Distributed Tracking of Leader-Follower Multiagent Systems Subject to Disturbed Leader’s Information

This paper considers the distributed tracking problem for leader-follower multiagent systems, where the followers are heterogenous multi-input multi-output systems which are subject to linearly parameterizable nonlinear matching uncertainties, and the leader’s information is disturbed during transmission. These two specific considerations make the problem investigated in this paper unsolvable by the existing control approaches. To tackle this problem, a novel distributed control approach is proposed. In particular, in order to recover the true value of the leader’s information in the presence of disturbances, an additive observability lemma has been established which facilitates the design of a new kind of filtering distributed observer. Moreover, an invariance-like lemma has been established to analyze the stability of a time-varying perturbed system, which helps to prove the stability of the closed-loop system. A numerical example is presented to validate the effectiveness of the proposed control approach.


I. INTRODUCTION
Over the past few years, motivated by the potential applications in vehicle formation and robot coordination [1]- [4], the cooperative control of multiagent systems has attracted a lot of attention and has been addressed by various control approaches, such as the robust control approach [5]- [8], the neural network approximation approach [9]- [11], the datadriven model-free approach [12]- [14], and so on. Particularly, the distributed tracking of leader-follower multiagent systems with linearly parameterizable nonlinear matching uncertainties was investigated in [15]- [17] by the adaptive control approach, where the first order systems, the second order systems and the more general systems with identical linear structure were considered in [15], [16] and [17], respectively.
One of the systematic and effective approaches to dealing with the cooperative control problem for leader-follower The associate editor coordinating the review of this manuscript and approving it for publication was Weiguo Xia . multiagent systems is the distributed observer approach [18], which consists of two parts. First, a distributed observer is designed for each follower to recover the leader's information. Second, based on the estimated information of the leader, a local tracking controller is synthesized to achieve the cooperative control objective. The distributed observer approach was first proposed in [19] for a linear leader system under fixed communication network, and then extended in [20] for jointly connected switching communication network. Since after, some other variants of the distributed observer have been studied in [21]- [23], which further reduced the information needed from the leader. Specifically, [21] estimated both the system matrices and the state of the leader system under fixed communication network, and [22] found that the same distributed observer in [21] also works for jointly connected switching communication network. [23] further considered the scenario of a fully unknown leader system, and the proposed distributed observer depends solely on the output of the leader without any structural information of the leader's dynamics. There are also some results focusing on distributed observer for the case of nonlinear leader system, such as the rigid body system [24], [25], and for the case of multiple leaders [26]- [28].
The distributed tracking problem can also be viewed as a special case of the cooperative output regulation problem [19]- [21], [29]- [32], where the leader is referred to as the exosystem. Roughly speaking, there are mainly two control frameworks for solving the cooperative output regulation problem, i.e., the distributed feedforward approach [19]- [21] and the distributed internal model approach [29]- [32]. The distributed feedforward approach has the advantage that it does not rely on the transmission zeros condition, and thus allows the system output dimension to be higher than the system input dimension. While, the distributed internal model approach is endowed with the property of robustness against uncertain system parameters.
In this paper, we further consider the distributed tracking problem for leader-follower multiagent systems, where the followers are heterogenous multi-input multi-output systems which are subject to linearly parameterizable nonlinear matching uncertainties, and the leader's information is disturbed during transmission. These two specific considerations make the problem investigated in this paper unsolvable by the existing control approaches. To tackle this problem, a novel distributed control approach is proposed. In particular, in order to recover the true value of the leader's information in the presence of disturbances, an additive observability lemma has been established which facilitates the design of a new kind of filtering distributed observer. Moreover, an invariance-like lemma has been established to analyze the stability of a time-varying perturbed system, which helps to prove the stability of the closed-loop system under the filtering distributed observer and the local tracking controller.
In contrast to the existing works, the key contributions of this paper are summarized as follows.
• In [5]- [10], the followers' dynamics have identical linear structure. In this work, by taking advantage of output regulation theory, the followers' dynamics can be heterogenous. Moreover, the coefficient for the control input is allowed to be unknown, which increases the robustness of the proposed control from the perspective of fault tolerance.
• In contrast to the distributed internal model approaches as in [29]- [31] which studied single-input single-output nonlinear systems, the proposed control approach in this work is capable of dealing with multi-input multi-output nonlinear systems.
• Different from most of the existing works studying leader-follower multiagent systems, in this paper, the leader's information obtained by the followers is subject to disturbances in a way that the disturbances cannot be pre-rejected by the leader itself and thus will propagate over the communication network. To tackle this issue, each follower is equipped with a novel filtering distributed observer to reject the disturbances and recover the leader's information locally.
The rest of this paper is organized as follows. Preliminaries are summarized in Section II. Problem formulation is given in Section III. Section IV presents the main results. A numerical example is shown in Section V. Section VI concludes the paper.

II. PRELIMINARIES
In this section, we will first introduce the notation adopted in this paper, and then establish three lemmas which will be used subsequently.

A. NOTATION
⊗ denotes the Kronecker product of matrices. 1 N denotes an N dimensional column vector whose components are all 1.
||x|| denotes the Euclidean norm of a vector x and ||A|| denotes the induced norm of a matrix A by the Euclidean norm. R and C denote the real and complex number sets, respectively. For any a ∈ R, S(a) = 0 1 −a 2 0 . For A is called neutrally stable if all the eigenvalues of A are semi-simple with zero real part, σ (A) = min 1≤i≤n { (λ i (A))}, where (a) denotes the real part of a complex number a. For a set of matrices A 1 , A 2 , . . . , A n , D(A 1 , A 2 , . . . , A n ) = block diag{A 1 , A 2 , . . . , A n }. For any column vector X ∈ R nq for some positive integers n and q, M q n (X ) = [X 1 , . . . , X q ], where X i ∈ R n , i = 1, . . . , q, and X = col(X 1 , . . . , X q ). If a function f (t) : R → R m×n satisfies ||f (t)|| ≤ βe −αt for some α, β > 0, then f (t) is said to decay to zero exponentially at the rate of α.
A digraph G is defined as G = (V, E) which consists of a node set V = {1, . . . , N } and an edge set An edge from node i to node j is denoted by (i, j), and node i is called the neighbor of node j. If the digraph G contains a sequence of edges of the form (i 1 , i 2 ), (i 2 , i 3 ), . . . , (i k , i k+1 ), then the set . . , (i k , i k+1 )} is called a path of G from node i 1 to node i k+1 , and node i k+1 is said to be reachable from node i 1 . A digraph is said to contain a spanning tree if there exists a node i such that any other node is reachable from it and node i is called the root of the spanning tree. The edge (i, j) is called undirected if (i, j) ∈ E implies (j, i) ∈ E. The digraph G is called undirected if every edge in E is undirected. The weighted adjacency matrix A = [a ij ] ∈ R N ×N of G is defined as a ii = 0, and for i = j, a ij > 0 ⇔ (j, i) ∈ E and a ij = 0 otherwise. Moreover, a ij = a ji if (j, i) is an undirected edge. The Laplacian matrix L = [l ij ] ∈ R N ×N of G is defined as l ii = N j=1 a ij and l ij = −a ij if i = j. VOLUME 8, 2020 B. LEMMAS Lemma 1: Consider two observable pairs (C 1 , A 1 ), By the well-known PBH test (Theorem 2.4-9, [33]), the pair (C, A) is observable if and only if for all λ ∈ C, R(H (λ)) = p + q. Let Since the pair (C 1 , A 1 ) is observable, for all λ ∈ C, the columns of H 1 (λ) are linear independent. Similarly, the columns of H 2 (λ) are also linear independent.
By invoking Lemma 1 recursively, the following result can be obtained immediately.
Corollary 1: Consider n observable pairs (C 1 , A 1 ), Remark 1: Lemma 1 presents an observability result for two additive systems, i.e., whether the state of the augmented system is observable from the addition of the outputs of the two subsystems. The sufficient condition σ (A 1 ) ∩ σ (A 2 ) = ∅ is easy to understand because if σ (A 1 ) ∩ σ (A 2 ) = ∅, then it is possible that the components of the outputs of the two subsystems having the same mode are mixed together, which, as a result, may disable the distinction of the original state. A simple example is the addition of two constant signals.
Lemma 2: Consider the following systeṁ where x ∈ R n , f : R n × R → R n is locally Lipschitz in x, piecewise continuous in t, and uniformly bounded if x is bounded for all t ≥ 0, : R → R n is bounded, piecewise continuous in t, and satisfies satisfying n a > 0, n b ≥ 0 with n a + n b = n, and moreover, there exists V (x) : R n → R being positive definite, radially unbounded, and satisfyingV where K ∈ R n a ×n a is positive definite and ∈ R n a ×n . Then lim t→∞ x a (t) = 0. Proof: Let κ = σ (K ), = || ||. By (7), we havė For all t ≥ 0, the right hand side of (8) achieves the maximum value if and only if ||x a (t)|| = 2κ || (t)||. Therefore, Since (t) is bounded and Since x is bounded,ẋ is bounded by (6) and so isẄ (t). Thus, by Barbalat's Lemma, lim t→∞Ẇ (t) = 0, which indicates that lim t→∞ x a (t) = 0.
Remark 2: Lemma 2 presents an invariance-like result which can be viewed as a generalization of Theorem 8.4 of [34] in that the system is subject to external perturbation. Note that in Theorem 8.4 of [34],V is negative semidefinite. While, in Lemma 2,V is indefinite.
Lemma 3: Let A ∈ R m×n , b ∈ R m , and rank(A) = rank(A, b) = k for some positive integers m, n, k ≥ 1, and let A(t) ∈ R m×n andb(t) ∈ R m be any bounded and piecewise continuous functions of t such thatÃ(t) → 0 as t → ∞ exponentially at the rate of α. Then, for any x(t 0 ) ∈ R n and ε > 0, the following systemẋ has a unique bounded solution for some a 1 > 0. Therefore, by the comparison lemma and SinceĀ has full column rank, there exists a uniquex * 1 ∈ R k such thatĀx * 1 = b. Letx * 2 be any column vector of dimension (n − k), and x * = P Then Next, letx = P T x. Then, where Clearly, (17) is in the same form as equation (16) of [21], and lim t→∞ d(t) = 0 exponentially at the rate of α. The rest of the proof is similar to the proof of Lemma 3 of [21] and thus is omitted.
Remark 3: Lemma 3 is a straightforward extension of Lemma 3 of [21] and thus the proof is similar.

III. PROBLEM FORMULATION
Consider a leader-follower multiagent system consisting of one leader and N followers. The dynamics of the leader are given byη where η 0 ∈ R q , y 0 ∈ R p are the state and output of the leader, respectively, and 0 ∈ R q×q , 0 ∈ R p×q are constant matrices. For i = 1, . . . , N , the dynamics of the ith follower are given byẋ is a known regression function which is assumed to be locally Lipschitz in x i , piecewise continuous in t, and uniformly bounded if x i is bounded for all t ≥ 0; θ i ∈ R r i is a constant vector consisting of uncertain system parameters. Some assumptions regarding systems (19) and (20) are listed as follows.
Assumption 1: Remark 4: Assumption 1 is a standard assumption for the control problem of systems embedded with linear structure. One may refer to [19]- [21], [27], [28], [32], [35], just to name a few. Regarding Assumption 2, in this paper, we consider a challenging case where only the output y 0 of the leader is available. In this scenario, Assumption 2 is necessary since without it, one can never recover the leader's state η 0 solely from the output y 0 . Note that for the case where the state η 0 is available, it turns out that 0 = I q and hence Assumption 2 shall always hold. Therefore, it becomes a special case of the case considered in this paper.
The communication network for the leader-follower multiagent system composed of (19) and (20)  of G and H = L + D(a 10 , . . . , a N 0 ). To ensure that the leader's information can pass to each follower through at least one communication path, the following standard assumption regarding the communication network is imposed.
Assumption 3:Ḡ contains a spanning tree with the node 0 as the root.
Remark 5: Under Assumption 3, by Lemma 1.6 of [4], For i = 1, . . . , N , define the tracking error as Then the distributed tracking problem considered in this paper is formulated as follows. Problem 1: Given systems (19), (20), and a digraphḠ, for each follower, design a distributed control law u i which utilizes only local measurement and neighboring information over the communication network such that the state of the closed-loop system starting from any initial condition exists and is bounded for all t ≥ 0, and for i = 1, . . . , N , In order to solve Problem 1, the following two assumptions are necessary.
Assumption 4: 0 is neutrally stable. Assumption 5: The following regulator equations Remark 6: In practical applications, the boundedness of the state, and thus the output, of the closed-loop system for a nonlinear control system is often considered as a basic control objective 2 as required by Problem 1. By the definition of the tracking error (21) and the control objective (22), it is obvious that if Problem 1 is solvable, then y i is bounded if and only if y 0 is bounded. Given this circumstance, Assumption 4 is necessary since given the linear leader system (19), under Assumption 2, y 0 is bounded, and, in the meanwhile, not decaying to zero, only if Assumption 4 holds. While, if we do not require the boundedness of the state or the output of the closed-loop system for some specific problem, then we no longer need Assumption 4.
Remark 7: Combine systems (19), (20) and (21) aṡ implies that there exists a control law u i solving Problem 1 with respect to systems (19), (20) and (21)  to system (24). Furthermore, by Theorem 1.7 of [35], there exists a control lawū i solving Problem 1 with respect to system (24) only if the regulator equations (23) admit a solution pair (X i , U i ). As a result, Problem 1 is solvable only if Assumption 5 holds. Note that the existence of the solution to the regulator equations is a standard assumption in the literature of output regulation theory, and one may refer to [18]- [21], [28], [32], [35].
In this paper, different from the existing results on the cooperative control problem of leader-follower multiagent system which assume perfect leader, we consider the situation where the leader is subject to external disturbance. In general, there are two ways that the leader could get disturbed. First, the disturbance might impose directly on the leader as shown by case (a) in Fig. 1, which, mathematically, can be described as followsḋ where d 0 ∈ R n d , y d 0 ∈ R p , ϒ 0 ∈ R n d ×n d and 0 ∈ R p×n d are constant matrices, and y m 0 ∈ R p denotes the disturbed output of the leader. Without loss of generality, it can be assumed that ( 0 , ϒ 0 ) is observable. Let 0 = ( 0 , 0 ) and 0 = D( 0 , ϒ 0 ). Then, according to Lemma 1,under Then, by designing the following local Luenberger observer, where L 0 is such that 0 − L 0 0 is Hurwitz, it follows that lim t→∞ (y 0 (t) −ŷ 0 (t)) = 0 whereŷ 0 (t) = 0η0 (t). Hence, instead of sending the disturbed output y m 0 (t), the leader can send the estimated output y 0 (t) which will converge to the true value y 0 (t) exponentially without making great impact on the followers. In other words, the impact of the external disturbance can be annihilated locally at the leader, which makes this case easy to deal with.
The second scenario, which will be the focus of this paper, is much more difficult to deal with than the first one. For simplicity, we use the generic symbol χ 0 ∈ R n χ to represent the leader's information that will be passed to each follower through the communication network. Note that, as will be seen later in the next section, χ 0 will not be limited to y 0 . Then, we consider the case that χ 0 is disturbed on the way of transmission as shown by case (b) in Fig. 1, which is no longer locally at the leader as in the first case. Specifically, it is assumed that where χ d 0 = X 0 d 0 with X 0 ∈ R n χ ×n d denoting disturbance imposed on χ 0 , and χ m 0 denoting the disturbed leader's information. Clearly, for this case, the disturbance χ d 0 can no longer be annihilated locally at the leader and thus has to be dealt with by each follower.

IV. MAIN RESULTS
To solve Problem 1, we adopt the distributed observer approach, which consists of two parts. The first part aims to find a distributed observer for each follower to estimate the leader's information, and the second part aims to design a local tracking controller based on the estimated leader's information obtained by the distributed observer. In what follows, we will show the design process of these two parts.
Note that for the distributed observer (29), the leader's information χ 0 can be detailed as However, in contrast to [37], in this paper, the acquisition of χ 0 cannot be made accurately, but is subject to equation (28). As a result, the distributed observer (29) is no longer feasible. In light of Lemma 1, in order to recover the leader's information χ 0 from the disturbed information χ m 0 , we impose the following assumption on the external disturbance χ d 0 . Assumption 6: Suppose χ d 0 = col(χ d 0,1 , . . . , χ d 0,n χ ), and with known frequencies ω k and unknown amplitudes α k,i and initial phases β k,i . Moreover, σ ( 0 ) ∩ = ∅ where = {±ω k i, k = 1, . . . , δ} with i 2 = −1. Remark 8: By Fourier series, any periodic disturbance satisfying the Dirichlet conditions with zero periodical integral can be approximately expressed in the form of (31) with arbitrarily prescribed accuracy. Thus, Assumption 6 can accommodate a large class of disturbances. While, the constant bias is not considered in (31) because it would make the estimation impossible. In particular, it is necessary for the distributed observer to estimate the system and output matrices of the leader, namely, 0 and 0 , since they contain the key structure information of the leader's dynamics. Note that the entries of 0 and 0 are all constants, and it is obviously impossible to separate two constants from their sum. Therefore, to make it possible to recover 0 and 0 from their disturbed values, constant bias in the disturbance should be ruled out.
Next, we show the design of the filtering distributed observer, which is able to filter out the external disturbance imposed on the leader's information.
Together, (37) and (51) constitute the filtering distributed observer which recovers 0 , 0 and η 0 (t) of the leader for each follower by i (t), i (t) and η i (t), respectively.
Remark 9: In Lemmas 5 and 6, the selection of the observer gains µ i , µ i , µ L i , µ ψ i depends on the communication graph. While, for a communication graph with fixed number of nodes, there exists a maximal σ (H ) −1 for all possible network topologies. Consequentially, the selection of the observer gains only depends on the number of the followers. It might also be interesting to develop adaptive gain techniques as in [5] to update the observer gain online. Moreover, in the filtering distributed observer design, all the VOLUME 8, 2020 entries of 0 , 0 and L 0 are estimated for completeness. While, in practice, it suffices to estimate the unknown entries of these matrices to lower down the dimension of the distributed observer.

B. LOCAL TRACKING CONTROL LAW AND STABILITY ANALYSIS
In this part, we will show how to design a local tracking controller based on the estimated leader's information i (t), i (t) and η i (t) obtained by the distributed observer (37) and (51). Let Then, under Assumption 5, by Theorem 1.9 of [35], the solution of the regulator equation (23) can be expressed as Then, by Lemma 5,Z i =Ẑ i − Z i andφ i =φ i − φ i would decay to zero exponentially. For i = 1, . . . , N , desigṅ . Then, following Lemmas 3 and 5, we can directly obtain the following result.
Lemma 7: Under Assumptions 3, 5 and 6, for i = 1, . . . , N , for any initial conditionζ i (t 0 ) and any µ ζ i > 0, the solution of (58) has a unique bounded solution over t ≥ t 0 and satisfies Then by Lemma 7, it follows that lim t→∞ i1 (t) = 0 and lim t→∞ i2 (t) = 0 exponentially. Moreover, which indicates thatζ i , and henceẊ i andU i will all decay to zero exponentially.
Under Assumption 1, by condition 5 of [38], there exists a unique positive definite matrix P i ∈ R n i ×n i satisfying given some positive definite matrix Q i ∈ R n i ×n i . Let For i = 1, . . . , N , design the following control laẇ We have the following result. Theorem 1: Given systems (19), (20) Then by (50a) and (51), it follows thaṫ and by Lemma (6), η di will decay to zero exponentially. Let Since η i ,X i are bounded and i1 ,Ẋ i ,¯ i and η di all tend to zero exponentially, we have i (t) → 0 exponentially as t → ∞.
Regarding systems (64) and (65), for i = 1, . . . , N , let Then along the trajectories of (64) and (65), we havė Since x i =x i +X i η i andX i , η i are bounded, ifx i is bounded, then x i is bounded. As a result, ifx i andθ i are bounded, the right hand sides of both (64) and (65) with i = 0 are bounded. Moreover, V i is positive definite and radially unbounded. Since i (t) → 0 exponentially, i (t) is bounded and ∞ 0 || i (t)||dt < ∞. Therefore, by Lemma 2,x i (t) → 0 as t → 0. Finally, noticing that gives lim t→∞ e i (t) = 0 and the proof is complete.
Remark 10: There are two main differences between this work and the work of [21]. First, [21] considered linear follower systems, while this paper considers a class of nonlinear follower systems subject to uncertain system parameters, which makes the local control law design much more complicated. In order to conduct the stability analysis of the closed-loop system, a new analysis tool, i.e., Lemma 2, has been established. Second, in contrast to [21] which assumed an ideal leader system, in this paper, the leader's information obtained by the followers is subject to disturbances in a way that the disturbances cannot be pre-rejected by the leader itself and thus will propagate over the communication network. To tackle with this issue, a new filtering distributed observer has been proposed which is capable of recovering the leader's information locally by each follower.

V. EXAMPLE
In this section, we present a numerical example to illustrate the proposed control approach. Consider a leader-follower multiagent system consisting of one leader and six followers, whose communication network is depicted by Fig. 2. Clearly, Assumption 3 is satisfied.
The dynamics of the leader are given by (19) with The dynamics of the ith follower are given by (20) with where   Regarding the disturbance (31), we assume δ = 1, The observer and control gains are selected as follows. Let 3 . The outputs and the estimated unknown parameters are shown by Figs. 3 and 4, respectively. It can be seen that the outputs of the followers have successfully tracked the output of the leader and the estimated unknown parameters converge to constant values.

VI. CONCLUSION AND FUTURE WORK
In this paper, the distributed tracking problem for a leader-follower multiagent system is considered, where the follower dynamics are heterogenous, multi-input, multioutput, and uncertain, and the leader's information is subject to disturbances. A novel distributed control approach has been proposed combining a new kind of filtering distributed observer and the local tracking controller. In the future, it might also be interesting to consider the scenario where the disturbances not only occur to the information transmission from the leader to the followers, but also to the information transmission among the followers.
There are also some interesting issues which are considered as our future directions. First, the communication network considered in this paper is assumed to be static and delay-free. In the future, one may further consider the more practical scenario where the communication network is subject to switched topology and uncertain time-delay as in [40], [41]. Second, in this paper, the sign of b i is a prerequisite for the control law design. By utilizing the Nussbaum-type gain method [42], [43], it might be possible to deal with the case where the sign of b i is also unknown. Third, the nonlinear uncertainties considered in this paper should satisfy the matching condition. It is also desirable to consider the distributed tracking problem of leader-follower multiagent systems with the followers being subject to mismatched nonlinear uncertainties as in [44], [45]. in 1998 and 2001, respectively. He is currently working with the School of Automation Science and Engineering, South China University of Technology. His current research interests include autonomous systems, cyber physical systems, iterative learning control, active disturbance rejection control, and networked control systems.