Robust L₂-L_∞ consensus control of second-order multi-agent systems

CUI Yan(崔艳), JIA Ying-min(贾英民)

(The Seventh Research Division, Beihang University (BUAA), Beijing 100191, China)

Abstract: Robust L₂-L_∞ consensus control is studied for second-order multi-agent systems with external disturbances and parameter uncertainties. By defining an appropriate controlled output, consensus problem of the systems is transformed into a normal L₂-L_∞ control problem, and then a distributed state feedback protocol with time-delay is proposed. Sufficient conditions are established for the convergence to consensus of the network on fixed or switching topology. Numerical simulations are provided to demonstrate the effectiveness of theoretical results.

Key words: consensus; external disturbance; parameter uncertainty; L₂-L_∞ control; multi-agent systems

CLC number: TP273             Document code: A            Article ID: 1672-7207(2011)S1-0365-05

1 Introduction

Coordination control of a group of agents has received compelling attentions within the control community. Its broad applications involve satellite clusters^[1], unmanned air vehicles^[2], formation control^[3], distributed sensor network^[4], rendezvous in space^[5] and so forth. Generally, consensus for the multi-agent systems means that all the agents could reach an agreement on certain quantities of interest by employing the appropriate control protocols based on local information, which is one of the most important issues in the coordination control.

In the past decades, consensus problems for the multi-agent systems have been studied by many researchers^[6-13]. However, the multi-agent systems are often subjected to external disturbances and model uncertainties in practical applications, such as actuator bias, measurement or calculation errors and the variation of communication links, which might usually destroy the convergence performance of the systems. For such cases, several researches have been done. For example, Ren^[9] considered the actuator saturation and the lack of relative velocity measurements and proposed several consensus algorithms for the second-order multi-agent systems in the absence or presence of a group reference velocity, respectively. Lin et al^[10] introduced H_∞ method into the consensus problem of multi-agent systems with external disturbances and model uncertainties for directed networks with zero and nonzero time-delay on fixed and switching topologies. Liu et al^[13] designed the H_∞ controller and obtained the consensus conditions with desired performance against the disturbances for the multi-agent systems with first-order, second-order, high-order and linear coupling dynamics. References [10] and [13] both employed the H_∞ control method to attenuate the external disturbance signal. Whereas, the peak value of the controlled output in many projects is required to be within a certain range, when the influence of external disturbances and time-delays on the performance of the system is taken into account. Here, we try to solve these problems by L₂-L_∞ control method. L₂-L_∞ control not only resembles H_∞ control that can attenuate the external disturbance signal but also minimizes the controlled output value for a multi-agent system.

2 Preliminaries and problem description

2.1 Graph theory

Let G=(V, ε, A) be a directed graph of order n with the set of nodes V={s₁, …, s_n}, set of edges εV×V, and a weighted adjacency matrix A=[a_ij] with a_ij≥0 (i≠j). The node indexes belong to a finite index set I={1, 2, …, n}. In particular, it is assumed that a_ii=0 for I. In graph G, node s_i represents the i-th agent, and the set of neighbors of node s_i is denoted by N_i={s_jV:(s_i, s_j)ε}.

The out-degree is defined as  The

Laplacian with the directed graph is defined as L=Δ-A, where Δ=[Δ_ij] is a diagonal matrix with Δ_ii=d_o(s_i). A directed path is a sequence of ordered edges of the form

 …, where  If there is a

directed path from every node to every other node, the graph is said to be strongly connected. Moreover, if there exists a node such that there is a directed path from every other node to this node, the directed graph is said to have spanning trees.

Lemma 1: If the graph G has a spanning tree, then its Laplacian L satisfies:

1) Zero is one eigenvalue of L, and 1_n is the corresponding eigenvector, i.e., L(1_n)=0.

2) The rest n-1 eigenvalues of L are all positive real-parts.

2.2 L₂-L_∞ control

Consider a multi-agent system consisting of n identical agents subjected to external disturbances. Suppose the i-th agent (iI) has the dynamics:

            (1)

where  and v_i(t)Rⁿ are the position and velocity vectors of the i-th agent, ω_i(t)L₂[0, ∞) is the external disturbance signal and L₂[0, ∞) denotes the space of square integrable vector functions over [0, ∞), and u_i(t) is control input. Rⁿ denotes n-dimensional Euclidean space.

According to the special control objective, we define an appropriate controlled output    to analyze the effect of system performance with external disturbances and parameter uncertainties as follows

         (2)

and the controlled output is  .

It is obvious that consensus for the second-order multi-agent system (1) can be achieved if and only if

 i.e.  

 i, jI. And the attenuating ability of the multi-agent system against external disturbances can be quantitatively measured by the L₂-L_∞ performance index of the closed-loop transfer function matrix T_zω from the external disturbance input ω(t) to the controlled output z(t) shown as

       (3)

where

Therefore, we should design the protocol u_i(t) meeting the following two conditions simultaneously:

1) The states of agents satisfy  i.e. the

second-order multi-agent system (1) achieves consensus;

2) Under the zero-valued initial condition, the protocol u_i(t) can make the closed-loop transfer function T_zω satisfy  here γ is a given positive scalar.

Lemma 2^[10]: Give the symmetry matrix L_c=I_n-

R^n×n, where I_n denotes the n-dimensional

identity matrix, then the following statements hold:

1) The eigenvalues of L_c are 1 with multiplicity n-1 and 0 with multiplicity 1. The vectors  and 1_n are the left and right eigenvectors of L_c associated with the zero eigenvalue, respectively.

2) There must exist an orthogonal matrix U with the

last column  such that Further-

more, let LR^n×n be the Laplacian matrix of any directed

graph, then  here R^n×(n-1).

* denotes the symmetric part of a symmetric matrix. For the convenience of discussion, denote U=[U₁  U₂], U₁

R^n×(n-1), R^n×1.

3 Control protocol and system dynamics

To solve consensus problem of the second-order multi-agent system, we design the following protocol:

    (4)

where K₁＞0, K₂＞0 are protocol parameters, τ is time-delay. a_ij is the weight of edge, and Δa_ij(t) denotes the uncertainty of a_ij with

and ψ_ij is a specified constant for i, jI.

Denote

Then, we can get the second-order multi-agent system dynamics with protocol (4):

(5)

where  the corresponding Laplacian matrix of the edge weight a_ij and the uncertainty Δa_ij(t) are denoted by L and ΔL. Then ΔL can be rewritten as E₁Σ(t)E₂, where ,  are the determined constant matrices. Σ(t) reflects the uncertainties of the edges and satisfies Σ^T(t)Σ(t)≤I.

4 Main results

Lemma 3^[14]: For any real matrices DR^n×m, E R^m×n with F(t)R^m×m satisfying ||F(t)||≤1, and any scalar ε>0, we have

Lemma 4 (Schur Complement Formula)^[15]: For given symmetry matrix SR^n×n with the form S=[S_ij],   i, j{1, 2}, S₁₁R^r×r, S₁₂R^r×(n-r), S₂₂R^(n-r)×(n-r), then S<0 if and only if S₁₁<0, or equivalently, S₂₂<0,

Theorem 1: Consider a directed network with fixed topology and time-delay τ. For the multi-agent system (5), consensus can be achieved with  (a given index γ>0), if the symmetric matrix P>0, Q>0, R>0 and P, Q, RR^{2(n-1)×2(n-1)} and positive scalars ε₁, ε₂, ε₃, ε₄, ε₅, ε₆ satisfy

          (6)

where X₁₁, X₁₂, X₂₂ are respectively

,

Here

 ,

diag{m₁, …, m_n} denotes a block-diagonal matrix whose diagonal blocks are given by {m₁, …, m_n}.

Note that the Laplacian matrix of switching topology G_σ(t) is denoted as L_σ(t), where σ(t) is the switching signal at time t that determines the topology.

Theorem 2: Consider a directed network with switching topology and time-delay τ. For the multi-agent system (5), consensus can be achieved with  (a given index γ>0) for any σ(t), if the symmetric matrix P>0, Q>0, R>0 and P, Q, R R^{2(n-1)×2(n-1)}  and positive scalars ε₁, ε₂, ε₃, ε₄, ε₅, ε₆ satisfy

           (7)

where M₁₁, M₁₂, M₂₂ are respectively

where

,

,

  .

,

,

,

5 Simulation results

We present some numerical simulations to show the theoretical results obtained in the previous sections. The simulations are performed with four agents, whose initial conditions are all zeros. Figure 1 depicts four different networks {G_a, G_b, G_c, G_d}. The switching mode starts at G_a and the order is G_a→G_b→G_c→G_d→G_a. Moreover, the topology of the multi-agent system switches every 0.01 s to the next state. It is assumed that the weights a_ij are all 1 and the uncertainty of each edge satisfies |Δa_ij|≤0.01. Let the performance index γ=1. In practice, external disturbances are unpredictable. Especially, let it be white noise w(t). Then, the disturbance ω(t)=[1  -1  2  3]^Tw(t).

Let K₁=5 and K₂=3 for the value of the control parameters K₁, K₂ little impacts on the simulation results. Then, the simulation results are given for the network with switching topology and time-delay. Applying Theorem 2, we can get time-delay τ=0.203 s. Figure 2 describes the position trajectories and velocity trajectories. Figure 3 depicts the corresponding peak value trajectory of controlled output z(t) and energy trajectory disturbance signal ω(t). It is easily to see that

consensus is asymptotically achieved with

Fig.1 Four directed graphs

Fig.2 Position trajectories and velocity trajectories of network with switching topology and time-delay

Fig.3 Peak value trajectory of controlled output z(t) and energy trajectory disturbance signal ω(t) of network with switching topology and time-delay

6 Conclusions

We have employed the L₂-L_∞ control method to solve the consensus problem of multi-agent system subjected to external disturbances and parameter uncertainties with fixed and switching topologies. Neighbor-based control protocols with time-delay have been proposed for each agent. And some conditions are derived to ensure the consensus of multi-agent system with the desired L₂-L_∞ performance. Finally, numerical simulations are provided to show the effectiveness of theoretical results.

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(Edited by YANG Bing)

Received date: 2011-04-15; Accepted date: 2011-06-15

Foundation item: Project(60921001) supported by the National Natural Science Foundation of China

Corresponding author: CUI Yan, PhD; Tel: +86-10-82317940; E-mail: cuiyan8013@163.com