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How to Cite | Author Information | Publication History
International Journal of Computer & Software Engineering Volume 3 (2018), Article ID 3:IJCSE-133, 9 pages
https://doi.org/10.15344/2456-4451/2018/133
Original Article
Distributed H2/H∞ Consensus Control for Multi-agent Systems with Directed Graph and Actuator Uncertainty

Yichao Ao and Yingmin Jia*

School of Automation Sciences & Electrical Engineering, BeiHang University, XueYuan Road No.37, HaiDian District, BeiJing, China
Prof. Yingmin Jia, School of Automation Sciences & Electrical Engineering, BeiHang University, XueYuan Road No.37, HaiDian District, BeiJing, China, Tel: +010-82-338683; E-mail: ymjia@buaa.edu.cn
01 March 2018; 18 April 2018; 20 April 2018
Ao Y, Jia Y (2018) Distributed H2/H∞ Consensus Control for Multi-agent Systems with Directed Graph and Actuator Uncertainty. Int J Comput Softw Eng 3: 133. doi: https://doi.org/10.15344/2456-4451/2018/133
© 2018 Ao et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Abstract

In this work, single-integrator multi-agent directed networks with energy bounded disturbances and Guassian white noises are considered in term of H2 and H∞ performance. Meanwhile actuator uncertainty is also the concern of H∞ performance. Distributed control law of undetermined parameters is proposed. By using H2/H∞ theory and linear algebra, the problem is transformed into a non-convex optimization problem on diagonal matrices. Numerical iterative algorithms with consideration on both feasibility and conservatism are presented.


1. Introduction

In last two decades, consensus control of multi-agent systems has been researched deeply and extensively, for its broad range of applications in several areas, such as: robotic teams cooperation, sensor networks, air vehicles formation flying, social networks and so on [20-24]. And it has attracted the attention from several fields including but not limited to engineering, biology, control theory and computer science.

For consensus problems, abundant achievements for first-order agents have been acquired by previous researchers [1-4]. The study of the multi-agent systems was from undirected graph to directed graph, from fixed communication topology to switched topology, and with time-delay or not. However, under real circumstances, the transmission errors, communication obstacles, actuator bias and other disturbances do exist. From now on, several researchers had investigated the disturbance rejection performance of multi-agent systems in different perspectives. P. Lin [5,6] firstly introduced H∞ control theory to multi-agent systems. In his work, An orthogonal transformation was utilized to guarantee the validity of H∞ control theory. And Y. Liu [7] investigated multi-agent systems of high-order integrator and general linear time-invariant dynamics by H∞ criterion. Consensus protocols involving a coefficient were raised by Z. Li and Y. Zhao [9,10], and the H∞ performance for both state feedback controlled systems and output feedback controlled systems was investigated by introducing H∞ performance region of the coefficient. It is worthy mentioning that the authors in [9,29] addressed H∞ and H2 consensus problems of linear multi-agent systems and proposed algorithms to satisfy the H∞ and H2 conditions separately. Y. Cui [8] employed L2 − L∞ theory to solve the peak bounded consensus problems of high-order multi-agent systems with external disturbances and parameter perturbations. In other perspectives, stochastic theory and probability limit theory were introduced to consensus problems of multi-agent systems [11,12], in both continuous-time and discretetime case. V. Gupta [13] and Y. Jia [15] considered the LQG problems of networks of dynamical agents. When the systems are disturbed by Gaussian noises, the designs were synthesized under H2 criteria for robust [25,26,31]. On another aspect of H2 control, the recent paper [30] studied guaranteed cost problem for multi-agent systems with actuator faults and uncertainty.

In last century, numerous results were obtained in H2/H∞ control. DS. Bernstein [14] proposed a LQG performance approach with an H∞ performance bound based on the solutions of three coupled Riccati equations. PP Khargonekar [16] simplified it to a convex suboptimization problem. D. Arzelier [15] gave a sufficient condition in the form of bilinear matrix inequalities (BMIs) to reduce the conservatism, and solved this non-convex problem by a numerical iterative algorithm. C. Scherer [27] considered the more universal case where the channels of the disturbance inputs and controlled outputs are different, when different performance indexes are concerned. However, H2 performance and H∞ performance have never been considered together in multi-agent systems.

In this paper, consensus problems of single-integrator multi-agent systems with fixed directed interaction topology will be studied. We address the consensus performance subject to external L2 disturbances and Guassian white noises by combining H∞ theory and H2 theory. Our objective is to search the linear feedback controller, based on relative states of its neighbors, to optimize the H2 performance of the multiagent systems, under two constraints. One is that the multi-agent systems reach asymptotically consensus with the absence of external signals, and the other is the constraint of H∞ performance against both energy bounded disturbances and actuator uncertainties. Firstly, we will present the algorithms to design the satisfactory controller, rather than proposing a theorem to verify the protocol is valid or not. Secondly, different from other work, the feedback gain matrices of the multi-agent systems are Laplacian matrices with a fixed graphical structure constraint, which has more complicated relationship between the entries. We transform it to an optimization problem in a set of diagonal matrix. To measure the consensus H2 performance, a novel output function is defined, which is linked to the LQR problem of linear-consensus. After a procedure of model transformation to guarantee the validity of H∞ theory, an original elimination lemma is derived. By using this lemma, the problem is simplified to a numerically solvable BMI constrained optimization problem on diagonal matrices. Finally, a previously proposed iterative method is used to solve the BMI optimizaiton problem.

The paper is organized as follows. In section 2, some basic frameworks of graph theory and H2/H∞ control are introduced. H2/H∞ consensus problem for first-order directed network is described in section 3. Section 4 gives the main results and the approaches to solving the H2/H∞ consensus problem. In section 5, numerical simulations are presented to verify our results. Finally, conclusions are drawn in section 6.

1.1 Notations

In this paper, In represents the identity matrix with n dimensions, and In represents n-dimension column vector [1,1,...,1]T . Denote sym(A) = A +AT , for a square matrix A. And diag{m1,m2,...,mn} is the diagonal matrix whose diagonal entries are given by m1,m2,...,mn. For a n-order digraph, Ln denotes the set of all n-dimensional Laplacian matrices of the graph. And for a digraph with m edges, Dm denotes the set of all m-dimensional positive definite diagonal matrices. L2[0,∞) represents the space of square integrable vector functions over [0,∞).

2. Preliminaries

2.1 Graph theory

Let G = (ν,ε,A) be a weighted directed graph of n orders, which consists of a set of nodes V= {v1,v2,...,vn}, a set of edges ε ⊆ V×V, And a weighted adjacency matrix A =[aij]. Node pair (vi,vj) Є ε implies that there is a formation flow from vj to vi, where vi and vj call the tail of the edge and the head of the edge, respectively. The neighbor of vi is defined by Ni={vj Є V: (vi, vj) Є ε}. And the elements of adjacency matrix A are nonnegative, in which aij>0 if vj Є Ni, and aij =0 otherwise. A sequence edges is called a directed path, if the tail of the latter edge is the same as the head of previous pair, such as ( v i 1 , v i 2 ),( v i 2 , v i 3 ),( v i 3 , v i 4 ),.... . A node is called vertex, if there is a directed path from every other nodes to it. And the graph which contains only one vertex and no circle is called spanning tree.

The Lapliacan matrix Σ of a graph is defined as L= [lij]m×n, where l ii = ∑ j=1 n a ij and lij =-aij if a≠j. Obviously, a Laplacian matrix will be a nonnegative definite matrix.

2.2 H2 and H∞ Theory

Consider the strictly proper system Tzw which yields to the following dynamics:

x ˙ =Ax+Bw Z=Cx

where A is a stable matrix.

Lemma 1

(Y. Jia [19]) The L2 norm of strict proper system (1) satisfies || T zw | | 2 2 =trace( B T L 0 B) , where Lo is the observability Gramian that satisfies

A T L 0 + L 0 A+ C T C=0

Lemma 2

(Y. Jia [19]) Consider the system (1), the necessary and sufficient condition for || T zw | | ∞ <ϒ is that the following Riccati equation has a positive definite solution

A T P+PA+ ϒ −2 PB B T P+ C T C<0

2.3 Linear algebra

Lemma 3

(P. Lin et al. [5]) For matrix

C= [ n−1 n − 1 n − 1 n ⋯ − 1 n n−1 n − 1 n ⋯ ⋮ ⋱ ⋱ ⋱ − 1 n ⋯ − 1 n n−1 n ] n×n

and an arbitrary n-order Laplacian matrix L, there exists an orthogonal matrix U=[ U 1 , U ¯ 1 ] , where U ¯ 1 = [1/ n ,...,1/ n ] T , such that U T CU=[ I n−1 0 0 0 ] and  U T LU=[ U 1 T L U 1 0 U ¯ 1 T L U 1 0 ] .

Lemma 4

(CE. De Souza et al. [28]) Assume that D and E are real matrices with compatible dimensions. For an arbitrary scalar ε, we can get

DE+ E T D T ≤ ∈ −1 D D T +∈ E T E

Lemma 5

(Schur complement) Given a symmetric matrix S Є Rn×n decomposed as S=[ S 11 S 12 S 21 S 22 ]

where S11 Є Rn×n, S12 Є R(n-r)×n and S22 Є R(n-r)×(n-r). Then S<0 if and only if S11<0 and S 22 − S 12 T S 11 −1 S 12 <0 , or equivalently S22 < 0 and S 11 − S 12 S 22 −1 S 12 T <0 .

3. Problem Formulation

In [26,32], multi-UAVs (unmanned aerial vehicles) and satellite formations were supposed to disturbed by zero-mean Gaussian white noise. However, mathematically saying, the statistical characteristics of stochastic process may vary according to the information we have. In other words, the zero mean condition can not be guaranteed all the time. So we consider the singleintegrator multi-agent systems with identical dynamics, which are driven by energy bounded disturbances and white Gaussian noises.

x ˙ i (t)= u i (t)+ w ∞i (t)+ b i w 0i (t)

where xi(t) is the state of the ith agent, w∞i(t) Є L2[0,∞) is the energy bounded disturbance, and w (t) is the standard white Gaussian noise.

We propose the following protocol based on the states of neighbors to achieve consensus. However, multi-agent systems are always in the face of calculation errors and variations of communication circumstances. Different from other consensus scheme, the communication perturbation is reflected on the multiplicative uncertainty of weight elements, so H∞ theory can be utilized.

u i (t)=− ∑ j∈ N i [ a ij (1+ ψ ij (t))][ x j (t)− x i (t)]

where aij is the parameter to be determined and ψij(t) is unknown multiplicative uncertainty of aij with

ψ ij (t)={ ≤ ψ ¯ ij if i≠j and  a ij ≠0 0 otherwise,

In our work, the communication topology of the network is fixed and known. For example, it is like the layout of submarine optical cables and sensor networks, or some social relationship, in which the interaction topologies will not be changed over a period of time. Our objective is to find an optimal protocol to make the closed-loop system possess a desired level of disturbance rejection. For the H∞ performance, on the one hand, the controlled output z∞i(t) is present to value the difference between its state and the average state of all agents.

z ∞i (t)= x i (t)− 1 n ∑ j=1 n x j (t)

On the other hand, we expect that there is a tradeoff between the better system performance and the lower energy consumption. To consider the H2 performance of the multi-agent system, a novel controlled output is given.

z 0i (t)= ∑ j=1 n q ij ( x i (t)− x j (t))+ r i u i (t)

where qij > 0 corresponding to an undirected complete graph and ri > 0 are H2 performance coefficients. On the one hand, denoting u t (t)= R −1 1 n 1 n T x(t)+v(t)+|| T z o w 0 | | 2 2 = ∫ 0 ∞ [ x T (t) Q 2 x(t)+ v T (t) R 2 v(t)]dt , which is the interaction free cost function of LQR-based optimization problem of linear-consensus in [17,18], where Q=[qij]n×n and R= diag{r1,...,rn}. On the another hand, when ri = 0 and Q=C, z 0 i can be used to measure the static output variance of consensus under Guassian white noises. While || T z ∞ w ∞ | | ∞ is adopted to guarantee the energy amplification from the expectation of the noises to the expectation of consensus output.

By utilizing protocol (7), the closed-loop dynamics of whole multiagent system can be written as

x ˙ (t)=−(L+ΔL)x(t)+ w ∞ (t)+ B w 0 (t) z ∞ (t)= C x (t) z 0 (t)=(Q−RL)x(t)

where B = diag{b1, . . . , bn}, and C is defined in equation (4).

Problem 1

Given an H∞ level γ, determine a Laplacian matrix L of the fixed interconnection of the multi-agent system (6) with multiplicative weight uncertainty satisfying (8), such that

J= inf L∈ L n sup | ψ ij (t)≤ ψ ¯ ij | || T z 0 w 0 | | 2

under

  1. the closed-loop multi-agent system (11) achieves asymptotically consensus for arbitrary ψij(t) satisfied (8) with the absence of w∞(t) and w0(t) and the arbitrary initial states x(0).
  2. || T z ∞ w ∞ | | ∞ <ϒ i.e  ∫ 0 ∞ z ∞ T z ∞ − ϒ 2 w ∞ T w ∞ dt<0 for w0(t) = 0, initial state x(0) = 0 and arbitrary w∞ Є L2[0,∞) and ψij(t) satisfied (8).

Lemma 6

(W. Ren and RW. Beard [4]) The Laplacian matrix L of a directed graph G has a single zero eigenvalue (with eigenvector 1n) if and only if the graph G contains a spanning tree. Moreover, other eigenvalues of L are strictly positive in this case.

In order to satisfy the first condition of Problem 1, it is a necessary condition in our work that the directed network of the multi-agent system contains a spanning tree.

Lemma 7

(P. Lin et al. [5]) Consider a directed graph G. Let E = [eij] and F = [fij] be the 01-matrix. The rows and columns of E are indexed by the nodes and edges, while the columns and rows of F are indexed by the nodes and edges. eij = 1 if the vertex i is the tail of the edge j, eij = 0 otherwise. And fij = 1 if the vertex j is the head of the edge i, fij = 0 otherwise. Let weight matrix W = diag{w1,w2, . . . ,w|ε|}, where wp (p = 1, . . . , |ε|) is the weight of the pth edge of G and |ε| is the number of the edges. Then the relationship of the Laplacian matrix can be obtained L = EWD, where D = ET − F.

4. Main Results

In this section, some main results of this paper are presented.

4.1 Model Transformation

Because of the singularity of the system matrix −(L + ΔL), which leads to H2/H∞ theory invalid, some model transformations need to be conducted.

x ^ (t)=x(t)− 1 n n ∑ i=1 n ∫ 0 t [ w ∞ (s)+ B w 0 (s)] , δ(t)= U 1 T x ^ (t) and  δ ¯ (t )= U ¯ 1 T x ^ (t)

Remark 1

Although w∞(t) is a deterministic function with bounded norm while w0(t) is a random signal, sharing a common integral operator ∫ 0 ∞ will not create any confusions.

Then we can obtain that

[ δ ˙ (t) δ ¯ ˙ (t) ]=−[ U 1 T (L+ΔL) U 1 0 U 2 T (L+ΔL) U 1 0 ][ δ(t) δ ¯ (t) ]+[ U 1 T w ∞ 0 ]+[ U 1 T w 0 0 ] z ∞ =[ U 1 0 ][ δ(t) δ ¯ ] z 0 =[Q U 1 −RL U 1 0][ δ(t) δ ¯ (t) ]

Consider the H2 and H∞ performance || T z 0 w 0 | | 2  and || T z ∞ w ∞ | | ∞ of the multi-agent system, by Lemma 7, we study the reduced dimension system

T zw f (s)=[ − U 1 T EWD U 1 − U 1 T EΔW(t)D U 1 U 1 T U 1 T B U 1 0 0 Q U 1 −REWD U 1 0 0 ]           =[ A f +Δ A f (t) B ∞f B 0f C ∞f 0 0 C of 0 0 ]

Lemma 8

The following conditions are equivalent:

  1. The protocol (7) can solve the Problem 1.
  2. For a multi-agent system with digraph which contains a spanning tree, there exist symmetric positive definite matrices X ˜ 2 , X ∞ and a |ε|-dimensional diagonal positive definite matrix W solving the following non-convex optimization problem:

Problem 2

J= min X ∞ , X ˜ 2  and L∈ L n sup ψ i (t)  trace( B of T X ˜ 2 B 0f )

under

A f T X ˜ 2 + X ˜ 2 A f + C f T C f =0

[ ( A f +Δ A f ) T X ∞ + X ∞ ( A f +Δ A f )+ C f T C f X ∞ B ∞f B ∞f T X ∞ − ϒ 2 I ]<0

And the cost function J is the Laplacian matrix L.

Proof Equation (16) can be obtained readily by the definition of L2 norm in Lemma 1. And the H∞ constraint can be transformed into inequlity (17) referred to [19].

4.2 Conditions for H2/H∞ Consensus

Lemma 9

(Elimination lemma) P=[ P ¯ ,−I],H are given matrices with compatible dimensions, and H is a symmetric matrix. Denote N P = [I, P ¯ T ] T .

Then, there exists a matrix X= [ X 1 T , X 2 ] T where X2 is a positive definite diagonal matrix, such that

H+ P T X T +XP<0

if and only if

N P T H N P <0

Proof Necessity can be obtained noticeably. Next, we use a constructive approach to prove the sufficiency. A full column rank matrix V ¯ = [0,I] T is introduced to expand NP into a group of basis of the entire space, i.e. V=[ N P , V ¯ ] is an invertible square matrix. So inequality (18) holds if and only if there exists X such that

V T =(H+ P T X T +XP)V<0

holds. According to the block of V , we assume that

V T HV=[ H 11 H 12 H 12 T H 22 ]

and obtain that

V T XPV=[ 0− X 1 − P ¯ T X 2 0 − X 2 ]

Substituting equations (21) and (22) into inequation (20) so that it can be rewritten as

[ H 11 H 12 − X 1 − P ¯ T X 2 H 12 T − X 1 T − X 2 P ¯ H 22 − X 2 − X 2 ]<0

By Schur complement lemma, inequation (23) holds if and only if H11 < 0 and

H 22 − X 2 − X 2 −( H 12 T − X 1 T − X 2 P ¯ ) H 11 −1 ( H 12 − X 1 − P ¯ T X 2 )<0

H11 < 0 is obtained by condition (19) and it is obvious that for matrix X 1 =− P ¯ T X 2 , there exists a positive diagonal matrix X2 such that inequality (24) holds. So the sufficiency is proofed.

To simplify the conditions of optimization Problem 2, the following notations are stated:

N( X 2 )=[ U 1 T Q 2 U 1 − X 2 U 1 T E− U 1 T QRE − E T U 1 X 2 − E T RQ U 1 E T R 2 E ] L( X ∞ )=[ μ U 1 T E 2 T E 2 U 1 +I X ∞ 0 − X ∞ U 1 T E 1 X ∞ − ϒ 2 I 0 0 0 0 −μ ψ −2 0 − E 1 T U 1 X ∞ 0 0 0 ]

Theorem 1

Assume the directed graph of multi-agent system (6) contains a spanning tree, distribute H2/H∞ synthesis problem can be transformed to the following non-convex constraint optimization problem:

Problem 3

If there exist symmetric positive definite matrices X∞, X2, matrices K21, K∞1, K∞2, K∞3, diagonal matrices Y , Z, F2, F∞ and a positive scalar μ, such that

J ^ = min μ, X 2 , X ∞ , K 21, K ∞1, K ∞2, K ∞3  and Y,Z, F 2 , F ∞ ∈ D |∈|  trace( B T U 1 X 2 U 1 T B)

under

Z− F ∞ F 2 −1 Y=0

N( X 2 )+sym( [ K 2 T −I ][YD U 1   F 2 ] )<0 L( X ∞ )+sym( [ K ∞1 T K ∞2 T K ∞3 T −I ][ZD U 1 −Z F ∞ ] )<0

It provides a sub-optimal solution to the H2/H∞ consensus problem and the upper bound J ^ of the L2 norm of the multi-agent system, by solving Problem 3.

At the optimum, the sub-optimal weight matrix can be given by

W * =− F ∞ *−1 Z * =− F 2 *−1 Y *

Proof By Lemma 4, the following inequality can be obtained for an arbitrary scalar μ > 0

Δ A f (t) T X ∞ + X ∞ Δ A f (t)≤μ U 1 T D T D U 1 + 1 μ X ∞ U 1 T EW ψ ¯ 2 W E T U 1 X ∞

By Lemma 5, the H∞ condition || T z ∞ w ∞ | | ∞ <ϒ holds, if there exists a positive definite matrix X∞ such that

[ − A f T X ∞ − X ∞ A f + I n−1 +μ U 1 T D T D U 1 X ∞ X ∞ U 1 T DW X ∞ −ϒI 0 W E T U 1 X ∞ 0 −μ ψ −2 ]<0

Substitute L(X∞) and N(X2) into the constraint (16) and (17). Then the H2/H∞ synthesis problem can be rewritten as

Problem 4

min μ, X 2 , X ∞ , and W∈ D |∈|  trace( B T U 1 X 2 U 1 T B)

under

[I U 1 T D T W]N( X 2 )[ I WD U 1 ]<0

[ I 0 0 U 1 T D T W 0 I 0 0 0 0 I −W ]L( X ∞ )[ I 0 0 0 I 0 0 0 I WD U 1 0 −W ]<0

It indicates that the feasible region of W of Problem 4 is contained in the feasible region of Problem 2 by inequality (29). And we can readily get that X 2 ≥ X ˜ 2 where X2 is governed by inequality (31) while X ˜ 2 yields to equation (16). So it gives a upper bound of the L2 norm of the multi-agent system by solving Problem 4.

Using Lemma 9 to slack some slack variables, we can obtain that Problem 4 equals to

min μ, X 2 , X ∞ , F 21 , F ∞1 , F ∞2 , F ∞3 , and W, F 22 , F ∞4 ∈ D |∈|  trace( B T U 1 X 2 U 1 T B)

under

N( X 2 )+sym( [ F 21 T F 22 ][WD U 1 −I] )<0 L( X ∞ )+sym( [ F ∞1 T F ∞2 T F ∞3 T F ∞4 ][WD U 1  0−W−I] )<0

where F22, F∞4 Є D|ε| are positive definite diagonal matrices and other slack variables are real matrix with compatible dimensions. State that

Y=− F 22 W Z=− F ∞4 W  K 2 =− F 22 −T F 21 K ∞1 =− F ∞4 −T F ∞1   K ∞2 =− F ∞4 −T F ∞2 K ∞3 =− F ∞4 −T F ∞3  and  F 2 = F 22   F ∞ = F ∞4

Since matrices Y , Z, F2 and F∞ are diagonal matrices with equality relationship (26), Problem 3 is obtained and Theorem 1 is proofed.

Remark 2

However, because of the nonlinear equational constraint (26), it is hard to utilize a numerical procedure to solve the constrained optimization Problem 3. We relax it by assuming that

F ∞ = F 2 =F and Y=Z

Then we solve the following BMI constrained optimization problem

Problem 5

If there exist symmetric positive definite matrices X∞, X2, matrices K2, K∞1, K∞2, K∞3, diagonal matrices Z, F and a positive scalar μ solution of the optimization problem:

min μ, X 2 , X ∞ , K 2 , K ∞1 , K ∞2 , K ∞3 , and Z,F∈ D |∈|  trace( B T U 1 X 2 U 1 T B)

under

N( X 2 )+sym( [ K 2 T −I ][ZD U 1 F] )<0 L( X ∞ )+sym( [ K ∞1 T K ∞2 T K ∞3 T −I ][ZD U 1  0−Z F] )<0

to get an upper bound on the L2 norm || T z 0 w 0 | | 2 . And optimal weight matrix W* = −F*-1Z*.

A coordinate-descent based iterative procedure similar to [15] is used to solve the BML optimal problem.

4.3 Numerical Algorithm Description

Algorithm 1

  1. Step 1: For a directed graph G which contains a spanning tree, choose the initial matrices K2 = K∞1 = W0DU1 and K∞2 = K∞3 = 0, where W0 is an appropriate matrix with all eigenvalues positive.
  2. Step 2k: Pick the optimal solution matrices K2, K∞1, K∞2 corresponding to the minimum situation in the last step. Then solve the optimization linear matrix inequality (LMI) problem β 2k 2 = min μ, X 2 , X ∞ , and Z,F∈ D |∈|  trace( B T U 1 X 2 U 1 T B) under inequality constraint (36)
  3. Step 2k+1: Pick the diagonal optimal solution matrices Z, F corresponding to the minimum situation in the last step. Then solve the convex optimization LMI problem β 2k+1 2 = min μ, X 2 , X ∞ ,  K 2 , K ∞1, K ∞2 , K ∞3  trace( B T U 1 X 2 U 1 T B) under inequality constraint (36)
  4. Final step: If β2k −β2k+1 < ε, then stop. The sub-optimal weight matrix of the digraph of the first-order multi-agent system is W β * =− F −1 Z obtained, and the upper bound on the L2 norm || T z 0 w 0 | | 2 is obtained as β 2k+1 1/2 ; else go on to step 2k+2.

Remark 3

From the description of the algorithm, it can be guaranteed that the target function will not increase at each iteration. It still will be a challenge task to proof the global convergency mathematically. However it has turned out effective in simulation and practice by choosing an appropriate initial value in the step 1 of Algorithm 1.

5. Numerical simulations

Consider a first-order multi-agent system with 5 agents. Their communication connection relationships are as Figure 1. Then we design a Laplacian matrix to solve the H2/H∞ sub-optimization problem. In other words, the weight of each edge need to be determined. By using Remark 2 and Algorithm 1, setting γ = 1, ε = 0.01 and ψ ¯ ij =0.05 then we can obtained the H2/H∞ sup-optimal solution W Є D7 and L Є L5, where W = diag{3.1043, 1.0915, 0.6263, 0.1846, 0.5366, 1.2051, 0.1403} and the parameters of protocol (7) and the upper bound β 2k+1 1/2 =2.4685 are also reaped.

figure 1
Figure 1: Topology 1 of the directed network of agents.

L=[ 0.6769 0 0 −0.5366 −0.1403 −3.1043 3.1043 0 0 0 0 −1.0915 1.0915 0 0 0 −0.6263 −0.1846 0.8108 0 −1.2051 0 0 0 1.2051 ]

It is revealed from Figure 2 that the 5 agents can achieve consensus with good performance against Guassian white noises. And H∞ performance is illustrated where w∞i are sinusoidal disturbances in Figure 3. Figure 4 reveals the trajectory of the sum of the impulse response energy of z0, which is equivalent to the L2 norm of the closedloop system || T z 0 ω 0 | | 2 , and it is found approaching the upper bound β 2k+1 1/2 we obtained hereinbefore.

figure 2
Figure 2: The state trajectories of the agents with Gaussian noises.
figure 3
Figure 3: Energy trajectories of the energy bounded disturbance w∞ and the output z∞.
figure 4
Figure 4: The sum of the impulse response energy of z0 by H2/H∞ approach.

Compared with [5], our work goes a step further, not only because of the design of the Laplacian matrix we proposed, but also the consideration of H2 performance. Using H∞ approach in [5], assuming that the weight of each edge in graph 1 is equivalent, we can verify that the below equation; minimizes the L2 norm of system (6) with || T z ∞ w ∞ | | ∞ <1 . And in this case, the sum of the impulse response energy of z0 is shown in Figure 5.

figure 5
Figure 5: The sum of the impulse response energy of z0 by H∞ approach in [5].

L=[ 2.762 0 0 −1.381 −1.381 −1.381 1.381 0 0 0 0 −1.381 −1.381 0 0 0 −1.381 −1.381 2.762 0 −1.381 0 0 0 1.381 ]

Two extra examples are provided for comparison. The topologies of multiagent systems are shown in Figure 6 and Figure 7. H2 performance results achieved by H2/H∞ approach and H∞ approach are illustrated by Table. 1, which indicates that H2/H∞ approach can achieve a better robust H2 performance compared with H∞ approach with || T z ∞ w ∞ | | ∞ <1 .

figure 6
Figure 6: Topology 2 of the directed network of agents.
figure 7
Figure 7: Topology 3 of the directed network of agents.
table 1
Table 1

6. Conclusion

This paper has studied the disturbance rejection problem of first-order multiagent system consensus with directed interaction topology. L2 norm and H∞ norm were utilized to measure the system consensus performance. Under the premise of satisfying given H∞ performance, a new type of measure output contained more generality was proposed, and a relaxation approach with lower conservatism was presented to search the sub-optimal solution of the H2/H∞. Notably, a novel elimination lemma was obtained. In future work, high-order multi-agent systems with parameter uncertain and time-delay will be considered. For these cases, measurement errors caused by noises and higher conservatism caused by the larger number of the agents will also be the challenges to overcome.

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

This work was supported by the NSFC (61520106010, 61327807, 61521091, 61134005) and the National Basic Research Program of China (973 Program: 2012CB821200, 2012CB821201).


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