Semi-global leader-following output consensus of heterogeneous systems subject to actuator position and rate saturation

We study in this paper a semi-global leader-following output consensus problem for multiple heterogeneous linear systems in the presence of actuator position and rate saturation over a directed topology. For each follower, via the low gain feedback design technique and output regulation theory, both a state feedback consensus protocol and an output feedback consensus protocol are constructed. In the output feedback case, different distributed observers are designed for the informed followers and uninformed followers to estimate the state of the leader and the follower itself. We show that the semi-global leader-following output consensus of heterogeneous linear systems can be achieved by the two consensus protocols if each follower is reachable from the leader in the directed communication topology.

spanning tree. The discrete-time counterparts of these results are also obtained in [18,19]. There are also some results on global consensus problem of multi-agent systems subject to actuator saturation. In particular, Zheng et al. [21] constructed a saturated consensus protocol for each second-order follower system under undirected connected graphs, but the input bound depends on the communication graph, which may not meet the requirements of real systems. It is shown by Meng et al. [22] that for neurally stable systems and double integrator systems, the global leader-following consensus can be achieved by linear local feedback laws over a static communication topology and nonlinear local feedback laws over a switching topology. As an extension of the results done by Meng et al. [22], Xie and Lin [23] constructed a bounded consensus protocol with intermittent directed communication to solve the global leader-following consensus problem for a group of agents described by a chain of integrators of an arbitrary length.
Besides the position saturation of actuators, actuator rate saturation may worsen the performance of the closedloop system, and may even lead to instability. As reported in [24], actuator saturation is exactly a contributing factor for the mishaps of YF-22 fighter aircraft. Therefore, it is crucial to take into account both the actuator position and rate saturation in the consensus problem for a multi-agent system. The position and rate-limited case is firstly studied by Lin [25] to solve the semi-global stabilization problem of a linear system if the open-loop system is stabilizable and all its poles located at the closed lefthalf complex plane. Lim and Ahn [26] are for a class of nonlinear interconnected systems where a decentralized state feedback controller is proposed based on linear matrix inequality conditions. In recent years, the methods of Lin [25] are extended to the coordination control of multiple linear systems subject to actuator position and rate saturation. The semi-global containment control and leader-following consensus problems are, respectively, considered by Zhao and Lin [27], and Zhao and Shi [28], where both state feedback control and output feedback control are proposed under connected undirected graphs.
To the best of our knowledge, there is no result on the output consensus problem for multiple heterogeneous systems with both actuator position and rate saturation, which is exactly the problem we consider in this paper. By the low gain approach and output regulation theory, we construct both a state feedback consensus protocol and output feedback consensus protocol for each follower over a directed network. In the state feedback type, the protocol is designed based on a distributed observer that estimates the state of the leader. In the output feedback type, different distributed observers that estimate the state of the leader and the follower itself are designed for the informed followers and uninformed followers which respectively have and do not have access to the output of the leader. It is worthy to note that the methods in the consensus or containment control for linear systems in [27,28] can not be used to solve the consensus of heterogeneous systems by a simple modification under output regulation theory. The nontrivial consensus protocols we designed allow the actuator rate converge to the value it should be when the consensus problem is solved. Moreover, our consensus protocols are applicable to directed topologies.
The outline of the rest of this article is as follows. Section 2 gives the definitions of both semi-global state feedback type and output feedback type leader-following output consensus of heterogeneous linear systems with position and rate-limited actuators. Two corresponding consensus protocols are respectively constructed in Subsections 3.1 and 3.2. We give illustrative examples in Section 4 to verify the effectiveness of the two control laws. Finally, we conclude our work by Section 5 with some remarks.
Throughout this paper, for a time constant T ≥ 0 and a signal x : denotes a vector with all elements being 1. I n ∈ R n×n is the identity matrix. Kronecker product is denoted by ⊗. X T stands for the transpose of the vector or matrix X. 0 represents a vector or matrix of zero with appropriate dimension.

Problem formulation and preliminaries
Consider a group of N + 1 heterogeneous systems consisting of a leader and N followers. The leader, labeled as 0, is described as where w ∈ R s , y 0 ∈ R m are the state and output, respectively. Similar to the system in [29], the dynamics of the i-th follower, i = 1, 2 · · · , N, is subject to actuator position and rate saturation and it is described by the following equation, where x i ∈ R n i , y i ∈ R m and u i ∈ R q i are respectively, the plant state, output and control input of the i-th follower. The second equation denotes the actuator dynamics with state v i ∈ R q i . The positive definite diagonal matricesT i = diag{τ i,1 , τ i,2 , · · · , τ i,q i } ∈ R q i ×q i represents the "time constants" of the actuators. e i ∈ R m , which is called the regulated output, denotes output tracking error between the i-th follower and the leader. W i w with W i ∈ R n i ×n represents external disturbances caused by the leader. σ p (·), σ r (·) : R q i → R q i represent vector valued saturation functions with p and r are known constants. For In this paper, we aim to design a distributed control law u i for each follower so that the output tracking error satisfies lim t→∞ e i = 0.
The communication topology among the leader and the followers is represented by a directed graph G = {V, E}, with V = {0, 1, · · · , N} being the node set and E = V × V being the edge set. For i, j ∈ V, (j, i) ∈ E if and only if node i have access to the information of node j. Then node j is called the neighbor of node i and node i is called a child of node j. We use F = {1, 2, · · · , N} to denote the set of followers, and use N i := {j : (j, i) ∈ E} to represent the set of neighbors of node i. Depending on whether or not the followers have access to the information of the leader, the followers are divided into two classes. The informed ones can obtain the information of the leader, while the uninformed ones can not, and we use F in and F un to represent the informed followers and the uninformed followers, respectively, that is, F in := {i : i ∈ F, (0, i) ∈ E} and F un := F \ F in . Without loss of generality, we assume the first l followers are the informed ones and the left N −l followers are the uninformed ones. If the graph contains a sequence of edges (i 1 , i 2 ), (i 2 , i 3 ), · · · , (i k−1 , i k ), then we say there is a directed path from node i 1 to i k , or i k is reachable from i 1 . For a directed graph G, the adjacency According to the classification of the leader, the informed followers and the uninformed followers, L can be partitioned as Assumption 1 There is a directed path from the leader node 0 to each follower node i.

Assumption 2 The following regulator equations
have a pair of solutions i ∈ R n i ×s and i ∈ R q i ×s .

Assumption 3
For each i = 1, 2, · · · , N, the pair (A i , B i ) is stabilizable, and all eigenvalues of A i have non-positive real parts.

Assumption 5
For each i = 1, 2, · · · , N, there exist a time T ≥ 0 and two positive constants δ p and δ r , such that (1), it implies that all eigenvalues of S have non-positive real parts, and those eigenvalues with zero real parts are semi-simple. Besides, Assumption 5 also implies p > i w T,∞ and r > i Sw T,∞ . i w and i Sw can be viewed as the generalized actuator position and rate of the leader. If the actuator position or rate of each follower is less than that of the leader, i.e., p < i w T,∞ or r < i Sw T,∞ , it is impossible for the followers to catch up the leader when it moves at its maximal pace. Thus, Assumption 5 is reasonable in real applications.

Remark 1 Assumption 5 means that w is bounded for all time t ≥ T. Since w is determined by
For the case that the states of all agents can be measured, a distributed observer is firstly designed for each follower to estimate the state of the leader. Consider the following distributed observer: where μ 1 is a positive constant such that (I N ⊗S −μ 1 L ff ⊗ I s ) is Hurwitz. Such a μ 1 exists because under Assumption 1, all eigenvalues of L ff have positive real parts (see the work of Hong et al. [30]).
The state feedback-based semi-global leader-following output consensus problem formed by followers (2) and leader (1) is defined as follows.

Problem 1 (State feedback-based semi-global leaderfollowing output consensus problem) Consider a multi-agent system consisting of leader (1) and followers (2). Assume that Assumptions 1-3 and 5 hold. Let
T , n = n 1 + n 2 + · · · + n N , and q = q 1 + q 2 + · · · + q N . For a priori given bounded sets However, in reality, the plant state information of the agents may not be available, while only the outputs can be measured. In such a case, output-based estimations of the states of the leader and followers should be designed for each follower. To this end, consider the following dynamic compensators: wherex i ∈ R n i andŵ i ∈ R s are respectively designed to estimate the state of the i-th follower itself and the state of the leader. L S,i and L A,i are observer gains such that S + L S,i Q and A i +L A,i C i are Hurwitz. μ 2 is a positive constant such that (I N−l ⊗ S − μ 2 L 5 ⊗ I s ) is Hurwitz. Such a μ 2 exists because all eigenvalues of L 5 have positive real part under Assumption 1 (see the work of Li et al. [31]).

Problem 2 (Output feedback-based semi-global leaderfollowing output consensus problem)
Consider a multi-agent system consisting of leader (1) and followers (2). Assume that

Output consensus over directed topologies
In this section, we will propose two consensus protocols, the state feedback type and the output feedback type, to solve the semi-global leader-following output consensus problems defined in Section 2. For each i = 1, 2, · · · , N, let P i ( ) be the solution of the parametric ARE in (4). According to Lemma 1, P i ( ) is unique and positive definite, and it satisfies lim →0 P i ( ) = 0. For notation convenience, we denote P i := P i ( ) hereafter.

Semi-global output consensus via state feedback
In this section, based on the distributed leader state observer (5), the following state feedback consensus protocol is constructed to solve Problem 1: where P i is the solution of (4) with γ = 2, i and i are a pair solution of the regulator Eq. (3), η i is the state of the observer (5). The first three terms of (9) follows from the fact that if lim t→∞ e i = 0, then lim t→∞ ( (9) is to make up for the term −T i v i of (2). The distributed control law (9) is a combination of a purely decentralized control law and a distributed observer.
Theorem 1 Consider a multi-agent system consisting of leader (1) and followers (2). Assume that Assumptions 1-3 and 5 hold. The state feedback consensus protocols (9) solve Problem 1. That is, for a priori given bounded sets X 0 , V 0 , W 0 and Z 0 , there exists an * ∈ (0, 1] such that for each ∈ (0, * ] and for all [ Proof Denote the estimation error byη i = η i − w, and η =[η T 1 , η T 2 , · · · ,η T N ] T . Followed by (5),η is determined by the following equation: where the last equality holds is due to the first equation of (3) (Assumption 2). 1 , · · · , A N }, B = diag{B 1 , · · · , B N }, P = diag{P 1 , · · · , P N }, = diag{ 1 , · · · , N }, = Zhou and Chen Autonomous Intelligent Systems (2021) 1:8 Page 5 of 13 diag{ 1 , · · · , N }. Then the compact form of (10) can be written aṡ and v follows the dynamic equatioṅ Define the Lyapunov function candidate Notice that V 1 is positive definite. According to Assumption 5, we have Recall that lim t→∞ηi = 0. For simplicity, we assume for all ∈ (0, 1] and all initial conditions ofη i (0). For any and v(T) belong to bounded setsX T and V T , respectively, independent of , since they are determined by linear differentiate equations with bounded inputs. According to Remark 1, w(t) is bounded. Therefore, there exists a bounded set W T such that w(T) ∈ W T . Let c 1 > 0 be a constant such that Such a c 1 exists because X T , V T and W T are bounded, and lim →0 P = 0. Define The existence of such an * is due to the fact that lim →0 P i = 0. The derivative of V 1 along the trajectories (1), (11) and According to (13) and (14), we have (i) We first consider the case that and Taking (17) and (18) into (16) giveṡ There exists an * 1 ∈ (0, 1] such that for any (ii) Next, we consider the case that It follows that (16) can be rewritten aṡ Since r − i Sw < δ r and θ i ≤ δ r 2 , there exists an * 2 ∈ (0, 1] such that for any ∈ (0, * 2 ], φ i + r + i Sw − θ i > 0. Thus, it follows that (19) holds.

Semi-global output consensus via output feedback
In this section, we construct the following output feedback consensus protocol based on the distributed observers (6)- (8): where P i is the solution of (4) with γ = 1, i and i are a pair solution of the regulator Eq. (3),x i andŵ i are the states of the distributed observers (6)- (8). The idea of designing control law (21) is similar to the design of control law (9).
Denotex i =x i − x i , then it follows from (8) that Let is independent of the control law, so it also holds.
For each follower, we define a Lyapunov function candidate Let c 2,i be a constant scalar such that Such a c 2,i exists because sets X 0 , V 0 , W 0 ,X 0 and W 0 are bounded, and lim →0 P i = 0. Define L V 2,i := The existence of such an * i is due to lim →0 P i = 0. The derivative of V 2,i iṡ Let Then, (26) can be written aṡ Moreover, Since σ p is a standard saturation function, we have Then, in the case of − 1 where we assume i w ≤ k ψ i because i w is bounded according to Assumption 5. In addition, The last term of (28) follows that Then, taking (29)-(31) into (28) giveṡ Choose N i large enough such that Hence, we havė Similarly, we can show thatV 2,i < 0 when

Illustrative examples
In this section, two examples are given to verify the effectiveness of the state feedback consensus protocol (9) and output feedback consensus protocol (21), which, respectively, solves the output consensus problem defined in Problem 1 and Problem 2. The control laws have also been applied successfully to formation control in practical scenarios in [32]. The multi-agent system consists of one leader (labeled as 0), one informed follower (labeled as 1) and four unin-formed followers (labeled as 2, 3, 4 and 5, respectively). The communication graph G shown in Fig. 1 is a directed network containing a loop. It is clear that Assumption 1 is satisfied. The corresponding Laplacian matrix L, L ff and L 5 are respectively Notice that all eigenvalues of L ff and L 5 have positive real parts.

Semi-global output consensus via state feedback
In this section, we present simulation results to verify state feedback consensus protocol (9) that solves Problem 1, under Assumptions 1-3 and 5. The initial states of the distributed observers (5) are set as random scalars between (0, 1) for i = 1, 2, . . . , 5, and the positive scalar μ 1 is chosen as μ 1 = 3. It is easy to verify that (I 5 ⊗ S − μ 1 L ff ⊗ I 3 ) is Hurwitz. As shown in Fig. 2, the states η i (t) asymptotically converge to the state of the leader w(t).
The initial system states and actuator positions of the followers are chosen as The low gain parameter is chosen as = 0.1. The simulation result is shown in Fig. 3. It can be seen that the outputs of the five followers converge to the output of the leader asymptotically, and the output consensus errors asymptotically converge to 0. Thus, the semi-global leader-following output consensus problem defined in Problem 1 is solved by the state feedback consensus control protocol (9) with = 0.1.

Semi-global output consensus via output feedback
In this section, simulation results are given to verify output feedback consensus protocol (21) in solving Problem 2, under Assumptions 1-5.
For i = 1, 2, . . . , 5, the initial states of the distributed observers (6)-(8) are chosen as random constants between (0, 1). The initial system states and actuator positions of the followers are chosen as The trajectories of w i ,ŵ i , x i ,x i , and estimation errorŝ w i − w i ,x i − x i , i = 1, 2, . . . , 5, are shown in Figs. 4 and 5. It is clear thatŵ i converge to w i andx i converge to x i asymptotically.
We consider low gain parameter = 0.01. It is easy to get solutions of the parametric ARE (4) with γ = 1.
The simulation result is shown in Fig. 6, from which we can conclude that the outputs of the followers y i , i = 1, 2, . . . , 5, are regulated to the output of the leader y 0 asymptotically and the output consensus errors e i converge to zero, which implies that the consensus protocol (21) achieves the semi-global leader-following output consensus problem with the followers subject to actuator position and rate saturation.

Conclusion
In this paper, we have investigated the semi-global output consensus problem for multiple heterogeneous linear systems subject to actuator position and rate saturation. Both a state feedback-based consensus protocol and an output feedback-based consensus protocol for each follower are constructed, using the information of the follower and its neighbors. It is proved that given any a priori given bounded conditions, the problem is solved by the consensus protocols if the low gain parameter is tuned small enough and the communication graph contains a spanning tree.