Observer-Based Consensus Control for Heterogeneous Multi-Agent Systems with Output Saturations

: This paper studies the consensus problem for heterogeneous multi-agent systems with output saturations. We consider the agents to have different dynamics and assume that the agents are neutrally stable and that the communication graph is undirected. The goal of this paper is to achieve the consensus for leaderless and leader-following cases. To solve this problem, we propose the observer-based distributed consensus algorithms, which consists of three parts: the nonlinear observer, the reference generator, and the regulator. Then, we analyze the consensus based on the Lasalle’s Invariance Principle and the input-to-state stability. Finally, we provide numerical examples to demonstrate the validity of the proposed algorithms.


Introduction
In the past decade, the consensus problem for multi-agent systems has received a lot of attention since it has wide applications such as formation control [1], and distributed filtering [2], cooperative control [3]. The goal of consensus is to achieve an agreement using local interactions between agents, and the consensus is one of the most studied methods to control the multi-agent systems in a distributed way.
Most pioneering works have considered the consensus problem for homogeneous agents, which have identical dynamics, with single-integrator [1,2,4,5], double-integrator [6][7][8], highorder linear [9][10][11][12], and nonlinear dynamics [13][14][15]. However, in real applications, it is often unrealistic for all agents to have an identical model. Therefore, in recent years, consensus for heterogeneous agents, which have nonidentical dynamics, has been widely studied [16][17][18][19]. Specifically, the necessary and sufficient condition for heterogeneous linear agents was studied in [16]. They showed that all agents must have a common internal model such that they can generate the same trajectory. Then, under this assumption, the observer-based consensus algorithm was constructed via the output regulation approach. In [17], the dynamic consensus protocol was proposed for leaderless and leader-following cases. In [18], the output consensus problem using the state feedback and the output feedback was studied. In [19], the dynamic event-triggered consensus protocol was developed based on input-to-state stability. In [20], heterogeneous oscillator were considered and the adaptive observer was developed.
Most actuators and sensors in real systems have saturation constraints owing to their limited capacity. Since the saturations lead to poor performance of the system, control problems for systems under saturations are an important issue in real applications. Specifically, the consensus problem for agents with input saturations has been widely studied in recent years [21][22][23][24][25][26][27][28]. For homogeneous agents with input saturations, semiglobal consensus and global consensus were studied in [21][22][23][24][25]. For heterogeneous agents with input saturations, the semi-global output consensus was studied in [26][27][28]. Although there are many results dealing with input saturations in the consensus problem, the consensus problem under the output saturations has rarely been studied [29][30][31][32]. Since the consensus algorithm uses relative information between two agents, consensus may not be realized when the outputs are saturated. In [29][30][31], the necessary and sufficient conditions for single-integrator agents with output saturations were investigated. They showed that the weighted average in a group should be bounded such that the consensus trajectory can be measured. In [32], leader-following consensus for homogeneous highorder linear agents with output saturations was studied. The authors developed an observer-based consensus algorithm inspired by the nonlinear observer [33] and analyzed the asymptotic convergence based on the Lyapunov stability theory. However, to the best of our knowledge, the existing works on the consensus problem under the output saturations have considered homogeneous agents.
Motivated by the above observations, this paper studies the consensus problem for the heterogeneous agents with output saturations. We assume that the agents are neutrally stable and the communication graph is undirected. Then, we propose a distributed observer-based consensus algorithm based on the output regulation approach. The main contributions of this paper are summarized as follows. First, the output consensus problem for the heterogeneous agents with output saturations is investigated, and the leaderless and leader-following cases are considered. Therefore, this paper is a generalized version of the previous papers, which considered homogeneous agents [29][30][31][32]. Second, we construct the observer-based algorithm considering the output saturations. The output regulation approach has been applied to solve the consensus problem for heterogeneous agents in [16][17][18]. Then, by solving the linear matrix equations, called the regulation equations, they developed the consensus algorithms, which consist of three parts: the first part is the state observer, the second part is the reference generator, and the third part is the regulator. Then, by choosing control gain matrices such that the error systems are Hurwitz, the consensus is analyzed. However, in the presence of output saturations, the analysis techniques of [16][17][18] cannot be applied, since the observer contains saturation nonlinearity. Therefore, inspired by the works [32,33], this paper proposes the nonlinear observer and analyzes the consensus based on the Lasalle's Invariance Principle and the input-to-state stability.
The rest of this paper is organized as follows. In Section 2, the mathematical background and problem formulation are presented. In Section 3, the observer-based consensus algorithms for the leaderless and leader-following cases are constructed. In Section 4, numerical examples are provided, and conclusions are made in Section 5.

Notations and Graph Theory
For a vector x ∈ R n , x (i) denotes the ith component of x. For a matrix A ∈ R n×n , A T and A −1 denote the transpose and the inverse of A, respectively, and λ i (A), i = 1, 2, ..., n, are the eigenvalues of A in ascending order, i.e., λ 1 (A) ≤ λ 2 (A) ≤ · · · ≤ λ n (A). We say that A ∈ R n×n is Hurwitz if every eigenvalue of A has strictly negative real part, and is neutrally stable if every eigenvalue of A has non-positive real part with those on the imaginary axis being simple. A ⊗ B denotes the Kronecker product of A and B. I N ∈ R N×N and 1 N ∈ R N denote the identity matrix and the column vector with all entries equal to 1. blkdiag(A i ) N i=1 represents a block-diagonal matrix with matrices A i , i = 1, ..., N, on its diagonal.
Let G = (V, E , A) be an undirected graph, which represents the communication between agents, with a set of nodes (or agents) V := {1, 2, ..., N}, a set of undirected edges E ⊆ V × V, and an weighted adjacency matrix A = [α ij ] ∈ R N×N . For the undirected graph, if (i, j) ∈ E , then (j, i) ∈ E , which means the agents i and j can communicate with each other. The weights α ij = α ji > 0 if and only if (i, j) ∈ E and α ij = α ji = 0 otherwise.
The Laplacian matrix of the graph G is denoted by L = [l ij ] ∈ R N×N , where l ii = ∑ N j=1,j =i α ij , l ij = −α ij , i = j, and, thus, L is a positive semi-definite matrix with 0 = λ 1 (L) ≤ λ 2 (L) ≤ · · · ≤ λ N (L). The undirected graph is connected if there exists a path between any two distinct nodes. For the connected graph, L has a simple zero eigenvalue that is 0 = λ 1 (L) < λ 2 (L).

Problem Formulation
This paper considers a heterogeneous multi-agent system. The dynamics of each agent is described byẋ where x i ∈ R n i , u i ∈ R m i , y i ∈ R q , and z i ∈ R q represent the state, control input, controlled(or real) output, and measured output of ith agent, respectively, and A i , B i , and C i are real constant matrices with compatible dimensions. The function σ(·) is a normalized standard saturation function defined by σ(y i ) = σ(y i(1) ), σ(y i(2) ), ..., σ(y i(q) ) T σ(y i(j) ) = sign(y i(j) ) min y i(j) , 1 . (2) Then, the goal of this paper is to achieve the output consensus, that is, To achieve the consensus, we require some standard assumptions. We first consider the existence of solution to the output consensus (3). It was shown in [16] that the necessary condition for the output consensus is the existence of a common internal model such that all agents can generate the same trajectory. This condition can be summarized as the following assumption [16]: Assumption 1. There exist matrices S ∈ R n 0 ×n 0 , R ∈ R q×n 0 , Π i ∈ R n i ×n 0 , and Γ i ∈ R m i ×n 0 for the following linear matrix equations: We next consider the following assumptions to control the agents under output saturations.

Assumption 2.
The agents satisfy the following conditions: The matrices A i , ∀i ∈ V, and S are neutrally stable.
Note that Condition 1 in Assumption 2 is the standard assumption to construct an observer-based controller. Moreover, Condition 2 requires controling the system in the global sense, since we cannot track the exponentially growing signals in the presence of saturation nonlinearities. Next, we suppose that the communication graph between the agents in (1) is given by G = (V, E , A). Then, we further consider the following assumption, which is the necessary condition for the consensus. Before we construct the consensus algorithm, we consider the dynamics of agents in (1). From Condition 2 of Assumption 2, there exits a non-singular matrix T i ∈ R n i ×n i such that [32] where A h,i ∈ R n h,i ×n h,i , and A s,i ∈ R (n i −n h,i )×(n i −n h,i ) are Hurwitz and skew-symmetric matrix, respectively, the pair (A s,i , B s,i ) is controllable, and (C s,i , A s,i ) is observable. Then, we consider the following lemma that will be used to construct the consensus algorithm [32].

Lemma 1.
For the matricesĀ i andC i in (5), there exists a positive definite matrix P i ∈ R n i ×n i given by with a symmetric positive semi-definite matrix We next consider the following lemmas that will play a crucial role to analyze the consensus [33,34].
, and no solution can stay identical in M except for x = 0; then, the origin is globally asymptotically stable.
is input-to-state stable and the origin of the systemż = f 2 (z) is globally asymptotically stable, then the origin of the cascade connectionẋ is globally asymptotically stable.

Main Results
In this section, we construct the consensus algorithms for heterogeneous agents with the output saturations and consider the cases of leaderless and leader-following. The proposed algorithms are composed of three parts, as shown in Figure 1. The first part is the nonlinear observer to measure the state using the measured output, the second part is the distributed reference generator to generate the common trajectory, and the third part is the controller to track the common trajectory based on the output regulation theory.

Leaderless Case
In this subsection, we consider the output consensus without the leader agent. We propose the observer-based consensus algorithm as follows: wherex i ∈ R n i and w i ∈ R n 0 are the states of the observer and the reference generator of the ith agent, respectively; H i , F and K i are the gain matrices with compatible dimensions that will be determined later; and S, Π i and Γ i are the constant matrices satisfying Assumption 1. Before we present our main result, we consider the following lemma [11].

Lemma 4.
Suppose that the pair (S, Q) is stabilizable. Then, there exists a symmetric positivedefinite matrix W such that Then, we can solve the leaderless output consensus problem from the following theorem. (1), and suppose that Assumptions 1-3 hold. Then, we can achieve the output consensus using the observer-based consensus algorithm (8) if the gain matrices satisfy the following conditions:

Theorem 1. Consider a group of N heterogeneous agents
where β i is any positive constant, T i andC i are given in (5), and P i is given in Lemma 1.

2.
F = τQQ T W, where τ is a positive constant such that τλ 2 (L) > 1, and Q and W are the solution of (9) in Lemma 4.

K i is a constant matrix such that
wherew is the solution of the following dynamics:w Proof of Theorem 1. To prove the consensus, we investigate the asymptotic stability of the error dynamics. Then, we first consider the reference generator (8b) and define the state vector w = [w T 1 , ..., w T N ] T . Then, from (8b), we havė Since the graph is undirected and connected, there exists an orthogonal matrix and U T 2 U 2 = I N−1 [14].
We next consider the observer (8a) and the regulator (8c) and define the observer error and the regulation error as e i = T i (x i −x i ) and i =x i − Π i w i , respectively. Then, from (5), the observer error dynamics can be written aṡ and, from Assumption 1, the regulation error dynamics is given bẏ Let (12), (13), and (14), the overall error dynamics can be written aṡζ It is clear that we can achieve the consensus if the error dynamics (15) is asymptotically stable, i.e., if (ζ, e, ) → (0, 0, 0), then y i − y j = 0, ∀i, j ∈ V. Then, to prove the consensus, we first analyze the asymptotic convergence ofζ and e to the origin, applying Lemma 2. We define the continuously differentiable functionV as follows: where W is the solution of (9) in Lemma 4 and P = blkdiag(P i ) N i=1 with P i given in Lemma 1.
Moreover,ζ = 0 implies w i =w, and, thus, from Condition 4 in Theorem 1, we have Then, for t ≥ t 1 , the observer error dynamics can be rewritten aṡ Since (Ā − βP −1CTC ) is Hurwitz from Lemma 1, e converges to 0. In summary, we have shown thatV ≤ 0 and e =ζ = 0 is a unique, asymptotically stable equilibrium point in M. Therefore, according to Lemma 2, we have lim t→∞ e i = 0 and lim t→∞ ζ i = 0, ∀i ∈ V. Moreover, since (A + BK) is Hurwitz, the regulation error dynamics is input-to-state stable. Then, from Lemma 3, we have lim t→∞ i = 0. Finally, we can conclude that (ζ, e, ) → (0, 0, 0) and lim t→∞ C i x i − C j x j = 0, ∀i, j ∈ V, which completes the proof.

Leader-Following Case
In this subsection, we consider the leader agent, which generates the reference trajectory. The dynamics of the leader is given bẏ where x 0 ∈ R n 0 and y 0 ∈ R q are the state and output, respectively, of the leader. S and R are the constant matrices satisfying Assumption 1. Then, to achieve the consensus, we consider the following assumption to control the follower agents into the leader's trajectory.

Assumption 4.
The leader agent (22) satisfies the following conditions: For the leader agent, there exists 0 < δ < 1, satisfying Then, to achieve the consensus, we propose the following consensus algorithm:x wherex i ∈ R n i and w i ∈ R n 0 are the states of the observer and the reference generator of the ith follower, respectively; H i , F and K i are the gain matrices with compatible dimensions that will be determined later; and Π i and Γ i are the solution of linear matrix equations in Assumption 1. We next consider the following lemma [10], which is the dual problem of Lemma 4: Lemma 5. Suppose that the pair (R, S) is detectable. Then, there exists a symmetric positivedefinite matrix W such that Then, we can solve the leader-following output consensus from the following theorem.
Theorem 2. Consider a group of N follower agents (1) with the leader (22). Suppose that Assumptions 1-4 hold, and there exists at least one follower that can receive the information from the leader. Then, we can achieve the leader-following output consensus using the observer-based consensus algorithm (24) if the gain matrices satisfy the following conditions: where β i is any positive constant, T i andC i are given in (5), and P i is given in Lemma 1. 2. F = τW −1 R T R, where τ is a positive constant such that τλ 1 (L) > 1, whereL = L + diag(α 10 , ..., α N0 ), and the positive definite matrix W is the solution of (25) in Lemma 5.

3.
K i is a constant matrix such that A i + B i K i is Hurwitz.
Proof of Theorem 2. We apply the same procedure as in the proof of Theorem 1. We define the tracking error of the reference generator, the observer error and the regulation error as ζ i = w i − x 0 , e i = T i (x i −x i ), and i =x i − Π i w i , respectively. Then, the tracking error dynamics is given bẏ and, from (13) and (14), the observer error dynamics and the regulation error dynamics are given byė where we have used the fact that (w j − w i ) = (ζ j − ζ i ).

Remark 1.
Note that Theorems 1 and 2 give the sufficient conditions to achieve the output consensus. The existence of control gains follows from Assumptions 1-4, which are the standard assumptions to achieve the consensus for the heterogeneous agents. Moreover, the control gains can be constructed from the knowledge of the system matrices except τ, which requires the global information, i.e., λ 2 (L) in Theorem 1 and λ 1 (L) in Theorem 2. However, by choosing τ as an arbitrarily large constant, the conditions can be satisfied.

Simulations
In this section, we present two numerical examples to demonstrate the theoretical results.

Leaderless Case
In this subsection, we consider the leaderless case with 10 agents modeled by harmonic oscillators. The dynamics of each agent is of the form (1) with, for i ∈ V = {1, 2, ..., 10}, Then, we can choose the non-singular matrix T i as follows: which gives, from (5), We next consider the linear matrix equations (4) in Assumption 1. The analytic solution of (4) is given in [20] as follows: In this simulation, we assume that the communication between agents is as ring topology given in Figure 2. We consider α ij = 1, if (i, j) ∈ E and α ij = 0 otherwise. Then, the Laplacian matrix is given by and λ 2 (L) = 0.3820. We next construct the observer-based consensus algorithm (8) applying Theorem 1. From the condition 1, we choose We next choose W = Q = I 2 such that (S, Q) is stabilize and Lemma 4 is satisfied. Then, from the condition 2, we choose The simulation results using the proposed algorithm are shown in Figures 3-6. Figure 3 shows the state trajectories of the reference generators. As we can see from Figure 3, the reference generators converge to the common trajectory. Figure 4 shows the square norms of the observer errors, which converge to zero. Thus, the states of the agents can be measured under the proposed nonlinear observer (8a). Figure 5 depicts the controlled and the measured output trajectories of agents. Although the measured outputs are saturated, the agents achieve the consensus under the proposed algorithm. Moreover, to investigate the effect of the control gain β i , we conduct the simulation with β i = β = 1, 5, 10, 100, and the square norms of the output errors between agents are shown in Figure 6. The simulation result shows that, as the control gain β i increases, the agents achieve a fast convergence speed.

Leader-Following Case
In this subsection, we consider a group of four agents (1), i.e., V = {1, 2, 3, 4}, and a leader agent (22) with [18] which satisfies Assumption 2 and Condition 1 in Assumption 4. Then, we can choose the non-singular matrix T i = I 2 , ∀i ∈ V and the positive semi-definite matrix P i satisfying Lemma 1 as follows: In this simulation, we consider the communication topology between the agents given in Figure 7. We consider α ij = 1, if (i, j) ∈ E and α ij = 0 otherwise. Then, the Laplacian matrix is given by and P i given in (42). We next choose W satisfying Lemma 5 as Then, from Condition 2, we choose F = τW −1 R T R with τ = 3 such that τλ 1 (L) = 1.2 > 1. Finally, we choose the gain matrix K i = −[100 100], ∀i ∈ V such that A i + B i K i is Hurwitz

Conclusions
In this paper, the output consensus problems for heterogeneous agents were studied under the output saturations. Applying the output regulation approach to solve the consensus problem, we have proposed the observer-based consensus algorithms considering leaderless and leader-following cases. Specifically, the proposed algorithm consists of three parts: the nonlinear observer, the reference generator, and the regulator. By defining the error dynamics, we have transformed the consensus problem into the stability problem of the error dynamics. Then, based on the Lasalle's Invariance Principle and the input-tostate stability, the stability of error dynamics and the existence of the control gains have been derived under the standard assumptions for the consensus. Finally, two numerical examples have been given to demonstrate the theoretical results. Although the effect of the control gain β i has been investigated by simulation, the performance in a group has not been addressed. Thus, the consensus control with the performance analysis would be worthwhile for a further study. Moreover, as mentioned in Remark 1, the proposed algorithm requires global information, i.e., λ 2 (L) and λ 1 (L). To solve this problem, the fully distributed algorithm has been widely used [14]. By using the state dependent control gain, the consensus can be solved without global information. Therefore, it would be interesting to extend the results of this paper to fully distributed consensus.

Conflicts of Interest:
The authors declare no conflict of interest.