CN110083066A - The fractional order iteration control method of multi-agent system - Google Patents

Abstract

The invention discloses the fractional order iteration control methods of multi-agent system, comprising the following steps: converts tracking error stability control problem within a certain period of time for the control problem that fractional order multi-Agent coordination is tracked；Design the PD type iterative learning controller with initial state learning ability；It is proved using close-coupled convergence of the norm to tracking error；Digital simulation verifying is carried out to the analytical proof of S30, theoretical validation is carried out to the fractional order multiple agent of a virtual leader of 4 intelligent body 1.The convergence of analytical proof multiple agent of the present invention derives the condition of convergence.Because the condition of convergence may insure the increase with the number of iterations, all tracking errors gradually decrease to sufficiently small value.

Description

The fractional order iteration control method of multi-agent system

Technical field

The invention belongs to multi-agent system control technology fields, are related to a kind of fractional order iteration control of multi-agent system Method and system processed.

Background technique

Research shows that when problem such as polymeric fluid, visco-elastic systems, submersible machine isodisperse in face of some complexity Rank calculus has better Memorability and science of heredity.Coordinate multi-agent system control answering extensively in each field in recent years With and proposing many problems on this basis, wherein fractional order multi-agent system is because of its wide applicability and preferably Stability becomes research focus.Domestic and foreign scholars have carried out a large amount of research to the Tracing Control of intelligent body, and in majority Research in the problem of being directed to progressive consistency, i.e., multiple states in system gradually become unified, but this is in weight It is unable to satisfy in the movement and production process of renaturation.

Iterative learning control is a kind of control algolithm to the system with repeatable motion, it is in previously control Data information being capable of precisely controlled effect by finding suitable control input in line interation.And fractional calculus has There are good memory function and hereditary capacity, fractional calculus is combined with iterative learning control, it will be a kind of feasible Control algolithm.

Summary of the invention

To solve the above problems, the present invention provides a kind of multiple agent system with leader based on fractional order iteration System.And the convergence of analytical proof multiple agent derives the condition of convergence.Because the condition of convergence may insure with the number of iterations Increase, all tracking errors gradually decrease to sufficiently small value.

To achieve the above object, the technical scheme is that the fractional order iteration control method of multi-agent system, packet Include following steps:

It is steady within a certain period of time to convert tracking error for the control problem that fractional order multi-Agent coordination is tracked by S10 Qualitative contrlol problem；

S20 designs the PD type iterative learning controller with initial state learning ability；

S30 is proved using close-coupled convergence of the norm to tracking error；

S40 carries out digital simulation verifying to the analytical proof of S30, to the fractional order of a virtual leader of 4 intelligent body 1 Multiple agent carries out theoretical validation.

Preferably, the S10 the following steps are included:

S11, the undirected weighted graph of connection between intelligent bodyIt indicates, schemes the Laplacian matrix L of G ∈R^n×nIt is defined as follows,

Wherein, D=diag [d₁,d₂,…,d_n],

Scheme the n node v={ v of G₁,v₂,…,v_nN intelligent body is represented,For the side of figure, the connection of i and j System indicates with weighted adjacency matrix A, diagonal entry a_ii=0, if being related between i and j, a_ij=a_ji> 0, and j ∈ N_i, N_i={ j } is the neighbor node collection of node i, when virtual leader is added, n intelligent body and figureComposition figure S=diag (s₁, s₂..., s_n), work as s_iIndicate that intelligent body obtains information from virtual leader when > 0；

S12, in section [t₀, t] in the α fractional order integration of function f (t) be defined as follows,

Wherein, α > 0,

In section [t₀, t] in the α fractional order differential of function f (t) be defined as follows,

Wherein, α > 0, [α] are the integer part of α；

If function f (x (t), t) be it is continuous, then,

Formula (3) is equivalent to following nonlinear Volterra equation,

S13, a linear fractional rank intelligent body are made of n intelligent body, and the dynamical equation of each intelligent body is as follows,

Wherein [0, T] t ∈, k are the number of iterations, x_{I, k}(t)∈R^mFor the state of intelligent body i, u_{I, k}(t)∈R^m1It is defeated to control Enter, y_{I, k}(t)∈R^m2For output, A, B, C is Changshu matrix with compatible size, D^αx_{I, k}It (t) is x_{I, k}(t)∈R^mα rank lead Number, as t≤0, x_{I, k}(t)=0；

Assuming that y_{0, d}(t)∈R^m2It is by virtual leader v₀The expection for the fractional order multiple agent amplified out refers to rail Mark, but y_0,d(t) information can not directly be shared with each intelligent body, only individual intelligent bodies can direct sharing information, virtually Leader is shown below,

The consistency problem being equivalent in distributed multi agent system finite time finds suitable control input u_i,k(t) meet following formula,

S14, convolution (5) and formula (6), tracking error is as follows,

e_i,k(t)=y_0,d(t)-y_i,k(t) (8)

e_{Ij, k}(t)=y_i,k(t)-y_{J, k}(t), j ∈ N_i (9)

Wherein, e_i,k(t) and e_{Ij, k}(t) indicate when kth time iteration i-th of intelligent body respectively with virtual leader and its neighbour Error between residence；

For convenient for Convergence analysis, formula (5) is transformed to,

Wherein I_n∈R^n×nFor unit matrix；

For Kronecker product.

Preferably, the S20 is specially；

To solve the problems, such as represented by formula (7), PD type iterative learning control method such as following formula,

Wherein, φ ∈ R^m1×m2WithFor learning gains matrix,WithIt is e_i,k(t) and e_ij,k (t) α Fractional Derivative；

Error equation (9) is transformed to,

e_ij,k(t)=y_{I, k}(t)-y_{J, k}(t)

=(y_0,d(t)-y_j,k(t))-(y_0,d(t)-y_i,k(t))

=e_j,k(t)-e_i,k(t) (12)。

Preferably, the S30 the following steps are included:

Formula (10) is transformed to by S31 according to formula (12),

Then the close-coupled of formula (10) is,

Wherein,

The norm lambda definition of phasor function f (t) is as follows,

Wherein, | | | | it can be any general phase norm, then release following formula from formula (16),

Assuming that 1, y_i,k(t), the α rank of i=1,2 ... n export, and desired output isIn section [0, T], there are y_i,d(t) =y_0,d(t) and control inputsMeet:

Assuming that 2, the desired output y of system_d(t) do not change as the number of iterations increases；

Assuming that 3, it is always constant in the original state of section [0, T] Chinese style (6), i.e., to any k and i, x_{I, k}(0)=0；

S32, convergence, it is assumed that

According to (11) and assume that 1 obtains,

According to formula (3) and formula (4), can obtain,

It exports again:

λ is brought into formula (21) and according to formula (15) and formula (16), can be obtained,

Wherein,

S33 brings formula (23) into formula (22) and obtains,

Sufficiently large λ is taken to make,

Then formula (24) is transformed to,

It is obtained according to formula (11) and formula (18),

Then formula (19) and formula (27) is brought into formula (28) to obtain,

S34 brings norm λ into formula (29) both sides and obtains,

Formula (26) is brought into formula (30) again to obtain,

Wherein,Due toThere are a sufficiently large λ satisfactions:

Therefore according to the following formula,

With the increase of the number of iterations k, k → ∞,

Make k=k+1, bring λ into formula (28) both sides and obtain,

Then it obtains,

It shows tracking error and tends to 0 in k → ∞, it was demonstrated that finish.

Preferably, S40 the following steps are included:

The dynamical equation of S41, a network including 4 fractional order intelligent bodies, 1 fractional order leader are as follows,

Wherein,α=0.9

Desired reference locus,

y_0,d=t+sin (t*2 π), t ∈ [0,1] (38)

Wherein, virtual leader and intelligent body marked as 0,1,2,3,4；

S42, weighting are bordered by matrix A,

D=diag [1.6 1.7 1.4 1.3]；

Laplacian matrix, intelligent body and the relationship virtually led are as follows,

S=diag [1.8 00 2]；

Wherein original state, initial input, initial tracking error are all 0, the gain matrix of fractional order iterative learning control, φ andAll meet the condition of convergence, i.e.,φ=0.16；

S43, the inspection condition of convergence, makes λ=100, obtains,

I.e. the condition of convergence meets.

Beneficial effects of the present invention are as follows: the present invention converts the control problem that fractional order multi-Agent coordination is tracked to and chases after Track error stability control problem within a certain period of time, better and more conveniently to design and establish with initial state learning energy The PD type iterative learning controller of power, and proved using close-coupled convergence of the norm to tracking error, and analysis is demonstrate,proved Bright progress digital simulation verifying carries out theoretical validation to the fractional order multiple agent of virtual leader of 4 intelligent body 1, and 4 points With the increase of the number of iterations, track reference track is completely coincident number rank multi-agent systems gradually, when the number of iterations is close to 150 Error norm is close to 0.

Detailed description of the invention

Fig. 1 is the step flow chart of the fractional order iteration control method of the multi-agent system of the embodiment of the present invention；

The network that Fig. 2 is step S41 in the fractional order iteration control method of the multi-agent system of the embodiment of the present invention is logical Believe topological digraph；

Fig. 3 is the tracking mode figure of the fractional order iteration control method of the multi-agent system of the embodiment of the present invention；

Fig. 4 is the maximum tracking error of the fractional order iteration control method of the multi-agent system of the embodiment of the present invention and changes Generation number comparison diagram；

Fig. 5 is the realization output-consistence frame of the fractional order iteration control method of the multi-agent system of the embodiment of the present invention Figure.

Specific embodiment

In order to make the objectives, technical solutions, and advantages of the present invention clearer, with reference to the accompanying drawings and embodiments, right The present invention is further elaborated.It should be appreciated that the specific embodiments described herein are merely illustrative of the present invention, and It is not used in the restriction present invention.

On the contrary, the present invention covers any substitution done on the essence and scope of the present invention being defined by the claims, repairs Change, equivalent method and scheme.Further, in order to make the public have a better understanding the present invention, below to of the invention thin It is detailed to describe some specific detail sections in section description.Part without these details for a person skilled in the art The present invention can also be understood completely in description.

It is the embodiment of the present invention the technical scheme is that the fractional order iteration control of multi-agent system referring to Fig. 1 The step flow chart of method processed, comprising the following steps:

It is steady within a certain period of time to convert tracking error for the control problem that fractional order multi-Agent coordination is tracked by S10 Qualitative contrlol problem；

S20 designs the PD type iterative learning controller with initial state learning ability；

S30 is proved using close-coupled convergence of the norm to tracking error；

S40 carries out digital simulation verifying to the analytical proof of S30, to the fractional order of a virtual leader of 4 intelligent body 1 Multiple agent carries out theoretical validation.

S10 the following steps are included:

S11, the undirected weighted graph of connection between intelligent bodyIt indicates, schemes the Laplacian matrix L of G ∈R^n×nIt is defined as follows,

Wherein, D=diag [d₁,d₂,…,d_n],

Scheme the n node v={ v of G₁,v₂,…,v_nN intelligent body is represented,For the side of figure, the connection of i and j System indicates with weighted adjacency matrix A, diagonal entry a_ii=0, if being related between i and j, a_ij=a_ji> 0, and j ∈ N_i, N_i={ j } is the neighbor node collection of node i, when virtual leader is added, n intelligent body and figureGroup At figure S=diag (s₁, s₂,…,s_n), work as s_iIndicate that intelligent body obtains information from virtual leader when > 0；

S12, in section [t₀, t] in the α fractional order integration of function f (t) be defined as follows,

Wherein, α > 0,

In section [t₀, t] in the α fractional order differential of function f (t) be defined as follows,

Wherein, α > 0, [α] are the integer part of α；

If function f (x (t), t) be it is continuous, then,

Formula (3) is equivalent to following nonlinear Volterra equation,

S13, a linear fractional rank intelligent body are made of n intelligent body, and the dynamical equation of each intelligent body is as follows,

Wherein [0, T] t ∈, k are the number of iterations, x_{I, k}(t)∈R^mFor the state of intelligent body i, u_i,k(t)∈R^m1It is defeated to control Enter, y_i,k(t)∈R^m2For output, A, B, C is Changshu matrix with compatible size, D^αx_i,kIt (t) is x_i,k(t)∈R^mα rank lead Number, as t≤0, x_i,k(t)=0；

Assuming that y_0,d(t)∈R^m2It is by virtual leader v₀The expection for the fractional order multiple agent amplified out refers to rail Mark, but y_0,d(t) information can not directly be shared with each intelligent body, only individual intelligent bodies can direct sharing information, virtually Leader is shown below,

It is run by iteration, finally searches out the suitable control input of each intelligent body, make the running track of each intelligent body In the finite time [0, T] with desired trajectory y_0,d(t) consistent.

The consistency problem being equivalent in distributed multi agent system finite time finds suitable control input u_i,k(t) meet following formula,

I.e. consistency target (8) means to find a suitable Iterative Learning Control Algorithm, with the number of iterations Increase, each intelligent body in multi-agent system is in Finite-time convergence in desired reference locus.

S14, convolution (5) and formula (6), tracking error is as follows,

e_i,k(t)=y_0,d(t)-y_i,k(t) (8)

e_ij,k(t)=y_i,k(t)-y_j,k(t),j∈N_i (9)

Wherein, e_i,k(t) and e_ij,k(t) indicate when kth time iteration i-th of intelligent body respectively with virtual leader and its neighbour Error between residence；

For convenient for Convergence analysis, formula (5) is transformed to,

Wherein I_n∈R^n×nFor unit matrix；

For Kronecker product.

S20 is specially；

To solve the problems, such as represented by formula (7), PD type iterative learning control method such as following formula,

Wherein, φ ∈ R^m1×m2WithFor learning gains matrix,WithIt is e_{I, k}(t) and e_{Ij, k} (t) α Fractional Derivative；

Error equation (9) is transformed to,

e_{Ij, k}(t)=y_{I, k}(t)-y_{J, k}(t)

=(y_{0, d}(t)-y_{J, k}(t))-(y_0,d(t)-y_i,k(t))

=e_j,k(t)-e_i,k(t) (12)。

S30 the following steps are included:

Formula (10) is transformed to by S31 according to formula (12),

Then the close-coupled of formula (10) is,

Wherein,

The norm lambda definition of phasor function f (t) is as follows,

Wherein, | | | | it can be any general phase norm, then release following formula from formula (16),

Assuming that 1, y_{I, k}(t), the α rank of i=1,2 ... n export, and desired output isIn section [0, T], there are y_i,d(t) =y_0,d(t) and control inputsMeet:

Assuming that 2, the desired output y of system_d(t) do not change as the number of iterations increases；

Assuming that 3, it is always constant in the original state of section [0, T] Chinese style (6), i.e., to any k and i, x_i,k(0)=0；

S32, convergence, it is assumed that

According to (11) and assume that 1 obtains,

According to formula (3) and formula (4), can obtain,

It exports again:

λ is brought into formula (21) and according to formula (15) and formula (16), can be obtained,

Wherein,

S33 brings formula (23) into formula (22) and obtains,

Sufficiently large λ is taken to make,

Then formula (24) is transformed to,

It is obtained according to formula (11) and formula (18),

Then formula (19) and formula (27) is brought into formula (28) to obtain,

S34 brings norm λ into formula (29) both sides and obtains,

Formula (26) is brought into formula (30) again to obtain,

Wherein,Due toThere are a sufficiently large λ satisfactions:

Therefore according to the following formula,

With the increase of the number of iterations k, k → ∞,

Make k=k+1, bring λ into formula (28) both sides and obtain,

Then it obtains,

It shows tracking error and tends to 0 in k → ∞, it was demonstrated that finish.

S40 the following steps are included:

The dynamical equation of S41, a network including 4 fractional order intelligent bodies, 1 fractional order leader are as follows,

Wherein,α=0.9

Desired reference locus,

y_0,d=t+sin (t*2 π), t ∈ [0,1] (38)

Wherein, virtual leader and intelligent body marked as 0,1,2,3,4；Network communication topology digraph referring to fig. 2,

S42, from Fig. 2 it can be gathered that weighting is bordered by matrix A,

D=diag [1.6 1.7 1.4 1.3]；

Laplacian matrix, intelligent body and the relationship virtually led are as follows,

S=diag [1.8 00 2]；

Wherein original state, initial input, initial tracking error are all 0, the gain matrix of fractional order iterative learning control, φ andAll meet the condition of convergence, i.e.,φ=0.16；

S43, the inspection condition of convergence, makes λ=100, obtains,

I.e. the condition of convergence meets.

As a result shown in Figure 3, it can be seen that 4 fractional order multi-agent system Agent1~4 are with iteration time Several increases is completely coincident gradually with the track reference track of virtual leader Leader.Fig. 4 is error norm convergence, can be with Error norm is close to 0 when finding out the number of iterations close to 150.

Output-consistence block diagram shown in fig. 5, control amount u next time_k+1(t) by this control amount u_k(t) and this Error e_k(t) withProduct, this error e_k(t) α rank leads e_k ^α(t) withProduct addition obtain. Wherein L is Laplacian Matrix, and S is leader and the relational matrix that is followed by, φ be proportionality coefficient andIt is differential coefficient, u_k+1(t) it acts on after fractional order multiagent system and is produced next time with fractional order leader by communication network topology Error e_k+1(t)。

The foregoing is merely illustrative of the preferred embodiments of the present invention, is not intended to limit the invention, all in essence of the invention Made any modifications, equivalent replacements, and improvements etc., should all be included in the protection scope of the present invention within mind and principle.

Claims (5)

1. a kind of fractional order iteration control method of multi-agent system, which comprises the following steps:

The control problem that fractional order multi-Agent coordination is tracked is converted tracking error stability within a certain period of time by S10 Control problem；

S20 designs the PD type iterative learning controller with initial state learning ability；

S30 is proved using close-coupled convergence of the norm to tracking error；

S40 carries out digital simulation verifying to the analytical proof of S30, to the more intelligence of fractional order of a virtual leader of 4 intelligent body 1 It can body progress theoretical validation.

2. the method according to claim 1, wherein the S10 the following steps are included:

S11, the undirected weighted graph of connection between intelligent bodyIt indicates, schemes the Laplacian matrix L ∈ R of G^{n ×n}It is defined as follows,

Wherein, D=diag [d₁,d₂,…,d_n],

Scheme the n node v={ v of G₁,v₂,…,v_nN intelligent body is represented,For the side of figure, i and j's contacts use Weighted adjacency matrix A expression, diagonal entry a_ii=0, if being related between i and j, a_ij=a_ji> 0, and j ∈ N_i, N_i= { j } is the neighbor node collection of node i, when virtual leader is added, n intelligent body and figureComposition figure S= diag(s₁,s₂,…,s_n), work as s_iIndicate that intelligent body obtains information from virtual leader when > 0；

S12, in section [t₀, t] in the α fractional order integration of function f (t) be defined as follows,

Wherein, α > 0,

In section [t₀, t] in the α fractional order differential of function f (t) be defined as follows,

Wherein, α > 0, [α] are the integer part of α；

If function f (x (t), t) be it is continuous, then,

Formula (3) is equivalent to following nonlinear Volterra equation,

S13, a linear fractional rank intelligent body are made of n intelligent body, and the dynamical equation of each intelligent body is as follows,

Wherein [0, T] t ∈, k are the number of iterations, x_i,k(t)∈R^mFor the state of intelligent body i, u_i,k(t)∈R^m1It is inputted for control, y_i,k(t)∈R^m2For output, A, B, C is Changshu matrix with compatible size, D^αx_{I, k}It (t) is x_{I, k}(t)∈R^mα order derivative, As t≤0, x_i,k(t)=0；

Assuming that y_0,d(t)∈R^m2It is the expection reference locus for the fractional order multiple agent amplified out by virtual leader v0, but y_0,d(t) information can not directly be shared with each intelligent body, only individual intelligent bodies can direct sharing information, it is virtual to lead Person is shown below,

The consistency problem being equivalent in distributed multi agent system finite time finds suitable control input u_i,k(t) Meet following formula,

S14, convolution (5) and formula (6), tracking error is as follows,

e_i,k(t)=y_0,d(t)-y_{I, k}(t) (8)

e_ij,k(t)=y_i,k(t)-y_j,k(t), j ∈ N_i (9)

Wherein, e_i,k(t) and e_{Ij, k}(t) indicate when kth time iteration i-th of intelligent body respectively between virtual leader and its neighbour Error；

For convenient for Convergence analysis, formula (5) is transformed to,

Wherein I_n∈R^n×nFor unit matrix；

For Kronecker product.

3. according to the method described in claim 2, it is characterized in that, the S20 is specially；

To solve the problems, such as represented by formula (7), PD type iterative learning control method such as following formula,

Wherein, φ ∈ R^m1×m2WithFor learning gains matrix,WithIt is e_i,k(t) and e_ij,k(t) α Fractional Derivative；

Error equation (9) is transformed to,

e_ij,k(t)=y_i,k(t)-y_j,k(t)

=(y_0,d(t)-y_j,k(t))-(y_0,d(t)-y_i,k(t))

=e_j,k(t)-e_i,k(t) (12)。

4. according to the method described in claim 3, it is characterized in that, the S30 the following steps are included: S31, by formula (10) basis Formula (12) is transformed to,

Then the close-coupled of formula (10) is,

Wherein,

The norm lambda definition of phasor function f (t) is as follows,

Wherein, | | | | it can be any general phase norm, then release following formula from formula (16),

Assuming that 1, y_i,k(t), the α rank of i=1,2 ... n export, and desired output isIn section [0, T], there are y_i,d(t) =y_0,d(t) and control inputsMeet:

Assuming that 2, the desired output y of system_d(t) do not change as the number of iterations increases；

Assuming that 3, it is always constant in the original state of section [0, T] Chinese style (6), i.e., to any k and i, x_{I, k}(0)=0；

S32, convergence, it is assumed that

According to (11) and assume that 1 obtains,

According to formula (3) and formula (4), can obtain,

It exports again:

λ is brought into formula (21) and according to formula (15) and formula (16), can be obtained,

Wherein,

S33 brings formula (23) into formula (22) and obtains,

Sufficiently large λ is taken to make,

Then formula (24) is transformed to,

It is obtained according to formula (11) and formula (18),

Then formula (19) and formula (27) is brought into formula (28) to obtain,

S34 brings norm λ into formula (29) both sides and obtains,

Formula (26) is brought into formula (30) again to obtain,

Wherein,Due toThere are a sufficiently large λ satisfactions:

Therefore according to the following formula,

With the increase of the number of iterations k, k → ∞,

Make k=k+1, bring λ into formula (28) both sides and obtain,

Then it obtains,

It shows tracking error and tends to 0 in k → ∞, it was demonstrated that finish.

5. according to the method described in claim 4, it is characterized in that, S40 the following steps are included:

The dynamical equation of S41, a network including 4 fractional order intelligent bodies, 1 fractional order leader are as follows,

Wherein,α=0.9

Desired reference locus,

y_0,d=t+sin (t*2 π), t ∈ [0,1] (38)

Wherein, virtual leader and intelligent body marked as 0,1,2,3,4；

S42, weighting are bordered by matrix A,

D=diag [1.6 1.7 1.4 1.3]；

Laplacian matrix, intelligent body and the relationship virtually led are as follows,

S=diag [1.8 00 2]；

Wherein original state, initial input, initial tracking error are all 0, fractional order iterative learning control gain matrix, φ andAll meet the condition of convergence, i.e.,φ=0.16；

S43, the inspection condition of convergence, makes λ=100, obtains,

I.e. the condition of convergence meets.