considered to model uncertainty especially with exponent number
The consistency problem of probabilistic fractional order multi-agent system is very real and necessary.If these above-mentioned problems
If not solving, it cannot realize the theoretical real application of fractional order multi-agent system and promote.
In addition, for traditional single controlled fractional order Interval System, when exponent number and other model parameters exist simultaneously not
It can utilize ripe robust control theory design controller that corresponding closed loop fractional order Interval System is realized during certainty
Robust stability.In consideration of it, traditional robust control theory and method are applied to the defeated of fractional order section multi-agent system
Go out in consistency control will be a feasible scheme.However, it is contemplated that the complexity of fractional order section multi-agent system
Property, the fractional order dynamics of intelligence individual and the coupling of network topology, robust consistency problem particularity etc., how should
Become the pass for solving fractional order section multi-agent system consistency control problem with existing robust control theory and method
Key.
Invention content
It is an object of the invention to be directed to the drawback of the prior art, a kind of fractional order section multi-agent system robust is provided
The control method of output-consistence thoroughly solves the robust output consistency control problem of the system.
Problem of the present invention is solved with following technical proposals：
A kind of control method of fractional order section multi-agent system robust output consistency, the method includes following steps
Suddenly：
A. the control problem of the robust output consistency of fractional order section multi-agent system is converted into fractional order section
The Stabilization of the state zero of multi-agent system：
Assuming that undirected topology fraction rank section multiple agent is formed by N number of fractional order intelligent body with bounded-but-unknown uncertainty
System, the dynamic model of i-th of intelligent body are：
yi(t)=Cxi(t),
Wherein i ∈ { 1,2, L, N }；xi(t)∈Rn, yi(t)∈RpAnd ui(t)∈RpRespectively i-th of intelligent body is in t moment
State, output and input；C∈Rp×nIt is output matrix；α0、A0And B0The corresponding normal parameter of nominal model for system；For the α defined using Caputo differential0+ Δ α order derivatives；The uncertain Δ α of exponent number is defined as：
Δ α=αMζ
Wherein, αMFor exponent number maximum perturbation range and meet α0+αM＜ 1 and α0-αM＞ 0, ζ are on section [- 1,1]
A random number；
The uncertain part Δ A and Δ B of sytem matrix meet respectively：
With
Wherein γijAnd βijIt is positive scalar constant, σijAnd ηijTo be in the random number of section [- 1,1], and AMAnd BMIt is
Known matrix with determining value, symbol " o " represent Hadamard products；
For the ease of handling uncertainty Δ A and Δ B, variable is introduced
WhereinWithRepresent that k-th of element is the column vectors that 1 other elements are 0, diag { σ11…
σ1n…σn1…σnnRepresent diagonal matrixTherefore Δ A=DAFAEAWith Δ B=DBFBEB。
Introduce new variable δi(t)=x1(t)-xi(t), the control problem of the robust output consistency of system is converted into
Fractional order section multi-agent system
State zero Stabilization；
B. distributed output feedback controller is designed
Wherein NiNeighborhood for intelligent body i；hijWeights for side in information exchange topology G；If intelligent body i can
The output information of intelligent body j is received, then hij=1；Otherwise, hij=0, F are feedback matrixes undetermined；Non-directed graph G's
Laplacian matrixes are denoted as L；
C. the Stabilization of the state zero of closed loop fractional order section multi-agent system is converted into N-1 fractional order
The stability analysis problem of the state zero of system：
DefinitionUsing orthogonal transformationΞ is the orthogonal moment of appropriate dimension
Battle array, then N-1 fractional order subsystem be：
Wherein,The wherein transposition of upper right footnote " T " representing matrix or vector, λi(i=2,
3 ..., N) it is L22+1N-1·βTCharacteristic value, βT=[h12,h13,…,h1N],
SymbolExpression Kronecker product, 1N-1∈RN-1It is all 1 column vector to represent all elements；
D. the condition for the state zero Simultaneous Stabilization that can ensure N-1 fractional order subsystem is provided：
Assuming that the singular value decomposition of output matrix C meets C=U [S 0] VT, U and V are the unitary matrice of appropriate dimension, and S is
Diagonal matrix, the element on leading diagonal is the singular value of C arranged in descending order.If there is a matrix X ∈ Rp×p,
Two symmetric positive definite matrix Q11∈Rp×p,Q22∈R(n-p)×(n-p)With 4 real constant εj＞ 0, ρj＞ 0 (j=1,2) makes following 4
Inequality
With
It sets up simultaneously, wherein sym (M) represents M+MT, then N-1 subsystem Simultaneous Stabilization, i.e., it is anti-in distribution output
Fractional order section multi-agent system can realize robust output consistency under feedback controller action, wherein,
I2Represent 2 × 2 unit matrix,
Represent 2n2×2n2Unit matrix,
E. the feedback matrix undetermined in output feedback controller is solved：
The computational methods of feedback matrix F are in controller：
Wherein
Output feedback controller proposed by the invention not only designs simply, and it is convenient to solve, and can resist due to rank
It is interfered caused by number and the bounded-but-unknown uncertainty of other model parameters, possesses good control effect, there is very strong practicality
Property, well solve the robust output consistency control problem of fractional order section multi-agent system.
Description of the drawings
The invention will be further described below in conjunction with the accompanying drawings.
Fig. 1 is the output feedback controller design cycle schematic diagram of mid-score rank section multi-agent system of the present invention；
Information exchange topological diagrams of the Fig. 2 between mid-score rank section intelligent body of the present invention；
Fig. 3 (a) is to be not added with randomly generated during any control 500 fractional order multi-agent systems for meeting given section
Position of the characteristic value in complex plane；
Fig. 3 (b) be not added with randomly generating and meet during any control given section fractional order multi-agent system it is defeated
Go out error locus；
Fig. 4 (a) is the 500 more intelligence of fractional order for meeting given section randomly generated under output feedback controller effect
Position of the characteristic value of energy system system in complex plane；
Fig. 4 (b) is the fractional order multiple agent that given section is randomly generated and met under output feedback controller effect
The output error track of system.
A symbol is in text：xi(t)∈Rn, yi(t)∈RpAnd ui(t)∈RpRespectively i-th of intelligent body is in the shape of t moment
State, output and input；C∈Rp×nIt is output matrix, the singular value decomposition of C meets C=U [S 0] VT, U and V are appropriate dimensions
Unitary matrice, S is diagonal matrix, and the element on leading diagonal is the singular value of C arranged in descending order；α0、A0And B0To be
The corresponding normal parameter of nominal model of system；For the α defined using Caputo differential0+ Δ α order derivatives；Δ α is rank
Several uncertainties；αMFor the maximum perturbation range of exponent number, ζ is a random number on section [- 1,1]；Δ A and Δ B
Uncertain part for sytem matrix；γijAnd βijIt is positive scalar constant, σijAnd ηijTo be in the random of section [- 1,1]
Number, and AMAnd BMIt is with the known matrix for determining value, symbol " o " expression Hadamard products；SymbolRepresent Kronecker
The transposition of product, upper right footnote " T " representing matrix or vector；WithIt is that 1 other elements are equal to represent k-th of element
It is 0 column vector, 1N-1∈RN-1Represent all elements be all 1 column vector, InRepresent the unit matrix of n × n；NiFor intelligent body
The neighborhood of i； hijWeights for side in information exchange topology G；F is feedback matrix undetermined；L is non-directed graph G's
Laplacian matrixes；Ξ is the orthogonal matrix of an appropriate dimension；Sym (M) represents M+MT；diag{σ1,σ2LσnRepresent diagonal
MatrixIf A is vector, | | A | | represent the Euclidean norms of vector A, if A is matrix, | | A | | table
Show 2 norm of induction of matrix A.
Specific embodiment
It is an object of the invention to the fractional order multi-agent system with bounded-but-unknown uncertainty, propose defeated based on part
Go out the output feedback ontrol method of information so that fractional order section multi-agent system can realize robust output consistency.
As shown in Figure 1, the technical solution of the present invention is realized as follows：
1. the control problem of the robust output consistency of fractional order section multi-agent system is converted into fractional order section
The Stabilization of the state zero of multi-agent system；
2. the distributed output feedback controller of design；
3. the Stabilization of the state zero of closed loop fractional order section multi-agent system is converted into N-1 fractional order
The stability analysis problem of the state zero of system；
4. provide the condition for the state zero Simultaneous Stabilization that can ensure N-1 fractional order subsystem；
5. solve the feedback matrix undetermined in output feedback controller.
The present invention has following some technical characteristics：
(1) by introducing an intermediate variable in step 1, by the robust consistency of fractional order section multi-agent system
Control problem is converted into the Stabilization of the state zero of fractional order section multi-agent system.
(2) designed in step 2 is an output feedback ontrol based on Local Interaction information between fractional order intelligent body
Device, and its feedback matrix is undetermined.
(3) the calm of state zero of closed loop fractional order section multi-agent system is asked using orthogonal transformation in step 3
Topic is converted into the stability analysis problem of the state zero of N-1 fractional order subsystem.
(4) N-1 fractional order subsystem is analyzed using existing robust control theory in step 4, obtaining can
Ensure the condition of N-1 fractional order subsystem Simultaneous Stabilization, that is, realize the condition of robust output consistency.
(5) the solution item of feedback matrix in output feedback controller is provided in step 5 in the form of linear matrix inequality
Part and calculation formula, the LMI tool boxes so as to use Matlab easily carry out Matrix Solving.
It is known：The undirected more intelligence in topology fraction rank section being made of N number of fractional order intelligent body with bounded-but-unknown uncertainty
Energy system system, the dynamic model of i-th of intelligent body are：
Wherein i ∈ { 1,2, L, N }.xi(t)∈Rn, yi(t)∈RpAnd ui(t)∈RpRespectively i-th of intelligent body is in t moment
State, output and input.C∈Rp×nIt is output matrix, α0、A0And B0The corresponding normal parameter of nominal model for system (1).For the α defined using Caputo differential0+ Δ α order derivatives.The uncertain Δ α of exponent number is defined as
Δ α=αMζ (2)
Wherein, αMFor exponent number maximum perturbation range and meet α0+αM＜ 1 and α0-αM＞ 0, ζ are on section [- 1,1]
A random number.
The uncertain part Δ A and Δ B of sytem matrix meet respectively
With
Wherein γijAnd βijIt is positive scalar constant, σijAnd ηijTo be in the random number of section [- 1,1], and AMAnd BMIt is
Known matrix with determining value, symbol " ο " represent Hadamard products.
For the ease of handling uncertainty Δ A and Δ B, variable is introduced
WhereinWithRepresent that k-th of element is the column vectors that 1 other elements are 0, diag { σ11…
σ1n…σn1…σnnRepresent diagonal matrixTherefore
Δ A=DAFAEAWith Δ B=DBFBEB (3)
It is an object of the present invention to：Have for fractional order section multi-agent system (1) design and resist bounded-but-unknown uncertainty
The distributed output feedback controller of ability so that closed loop fractional order section multi-agent system can realize output-consistence
Wherein | | yj(t)-yi(t) | | represent vector yj(t)-yi(t) Euclidean norms.
With reference to Fig. 1, it is of the invention the specific implementation process is as follows：
Step 1：The robust output consistency control problem of fractional order section multi-agent system is converted into fractional order section
The Stabilization of the state zero of multi-agent system
Due to | | yj(t)-yi(t) | |=| | C (xj(t)-xi(t))||≤||C||||xj(t)-xi(t) | |, output is consistent
Property
It is equivalent to
Introduce new variable δi(t)=x1(t)-xi(t), (2) and (3) are utilized, by the robust output consistency of system (1)
Control problem be converted into fractional order section multi-agent system
State zero Stabilization.
Step 2：Output feedback controller designs
For the distributed output feedback controller of Stabilization design of the state zero of (4)：
Wherein NiNeighborhood for intelligent body i；hijWeights for side in information exchange topology G.If intelligent body i can
The output information of intelligent body j is received, then hij=1；Otherwise, hij=0.F is feedback matrix undetermined.Non-directed graph G's
Laplacian matrixes are denoted as L.
Step 3：The Stabilization of the state zero of closed loop fractional order section multi-agent system is converted into N-1 score
The stability analysis problem of the state zero of rank subsystem
Under above-mentioned output feedback controller (5) effect, (4) can be rewritten as
DefinitionThe transposition of upper right footnote " T " representing matrix or vector, then (6) become
Wherein βT=[h12,h13,…,h1N] andSymbolRepresent Crow
Interior gram of product.
To (7) using orthogonal transformation
Wherein, Ξ is the orthogonal matrix of an appropriate dimension, Λ@ΞT(L22+1N-1·βT) Ξ=diag { λ2,λ3,L,λN}。
Due toElement be block diagonalization, fractional order area
Between the Stabilization of state zero of multi-agent system (8) be equivalent to N-1 subsystem
State zero stability analysis problem.Wherein, λi(i=2,3 ..., N) it is L22+1N-1·βTCharacteristic value,
Step 4：The analysis of robust output condition for consistence
The key for providing robust output condition for consistence is to be determined to make in (9) N-1 subsystem Simultaneous Stabilization
Condition.(9) condition of N-1 subsystem Simultaneous Stabilization is as follows in：
Assuming that the singular value decomposition of output matrix C meets C=U [S 0] VT, U and V are the unitary matrice of appropriate dimension, and S is
Diagonal matrix, the element on leading diagonal is the singular value of C arranged in descending order.If there is a matrix X ∈ Rp×p, two
A symmetric positive definite matrix Q11∈Rp×p,Q22∈R(n-p)×(n-p)With 4 real constant εj＞ 0, ρj＞ 0 (j=1,2) makes following 4 not
Equation
With
It sets up simultaneously, then (9) N-1 subsystem Simultaneous Stabilization in is acted in distributed output feedback controller (5)
Lower fractional order section multi-agent system (1) can realize robust output consistency.
Wherein,
I2Represent 2 × 2 unit matrix,
Represent 2n2×2n2Unit matrix,
Step 5：The solution of feedback matrix
Based on the robust output condition for consistence in step 4, the computational methods for providing feedback matrix F in controller are：
F=XUSQ11 -1S-1U-1 (11)
Wherein
The effect of the present invention can be further illustrated by following emulation：
Emulation content：Consider the fractional order section multi-agent system (1) being made of four fractional order section intelligent bodies,
Parameter meets
Assuming that the information exchange topology between fractional order intelligent body is as shown in Figure 2.Its Laplacian matrixes
ThereforeWithC's is unusual
Value is decomposed intoThe original state of fractional order intelligent body
It is set as 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]T
And x5(0)=[- 7, -3,0.9]T.By Matlab LMI tool boxes solve withWithCorresponding linear matrix inequality technique
Formula (10), we obtain X=-2.5370, Q11=0.5589,ε1=ε2=131.5959, ρ1
=ρ2=134.0178. can obtain F=XUSQ by (11)11 -1S-1U-1=-4.5396.
Fig. 3 (a) describes the 500 fractional order multiple agents for meeting (12) randomly generated during no any control action
Position of the characteristic value of system (9) in complex plane, from Fig. 3 (a) it can be seen that there is some characteristic values to be distributed in range of instability
Domain.Therefore the random experiment in Fig. 3 (a) the result shows that during without any control action fractional order multi-agent system (9) it is defeated
Go out is not robust stability.Fig. 3 (b) randomly generated when giving no any control action and meet (12) fractional order it is more
The output error track of multiagent system (9), fractional order multiple agent system when can be seen that no any control action from Fig. 3 (b)
The output error track of system (9) is not restrained, this further demonstrates that the conclusion obtained from Fig. 3 (a).Fig. 4 (a) gives
The characteristic value of 500 fractional order multi-agent systems (9) for meeting (12) randomly generated under output feedback controller effect exists
Position in complex plane, from Fig. 4 (a) it can be seen that all characteristic values are distributed in stability region.Therefore it is random in Fig. 4 (a)
Result of the test shows that the output of the fractional order multi-agent system (9) in the case where output feedback controller acts on is robust stability.Figure
4 (b) give output feedback controller effect under randomly generate and meet (12) fractional order multiagent system (9) it is defeated
Go out error locus, the output that can be seen that the fractional order multi-agent system (9) under output feedback controller effect from Fig. 4 (b) misses
Poor track asymptotic convergence, this further demonstrates that the conclusion obtained from Fig. 4 (a).Therefore, it is as can be seen from Figures 3 and 4 that of the invention
In distributed output feedback ontrol method be effective and with robustness.
Claims (1)
1. a kind of control method of fractional order section multi-agent system robust output consistency, it is characterized in that, the method packet
Include following steps：
A. the control problem of the robust output consistency of fractional order section multi-agent system is converted into the more intelligence in fractional order section
The Stabilization of the state zero of energy system system：
Assuming that undirected topology fraction rank section multiple agent system is formed by N number of fractional order intelligent body with bounded-but-unknown uncertainty
System, the dynamic model of i-th of intelligent body are：
yi(t)=Cxi(t),
Wherein i ∈ { 1,2, L, N }；xi(t)∈Rn, yi(t)∈RpAnd ui(t)∈RpRespectively i-th of intelligent body is in the shape of t moment
State, output and input；C∈Rp×nIt is output matrix；α0、A0And B0The corresponding normal parameter of nominal model for system；
For the α defined using Caputo differential0+ Δ α order derivatives；The uncertain Δ α of exponent number is defined as：
Δ α=αMζ
Wherein, αMFor exponent number maximum perturbation range and meet α0+αM＜ 1 and α0-αM＞ 0, ζ are one on section [- 1,1]
A random number；
The uncertain part Δ A and Δ B of sytem matrix meet respectively：
Δ A=AMo[σij]n×n=[γij]n×no[σij]n×n=[γijσij]n×nWith Δ B=BMo[ηij]n×n=[βij]n×no
[ηij]n×n=[βijηij]n×n,
Wherein γijAnd βijIt is positive scalar constant, σijAnd ηijTo be in the random number of section [- 1,1], and AMAnd BMIt is to have really
The known matrix of definite value, symbol " o " represent Hadamard products；
For the ease of handling uncertainty Δ A and Δ B, variable is introduced
WhereinWithRepresent that k-th of element is the column vectors that 1 other elements are 0, diag { σ11Lσ1nLσn1L
σnnRepresent diagonal matrixTherefore Δ A=DAFAEAWith Δ B=DBFBEB。
Introduce new variable δi(t)=x1(t)-xi(t), the control problem of the robust output consistency of system is converted into score
Rank section multi-agent system
State zero Stabilization；
B. distributed output feedback controller is designed
Wherein NiNeighborhood for intelligent body i；hijWeights for side in information exchange topology G；If intelligent body i can be received
To the output information of intelligent body j, then hij=1；Otherwise, hij=0, F are feedback matrixes undetermined；The Laplacian of non-directed graph G
Matrix is denoted as L；
C. the Stabilization of the state zero of closed loop fractional order section multi-agent system is converted into N-1 fractional order subsystem
State zero stability analysis problem：
DefinitionUsing orthogonal transformationΞ is the orthogonal matrix of appropriate dimension, that
N-1 fractional order subsystem be：
Wherein,The wherein transposition of upper right footnote " T " representing matrix or vector；λi(i=2,3, L,
N) it is L22+1N-1·βTCharacteristic value, βT=[h12,h13,L,h1N]；1N-1∈RN-1
Represent all elements be all 1 column vector, symbolRepresent Kronecker product；
D. the condition for the state zero Simultaneous Stabilization that can ensure N-1 fractional order subsystem is provided：
Assuming that the singular value decomposition of output matrix C meets C=U [S0] VT, U and V are the unitary matrice of appropriate dimension, and S is to angular moment
Gust, the element on leading diagonal is the singular value of C arranged in descending order.If there is a matrix X ∈ Rp×p, two symmetrical
Positive definite matrix Q11∈Rp×p,Q22∈R(n-p)×(n-p)With 4 real constant εj＞ 0, ρj＞ 0 (j=1,2) makes following 4 inequality
With
It sets up simultaneously, then N-1 subsystem Simultaneous Stabilization, i.e., the fractional order section under the effect of distributed output feedback controller
Multi-agent system can realize robust output consistency, wherein,
I2Represent 2 × 2 unit matrix,
Represent 2n2×2n2Unit matrix,
E. the feedback matrix undetermined in output feedback controller is solved：
The computational methods of feedback matrix F are in controller：
F=XUSQ11 -1S-1U-1
Wherein
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Cited By (5)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN109031958A (en) * | 2018-10-16 | 2018-12-18 | 廊坊师范学院 | The iterative learning control method of fractional order multiple agent tracking consistency |
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