CN103279031A - Robust convergence control method of uncertain multi-agent system - Google Patents

Abstract

The invention provides a distributed convergence control method with robust anti-interference capability by aiming at a completed multi-agent system with external interference and mold uncertainty at the same time, and belongs to the field of coordination control of the uncertain multi-agent system. The method comprises the following realization processes that (1) the uncertain system convergence control is converted into the robust H infinite control problem; (2) a distributed state feedback convergence controller is provided; (3) model transformation and robust H infinite theories are applied to analyze the robust convergence condition of a closed loop multi-agent system; and (4) a solving method of matrixes to be fed back in the convergence controller is built on the basis of the robust convergence condition. The robust convergence controller provided by the invention has the advantages that the design and the solving are simple, in addition, the preset robust anti-interference performance can be reached, high practicability is realized, and the completed multi-agent system is suitable for the convergence control of uncertain system objects in civilian use and military affairs.

Description

The convergent control method of a kind of robust of uncertain multi-agent system

Technical field

The present invention relates to multi-agent system and coordinate the control technology field, having roughly the same the time at random at one, the multi-agent system of external disturbance and model uncertainty proposes to have the distributed convergent control method of robust antijamming capability.

Background technology

Multi-agent system is a research focus in current system control field in engineering, biology and the socic-economic field background that has a wide range of applications, and is specifically related to subjects such as control, physics, mathematics, computing machine, biology.From the control angle, the researchist mainly pays close attention to and how to cooperate alternately and mutually based on the local message between the intelligent individuality, and the macroscopic view of common realization expectation is emerged in large numbers behavior, i.e. so-called multi-agent system distributed coordination control.Wherein, convergent control is the most most typical one of study a question, it is the basis of complicated group behavior such as realize Flocking, form into columns, troop, all have wide application background in every field such as industry, space flight, military affairs, controlling as the coordination of multi-robot system, the scheduling of robotization highway, multi-satellite system, wireless sensor network all is its typical application example.

In recent years, lot of domestic and foreign scholar has carried out big quantity research and has obtained a series of progress the convergent control problem of multiple agent, but in existing document, for discussing conveniently, supposes that generally the dynamic model of intelligent body is determined known and linear permanent fully.Yet in practical engineering application, most of controlled devices are not desirable linear stational systems, and have the distributivity of non-linear and systematic parameter to a certain extent.Ignoring secondary cause, keeping and to cause inevitably between the system model that can obtain and practical object, having error when linear principal part carries out system modelling, be i.e. model uncertainty.Therefore, the convergent control problem of multi-agent system of considering to contain model uncertainty is very real and necessary.On the other hand, there are various unpredictable external disturbance usually in the residing actual environment of intelligent body.In actual traveling process, can be subjected to the external disturbance that causes because of pavement roughness, with factors such as barrier collision, friction force inevitably such as: robot.Therefore, the disturbed problem of necessary taking into account system and interference inhibition problem.If above-mentioned these problems do not solve, just can not realize real application and the popularization of multi-agent system theory, Here it is motivation of the present invention and starting point.

In fact, the coupling that is interweaved of the distributed information framework of the uncertainty that various factors causes and system has brought difficulty to the analysis and synthesis of multi-agent system under the uncertain environment.As everyone knows, for traditional single controlled device, when existing uncertain and external interference, model can utilize ripe robust H _∞Theoretical CONTROLLER DESIGN makes corresponding closed-loop system have the robust interference free performance of expectation.Given this, with traditional robust H _∞It will be a feasible scheme that theory and method are applied in the advolution control of uncertain multi-agent system.Yet, consider between the complicacy, distributivity, intelligent individuality of multi-agent system the information coupling and consider the singularity of convergent problem, how to use existing robust H _∞Control method becomes the key of dealing with problems.

The present invention proposes to utilize robust H _∞Method solves the convergent control problem of the multi-agent system that contains model uncertainty and external disturbance, has provided the distributed convergent controller design method with robust antijamming capability.

Summary of the invention

The objective of the invention is to containing the general linear multi-agent system of model uncertainty and external disturbance, propose to make multi-agent system can realize having expectation H based on the convergent control method of the robust of local state information _∞The state advolution of interference attenuation index.

As shown in Figure 1, technical solution of the present invention realizes as follows:

1. advolution control is converted into robust H _∞Control problem;

2. design the convergent controller of distributed feedback of status;

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

4. find the solution the feedback matrix undetermined in the convergent controller.

The present invention has following technical characterictics:

(1) passes through appropriate controlled output function of definition in the step 1, the advolution of multi-agent system is controlled be converted into a robust H ∞ control problem.

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

(3) utilize model transferring that the convergent track of the non-zero of closed loop multi-agent system is changed into the zero condition equilibrium point of a price reduction of equal value system in the step 3, and then use existing robust H _∞Theory is carried out performance evaluation to this reduced order system, obtains having the convergent condition of robust of expectation antijamming capability.

(4) provide solving condition and the computing formula of feedback matrix in the convergent controller in the step 4 with the LMI form, can use Matlab LMI tool box to carry out Matrix Solving easily.

The present invention's advantage compared with prior art is:

(1) the present invention has considered in the practical application, and dynamically there is the convergent control problem of robust under the situation in intelligent system not modeling simultaneously with external disturbance at random.Enrich the research range of convergent control, widened its engineering application category.

(2) the convergent controller of robust proposed by the invention not only designs and finds the solution simply, and can reach predefined control accuracy and effect, has very strong practicality.

Description of drawings

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

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

Fig. 3 is the information interaction topology of emulation multi-agent system among the present invention;

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

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

Embodiment

Known:

● by the undirected variable topological multi-agent system that n intelligent body formed, the dynamic model of i intelligent body is:

${\stackrel{\&CenterDot;}{x}}_i\left(t\right)={\mathrm{Ax}}_i\left(t\right)+{\mathrm{<figure-callout\; id="1"\; label="B"\; filenames="2013101595429100003DEST\_PATH\_IMAGE001.png,HDA00003138532300021.png"\; state="\{\{state\}\}">B</figure-callout>}}_1{\mathrm{\&omega;}}_i\left(t\right)+B_2{u}_i\left(t\right),i=1,2,\&CenterDot;\&CenterDot;\&CenterDot;,n---\left(1\right)$

Wherein,

With

Be respectively i intelligent body at the t internal state in the moment, external disturbance and the input signal of finite energy.The uncertain system matrix is as follows:

A=A ₀+ΔA(t),B ₁=B ₁₀+ΔB ₁(t),B ₂=B ₂₀+ΔB ₂(t)

In the formula, A ₀, B ₁₀, B ₂₀Be permanent nominal model matrix, and (A ₀, B ₂₀) can stablize; The uncertain part Δ A (t) of system matrix, Δ B ₁(t), Δ B ₂(t) Satisfying Matching Conditions [Δ A (t) Δ B ₁(t) Δ B ₂(t)]=E Σ (t) [F ₁F ₂F ₃], wherein

● given H _∞Interference free performance index γ〉0.

Target of the present invention is: design has the distributed convergent controller of robust antijamming capability for uncertain multi-agent system (1), makes the closed loop multi-agent system can realize having H _∞The state advolution of interference attenuation index γ.With reference to Fig. 1, specific implementation process of the present invention is as follows:

Step 1: problem transforms

Define controlled output function

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

The advolution control of system (1) is converted into robust H _∞Control problem:

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

In the formula, x (t), ω (t), u (t), z (t) are respectively in the system all intelligent bodies to dependent variable x _i(t), ω _i(t), u _i(t), z _i(t) (i=1,2 ..., the n) column vector of Gou Chenging; Matrix L _cBe a symmetric 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 are realized the state advolution, therefore control target and become: CONTROLLER DESIGN makes that the convergent track of state is asymptotic stable, and from external disturbance ω (t) to the controlled H that exports z (t) transfer function matrix _∞Norm is less than disturbing sorrow to subtract index γ, namely || T _{Z ω}(s) || _∞＜γ.

Step 2: controller design

At above-mentioned robust H _∞Control problem design distributed director:

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

N wherein _i(t) be t neighbours' set of intelligent body i constantly; a _Ij(t) be information interaction figure G _{σ (t)}The limit on weights (σ (t) is the communication topology switching signal of the normal value of segmentation), a _Ij(t) ≠ 0 and if only if, and intelligent body i can observe or receive the status information of intelligent body j;

Be feedback matrix undetermined.Note L _{σ (t)}=diag{d _{1 σ (t),}, d _{N σ (t)}}-Λ _{σ (t)}Be non-directed graph G _{σ (t)}The Laplacian matrix, all handover networks the minimum and maximum nonzero eigenvalue of corresponding Laplacian matrix remember conduct respectively

With

Wherein

$A_{\mathrm{\&sigma;}\left(t\right)}={\left[a_{\mathrm{ij}}\left(t\right)\right]}_{i,j=1}^n.$

Step 3: convergent condition analysis

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

The zero condition equilibrium point, and satisfy

And then obtain the design criteria of following convergent condition and controller feedback matrix:

If exist feedback matrix K to make that the zero point of reduced order system (2) is asymptotic stable, and satisfy

So uncertain multi-agent system (1) can realize having H _∞The state advolution of interference attenuation index γ.

Matrix L in the formula (2) _{1 σ (t)}By generating with down conversion: necessarily there is orthogonal matrix

Make

$U^T{L}_{c}U=\left[\begin{array}{cc}{I}_{n-1}& 0\\ 0& 0\end{array}\right],$ $U^T{L}_{\mathrm{\&sigma;}\left(t\right)}U=\left[\begin{array}{cc}{\mathrm{<figure-callout\; id="1"\; label="L"\; filenames="2013101595429100003DEST\_PATH\_IMAGE001.png,HDA00003138532300021.png"\; state="\{\{state\}\}">L</figure-callout>}}_{1\mathrm{\&sigma;}\left(t\right)}& 0\\ 0& 0\end{array}\right]$

Set up simultaneously.

Step 4: feedback matrix is found the solution

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

● for scalar ce〉0, if there is positive definite matrix

And matrix

Make LMI

$\left[\begin{array}{cccc}{\mathrm{PA}}_0^T+A_{0}P{+\mathrm{\&lambda;Q}}^T{B}_{20}^T+{\mathrm{\&lambda;B}}_{20}Q+{\mathrm{\&alpha;}}^2{\mathrm{EE}}^T& B_{10}& \mathrm{<figure-callout\; id="1"\; label="P"\; filenames="2013101595429100003DEST\_PATH\_IMAGE001.png,HDA00003138532300021.png"\; state="\{\{state\}\}">P</figure-callout>}& \frac{1}{\mathrm{\&alpha;}}{(F_{1}P+{\mathrm{\&lambda;F}}_{3}Q)}^T\\ B_{10}^T& {-\mathrm{\&gamma;}}^{-2}\mathrm{<figure-callout\; id="0"\; label="I"\; filenames="2013101595429100003DEST\_PATH\_IMAGE001.png,2013101595429100003DEST\_PATH\_IMAGE002.png"\; state="\{\{state\}\}">I</figure-callout>}& 0& \frac{1}{\mathrm{\&alpha;}}{F_2}^T\\ P& 0& -\mathrm{<figure-callout\; id="0"\; label="I"\; filenames="2013101595429100003DEST\_PATH\_IMAGE001.png,2013101595429100003DEST\_PATH\_IMAGE002.png"\; state="\{\{state\}\}">I</figure-callout>}& 0\\ \frac{1}{\mathrm{\&alpha;}}(F_{1}P+{\mathrm{\&lambda;F}}_{3}Q)& \frac{1}{\mathrm{\&alpha;}}{\mathrm{<figure-callout\; id="2"\; label="F"\; filenames="2013101595429100003DEST\_PATH\_IMAGE001.png,HDA00003138532300021.png"\; state="\{\{state\}\}">F</figure-callout>}}_2& 0& -I\end{array}\right]<0$

For

Set up simultaneously, the solution formula of K is K=QP so ^-1

Especially, for the multi-agent system that does not have the system model indeterminate (be system matrix A in the model (1), B ₁, B ₂Permanent and known), can calculate feedback matrix as follows:

If ● there is positive definite matrix And matrix

Make 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 simultaneously, the solution formula of K is K=QP so ^-1

Effect of the present invention can further specify by following emulation:

The emulation content: the uncertain multi-agent system (1) that consideration is made up of four intelligent bodies, its system matrix is

$A=\left[\begin{array}{cc}0& -1\\ 2& 1\end{array}\right],$ ${\mathrm{<figure-callout\; id="1"\; label="B"\; filenames="2013101595429100003DEST\_PATH\_IMAGE001.png,HDA00003138532300021.png"\; state="\{\{state\}\}">B</figure-callout>}}_1=\left[\begin{array}{c}0\\ 1\end{array}\right],$ ${\mathrm{<figure-callout\; id="2"\; label="B"\; filenames="2013101595429100003DEST\_PATH\_IMAGE001.png,HDA00003138532300021.png"\; state="\{\{state\}\}">B</figure-callout>}}_2=\left[\begin{array}{cc}1& 0\\ 0& 2\end{array}\right],$ $E=\left[\begin{array}{cc}0& 0\\ 0.8& 0.8\end{array}\right],$

$\mathrm{\&Sigma;}\left(t\right)=\left[\begin{array}{cc}\sin \left(10t\right)& 0\\ 0& 0\end{array}\right],$ ${\mathrm{<figure-callout\; id="1"\; label="F"\; filenames="2013101595429100003DEST\_PATH\_IMAGE001.png,HDA00003138532300021.png"\; state="\{\{state\}\}">F</figure-callout>}}_1=\left[\begin{array}{cc}1.2& 0\\ 0& 1.2\end{array}\right],$ ${\mathrm{<figure-callout\; id="2"\; label="F"\; filenames="2013101595429100003DEST\_PATH\_IMAGE001.png,HDA00003138532300021.png"\; state="\{\{state\}\}">F</figure-callout>}}_2=\left[\begin{array}{c}0.5\\ 0.5\end{array}\right],$ ${\mathrm{<figure-callout\; id="3"\; label="F"\; filenames="HDA00003138532300021.png"\; state="\{\{state\}\}">F</figure-callout>}}_3=\left[\begin{array}{cc}0.8& 0\\ 0& 0.8\end{array}\right];$

The interference attenuation index is taken as γ=1.For reflect the interference free performance of system visual and clearly, be t ∈ [0,10] action time that case of external is disturbed, and the formula of embodying is ω (t)=[ω ₁(t) ω ₂(t) ω ₃(t) ω ₄(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.Suppose that the information interaction figure between intelligent body is gathering { G ₁, G ₂, G ₃In switch at random, as shown in Figure 3.

Fig. 4 has described the state trajectory of four uncertain intelligent bodies under the convergent controller action of robust, wherein respectively corresponding first, second state component of (a) and (b); Fig. 5 has described the energy relationship between controlled output z (t) and the external disturbance ω (t).By Figure 4 and 5 as can be seen, the distributed convergent control method that proposes among the present invention has antijamming capability γ=1 of robustness and expectation.

Claims (5)

1. the convergent control method of the robust of a uncertain multi-agent system is characterized in that may further comprise the steps:

(1) advolution control is converted into robust H _∞Control problem;

(2) the convergent controller of the distributed feedback of status of design;

(3) the convergent condition of the robust of analysis closed loop multi-agent system;

(4) find the solution feedback matrix undetermined in the convergent controller.

2. according to the method described in claim 1 step (1), it is characterized in that, by defining an appropriate controlled output function, the advolution control of multi-agent system is converted into a robust H _∞Control problem.

3. according to the method described in claim 1 step (2), it is characterized in that, designed is one based on the static state feedback controller of local interactive information between intelligent body, and its feedback matrix is undetermined.

4. according to the method described in claim 1 step (3), it is characterized in that, utilize model transferring that the convergent track of the non-zero of closed loop multi-agent system is changed into the zero balancing point of a price reduction of equal value system, and then use existing robust H _∞Theory is carried out performance evaluation to this reduced order system, obtains having the convergent condition of robust of expectation antijamming capability.

5. according to the method described in claim 1 step (4), it is characterized in that, provide solving condition and the computing formula of feedback matrix in the convergent controller with the form of LMI, thereby can use Matlab LMI tool box to carry out Matrix Solving easily.

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