Semi-Global Leaderless Output Consensus of Heterogeneous Multi-Agent Systems With Input Saturations

This paper studies the semi-global leaderless output consensus problem for heterogeneous multi-agent systems with input saturations. To solve the consensus problem under the input saturations, the consensus trajectory should be bounded such that the control inputs of all agents are not saturated. However, generating a bounded trajectory satisfying this condition is not an easy problem. Accordingly, we construct the modified reference generator to generate the bounded trajectory for any bounded initial states, and two types of distributed controllers depending on the available information are proposed to track the generated trajectory. Specifically, we construct the distributed controllers using the state feedback and the output feedback based on the output regulation approach, respectively. Then, utilizing the low gain method, we show the existence of the solution for the semi-global output consensus problem. Finally, we provide numerical examples to demonstrate the theoretical results.


I. INTRODUCTION
Consensus problem for multi-agent systems has been widely studied over several decades since it has been extensively applied in many fields, such as formation control [1], [2], cooperative control [3], distributed filtering [4], [5], etc. The goal of consensus is to achieve an agreement among agents using local information exchange [6].
In the consensus problem, most of the pioneer works have focused on homogeneous agents, whose dynamics are identical, for instance, single integrator [6], [7], double integrator [8], and higher order dynamics [9], [10], etc. Moreover, the consensus problem for agents with unknown nonlinearities has been studied in [11] and [18] using the neural network control and the fuzzy adaptive control, respectively. In practice, the agents' dynamics contain different characteristics, The associate editor coordinating the review of this manuscript and approving it for publication was Haixia Cui . and, thus, the consensus problem for heterogeneous agents, whose dynamics are nonidentical, has been widely studied in recent years. The necessary and sufficient conditions for the consensus of the heterogeneous agents can be summarized as the internal model principle for synchronization, which implies the existence of a common model in the agents with local controllers [13]. Then, applying the output regulation approach, the dynamic consensus algorithms were proposed in [13], [14], [15], and [16]. Moreover, in [17] and [18], the quasi-synchronization problem has been studied since it is difficult to guarantee the existence of the solution given by the internal model principle.
Meanwhile, the problem of input saturation is an essential issue since input saturations can lead to not only performance degradation but also the instability of systems. In the presence of input saturation, it is well known that global asymptotic stabilization cannot be achieved using a linear feedback control [19], [20]. On the other hand, applying a low gain approach, a semi-global stabilization can be achieved for a linear system that is asymptotic null controllable with bounded controls(ANCBC) [20]. Therefore, most of the existing works considering the input saturations have studied a semi-global consensus problem for ANCBC heterogeneous agents applying the output regulation theory and the low gain approach [21], [22], [23], [24], [27]. The goal of semi-global consensus is to achieve an agreement for any initial conditions in a priori given arbitrarily large bounded set by low gain feedback such that the control input will not saturate. Note that the distributed controller based on the output regulation approach contains the feedforward control to render the consensus trajectory invariant. Therefore, the consensus trajectory should be bounded such that the control inputs of agents do not saturate. Then, under the assumption that the leader's trajectory is bounded within the prescribed bounded set, the leader-following consensus problem for ANCBC heterogeneous agents with input saturations have been studied in [21], [22], and [23]. In [21] and [22], the semi-global leader-following consensus problem was studied with fixed and switching topologies, respectively. In [23], the control input of the leader agent was considered and the fully distributed controller with the discontinuous functions was proposed. In [24], the suboptimal consensus algorithm was proposed using the model-based and data-driven methods.
Different from the leader-following consensus problem with input saturations, whose reference trajectory given by the leader agent is bounded, we need to generate the bounded trajectory in the leaderless consensus problem since the reference trajectory is not given. However, it is difficult to determine the bounded trajectory such that the control inputs of all agents are not saturated. Therefore, the leaderless consensus problem, which is more challenging, has rarely been addressed. In [25], the agents consisting of single and double integrators were considered. In [26], the second-order heterogeneous agents were considered. In [27], the semi-global leaderless state consensus problem for general ANCBC agents was studied and the modified reference generator based on the anti-windup technique was proposed to generate the bounded trajectory for any bounded initial conditions.
In this paper, we study the semi-global leaderless output consensus problem for heterogeneous multi-agent systems with input saturations. Depending on the available information, two types of distributed controllers are proposed to solve the leaderless output consensus problem by applying the regulation approach and the modified reference generator [27]. Specifically, when the state information is available, the distributed controller using the state feedback is proposed. When the output information is available, the distributed controller with the state observer is designed for each agent. The main contributions of this paper can be summarized as follows. First, the leaderless output consensus problem for the heterogeneous agents with input saturations is investigated. Although [27] studied the leaderless consensus problem with the relaxed initial conditions of the reference generators, they focused on the state consensus problem, which requires the identical state dimensions of agents. However, in practical applications, the agents may have different dimensions, and, thus, the output consensus problem is more realistic and challenging. Second, by utilizing the low gain approach, we show the existence of the solution for the semi-global output consensus problem. Although the control gain requires global information, by choosing the sufficiently small control gain, we can achieve the output consensus. Moreover, the control gain can be designed in a distributed way by applying the min-consensus algorithm [28].
The rest of this paper is organized as follows. In Section II, preliminaries and problem formulation are presented. In Section III, the semi-global leaderless output consensus problem is solved by constructing two consensus algorithms. In Section IV, the simulation results are presented, and the conclusion and future work are addressed in Section V.

II. PRELIMINARIES AND PROBLEM STATEMENT A. NOTATIONS AND GRAPH THEORY
For a vector x, we denote the Euclidean norm as x and the infinity norm as x ∞ . For a matrix A, we denote the largest eigenvalue as λ M (A). For matrices A and B, (AB) k denote the kth row of AB. ⊗ denotes the Kronecker product.
Let G = (V, E, A) be an weighted undirected graph of order N , with V = {1, 2, . . . , N } representing the set of nodes, E ⊆ V × V the set of undirected edges, and A = [a ij ] ∈ R N ×N the underlying weighted adjacency matrix. When (i, j) ∈ E, the nodes i and j are called adjacent, which means two nodes can exchange information with each other. The weights a ij = a ji > 0 if and only if (i, j) ∈ E; otherwise a ij = a ji = 0.
The Laplacian matrix L = [l ij ] ∈ R N ×N is defined as l ii = N j=1,j =i a ij , l ij = −a ij , i = j. The Laplacian matrix of the undirected graph is a positive semi-definite real symmetric matrix, thus all eigenvalues of L are non-negative real. An undirected graph G is called connected if between any pair of distinct nodes i and j there exists a path between i and j, i, j ∈ V. An undirected graph is connected if and only if its Laplacian has rank N − 1. In this case, the zero eigenvalue has a multiplicity one, and all other eigenvalues are positive. The remaining N − 1 eigenvalues are ordered in increasing order as 0 < λ 2 ≤ λ 3 ≤ · · · ≤ λ N .
Lemma 1 [8]: For an undirected graph and any x i ∈ R, i = 1, . . . , N , we have In this paper, we consider a group of N agents described by for i ∈ V := {1, . . . , N }: where x i ∈ R n i , u i ∈ R m i , and y i ∈ R p are the state, input, and output of agent i, respectively, and A i ∈ R n i ×n i , B i ∈ R n i ×m i , C i ∈ R p×n i . The function sat : R m i → R m i is a normalized vector valued saturation function described by The communication topology among the agents is described by an weighted undirected graph G = (V, E, A). Then, in this paper, we construct the distributed controller to solve the output consensus problem for the heterogeneous agents (1), i.e., lim t→∞ (y i − y j ) = 0, ∀i, j ∈ V. The output consensus implies the agents can generate a common trajectory. Therefore, we assume that the existence of common trajectory, which is given by the following internal model principle for synchronization [13]: Assumption 1: There exist matrices S ∈ R q×q , i ∈ R n i ×q , i ∈ R m i ×q , and R ∈ R p×q satisfying the following equations: Remark 1: Note that Assumption 1 is a necessary condition for the output consensus of the heterogeneous agents [13]. The condition (3) is known as the regulator equation, which implies that all agents can track the trajectory of the virtual exosystem described by the system matrix S and the output matrix R. In other words, the output consensus can be reached by regulating each agent to track the trajectory of the virtual exosystem, and thus the regulation equation (3) should be satisfied. The solvability of (3) is discussed in [29].
Remark 2: In practice, the equation (3) does not always have a solution. Accordingly, the quasi-synchronization problem without Assumption 1 has been studied in [17] and [18]. However, the goal of this paper is to achieve the perfect consensus, and thus Assumption 1 is required.
We next briefly introduce the standard approach to the output consensus problem without the input saturations, i.e., sat(u i ) = u i . Based on the regulation approach, the output consensus problem can be solved in two steps. First, the distributed reference generators are designed to generate the common trajectory using local information exchanges. Second, the observer-based controllers are designed to track the trajectory of the local reference generator. Then, the following distributed controller can solve the output consensus problem without the input saturations: As mentioned above, the distributed reference generator (4a) converges to the common trajectory generated by the following dynamics: Then, by regulating the reference signal w i using the state observer (4b) and the feedback controller (4c), we can achieve the output consensus, i.e., lim t→∞ z i − x i = 0 and Note that, when the agents achieve the consensus, the control input becomes u i = iw which makes the consensus trajectory invariant from Assumption 1. Thus, in the presence of input saturations, the trajectory generated by (5) should be bounded such that the control input of each agent does not saturate, i.e., sat( iw ) = iw . To generate the bounded trajectory, the initial states of the reference generators (4a) should be bounded. However, it is difficult to determine the initial states that satisfy this condition. To relax this condition, [27] proposed the modified reference generator to generate the bounded trajectory for any initial conditions in any priori given bounded set. Inspired by the work [27], this paper studies the semi-global output consensus problem by constructing two types of distributed controllers depending on the available information. Then, the problems studied in this paper can be summarized as follows: Problem 1 (Semi-Global Output Consensus Using State Feedback): Consider a group of N agents given by (1). Construct a distributed controller using state feedback such that, for any priori given bounded initial sets as long as ( In Problem II-B, it is assumed that each agent can measure the full state information. However, in many real applications, the full state information is not always available, and instead, the output information is most accessible. Therefore, we next consider the consensus problem using output feedback control, which can be summarized as follows: Problem 2 (Semi-Global Output Consensus Using Output Feedback): Consider a group of N agents given by (1). Construct the distributed controller using output feedback such that, for any priori given bounded initial sets X, as long as ( Note that the global consensus problem under input saturations can not solve in general since the control input can remain saturated for large initial conditions. However, the consensus can be achieved semi-globally, that is, the initial states of agents and controllers are bounded within a priori given bounded sets such that the control input remains unsaturated based on the low gain approach. It is worth mentioning that the priori given bounded sets can be arbitrarily large.
Throughout this paper, we impose the following assumptions: Assumption 2: The agents (1) satisfy the following conditions: The undirected graph G is connected. Note that Assumption 2-4 are the standard assumptions to solve the semi-global consensus in the presence of the input saturations [21], [22], [23].
Before discussing the main results, several preliminary lemmas are provided.
Lemma 2 [20]: Let Assumption 2 holds. Then, for each ∈ (0, 1], there exists a positive real number k * > 0 such that the following algebraic Riccati equations (AREs) have unique positive definite solutions P (k, ),i for any k ≤ k * : Moreover, E can be arbitrarily chosen, and A − BB T P (k, ),i is Hurwitz for any ∈ (0, 1]. Throughout this paper, we will dropout subscript (k, ) and write P i without ambiguity.

A. SEMI-GLOBAL OUTPUT CONSENSUS USING STATE FEEDBACK
In this subsection, we solve Problem II-B. State feedback distributed controller can be constructed based on the following three steps: Step 1) Solve the equations in (3) for ∀i ∈ V.
Step 2) Solve the following AREs for ∀i ∈ V where i ∈ (0, 1] and k * i > 0, ∀i ∈ V, can be chosen arbitrarily small such that the solution of (11) exists.
Step 3) Construct the following distributed controller for agent i where w i ∈ R q , dz δ (·) = (·) − sat δ (·), δ ∈ (0, 1), sat δ (·) = sign(·) min{| · |, δ}, and k is a positive constant satisfying Note that the existence of solution in Step 2) is given by Lemma 2. The dead zone function dz δ ( i w i ) in (12a) is used to generate the bounded trajectory. When i w i ∞ > δ, the dead zone function dz δ ( i w i ) becomes active and brings the states of the reference generators w i back into the bounded region. Therefore, as we will see below, the trajectory generated by the distributed controller (12a) will be bounded as lim t→∞ i w i ∞ ≤ δ, and δ ∈ (0, 1) determines the maximum value of the consensus trajectory. Moreover, by using sat δ ( i w i ) in (12b), the control input can be rewritten from Lemma 4 as where Then, we can determine the bounded sets X and W to apply the low gain approach such that the control input remains unsaturated, i.e., sat Then, the following theorem shows the proposed controller solves the semi-global output consensus problem.
Theorem 1: Consider a group of N agents (1) with the distributed controller (12), and let Assumption 1-3 hold. Then, for any priori given bounded sets X ⊂ R nN and W ⊂ R qN , there exist * i ∈ (0, 1] such that, for each i ∈ (0, * i ] and (x(t 0 ), w(t 0 )) ∈ X × W, the distributed controller (12) solves the semi-global output consensus problem.
Proof: We first define the error states as e i = x i − i w i , ∀i ∈ V. Then, from (14), we have where Then, from Assumption 1, we have the error dynamics as follows: where It is clear that the output consensus implies lim t→∞ e i = 0, lim t→∞ w i − w j = 0 ∀i, j ∈ V. Then, to prove the output consensus, we next choose the following Lyapunov function candidate: where P i is the solution of AREs (11), and α 1 and α 2 are the positive constants satisfying with L being the Laplacian matrix and = diag( 1 , . . . , Such a c exists since X and W are bounded and lim i →0 P i = 0, ∀i ∈ V. Let * i ∈ (0, 1] be such that, for each i ∈ (0, where e = [e T 1 , . . . , e T N ] T and w = [w T 1 , . . . , w T N ] T , implies that Then, within L V (c), the input of the error dynamics (16) does not saturate. We next consider the time derivative of V e for (e, w) ∈ L V (c) and i ∈ (0, * i ] aṡ Then, from AREs (11), we havė where Note that, from (17), it follows that where P = diag(P 1 , .., P N ) and = diag( 1 , . . . , N ). Therefore, we have from (24) and (25) thaṫ where we have used the fact that (k − k * i )|| i T i P i e i || 2 ≤ 0 since 0 < k ≤ min i∈V k * i . We next consider the time derivative of V w . Since S +S T = 0 from Assumption 3, we havė Therefore, from (26) and (27), the time derivative of V can be written aṡ Note that, from the condition of α 1 in (19), i.e., α 1 (1+k) < 2α 2 k, and Lemma 3, we have Moreover, the conditions of α 1 and α 2 in (19) give Therefore, the time derivative of V can be rewritten aṡ Therefore, the agents achieve the output consensus. Moreover, from the definition of d i (17), it follows that dz δ ( i w i ) = 0 in M, which implies lim t→∞ i w i ∞ ≤ δ, ∀i ∈ V.

B. SEMI-GLOBAL OUTPUT CONSENSUS USING OUTPUT FEEDBACK
In this subsection, we solve Problem II-B. To construct the distributed controller using the output feedback, the state observer is designed to estimate the states of agents. Then, the proposed controller can be constructed based on the following three steps: Step 1-2) Step 1-2 are the same as Step 1-2 in Section III-A.
Step 3) Construct the following distributed controller for agent i where w i ∈ R q , z i ∈ R n i , k is a positive constant satisfying (13), and H i ∈ R n i ×p such that The following theorem shows that Problem II-B can be solved based on the output feedback distributed controller (32).
Proof: Let us define the error states as e i = z i − i w i and ζ i = x i − z i . From (15) and Assumption 1, we have the error dynamics as follows: where We next show the output consensus under the proposed controller, that means lim t→∞ e i = 0, lim t→∞ ζ i = 0, lim t→∞ w i − w j = 0, ∀i, j ∈ V. Then, we choose the following Lyapunov function candidate: where P i and Q i are the solutions of AREs (11) and (33), respectively, and α 1 , α 2 , α 3 are the positive constants satisfying . . . , N ). Let c > 0 be such that Such a c exists since X, Z, and W are bounded and lim i →0 P i = 0, ∀i ∈ V.
Let * i ∈ (0, 1] be such that, for each i ∈ (0, VOLUME 10, 2022 where e = [e T 1 , . . . , e T N ] T and ζ = [ζ T 1 , . . . , ζ T N ] T , implies that implies that Then, within L V (c), the input of the error dynamics (34) does not saturate. We next consider the time derivative of V e for (e, w, ζ ) ∈ L V (c) and i ∈ (0, * i ] aṡ Then, from AREs (11), we havė where v i = B T i P i e i . Note that, from (35), it follows that Therefore, we have from (43) and (44) thaṫ where we have used the fact that (k − k * i )|| i T i P i e i || 2 ≤ 0 since 0 < k ≤ min i∈V k * i . We next consider the time derivative of V w . Since S +S T = 0 from Assumption 3, we havė Moreover, from (33), the time derivative of V ζ is given bẏ Therefore, from (45), (46), and (47), the time derivative of V can be written aṡ Note that, from the condition of α 1 in (38), i.e., α 1 (1+k) < 2α 2 k, and Lemma 3, we have Moreover, from the conditions of α 1 , α 2 , and α 3 in (38), it follows for ∀i ∈ V that Therefore, the time derivative of V can be rewritten aṡ Remark 3: Note that the condition in (13) requires the global information to choose the control gain k, i.e., k * i , ∀i ∈ V. However, by choosing the sufficiently small control gain k, the condition can always be satisfied. The control gain k also can be chosen in a distributed way by applying the min-consensus algorithm [28].

IV. SIMULATION RESULTS
In this section, we present numerical examples to demonstrate the theoretical results. Consider a group of twelve agents. The communication topology among the agents is given in Fig. 1, and we consider a ij = 1 if (i, j) ∈ E and a ij = 0 otherwise. The dynamics of agents are described in (1) with i ∈ {1, . . . , 4} : We next solve the AREs (11) in Step 2 for i = 0.01 with k * i = 1, ∀i ∈ V, to obtain P i as follows: Then, we construct the distributed controller (12) in Step 3 with k = 1 = min i∈V k * i and δ = 0.5. In this simulation, we randomly choose the initial conditions of each states of the agents and the reference generators    within the intervals [−5, 5] and [−10, 10], respectively. The simulation results are given in Fig. 2-5. Fig. 2 and 3 show the evolutions of the outputs, and Fig. 4 show the trajectories of the inputs. The simulation results show the outputs converge to the common trajectory generated by the reference generators under the input saturations. Moreover, Fig. 5 shows the proposed reference generator (12a) generates the bounded common trajectory.
We next compare the proposed reference generator (12a) with the standard reference generator (4a) in [13], [14], [15], and [16] under the same initial conditions. As discussed in Section II-B, the feedforward term i w i generated by the standard reference generator (4a) can exceed the saturation   level as shown in Fig. 6. Therefore, the control input using the standard distributed controller in the form of (4) remains saturated, and the semi-global consenseus problem can not be solved in the presence of input saturations. However, the proposed reference generator (12a) generates the bounded trajectory such that the control inputs do not saturate as shown in Fig. 7.
Example 2: In this example, we construct the distributed controller using output feedback in Section III-B. From The initial conditions are chosen the same as in Example 1 with z i = 0, ∀i ∈ V. Then, the simulation results are shown in Fig. 8-11. We can see from Fig. 8-10 that the proposed controller can solve the output consensus problem under the input saturations. In Fig. 11, we can see that the observer errors ζ i = x i − z i converge to zero.

V. CONCLUSION
This paper has studied the leaderless output consensus problem for heterogeneous multi-agent systems with input saturations. We have constructed the modified reference generator which generates the bounded trajectory to avoid input saturation. Moreover, applying the output regulation theory, we have constructed the state feedback and the output feedback controller, respectively, to achieve the output consensus. Then, we have analyzed the existence of the solution that solves the semi-global leaderless output consensus problem under input saturations. Although we have shown that the proposed controllers can solve the leaderless consensus problem under input saturations, the performance of the closed-loop system has not been considered. Therefore, future work will be focused on constructing distributed controllers to improve the convergence speed of overall networked agents. He is currently an Associate Professor with the Department of Convergence Electronic Engineering, Gyeongsang National University. His research interests include saturation constraints and multiagent systems. VOLUME 10, 2022