Leader-Following Consensus of Linear Multiagent Systems with State Observer under Switching Topologies

Abstract

The leader-following consensus problem of higher order multiagent systems is considered. The dynamics of each agent are given in general form of linear system, and the communication topology among the agents is assumed to be directed and switching. To track the leader, two kinds of distributed observer-based consensus protocols are proposed for each following agent, whose distributed observers are used to estimate the leader’s state and tracking error based on the relative outputs of the neighboring agents, respectively. Some sufficient consensus conditions are established by using parameter-dependent Lyapunov function method under a class of directed interaction topologies. As special cases, the consensus conditions for balanced and undirected interconnection topology cases can be obtained directly. The protocol design technique is based on algebraic graph theory, Riccati equation, and Sylvester equation. Finally, a simulation example is given to illustrate our obtained result.

1. Introduction

In recent years, the coordination control of the multi-agent systems has attracted a great number of researchers. The applications of the multi-agent systems include formation control, flocking, unmanned air vehicles, rendezvous, and distributed computations [1, 2]. The consensus problem has become a hot topic in the fields of coordination control for multi-agent systems. The main idea of multi-agent consensus is to design the distributed control protocol that enables a group of agents to reach an agreement on certain quantities.

The dynamics model of individual agent and the interacting topology of multi-agent systems are two key factors to achieve consensus. Usually, the stochastic matrix analysis method is used to solve the consensus problems with first-order agent dynamics under the switching interacting topology [3–6]. In [5], we have pointed out that the first-order continuous-time consensus problem investigated by [7, 8] can also be analyzed via the stochastic matrix method.

In many applications, the dynamics of agents are usually modeled by double integrator dynamics (second order). Although some authors tried to solve the second-order discrete-time consensus problem by using stochastic matrix-based method [9], as [10] has pointed out, the stochastic analysis method may not be applied directly to multi-agent systems with second-order dynamics. From this point, it is nontrivial from first-order consensus to the second-order consensus, and the Lyapunov-based approach is often chosen to solve second-order consensus problem [10–13]. A general framework was introduced by [14] to analyze the consensus problems of multi-agent systems in high-dimensional state space. Compared with the switching interacting topology, it is relatively easy to handle fixed topology by using the eigenvalue decomposition approach [15]. For the switching interacting topology case, the common Lyapunov function (CLF) method is a good substituted way to analyze consensus of multi-agent systems [10, 12, 16, 17]. Some other relevant research topics have also been addressed, such as consensus filtering [18], synchronization [19, 20], swarm stability [21, 22], neural network [23], time-delay [24], finite-time consensus [25], and communication constraint [26].

The leader-following configuration can be found in many biological systems [27, 28], which is very useful to design the multi-agent systems. leader-following consensus problems under jointly connected interacting topology were considered in [3]. In [29], the authors probed the controllability of a Leader-follower dynamic network. To track the active leader, the neighbor-based consensus protocol for each following agent was investigated by [10, 12, 17]. Leader-following consensus problem for multi-agent systems with general linear dynamics was investigated in [30].

In many practical systems especially in sensor networks and robot networks, some variables, which may lead the system to achieve a prescribed group behavior, cannot be obtained directly. To achieve the control goal, the control protocol often contains an observer to estimate those unmeasurable variables. In [31], the authors addressed the problem of output feedback control for networked control systems with limited communication capacity. In [10], the authors proposed an neighbor-based estimation rule for each first-order follower agent to estimate the active leaders’unmeasurable velocity. [16] assumed that each agent can also obtain its neighbor’s estimation value and use it in local control rule directly. To track the active leader [12], proposed distributed observer-based control laws for the second-order follower agents under the assumption that the velocity of the active leader cannot be measured. In [32], Abdessameud and Tayebi studied the observer-based consensus algorithm for the second-order agents to measure velocity under input constraints. To track the accelerated motion leader, [17] proposed a neighbor-based estimation rule to estimate the acceleration of the leader. The distributed observer-based cooperative control for multiple nonholonomic mobile agents was addressed by [33]. A distributed algorithm was proposed for the distributed estimation of a general active leader's unmeasurable state variables in [34], and [24] extended the results of [34] to the case of communication delays among agents. Consensus of high-order linear systems was solved by using dynamic output feedback compensator in [35]. A distributed observer-type consensus protocol to solve consensus problem with general linear or linearized agent dynamics under fixed communication topologies was referred to in [36]. A unified framework was introduced in [37] to address the consensus of multi-agent systems and the synchronization of complex networks, which proposed an observer-type consensus protocol using only the relative outputs of the neighboring agents under fixed communication topologies. [38] addressed the linear multi-agent consensus problem with discontinuous observations over a time-invariant undirected communication topology. Till now, the problem of observer-based consensus design has become an important topic in the study of multi-agent networks and is attracting more and more researchers.

Motivated by the above works, we study leader-following consensus problem of multi-agent systems with high-dimensional linear coupling dynamics under directed switching topology. The main contribution of this paper is that we propose two kinds of distributed observer-based consensus protocols to solve the leader-following consensus problem. The involved observers are used to estimate the leader's state and tracking error based on the relative outputs of neighboring agents, respectively. To construct the consensus protocols, an algorithm based on Riccati equation and Sylvester equation is proposed to design the protocol parameter matrices. By applying the proposed consensus protocols, we prove that the multi-agent system achieves consensus under any directed fixed topology. However, it becomes a challenging problem, when the interaction topology is time varying. By constructing a parameter-dependent common Lyapunov function, we prove that the multi-agent system achieves consensus under a class of directed interaction topologies. Obviously, the consensus conditions for some special cases such as balanced and undirected interconnection topology cases can be obtained directly.

The rest of the paper is organized as follows. In Section 2, the formulation of the consensus problem is given with the help of graph theory. Then in Section 3, the distributed consensus protocol based on distributed observer to estimate leader's state is investigated. Similarly, the distributed consensus protocol based on distributed observer to estimate tracking error is considered in Section 4. Following that, Section 5 provides a simulation example, and finally, the concluding remarks are given in Section 6.

2. Preliminaries and Problem Formulation

2.1. Problem Formulation

Consider a multi-agent system consisting of 𝑛 following agents and a leader. The dynamics of each following agent are modeled by the following linear system: 𝑥̇_𝑖=𝐴𝑥_𝑖+𝐵𝑢_𝑖,𝑦_𝑖=𝐶𝑥_𝑖,𝑖=1,…,𝑛, (1) where 𝑥_𝑖∈𝑅^𝑚 is the agent 𝑖's state, 𝑢_𝑖∈𝑅^𝑝 is the agent 𝑖's control input, and 𝑦_𝑖∈𝑅^𝑞 is the agent 𝑖's measured output. 𝐴, 𝐵, and  𝐶 are constant matrices with appropriate dimensions.

The leader, labeled as 𝑖=0, has linear dynamics as 𝑥̇₀=𝐴𝑥₀+𝐵𝑢₀(𝑡),𝑦₀=𝐶𝑥₀, (2) where 𝑥₀∈𝑅^𝑚 is the leader's state and 𝑦₀∈𝑅^𝑞 is the leader's measured output; the input 𝑢₀(𝑡) can be regarded as the common policy which is known by all following agents.

Remark 1. This leader-following consensus problem has been investigated by [30], and the special second-order leader-following consensus problem was studied by [11, 12]. [30] assumed that every following agent can obtain the state variables of its neighbors directly. Here, we assume that every following agent can only obtain the measured output of its neighbors directly. As [30] has pointed out, the system matrices for all the agents and the leader were taken to be identical because of their practical background such as group of birds and school of fishes.

The following assumption is used throughout the paper.

Assumption 1. (𝐴,𝐵) is stabilizable and (𝐴,𝐶) is detectable.

The leader-following multi-agent system is said to be achieved consensus, if the state of any following agent satisfies lim_𝑡→∞(𝑥_𝑖(𝑡)−𝑥₀(𝑡))=0, for any initial state 𝑥_𝑖(0),𝑖=0,1,…,𝑛. We say that the protocol 𝑢_𝑖(𝑘) can solve the consensus control problem, if the closed-loop feedback system achieves consensus.

2.2. Notations and Concepts

Let 𝑅^𝑚×𝑛 and 𝐶^𝑚×𝑛 be the set of 𝑚×𝑛 real matrices and complex matrices, respectively. 𝐼 is the identity matrix with compatible dimension. 𝐴^𝑇 and 𝐴^𝐻 represent transpose and conjugate transpose of matrix 𝐴∈𝐶^𝑚×𝑛, respectively. 𝟏_𝑛=[1,…,1]^𝑇∈𝑅^𝑛. ℐ denotes the set {1,2,…,𝑛}. For symmetric matrices 𝐴 and 𝐵, 𝐴>(≥)𝐵 means that 𝐴−𝐵 is positive (semi)definite. ‖⋅‖ denotes Euclidean norm. ⊗ denotes the Kronecker product, which satisfies (1)(𝐴⊗𝐵)(𝐶⊗𝐷)=(𝐴𝐶)⊗(𝐵𝐷); (2)if 𝐴≥0 and 𝐵≥0, then 𝐴⊗𝐵≥0.

To model interconnection topology, some preliminary knowledge of graph theory is introduced. More details are available in [39]. A weighted digraph is denoted by 𝒢={𝒱,𝜀,𝐴}, where 𝒱={𝑣₁,𝑣₂,…,𝑣_𝑛} is the set of vertices, 𝜀⊂𝒱×𝒱 is the set of edges, and a weighted adjacency matrix 𝐴=[𝑎_𝑖𝑗] has nonnegative adjacency elements 𝑎_𝑖𝑗 and 𝑎_𝑖𝑖=0. The set of all neighbor nodes of node 𝑣_𝑖 is defined by 𝒩_𝑖={𝑗∣(𝑣_𝑖,𝑣_𝑗)∈𝜀}. The degree matrix 𝐷={𝑑₁,𝑑₂,…,𝑑_𝑛}∈ℛ^𝑛×𝑛 of digraph 𝒢 is a diagonal matrix with diagonal elements 𝑑_𝑖=∑_{𝑗∈𝒩𝑖}𝑎_𝑖𝑗. Then the Laplacian matrix of 𝒢 is defined as 𝐿=𝐷−𝐴∈𝑅^𝑛×𝑛.

The in-degree and out-degree of node 𝑣_𝑖 are denoted as 𝑑_in(𝑖)=∑^𝑛_𝑗=1𝑎_𝑗𝑖 and 𝑑_out(𝑖)=∑^𝑛_𝑗=1𝑎_𝑖𝑗, respectively. A weighted digraph 𝒢 is said to be balanced if and only if 𝑑_in(𝑖)=𝑑_out(𝑖), for 𝑖=1,2,…,𝑛. Moreover, a weighted graph 𝒢 is balanced if and only if 𝟏^𝑇𝐿=0, where 𝟏=(1,1,…,1)^𝑇∈𝑅^𝑛. (see [7]). Certainly, any undirected weighted graph is balanced.

If there is a directed path from node 𝑣_𝑖 to node 𝑣_𝑗, then 𝑣_𝑗 is said to be reachable from 𝑣_𝑖. Node 𝑣_𝑖 is said to be globally reachable if there is a directed path from every other node to node 𝑣_𝑖 in digraph 𝒢. A directed graph 𝒢 has a globally reachable node if and only if there exists a directed spanning tree in 𝒢 (see [4]).

In what follows, we use digraph 𝒢 of order 𝑛+1 to model information topology relation of the multi-agent system that consisted of 𝑛 agents (labeled as 𝑣_𝑖,𝑖=1,2,…,𝑛) and one leader (labeled as  𝑣₀) and directed graph 𝒢 to model the topology relation of these 𝑛 followers. In fact, 𝒢 contains graph 𝒢, and 𝑣₀ with the directed edges from some agents to the leader describes the topology relation among all agents. To describe the variable interconnection topology, the set of all possible topology digraphs is denoted as 𝑆={𝒢₁,𝒢₂,…,𝒢_𝑁} with index set 𝒫={1,2,…,𝑁}. The switching signal 𝜎∶[0,∞)→𝒫 is used to express the index of topology digraph. Let 0=𝑡₁,𝑡₂,𝑡₃,… be an infinite time sequence at which the interconnection graph of the considered multi-agent system switches. Therefore, 𝒩_𝑖(𝑡) and the connection weights 𝑎_𝑖𝑗(𝑡),𝑏_𝑖(𝑡)(𝑖,𝑗=1,…,𝑛) are time invariant in any interval [𝑡_𝑖,𝑡_𝑖+1). Assume that there is a constant 𝜏₀>0, often called dwell time, with 𝑡_𝑖+1−𝑡_𝑖≥𝜏₀, for all 𝑖=1,2,3,….

For simplicity, the weights 𝑎_𝑖𝑗(𝑡) and 𝑏_𝑖(𝑡) are chosen as follows in our problem: 𝑎_𝑖𝑗(𝑡)=𝛼_𝑖𝑗𝑏ifagent𝑖isconnectedtoagent𝑗attime𝑡0otherwise,_𝑖(𝑡)=𝛽_𝑖ifagent𝑖isconnectedtotheleaderattime𝑡0otherwise, (3)(4) where 𝛼_𝑖𝑗>0(𝑖,𝑗=1,…,𝑛) is the connection weight constant between agent 𝑖 and agent 𝑗, and 𝛽_𝑖>0(𝑖=1,…,𝑛) is the connection weight constant between agent 𝑖 and the leader.

Let 𝐿_𝜎(𝑡) be the Laplacian matrix of the interaction graph 𝐺_𝜎(𝑡). 𝐵_𝜎(𝑡) is an 𝑛×𝑛 diagonal matrix whose 𝑖th diagonal element is 𝑏_𝑖(𝑡) at time 𝑡. For convenience, let 𝐻_𝜎(𝑡)=𝐿_𝜎(𝑡)+𝐵_𝜎(𝑡).

2.3. Preliminary Results

Before establishing our main results, some preliminary results are introduced, which will be used later.

Note that matrix 𝐻=𝐿+𝐵 plays a key role in the convergence analysis of the system. A matrix is said to be positive stable if all its eigenvalues have positive real parts. The following lemma, which is found in [11], shows a relationship between 𝐻 and the connectedness of digraph 𝒢.

Lemma 2. Matrix 𝐻=𝐿+𝐵 is positive stable if and only if node 0 is globally reachable in 𝒢.

The next two lemmas are well known, whose different versions can be found in many books.

Lemma 3 (see [40]). Let 𝑆 be a symmetric matrix partitioned as 𝑆=^𝑆11𝑆12_𝑆21𝑆22, where 𝑆₁₁∈𝑅^𝑟×𝑟, 𝑆₁₂∈𝑅^{𝑟×(𝑛−𝑟)}, and 𝑆₂₂∈𝑅^{(𝑛−𝑟)×(𝑛−𝑟)}. Then 𝑆<0 if and only if 𝑆₁₁<0,𝑆₂₂−𝑆₂₁𝑆⁻¹₁₁𝑆₁₂<0 (5) or equivalently 𝑆₂₂<0,𝑆₁₁−𝑆₁₂𝑆⁻¹₂₂𝑆₂₁<0. (6)

Lemma 4 (see [41]). If (𝐴,𝐶) is detectable and 𝑄 is a symmetric positive definite matrix, then there is a unique positive definite matrix 𝑃 to satisfy the Riccati equation 𝑃𝐴^𝑇+𝐴𝑃−𝑃𝐶^𝑇𝐶𝑃+𝑄=0. (7)

Furthermore, the real parts of all the eigenvalues of 𝐴^𝑇−𝐶^𝑇𝐶𝑃 are negative.

3. Distributed State Observer to Estimate the Leader's State

To solve the leader-following consensus problem, the relative measurement is involved. Let 𝑧_𝑖 be the relative output error of agent 𝑖 with its neighbor agent as follows: 𝑧_𝑖=_{𝑗∈𝒩𝑖}𝑎_𝑖𝑗(𝑡)𝑦_𝑖−𝑦_𝑗+𝛽_𝑖(𝑡)𝑦_𝑖−𝑦₀, (8) where 𝑦_𝑖=𝐶𝑥_𝑖 is the output of the estimator, the connection weights 𝑎_𝑖𝑗(𝑡) and 𝑏_𝑖(𝑡) are defined as (3) and (4), respectively. Then, we propose a distributed control protocol for agent 𝑖 as follows, which consists of a distributed estimation law and a feedback control law: (i)distributed estimation law for agent 𝑖: 𝑥̇‌_𝑖=𝐴𝑥_𝑖−𝜅𝐿𝑧_𝑖+𝐵𝑢₀(𝑡),𝑖=1,…,𝑛, (9) where 𝑥_𝑖∈𝑅^𝑚 is the protocol state, 𝜅 is the coupling strength, and 𝐿∈𝑅^𝑚×𝑞 is a given gain matrix;(ii)feedback control law for agent 𝑖: 𝑢_𝑖=𝑢₀(𝑡)−𝐾𝑥_𝑖−𝑥_𝑖𝑖=1,2,…,𝑛, (10) where 𝐾 is a given feedback gain matrix.

In fact, estimation law (9) plays the role of state observer for agent 𝑖 to estimate the leader's state variables. From (9), each agent relies only on the locally available information at every moment. A following agent cannot “observe” or “estimate” the leader directly based on the measured information of the leader if it is not connected to the leader. Thus, it has to collect the information of the leader in a distributed way from its neighbor agents.

Our objective is to design 𝜅,𝐿,𝐾 to make the leader-following multi-agent system achieve consensus. To this end, the following algorithm is presented to construct the gain matrix 𝐿 and the feedback matrix 𝐾 in state estimation law (9) and control law (10).

Algorithm 5. Given that (𝐴,𝐵,𝐶) is stabilizable and detectable, the gain matrix 𝐿 and feedback matrix 𝐾 are constructed as follows:(1)for a given positive definite matrix 𝑄, solve the following Riccati equation 𝐴𝑃+𝑃𝐴^𝑇−𝑃𝐶^𝑇𝐶𝑃+𝑄=0 (11) to obtain the unique positive definite matrix 𝑃. Then, the gain matrix 𝐿 is chosen by 𝐿=𝑃𝐶^𝑇;(2)choose 𝐾 such that 𝐴−𝐵𝐾 is stable.

Remark 6. One method to construct the feedback matrix 𝐾 is introduced as follows:(2.1) select a stable 𝑛×𝑛 matrix 𝐹 with a set of desired eigenvalues that contains no eigenvalues in common with those of 𝐴;(2.2) select 𝐾 randomly such that (𝐹,𝐾) is observable;(2.3) solve Sylvester equation 𝐴𝑇−𝑇𝐹=𝐵𝐾 (12) to get a nonsingular solution 𝑇. If 𝑇 is singular, select another 𝐾, until 𝑇 is nonsingular;(2.4) compute 𝑇⁻¹ and take 𝐾=𝐾𝑇⁻¹. From (2.1)–(2.4) of Algorithm 5, it's easy to get 𝐴−𝐵𝐾=𝐴−𝐵𝐾𝑇⁻¹=𝐴𝑇−𝐵𝐾𝑇⁻¹=𝑇𝐹𝑇⁻¹ (13)which means that 𝐴−𝐵𝐾 is stable. The above method, to construct feedback matrix 𝐾, can be found in [42]. Of course, there are several other methods to, construct matrix 𝐾 if (𝐴,𝐵) is stabilizable. From Lemma 4, the Riccati equation (11) is soluble if (𝐴,𝐶) is detectable. Thus, a sufficient condition for Algorithm 5 to construct protocols (9) and (10) successfully is that (𝐴,𝐵) is stabilizable and (𝐴,𝐶) is detectable.

Denote 𝑒_𝑖=𝑥_𝑖−𝑥₀ and 𝑒=(𝑒^𝑇₁,𝑒^𝑇₂,…,𝑒^𝑇_𝑛)^𝑇. Then, after manipulations with combining (1), (2), and (8), we have 𝑒_𝑖̇‌=𝐴𝑒_𝑖×−𝜅𝐿_{𝑗∈𝒩𝑖}𝑎_𝑖𝑗(𝑡)𝐶𝑒_𝑖−𝑒_𝑗+𝛽_𝑖(𝑡)𝐶𝑒_𝑖, (14) which can be written in stack vector form: 𝑒̇‌=(𝐼⊗𝐴)−𝐿_𝜎(𝑡)+𝐵_𝜎(𝑡)⊗(𝜅𝐿𝐶)𝑒. (15) Similarly, taking 𝑒_𝑖=𝑥_𝑖−𝑥₀ and 𝑒=(𝑒^𝑇₁,𝑒^𝑇₂,…,𝑒^𝑇_𝑁)^𝑇, we have 𝑒_𝑖̇=𝑥̇_𝑖−𝑥̇₀=𝐴𝑥_𝑖−𝑥₀+𝐵𝐾𝑥_𝑖−𝑥₀+𝑥₀−𝑥_𝑖=(𝐴−𝐵𝐾)𝑒_𝑖+𝐵𝐾𝑒_𝑖 (16) or equivalently 𝑒̇=𝐼⊗(𝐴−𝐵𝐾)𝑒+𝐼⊗𝐵𝐾𝑒. (17) From (15) and (17), the error dynamics system will be expressed in a compact form as follows: 𝜀̇(𝑡)=𝐹_𝜎𝜀(𝑡), (18) where 𝜀(𝑡)=𝑒𝑒,𝐹_𝜎=𝐼⊗(𝐴−𝐵𝐾)𝐼⊗(𝐵𝐾)0𝐼⊗𝐴−𝐻_𝜎⊗().𝜅𝐿𝐶 (19) Obviously, the multi-agent system achieves consensus if lim_𝑡→∞𝜀(𝑡)=0. Thus, the leader-following consensus problem of multi-agent system is transformed into the stability problem of error dynamic system (18).

3.1. Fixed Interconnection Topology Case

In this subsection, the leader-following consensus problem under fixed interconnection topology is investigated. In this case, the error system can be rewritten as follows by dropping the subscript 𝜎(𝑡): 𝜀̇(𝑡)=()()()𝜀(𝑡)(𝑡).𝐼⊗𝐴−𝐵𝐾𝐼⊗𝐵𝐾0𝐼⊗𝐴−𝐻⊗𝜅𝐿𝐶≜𝐹𝜀 (20)

Now, we present the following result for the fixed interconnection topology case.

Theorem 7. Suppose that the interconnection topology 𝒢 is fixed with globally reachable node 𝑣₀ and the matrices 𝐿, 𝐾 used in control protocol are constructed by Algorithm 5. Take the coupling strength 𝜅 satisfying 1𝜅≥2min_𝑖∈ℐRe𝜆_𝑖, (21) where 𝜆_𝑖 is 𝑖th eigenvalue of 𝐻. Then, the distributed control protocols (9) and (10) can guarantee that all following agents track the leader from any initial condition.

Proof. By applying Schur orthogonal decomposition to matrix 𝐻, there exists a unitary matrix 𝑈 such that 𝑈𝐻𝑈^𝐻=𝜆₁∗⋯∗0𝜆₂⋯∗⋮⋮⋮⋮00⋯𝜆_𝑛≜Δ. (22) From (22), we have 𝐼_𝑛⊗𝐼_𝑚𝑈⊗𝐼_𝑚𝐹𝐼_𝑛⊗𝐼_𝑚𝑈⊗𝐼_𝑚^𝐻=()()().𝐼⊗𝐴−𝐵𝐾𝐼⊗𝐵𝐾0𝐼⊗𝐴−Δ⊗𝜅𝐿𝐶 (23) Then, it is easy to see that 𝐹 is stable if and only if 𝐴−𝜆_𝑖𝜅𝐿𝐶 is stable for any 𝑖=1,2,…,𝑛. From the fact that positive definite matrix 𝑃 is a unique solution of (11) and 𝐿=𝑃𝐶^𝑇, we can obtain 𝑃𝐴−𝜆_𝑖𝜅𝐿𝐶^𝐻+𝐴−𝜆_𝑖𝜅𝐿𝐶𝑃=−𝑄+𝑃𝐶^𝑇𝐶𝑃−2𝑅𝑒𝜆_𝑖𝜅𝑃𝐶^𝑇𝐶𝑃≤−𝑄, (24) which implies that 𝐴−𝜆_𝑖𝜅𝐿𝐶 is stable. Thus, we have lim_𝑡→∞𝜀(𝑡)=0. The proof is now completed.

Remark 8. Since the interconnection topology 𝒢 is fixed with globally reachable node 𝑣₀, matrix 𝐻 is positive stable according to Lemma 2. Thus, min_𝑖∈ℐRe(𝜆_𝑖) is well defined and greater than zero. On the other hand, if the node 𝑣₀ is not globally reachable, at least a node must exist from which there is no directed path to node 𝑣₀ in graph 𝒢. This means that some following agents always do not get the state information of the leader directly or indirectly. Certainly, the multi-agent system may not achieve consensus for any given initial condition in this case. Thus, the condition that node 0 is globally reachable in 𝒢 is also necessary to achieve consensus under fixed interconnection topology.

3.2. Switching Topology Case

Now, we discuss the convergence analysis of system (18) under switching interconnection topology. For convenience, a class of interconnection topology graphs is defined by the following: Γ=𝒢∣𝑣₀isagloballyreachablenodeingraph𝒢and𝐻^𝑇𝒢+𝐻𝒢ispositivedefinite. (25) Therefore, define 𝒢𝜆∶=min_∈𝑆∩Γ𝜆𝐻^𝑇𝒢+𝐻𝒢. (26) Noticing that the set 𝑆∩Γ is a finite set and 𝐻^𝑇(𝒢)+𝐻(𝒢) is a positive definite, we know that 𝜆 is well defined, which is positive and depends directly on the constants 𝑎_𝑖𝑗 and 𝛽_𝑖 (𝑖,𝑗=1,2,…,𝑛) given in (3) and (4).

The convergence analysis result for multi-agent consensus under switching interconnection topology by using the parameter-dependent Lyapunov function method is given in the following theorem.

Theorem 9. Assume that 𝒢_𝜎(𝑡)∈𝑆⊂Γ in any interval [𝑡_𝑗,𝑡_𝑗+1) and the matrices 𝐿, 𝐾 used in control protocol are constructed by Algorithm 5. Take the coupling strength 𝜅 satisfying 1𝜅≥𝜆. (27) The distributed control protocols (9) and (10) can guarantee that all following agents track the leader from any initial condition.

Proof. To prove the theorem, we first consider the error dynamics in each interval. In any interval [𝑡_𝑖,𝑡_𝑖+1), the topology graph is fixed and the system matrices are time invariant with some fixed 𝜎(𝑡)=𝑝∈𝒫. Let 𝑈_𝑝 be an orthogonal transformation such that 𝑈_𝑝(𝐻^𝑇_𝑝+𝐻_𝑝)𝑈^𝑇_𝑝 is a diagonal matrix Λ_𝑝=diag{𝜆_1𝑝,𝜆_2𝑝,…,𝜆_𝑛𝑝}, where 𝜆_𝑖𝑝 is the 𝑖th eigenvalue of matrix 𝐻^𝑇_𝑝+𝐻_𝑝.
According to Algorithm 5 and condition (27), we can know that the unique solution 𝑃>0 of Riccati equation satisfies 𝑃1𝐴−2𝜆_𝑖𝑝𝜅𝐿𝐶^𝑇+1𝐴−2𝜆_𝑖𝑝𝜅𝐿𝐶𝑃=−𝑄+𝑃𝐶^𝑇𝐶𝑃−𝜆_𝑖𝑝𝜅𝑃𝐶^𝑇𝐶𝑃≤−𝑄, (28) which implies the following inequality: ()1𝐼⊗𝑃𝐼⊗𝐴−2Λ_𝑝⊗(𝜅𝐿𝐶)^𝑇+1𝐼⊗𝐴−2Λ_𝑝⊗(𝜅𝐿𝐶)(𝐼⊗𝑃)≤−𝐼⊗𝑄<0. (29) By pre- and postmultiplying the above inequality (29) with 𝑈_𝑝⊗𝐼 and its transpose, respectively, we have ()1𝐼⊗𝑃𝐼⊗𝐴−2𝐻^𝑇_𝑝+𝐻_𝑝⊗(𝜅𝐿𝐶)^𝑇+1𝐼⊗𝐴−2𝐻^𝑇_𝑝+𝐻_𝑝⊗(𝜅𝐿𝐶)(𝐼⊗𝑃)≤−𝐼⊗𝑄<0. (30) Thus, the following inequality holds by noting that 𝐿𝐶𝑃 is a symmetric matrix: (𝐼⊗𝑃)𝐼⊗𝐴−𝐻_𝑝⊗(𝜅𝐿𝐶)^𝑇+𝐼⊗𝐴−𝐻_𝑝⊗()()=(1𝜅𝐿𝐶𝐼⊗𝑃𝐼⊗𝑃)𝐼⊗𝐴−2𝐻^𝑇_𝑝+𝐻_𝑝⊗(𝜅𝐿𝐶)^𝑇+1𝐼⊗𝐴−2𝐻^𝑇_𝑝+𝐻_𝑝⊗(𝜅𝐿𝐶)(𝐼⊗𝑃)≤−𝐼⊗𝑄<0. (31) Set 𝑃₁=𝑃⁻¹ and 𝑄₁=𝑃₁𝑄𝑃₁. 𝑃₁ and 𝑄₁ are both positive definite matrices. From (33), it is not hard to obtain the following inequality: 𝐼⊗𝑃₁𝐼⊗𝐴−𝐻_𝑝⊗(+𝜅𝐿𝐶)𝐼⊗𝐴−𝐻_𝑝⊗(𝜅𝐿𝐶)^𝑇𝐼⊗𝑃₁≤−𝐼⊗𝑄₁<0. (32)
According to step (2) of Algorithm 5, 𝐴−𝐵𝐾 is stable; that is, there exist positive definite matrices 𝑄₂ and 𝑃₂ satisfying the Lyapunov equation 𝑃₂(𝐴−𝐵𝐾)+(𝐴−𝐵𝐾)^𝑇𝑃₂=−𝑄₂. (33)
Choose the following parameter-dependent Lyapunov matrix: 𝑃^∼=1𝜔𝐼⊗𝑃₂00𝐼⊗𝑃₁, (34) where 𝜔 is positive parameter. Obviously, 𝑃^∼ is positive matrix. Then, consider the following common Lyapunov function for error dynamic system (18): 𝑉(𝜀(𝑡))=𝜀(𝑡)𝑃^𝑇∼𝜀(𝑡). (35) For any interval [𝑡_𝑖,𝑡_𝑖+1), the time derivative of this Lyapunov function along the trajectory of system (18) is 𝑑𝑉(𝜀)𝑑𝑡=𝜀^𝑇𝐹𝑃^𝑇_𝜎^∼+𝑃^∼𝐹_𝜎𝜀=𝜀^𝑇𝑄_𝜎𝜀, (36) where 𝑄_𝜎=1𝜔𝐼⊗𝑃₂(𝐴−𝐵𝐾)+(𝐴−𝐵𝐾)^𝑇𝑃₂1𝜔𝐼⊗𝑃₂1𝐵𝐾𝜔𝐼⊗𝐾^𝑇𝐵^𝑇𝑃₂𝑄_2𝜎,𝑄_2𝜎=𝐼⊗𝑃₁𝐼⊗𝐴−𝐻_𝜎⊗()+𝜅𝐿𝐶𝐼⊗𝐴−𝐻_𝜎⊗(𝜅𝐿𝐶)^𝑇𝐼⊗𝑃₁. (37)
According to (30) and (33), we have 𝑄_𝜎≤−1𝜔𝐼⊗𝑄₂1𝜔𝐼⊗𝑃₂1𝐵𝐾𝜔𝐼⊗𝐾^𝑇𝐵^𝑇𝑃₂−𝐼⊗𝑄₁∶=−𝑄. (38) Choose 𝜔 satisfying 𝜔>𝜆_max𝑄₁⁻¹𝑃₂𝐵𝐾^𝑇𝑄₂⁻¹𝑃₂𝐵𝐾, (39) which implies that 𝜔𝑄₁>𝑃₂𝐵𝐾^𝑇𝑄₂⁻¹𝑃₂𝐵𝐾. (40) According to Lemma 3, we know that matrix 𝑄 is positive definite while condition (39) is satisfied.
It is well known that Lyapunov function 𝑉(𝜀) satisfies 𝜆𝑃_min^∼‖𝜀‖²≤𝑉(𝜀)≤𝜆𝑃_max^∼‖𝜀‖². (41) Therefore, we have ‖𝜀‖≤𝑉(𝜀)/𝜆_min(𝑃^∼). On the other hand, we know that 𝜀min^𝑇𝑄𝜀𝜀𝑃^𝑇∼𝜀≥𝜆_min𝑄𝜆𝑃_max^∼. (42) Let 𝛽=𝜆_min(𝑄)/𝜆_max(𝑃^∼). Therefore, from (36), we have (𝑑/𝑑𝑡)𝑉(𝜀)≤−𝛽𝑉(𝜀) or equivalently 𝑉(𝜀)≤𝑉(𝜀(0))𝑒^−𝛽𝑡. Thus, lim_𝑡→∞𝜀(𝑡)=0 is satisfied, which means that the leader-following consensus problem is solved by control law (10) together with state estimation law (9). The proof is now completed.

Remark 10. For the special case that the graph 𝒢_𝜎 associated with all followers is balanced, matrix 𝐿_𝜎+𝐿^𝑇_𝜎 is positive semidefinite in this case [7]. Moreover, suppose that 𝒢_𝑙 is balanced. Then 𝐻^𝑇_𝑙+𝐻_𝑙 is positive definite if and only if 𝑣₀ is globally reachable node in 𝒢_𝑙 [11]. Thus, Γ is not empty and at least concludes a class of interconnection topology 𝒢 whose 𝒢 associated with all followers is balanced and 𝑣₀ is globally reachable node in 𝒢. The undirected interconnection topology considered in [10, 24, 30] also belongs to Γ. Therefore, our established results can be applied to those special cases directly.

4. Distributed State Observer to Estimate Tracking Error

In this section, we propose another distributed control protocol for agent 𝑖, which also consists of a distributed estimation law and a feedback control law. The distributed estimation law plays the role of state observer to estimate tracking error instead of leader's state. The involved relative output error of agent 𝑖 with its neighbor agent is taken as 𝑧^∼_𝑖=_{𝑗∈𝒩𝑖}𝑎_𝑖𝑗(𝑡)𝑦^∼_𝑖−𝑦^∼_𝑗−𝑦_𝑖−𝑦_𝑗+𝛽_𝑖(𝑡)𝑦^∼_𝑖−𝑦_𝑖−𝑦₀, (43) where 𝑦^∼_𝑖=𝐶𝑥^∼_𝑖 and the connection weights 𝑎_𝑖𝑗(𝑡) and 𝑏_𝑖(𝑡) are taken as (3) and (4), respectively.

Then, another kind of distributed control protocol for agent 𝑖 is given as follows: (i)distributed estimation law for agent 𝑖 to estimate tracking error: 𝑥̇‌^∼_𝑖=(𝐴−𝐵𝐾)𝑥^∼_𝑖−𝜅𝐿𝑧^∼_𝑖,𝑖=1,…,𝑛, (44) where 𝑥^∼_𝑖∈𝑅^𝑚 is the protocol state; (ii)feedback control law for agent 𝑖: 𝑢_𝑖=𝑢₀(𝑡)−𝐾𝑥^∼_𝑖𝑖=1,2,…,𝑛. (45)

Here, the control parameter 𝜅, gain matrix 𝐿, and feedback matrix 𝐾 are defined and selected as the previous section.

Denote 𝑒^∼_𝑖=𝑥^∼_𝑖−(𝑥_𝑖−𝑥₀) and 𝑒^∼=(𝑒^∼𝑇₁,𝑒^∼𝑇₂,…,𝑒^∼𝑇_𝑛)^𝑇. From (1), (2), (43), and (45), we have 𝑒_𝑖̇‌^∼=𝐴𝑒^∼_𝑖−𝜅𝐿_{𝑗∈𝒩𝑖}𝑎_𝑖𝑗(𝑡)𝐶𝑒^∼_𝑖−𝑒^∼_𝑗+𝛽_𝑖(𝑡)𝐶𝑒^∼_𝑖, (46) which can be written in stack vector form: 𝑒̇‌^∼=(𝐼⊗𝐴)−𝐿_𝜎(𝑡)+𝐵_𝜎(𝑡)⊗(𝐿𝐶)𝑒^∼=(𝐼⊗𝐴)−𝐻_𝜎(𝑡)⊗(𝐿𝐶)𝑒^∼. (47)

Similarly, for the tracking error 𝑒_𝑖=𝑥_𝑖−𝑥₀ and 𝑒=(𝑒^𝑇₁,𝑒^𝑇₂,…,𝑒^𝑇_𝑁)^𝑇, we can obtain 𝑒_𝑖̇=𝑥̇_𝑖−𝑥̇₀=𝐴𝑥_𝑖−𝑥₀+𝐵𝑢_𝑖−𝑢₀=𝐴𝑥_𝑖−𝑥₀−𝐵𝐾𝑥^∼_𝑖=(𝐴−𝐵𝐾)𝑒_𝑖−𝐵𝐾𝑒^∼_𝑖 (48) or equivalently 𝑒̇=𝐼⊗(𝐴−𝐵𝐾)𝑒−𝐼⊗(𝐵𝐾)𝑒^∼. (49) From (47) and (49), the error dynamics system will be expressed in a compact form as follows: ̇‌^∼𝜀(𝑡)=𝐹^∼_𝜎^∼𝜀(𝑡), (50) where ^∼𝜀(𝑡)=𝑒_∼𝑒_∼,𝐹_𝜎=𝐼⊗(𝐴−𝐵𝐾)−𝐼⊗(𝐵𝐾)0𝐼⊗𝐴−𝐻_𝜎.⊗𝐿𝐶 (51) Obviously, the multi-agent system achieves consensus if lim^∼_𝑡→∞𝜀(𝑡)=0. Thus, the leader-following consensus problem of multi-agent system discussed in this paper is transformed into the stability problem of error dynamic system (50).

Although we use different control protocol for each agent, we get similar error dynamic system. There is just one difference between error dynamic system (50) and error dynamic system (18). By using similar analysis approach, we can also solve the consensus problem under fixed and switching topology cases. The proofs are omitted because it is quite similar to the proofs of Theorems 7 and 9, respectively.

Theorem 11. Suppose that the interconnection topology 𝒢 is fixed with globally reachable node 𝑣₀ and the matrices 𝐿, 𝐾 used in control protocol are constructed by Algorithm 5. Take the coupling strength 𝜅 satisfying (21). Then, the distributed control protocols (44) and (45) can guarantee that all following agents track the leader from any initial condition.

Theorem 12. Assume that 𝒢_𝜎(𝑡)∈𝑆⊂Γ in any interval [𝑡_𝑗,𝑡_𝑗+1) and the matrices 𝐿, 𝐾 used in control protocol are constructed by Algorithm 5. Take the coupling strength 𝜅 satisfying (27). The distributed control protocols (44) and (45) can guarantee that all following agents track the leader from any initial condition.

Remark 13. Of course, the established results of this section can be also applied to the balanced interconnection topology and undirected interconnection topology cases directly. Compared with the system matrix 𝐴 used in estimation (9), the system matrix 𝐴−𝐵𝐾 used in estimation law (44) must be stable, and it does not use the agent 𝑖's state 𝑥_𝑖 in feedback control law (45). Thus, the distributed control protocols (44) and (45) may be more accepted in applications.

5. Simulation Example

In this section, a numerical simulation is given to illustrate the theoretical results obtained in the previous sections. The multi-agent system is consisted of one leader and six followers. The system matrices of agent dynamics in (1) and (2) are given by ,,.𝐴=−1−2−3−2−21−311𝐵=101−121𝐶=10−103−2 (52) Without loss of generality, we take 𝑢₀(𝑡)=0 in numerical simulation. The interconnection topologies are arbitrarily switched among three graphs 𝒢_𝑖(𝑖=1,2,3,4), which are shown in Figure 1.

(a)

(b)

(c)

(d)

Figure 1 

Four interconnection topology graphs.

The Laplacian matrices 𝐿_𝑖(𝑖=1,2,3,4) for the four subgraphs 𝒢_𝑖(𝑖=1,2,3,4) are 𝐿₁=,𝐿3.5−1.5000−2−15.5−2.50−200−12−10000−25−1−20−20−15−2−100−1−24₂=,𝐿5−1−20−20−1.54.5−100−2−2−140−1000020−2−10−10200−20−204₃=,𝐿300−20−1020−1−1000200−2−2−105−200−10−230−20−2004₄=200−1−1004−20−200−25−20−1−20−2400−1−2005−200−10−23 (53) and the diagonal matrices for the interconnection relationship between the leader and the followers are 𝐵₁(),𝐵=diag1,0,0,1,0,0₂(),𝐵=diag0,1,0,1,0,0₃(),𝐵=diag1,0,0,0,1,1₄=diag(0,0,1,1,0,0). (54) All 𝐻_𝑖=𝐿_𝑖+𝐵_𝑖, 𝑖=1,2,3,4, is not symmetric and satisfies 𝐻_𝑖+𝐻^𝑇_𝑖>0. Then, we know that 𝜆=0.095. Take 𝜅=1>1/𝜆.

Take  𝑄=𝐼 and then solve the Riccati equation (11) to get the unique positive define solution 𝑃. Thus, the gain matrix 𝐿 can be constructed by 𝐿=𝑃𝐶^𝑇=3.40190.7514−2.0167−0.0430−4.0747−1.3478. (55) Matrix 𝐴 is not stable. Select feedback matrix 𝐾=−1.64290.21431.21430.2143−0.0714−1.0714, (56) such that 𝐴−𝐵𝐾 is stable.

The initial state of all agents is randomly produced. The state errors showed in Figures 2 and 3 are 𝑥_𝑖1−𝑥₀₁, 𝑥_𝑖2−𝑥₀₂ and 𝑥_𝑖3−𝑥₀₃, respectively.

(a)

(b)

(c)

Figure 2 

The error trajectories between the leader and each agent.

(a)

(b)

(c)

Figure 3 

The error trajectories between the leader and each agent.

We first use the approach proposed in Section 3 to solve the consensus problem; that is, each agent uses the feedback control law (10) together with the distributed estimation law (9). The trajectories of 𝑥_𝑖𝑗−𝑥_0𝑗, 𝑗=1,2,3, are depicted in Figure 2, which shows that the follower agents can track the leader agent.

Next, we use the feedback control law (10) together with the distributed estimation law (9) for agent 𝑖 to solve the consensus problem. The trajectories of 𝑥_𝑖𝑗−𝑥_0𝑗, 𝑗=1,2,3, are depicted in Figure 3, which also shows that the follower-agents can track the leader agent.

6. Conclusions

In this paper, the leader-following consensus problem for multi-agent systems with general form of linear dynamics and undirected switching topologies has been investigated. Based on the relative outputs of neighboring agents, a distributed observer-based consensus protocol is proposed to each following agent to track the leader. A multialgorithm has been proposed to construct the consensus protocol, and the control gain matrices used in the consensus protocol are obtained by solving the Riccati equation and the Sylvester equation. A sufficient consensus condition is established by using parameter-dependent lyapunov function method under switching topologies. By using the analysis method of this paper, it is easy to establish the distributed observer-based consensus protocol for multi-agent systems under leaderless case. We also will probe multi-agent robust consensus control problems with external disturbance under time-delay switching topologies in our future work.

Acknowledgments

This work was supported by the National natural Science Foundation of China under Grants 61074123 and 61174063 and the open project of State Key Laboratory of Industrial Control Technology in Zhejiang University, China, under Grant no. ICT1218.