CN103279031B - A kind of robust convergent control method of uncertain multi-agent system - Google Patents

Abstract

The present invention is directed to the complicated multi-agent system with external disturbance and model uncertainty, it is proposed that a kind of distributed convergent control method with Robust interference performance, belong to multi-agent system and coordinate control field.The process that realizes of the method includes: convergent for uncertain system control is converted into robust H by (1)_∞Control problem；(2) propose distributions and feed back convergent controller；(3) application model conversion and robust H_∞The convergent condition of robust of theory analysis closed loop multi-agent system；(4) based on the convergent condition of robust, the method for solving of feedback matrix undetermined in convergent controller is set up.The convergent controller of robust proposed by the invention not only designs and solves simple, and can reach Robust jamming performance set in advance, have the strongest practicality, it is adaptable to convergent control to uncertain system object in civilian and military.

Description

A kind of robust convergent control method of uncertain multi-agent system

Technical field

The present invention relates to multi-agent system Coordinated Control field, propose the distributed convergent control method with Robust interference performance for multi-agent system time similar with random external interference and model uncertainty.

Background technology

Multi-agent system has a wide range of applications background in engineering, biology and socic-economic field, is a study hotspot of current system control field, is specifically related to the subjects such as control, physics, mathematics, computer, biology.From controlling angle, research worker is primarily upon the most mutual based on the local message between intelligence individuality and cooperates, and jointly realizes desired macroscopic view emerging behavior, the most so-called multi-agent system distributed and coordinated control.Wherein, convergent control is most basic most typical one of study a question, it is the basis realizing the complicated group behavior such as Flocking, formation, cluster, in every field such as industry, space flight, military affairs, all there is wide application background, as the coordination of multi-robot system, automatization's highway scheduling, multi-satellite system, wireless sensor network controls to be all its typical application example.

In recent years, lot of domestic and foreign scholar control problem convergent to multiple agent has carried out numerous studies and has achieved a series of progress, but in existing document, convenient for discussing, generally assume that the dynamic model of intelligent body determines known and Linear Time Invariant completely.But in practical engineering application, most of controlled devices are not the most preferable Linear Time-Invariant Systems, and there is the distributivity of non-linear and systematic parameter to a certain extent.Ignore secondary cause, retain linear part carry out system modelling time inevitably lead to there is error between the system model being obtained in that and practical object, i.e. model uncertainty.Accordingly, it is considered to the convergent control problem of multi-agent system containing model uncertainty is the most real and necessary.On the other hand, the actual environment residing for intelligent body is usually present various unpredictable external disturbance.Such as: robot in actual traveling process inevitably by the external disturbance caused because of factors such as pavement roughness and barrier collision, frictional force.Therefore, it is necessary to consider disturbed problem and the AF panel problem of system.If these problems above-mentioned do not solve, the theoretical real application of multi-agent system cannot be realized and promote, here it is the motivation of the present invention and starting point.

The coupling it is true that the distributed information framework of the uncertainty that caused of various factors and system is interweaved, brings difficulty to the analysis and synthesis of multi-agent system under uncertain environment.It is known that for traditional single controlled device, the robust H of maturation can be utilized when model exists uncertain and external interference_∞Design Theory controller makes corresponding closed loop system have desired Robust jamming performance.In consideration of it, by traditional robust H_∞Theoretical and method is applied in the convergent control of uncertain multi-agent system would is that a feasible scheme.However, it is contemplated that information coupling between the complexity of multi-agent system, distributivity, intelligence individuality and considered the particularity of convergent problem, how to apply existing robust H_∞Control method becomes the key of solution problem.

The present invention proposes to utilize robust H_∞Method solves, containing model uncertainty and the convergent control problem of multi-agent system of external disturbance, to give the distributed convergent controller design method with Robust interference performance.

Summary of the invention

It is an object of the invention to the general linear multi-agent system containing model uncertainty and external disturbance, propose the convergent control method of robust based on local state information so that multi-agent system is capable of having expectation H_∞The state of interference attenuation index is convergent.

As it is shown in figure 1, the technical solution of the present invention realizes as follows:

1. convergent control is converted into robust H_∞Control problem；

2. design distributions feeds back convergent controller；

3. analyze the convergent condition of robust of closed loop multi-agent system；

4. solve the feedback matrix undetermined in convergent controller.

The present invention has following technical characteristics:

(1) by one appropriate controlled output function of definition in step 1, the convergent control of multi-agent system is converted into a Robust H-∞ control problem.

(2) design in step 2 is that its feedback matrix is undetermined based on the static state-feedback controller of Local Interaction information between intelligent body.

(3) step 3 utilizes model conversion that the convergent track of non-zero of closed loop multi-agent system changes into the zero state equilibrium point of a price reduction system of equal value, and then apply existing robust H_∞Theory carries out performance evaluation to this reduced order system, obtains the convergent condition of robust with expectation capacity of resisting disturbance.

(4) step 4 gives with LMI form solving condition and the computing formula of feedback matrix in convergent controller, it is possible to use Matlab LMI workbox carries out Matrix Solving easily.

Present invention advantage compared with prior art is:

(1), during the present invention considers actual application, multiagent system Unmarried pregnancy disturbs the convergent control problem of robust in the case of simultaneously existing with random external.Enrich the research range of convergent control, widen its engineer applied category.

(2) the convergent controller of robust proposed by the invention not only designs and solves simply, and can reach control accuracy set in advance and effect, has the strongest practicality.

Accompanying drawing explanation

Fig. 1 is the robust of uncertain multi-agent system convergent controller design cycle schematic diagram in the present invention；

Fig. 2 is external interference signals w (t) of analogue system in the present invention；

Fig. 3 is that to emulate the information of multi-agent system in the present invention the most topological；

Fig. 4 is the state trajectory of lower four the uncertain intelligent bodies of the convergent controller action of robust；

Fig. 5 is the energy relationship schematic diagram between the convergent error of state and external interference signals.

Detailed description of the invention

Known:

● the undirected variable topological multi-agent system being made up of n intelligent body, the dynamic model of i-th intelligent body is:

${\stackrel{\·}{x}}_i\left(t\right)={\mathrm{Ax}}_i\left(t\right)+B_1{\mathrm{\ω}}_i\left(t\right)+B_2{u}_i\left(t\right),i=1,2,\·\·\·,n---\left(1\right)$

Wherein, WithIt is respectively i-th intelligent body in the internal state of t, the external disturbance of finite energy and input signal.Uncertain system matrix is as follows:

A=A₀+ΔA(t),B₁=B₁₀+ΔB₁(t),B₂=B₂₀+ΔB₂(t)

In formula, A₀,B₁₀,B₂₀For permanent nominal model matrix, and (A₀,B₂₀) can stablize；Uncertain part Δ A (t) of sytem matrix, Δ B₁(t),ΔB₂T () meets matching condition [Δ A (t) Δ B₁(t)ΔB₂(t)]=EΣ(t)[F₁F₂F₃], wherein

● given H_∞Interference free performance index γ > 0.

It is an object of the present invention to: uncertain multi-agent system (1) design is had to the distributed convergent controller of Robust interference performance so that closed loop multi-agent system is capable of having H_∞The state of interference attenuation index γ is convergent.With reference to Fig. 1, the present invention to implement process as follows:

Step 1: problem converts

Define controlled output function

$z_i\left(t\right)=x_i\left(t\right)-\frac{1}{n}\underset{j=1}{\overset{n}{\mathrm{\Σ}}}{x}_j\left(t\right),i=\mathrm{1,2},\·\·\·,n$

The convergent control of system (1) is converted into robust H_∞Control problem:

$\left\{\begin{array}{c}\stackrel{\·}{x}\left(t\right)=(I_n\⊗A)x\left(t\right)+(I_n\⊗B_1)\mathrm{\ω}\left(t\right)+(I_n\⊗B_2)u\left(t\right)\\ z\left(t\right)=(L_c\⊗I_m)x\left(t\right)\end{array}\right.$

In formula, in x (t), ω (t), u (t), z (t) respectively system, all intelligent bodies are to dependent variable x_i(t),ω_i(t),u_i(t),z_i(t) (i=1,2 ..., column vector n) constituted；Matrix L_cBeing a symmetrical matrix, its diagonal element is (n-1)/n, and off-diagonal element is-1/n.Notice z (t)=0 and if only if that all n intelligent bodies realize state convergent, therefore control target to become: design controller makes state convergent track Asymptotic Stability, and from external disturbance ω (t) to the H of controlled output z (t) transfer function matrix_∞Norm subtracts index γ, i.e. less than interference sorrow | | T_{z ω}(s)||_∞<γ。

Step 2: controller designs

For above-mentioned robust H_∞Control problem design distributed director:

$u_i\left(t\right)=K\underset{j\&Element;N_i\left(t\right)}{\mathrm{\&Sigma;}}{a}_{\mathrm{ij}}\left(t\right)[x_i\left(t\right)-x_j\left(t\right)]$

Wherein N_iT () is the neighborhood of t intelligent body i；a_ijT () is information interaction figure G_{σ (t)}Limit on the weights communication topology switching signal of piecewise constant (σ (t) be), a_ijThe status information of t and if only if in () ≠ 0 intelligent body i can observe or receive intelligent body j；For feedback matrix undetermined.Note L_{σ (t)}=diag{d_{1 σ (t),}…,d_{n σ (t)}}-Λ_{σ (t)}For non-directed graph G_{σ (t)}Laplacian matrix, the minimum of Laplacian matrix corresponding to all handover networks and maximum nonzero eigenvalue be denoted as respectively intoWithWherein $A_{\mathrm{\&sigma;}\left(t\right)}={\left[a_{\mathrm{ij}}\left(t\right)\right]}_{i,j=1}^n.$

Step 3: convergent condition analysis

Utilize model to convert, the convergent track of the non-zero of closed loop multi-agent system under distributed director effect is changed into price reduction system of equal value

Zero state equilibrium point, and meet.And then obtain the design criteria of following convergent condition and controller feedback matrix:

Make the zero point Asymptotic Stability of reduced order system (2) if there is feedback matrix K, and meetThe most uncertain multi-agent system (1) is capable of having H_∞The state of interference attenuation index γ is convergent.

Matrix L in formula (2)_{1 σ (t)}Generated by with down conversion: certainly exist orthogonal matrixMake

$U^T{L}_{c}U=\left[\begin{array}{cc}{I}_{n-1}& 0\\ 0& 0\end{array}\right],$ $U^T{L}_{\mathrm{\&sigma;}\left(t\right)}U=\left[\begin{array}{cc}{L}_{1\mathrm{\&sigma;}\left(t\right)}& 0\\ 0& 0\end{array}\right]$

Set up simultaneously.

Step 4: feedback matrix solves

Based on the convergent condition of the robust in step 3, provide LMI solving condition and the computational methods of feedback matrix K in controller:

● for scalar ce > 0, if there is positive definite matrixAnd matrixMake LMI

$\left[\begin{array}{cccc}{\mathrm{PA}}_0^T+A_{0}P{+\mathrm{\&lambda;Q}}^T{B}_{20}^T+{\mathrm{\&lambda;B}}_{20}Q+{\mathrm{\&alpha;}}^2{\mathrm{EE}}^T& B_{10}& P& \frac{1}{\mathrm{\&alpha;}}{(F_{1}P+{\mathrm{\&lambda;F}}_{3}Q)}^T\\ B_{10}^T& {-\mathrm{\&gamma;}}^{-2}I& 0& \frac{1}{\mathrm{\&alpha;}}{F_2}^T\\ P& 0& -I& 0\\ \frac{1}{\mathrm{\&alpha;}}(F_{1}P+{\mathrm{\&lambda;F}}_{3}Q)& \frac{1}{\mathrm{\&alpha;}}{F}_2& 0& -I\end{array}\right]<0$

For Set up, then the solution formula of K is K=QP simultaneously^-1。

Especially, for there is no multi-agent system (the i.e. sytem matrix A, B in model (1) of system model indeterminate₁,B₂Permanent and known), can calculate feedback matrix as follows:

● if there is positive definite matrixAnd matrixMake LMI

$\left[\begin{array}{cc}{\mathrm{PA}}^T+\mathrm{AP}+{\mathrm{\&lambda;Q}}^T{B}_2^T+{\mathrm{\&lambda;B}}_{2}Q+{\mathrm{\&gamma;}}^{-2}{B}_1{B}_1^T& P\\ P& -I\end{array}\right]<0$

For Set up, then the solution formula of K is K=QP simultaneously^-1。

The effect of the present invention can be further illustrated by following emulation:

Emulation content: considering the uncertain multi-agent system (1) being made up of four intelligent bodies, its sytem matrix is

$A=\left[\begin{array}{cc}0& -1\\ 2& 1\end{array}\right],$ $B_1=\left[\begin{array}{c}0\\ 1\end{array}\right],$ $B_2=\left[\begin{array}{cc}1& 0\\ 0& 2\end{array}\right],$ $E=\left[\begin{array}{cc}0& 0\\ 0.8& 0.8\end{array}\right],$

$\mathrm{\&Sigma;}\left(t\right)=\left[\begin{array}{cc}\sin \left(10t\right)& 0\\ 0& 0\end{array}\right],$ $F_1=\left[\begin{array}{cc}1.2& 0\\ 0& 1.2\end{array}\right],$ $F_2=\left[\begin{array}{c}0.5\\ 0.5\end{array}\right],$ $F_3=\left[\begin{array}{cc}0.8& 0\\ 0& 0.8\end{array}\right];$

Interference attenuation index is taken as γ=1.For visual and clear the interference free performance reflecting system, it is assumed that the action time of external disturbance is t ∈ [0,10], and expression is ω (t)=[ω₁(t)ω₂(t)ω₃(t)ω₄(t)]^Τ=[2w(t)-1.5w(t)1.2w(t)1.8w(t)]^Τ, w (t) is the band-limited white noise on interval [0,10], as shown in Figure 2.Assume that the information interaction figure between intelligent body is at set { G₁,G₂,G₃Switch at random in }, as shown in Figure 3.

Fig. 4 describes the state trajectory of four uncertain intelligent bodies under the convergent controller action of robust, wherein (a), (b) the most corresponding first, second state component；Fig. 5 describes the energy relationship between controlled output z (t) and external disturbance ω (t).By Figure 4 and 5 it can be seen that the distributed convergent control method proposed in the present invention has robustness and desired capacity of resisting disturbance γ=1.

Claims (4)

1. the convergent control method of the robust of a uncertain multi-agent system, it is characterised in that bag Include following steps:

The undirected variable topological multi-agent system being made up of n intelligent body, i-th intelligent body Dynamic model is:

${\stackrel{\&CenterDot;}{x}}_i\left(t\right)={\mathrm{Ax}}_i\left(t\right)+B_1{\mathrm{\&omega;}}_i\left(t\right)+B_2{u}_i\left(t\right),i=1,2,...,n---\left(1\right)$

Wherein,It is respectively i-th intelligent body in t Internal state, the external disturbance of finite energy and input signal, carried out uncertain system matrix It is expressed as:

A=A₀+ΔA(t),B₁=B₁₀+ΔB₁(t),B₂=B₂₀+ΔB₂(t)

In formula, A₀,B₁₀,B₂₀For permanent nominal model matrix, and (A₀,B₂₀) controlled, sytem matrix Uncertain part Δ A (t), Δ B₁(t),ΔB₂T () meets matching condition

[ΔA(t)ΔB₁(t)ΔB₂(t)]=E ∑ (t) [F₁F₂F₃]

Wherein

(1) convergent control is converted into robust H_∞Control problem；Define controlled output function

$z_i\left(t\right)=x_i\left(t\right)-\frac{1}{n}\underset{j=1}{\overset{n}{\&Sigma;}}{x}_j\left(t\right),i=1,2,...,n$

Wherein, z_iT () is controlled output function, x_i(t) and x_jT () is respectively the state of intelligent body i and j and becomes Amount；

The convergent control of uncertain multi-agent system is converted into following robust H_∞Control is asked Topic:

$\left\{\begin{array}{c}\stackrel{\&CenterDot;}{x}\left(t\right)=(I_n\&CircleTimes;A)x\left(t\right)+(I_n\&CircleTimes;B_1)\mathrm{\&omega;}\left(t\right)+(I_n\&CircleTimes;B_2)u\left(t\right)\\ z\left(t\right)=(I_c\&CircleTimes;I_m)x\left(t\right)\end{array}\right.$

In formula, in x (t), ω (t), u (t), z (t) respectively system, all intelligent bodies are to dependent variable x_i(t),ω_i(t),u_i(t),z_iT column vector that () is constituted, wherein i=1,2 ..., n；Matrix L_cIt it is a symmetry Matrix, diagonal element is (n-1)/n, and off-diagonal element is-1/n；Notice z (t)=0 when and only When all n intelligent bodies, to realize state convergent, therefore controls target and becomes: design controller makes Obtain state convergent track Asymptotic Stability, and transmit to controlled output z (t) from external disturbance ω (t) The H of Jacobian matrix_∞Norm subtracts index γ, i.e. less than interference sorrow | | T_zω(s)||_∞<γ；

(2) design distributions feeds back convergent controller；For above-mentioned robust H_∞Control problem sets Meter distributed director:

$u_i\left(t\right)=K\underset{j\&Element;N_i\left(t\right)}{\&Sigma;}{a}_{ij}\left(t\right)\&lsqb;x_i\left(t\right)-x_j\left(t\right)\&rsqb;$

Wherein N_iT () is the neighborhood of t intelligent body i；a_ijT () is information interaction figure G_σ(t)Limit Upper weights, σ (t) is the communication topology switching signal of piecewise constant, a_ijT and if only if in () ≠ 0 intelligence Body i can observe or receive the status information of intelligent body j；For feedback square undetermined Battle array；Note L_σ(t)=diag{d_1σ(t),…,d_nσ(t)}-Λ_σ(t)For non-directed graph G_σ(t)Laplacian matrix, Minimum and the maximum nonzero eigenvalue of Laplacian matrix corresponding to all handover networks are remembered respectively AsWithWherein

(3) the convergent condition of robust of closed loop multi-agent system is analyzed；Utilize model to convert, will divide Under cloth controller action, the convergent track of the non-zero of closed loop multi-agent system changes into fall of equal value Level is united

Zero state equilibrium point, and meetMatrix L in formula (2)_1σ(t)By with Down conversion generates: certainly exist orthogonal matrixMake

$U^T{L}_{c}U=\left[\begin{array}{cc}{I}_{n-1}& 0\\ 0& 0\end{array}\right],U^T{L}_{\mathrm{\&sigma;}\left(t\right)}U=\left[\begin{array}{cc}{L}_{1\mathrm{\&sigma;}\left(t\right)}& 0\\ 0& 0\end{array}\right]$

Set up simultaneously；And then obtain the design criteria of following convergent condition and controller feedback matrix:

The zero point Asymptotic Stability of described reduced order system of equal value is made if there is feedback matrix K, And meetThe most uncertain multi-agent system is capable of having H_∞Interference declines The state subtracting index γ is convergent；

(4) feedback matrix undetermined in convergent controller is solved；

Based on the convergent condition of robust in step (3), provide feedback matrix K in controller LMI solving condition and computational methods:

For scalar ce > 0, if there is positive definite matrixAnd matrixMake line Property MATRIX INEQUALITIES

$\left[\begin{array}{cccc}{{\mathrm{PA}}_0}^T+A_{0}P+{\mathrm{\&lambda;Q}}^T{B_{20}}^T+{\mathrm{\&lambda;B}}_{20}Q+{\mathrm{\&alpha;}}^2{\mathrm{EE}}^T& B_{10}& P& \frac{1}{\mathrm{\&alpha;}}{(F_{1}P+{\mathrm{\&lambda;F}}_{3}Q)}^T\\ {B_{10}}^T& -{\mathrm{\&gamma;}}^{-2}I& 0& \frac{1}{\mathrm{\&alpha;}}{F}_2^T\\ P& 0& -I& 0\\ \frac{1}{\mathrm{\&alpha;}}(F_{1}P+{\mathrm{\&lambda;F}}_{3}Q)& \frac{1}{\mathrm{\&alpha;}}{F}_2& 0& -I\end{array}\right]<0$

ForSet up, then the solution formula of feedback matrix K is K=QP simultaneously^-1；

For there is no the multi-agent system of system model indeterminate, i.e. i-th intelligent body Sytem matrix A, B in dynamic model (1)₁,B₂Permanent and it is known that can calculate anti-as follows Feedback matrix:

If there is positive definite matrixAnd matrixMake LMI

$\left[\begin{array}{cc}{\mathrm{PA}}^T+AP+{\mathrm{\&lambda;Q}}^T{B_2}^T+{\mathrm{\&lambda;B}}_{2}Q+{\mathrm{\&gamma;}}^{-2}{B}_1{B}_1^T& P\\ P& -I\end{array}\right]<0$

ForSet up, then the solution formula of feedback matrix K is K=QP simultaneously^-1。

The convergent control of robust of a kind of uncertain multi-agent system the most according to claim 1 Method processed, it is characterised in that in described step (2) designed be one based on intelligent body Between the static state-feedback controller of Local Interaction information, and its feedback matrix is undetermined.

The convergent control of robust of a kind of uncertain multi-agent system the most according to claim 1 Method processed, it is characterised in that utilize model to convert how intelligent for closed loop in described step (3) The convergent track of non-zero of system system changes into the zero balancing point of a reduced order system of equal value, Jin Erying With existing robust H_∞Theory carries out performance evaluation to this reduced order system, obtains system and has the phase Hope the convergent condition of robust of capacity of resisting disturbance.

The convergent control of robust of a kind of uncertain multi-agent system the most according to claim 1 Method processed, it is characterised in that described step (4) is given with the form of LMI The solving condition of feedback matrix and computing formula in convergent controller, such that it is able to use Matlab LMI workbox carries out Matrix Solving easily.