Distributed Robust Containment Control of Linear Heterogeneous Multi-Agent Systems: An Output Regulation Approach

I. Introduction

RECENTLY, multi-agent techniques have received increasing attention due to their potential applications in industrial, agricultural and military fields [1]–[4]. The cooperative problem is one of the basic research directions of multi-agent systems, and considerable research on this topic have been presented in the literature [5]–[11]. The main focus of cooperative control is the design of distributed strategies under which the agents reach control objectives accessing only local information from itself and its neighbors rather than global information. Distributed cooperative control has been successfully applied in large-scale complex system control problems that are difficult or even impossible to solve via centralized or decentralized strategies.

Containment control, as a typical distributed cooperative control problem, has also attracted substantial attention. The control purpose is to drive followers to asymptotically converge to the convex hull spanned by the leaders [8], [10]-[14]. The containment control problem originates from natural phenomena and has broad practical applications, for example, when a team of autonomous vehicles or robots has to move from one area to another but only some individuals have the ability to detect dangerous areas. The method adopted is to treat autonomous vehicles or robots with detection capability as leaders and the rest as followers to drive the followers to enter the convex hull spanned by the leaders and move together, so as to ensure group security in the transfer process [12], [13].

Many theoretical studies on the containment problem have been conducted, and the integrator system has played an important role. Among them, containment problems with single-integrator and double-integrator agents were investigated in [8], and the necessary and sufficient conditions were provided with continuous-time and sampled-data based protocols under a fixed directed topology. Moreover, the final output of the follower systems were found to be completely dependent on topology structure and the initial states of the leaders. Subsequently, the containment problem of integrator agents with switching topology was considered in [15]–[17], and sufficient conditions for containment control were given utilizing different control approaches. Moreover, the containment problem of the integrator system subject to complex situations has begun to receive attention, with many meaningful results, including nonlinear system containment [18], [19], event-triggered containment [20], [21], finite-time containment [22]–[24], communication-delay containment [25]–[28], and input-saturation containment [29]-[31]. Due to the simplicity of the integrator system structure, it can give full play to the mathematical operation function based on the topology network, and we can obtain many meaningful results. However, it is difficult to directly extend this to problem solving of high-order systems.

Compared with the integrator system, the high-order system has complex system dynamics, so it is more difficult to solve the containment problem, but it is more in line with the needs of practical applications. The forementioned achievements have been reconsidered for high-order systems. For example, the distributed output and state feedback approaches were adopted for state containment [12], [32], and the distributed adaptive approach with agent output was proposed for output containment [33]. The containment problem for communication delay was investigated in [34], [35]. The containment problems for switching topology and performance optimization were considered in [11] simultaneously. To explore the impact of parameter uncertainty of system plants, the robust containment problem was discussed utilizing diverse control ideas [36]–[39]. In [37], a composite feedback containment controller was synthesized to solve the containment problem of unknown nonlinear dynamics based on neural network approximation theory and the Lyapunov theory. In [38], the adaptive factor was embedded in the robust containment protocol to address uncertain leaders utilizing an adaptive internal model and a recursive stabilization approach. In [39], the robust containment problem of fractional-order system dynamics was investigated, and the sufficient condition to state containment was provided according to the linear matrix inequalities technique. The containment problem with input saturation was considered in [40]. The bipartite containment problem was studied in [41] and [42] for linear multi-agent systems and descriptor multi-agent systems, respectively.

Notably, a common drawback associated with most of the forementioned results is that each agent must have the relative values of the state with respect to its neighbors. One major reason is that many containment methods involve the differential term of containment error, and containment error itself is an expression of variable coupling [15], [32], [39]–[42]. Therefore, these methods can only be applied in the case where the agents have identical dynamics or state dimensions and absolutely rules out the possibility of agents with nonidentical state dimensions. However, multi-agent systems with nonidentical state dimensions are common in practical applications. For example, different types of robots must work cooperatively to complete complex production tasks in industry.

In addition, the method in this paper is based upon output regulation theory, which can realize asymptotic tracking of the reference input [43] and has been broadly used in the cooperative control problem of multi-agent systems with leaders [5], [12], [13], [38], [44]–[46]. Under the framework of output regulation, the design of the compensator is an important issue. For example, a cooperative output regulation protocol was introduced to solve the containment problem of linear heterogeneous multi-agent systems using a dynamic compensator based on the leaders’ state information [12] (Details can be found in Remark 3). Output regulation protocols with similar compensators can be found in [5], [13], [44], [47]. However, the dynamic compensators are associated with the leaders’ state information: However, the state information is not always available due to various practical factors, such as high acquisition cost and difficult acquisition.

Recently, an interesting design idea was adopted in [11] to realize output containment by driving all agents to track embedded compensator systems. Inspired by [11], we have solved the robust output containment control problem of linear multi-agent systems with a directed network. The main differences are that the problem we considered involves robust containment, and the control strategies do not take measures against the leaders. The main contributions of this paper include the following three points. First, we modify the conventional state observer to produce an estimate of the convex combination of the leaders’ states by applying a directed network, which lends itself to the design of the distributed protocols. Moreover, the distributed observer can also be viewed as an extension of the compensators associated with the leaders’ states in [5], [12], [13], [44], [47] as these compensators can be regarded as a special case of our observer when the output matrix of the leader systems is of full column rank.

Second, based on the internal model principle and the compensator technique, distributed dynamic state and output feedback control laws were introduced to drive the containment errors to converge to the origin asymptotically, such that uncertain followers with identical nominal dynamics entered the convex hull spanned by the leaders under the output regulation framework. In addition, for the closed-loop stability analysis, the closed-loop system was divided into multiple subsystems via nonsingular transformation, and a Lyapunov inequality method was used to make multiple subsystems reach a stable state simultaneously, thereby stabilizing the closed-loop system.

Finally, we extended the theoretical results to a more general case where the followers have nonidentical state dimensions, where the robust containment problem was converted into a new tracking problem between the distributed observer systems and the follower systems by constructing a virtual error vector. A distributed dynamic output feedback control law was further devised by modifying the state-coupled item in the previous method to drive virtual error to converge to the origin asymptotically, such that multi-agent systems achieved output containment control. In this way, we have avoided the dependence of most existing containment protocols on the relative states of followers, such that the distributed control law is capable of solving the robust output containment problem for linear heterogeneous multi-agent systems with nonidentical state dimensions.

The remainder of this paper is organized as follows. Section II presents the preliminary knowledge and the problem formulation. Section III provides the main result, where the robust output containment control problem was solved for multi-agent systems with followers with identical state dimensions. Section IV presents further results about heterogeneous followers with nonidentical state dimensions. Section V discusses two numerical simulation examples to verify the effectiveness of the designed schemes. Finally, Section VI concludes this paper.

Notation: $\otimes$ denotes the kronecker product. Given any matrices A, B, C, D of compatible dimensions, the following hold: $(A+B)\otimes C = A\otimes C+B\otimes C$, $(A\otimes B)^{{T}} = A^{{T}}\otimes B^{{T}}$, $(A\otimes B)\times (C\otimes D) = AC \otimes BD$. $I_{N}$ denotes the N-dimensional identity matrix. I denotes the dimensionally-compatible identity matrix. ${\bf{1}}_{N}$ denotes the N-dimensional column vector with all 1 elements. ${\bf{0}}_{N\times M}$ denotes a $N\times M$-dimensional matrix with all 0 elements. 0 denotes a dimensionally-compatible matrix or vector with all 0 elements. ∅ denotes the null set. $\sigma(S)$ denotes the spectrum of the square matrix S. Given a vector set $\{x_{1},\ldots,x_{N}\}$, denote $\mathrm{col}(x_{1},\ldots,x_{N}) = [x_{1}^{{T}},\ldots,x_{N}^{{T}}]^{{T}}$. Given any matrix $A\in \mathbb{R}^{m\times n}$, denote $Vec(A) = \mathrm{col}(A_{1},\ldots,A_{n})$ where $A_{i}$, $i = 1,\ldots,n$ is the i-th column of A. $\mathbb{R}$ denotes the set of real numbers, and $\mathbb{C}$ denotes the set of complex numbers. $\parallel x\parallel$ denotes the Euclidean norm of the finite dimensional vector x. $\mathrm{dist}(x,{\cal{C)}}$ denotes the distance from $x\in\mathbb{R}^{n}$ to the set ${\cal{C}}\subseteq\mathbb{R}^{n}$ in the sense of Euclidean norm, that is, $\mathrm{dist}(x,{\cal{C}}) = \mathop{\mathrm{inf}}_{y\in {\cal{C}}}\parallel x-y\parallel$.

Definition 1 (Convex hull): A vector set ${\cal{C}}\subseteq\mathbb{R}^{N}$ is convex if $(1-\lambda)x+\lambda y\in {\cal{C}}$ for any $x,y\in {\cal{C}}$ and any $\lambda\in[0,1]$. The convex hull of a finite set of points $X = \{x_{1},\ldots,x_{q}\}$ is defined as $Co(X) = \left\{\sum_{i = 1}^{q}\alpha_{i}x_{i}|x_{i}\in X,\; \alpha_{i}\in \mathbb{R},\; \alpha_{i} \geq 0,\; \sum_{i = 1}^{q}\alpha_{i} = 1 \right\}$.

II. Preliminaries and Problem Formulation

A Algebraic Graph Theory

A group of agents, including N followers marked as $1,2,\ldots,N$ and M leaders marked as $N+1,N+2,\ldots,N+M$, is considered. The information communication relationship between all agents can be denoted by a directed graph ${\cal{G}} = ({\cal{V}},{\cal{E}},{\cal{A}})$, where ${\cal{V}} = \{{\cal{V}}_{1},{\cal{V}}_{2},\ldots,{\cal{V}}_{N+M}\}$ is the node set, ${\cal{E}} \;\subseteq \;\{(i,j)|i,j \;\in\; {\cal{V}} \}$ is the edge set, and ${\cal{A}} \;=\; [a_{ij}]\;\in \mathbb{R}^{(N+M)\times(N+M)}$ is the weight adjacency matrix. If an edge $(i,j)\in {\cal{E}}$, then node j can access the information from node i, and $a_{ji}>0$; otherwise, $a_{ji} = 0$. Assume no self-loops exist, i.e., $a_{ii} = 0,\;i\in{\cal{V}}$. The neighbor set of node j is denoted as ${\cal{N}}_{j}\; =\; \{i|i,j\;\in\; {\cal{V}},\;(i,j)\;\in\; {\cal{E}}\}$. The edge sequence $(i_{1},i_{2}),\ldots, (i_{k-1},i_{k})$ represents a directed path from node $i_{1}$ to $i_{k}$, and node $i_{k}$ is said to be reachable from node $i_{1}$. The graph ${\cal{G}}$ is said to contain a spanning tree if and only if there exists at least one node from which every other node is reachable. The graph ${\cal{G}}$ is said to have a united spanning tree if all the leaders have no neighbors and each follower has at least one directed path from the leaders. The Laplacian matrix $L = [l_{ij}]\in$ $\mathbb{R}^{(N+M)\times(N+M)}$ of the graph ${\cal{G}}$ is defined as $l_{ii} = \sum_{j = 1}^{N+M}a_{ij}$, and $l_{ij} = -a_{ij},\ i\neq j$. The Laplacian matrix of ${\cal{G}}$ has the following form:

where $L_{1}\in\mathbb{R}^{N\times N}$, $L_{2}\in\mathbb{R}^{N\times M}$ satisfy $L\cdot {\bf{1}}_{N+M} = {\bf{0}}$, $L_{1}\cdot {\bf{1}}_{N}+ L_{2}\cdot \boldsymbol{1}_{M} = {\bf{0}}$. The matrix $-L_{1}^{-1}L_{2}$ is nonnegative and satisfy $-L_{1}^{-1}L_{2}\cdot{\bf{1}}_{M} = {\bf{1}}_{N}$ when the graph ${\cal{G}}$ has a united spanning tree [8].

For convenience, denote ${\cal{V}}_{{\cal{F}}} = \{1,\ldots,N\}$ as the follower set, ${\cal{V}}_{{\cal{L}}} = \{N+1,\ldots,N+M\}$ as the leader set.

B System Model

In this paper, we consider the robust output containment problem of linear heterogeneous multi-agent systems with uncertain followers. Assume the N follower subsystems have the following dynamics:

where $x_{i}\in\mathbb{R}^{n}$, $u_{i}\in\mathbb{R}^{m}$, $y_{i}\in\mathbb{R}^{p}$, and $y_{mi}\in\mathbb{R}^{p_{m}}$ are the state, control input, regulated output and measurable output of the i-th follower, respectively. $\bar{A}_{i}$, $\bar{B}_{i}$, $\bar{C}_{i}$, and $\bar{C}_{mi}$ are uncertain constant matrices with compatible dimensions

where A, B, C, and $C_{m}$ are the nominal values of these matrices and $\Delta A_{i}$, $\Delta B_{i}$, $\Delta C_{i}$, and $\Delta C_{mi}$ are the uncertain parts. For convenience, denote

Assume the M leader subsystems have the following dynamics:

where $w_{k}\in\mathbb{R}^{q}$, $y_{k}\in\mathbb{R}^{p}$, and $y_{mk}\in \mathbb{R}^{p_{m}}$ are the state, tracked output and measurable output of the k-th leader, respectively. S, Q, and $Q_{m}$ are constant matrices with compatible dimensions.

The problem of robust output containment control is: Under appropriate control laws $u_{i},\ i\in{\cal{V}}_{{\cal{F}}}$, multi-agent systems (1) and (2) subject to uncertainty Δ are said to have achieved robust output containment control if the output of each follower asymptotically convergences to the convex hull spanned by the leaders’ output; that is, $\mathop{\mathrm{lim}}\nolimits_{t\rightarrow \infty}\mathrm{dist}(y_{i},Co(y_{N+1},\ldots, y_{N+M})) = 0$, $i\in {\cal{V}}_{{\cal{F}}}$.

Here, we introduce the output containment error vectors for the followers:

We can rewrite (3) in a compact form

where $e = \mathrm{col}(e_{1},\ldots,e_{N}),\;$ $y_{{\cal{F}}} = \mathrm{col}(y_{1},\ldots,y_{N}),\;$ $y_{{\cal{L}}} = \mathrm{col}(y_{N+1},\ldots, y_{N+M})$.

If digraph ${\cal{G}}$ has a united spanning tree and the containment error e converges to the origin asymptotically, the output of each follower is contained in the convex hull spanned by the leaders’ output. The final outputs of the followers are determined by both the topology structure and the leaders’ initial state and satisfy $\mathop{\mathrm{lim}}\nolimits_{t\rightarrow \infty}y_{{\cal{F}}} = (-L_{1}^{-1}L_{2}\otimes I_{p})y_{{\cal{L}}}$, which is in line with Definition 1 with the nonnegative matrix $-L_{1}^{-1}L_{2}$ satisfying $-L_{1}^{-1}L_{2}\cdot{\bf{1}}_{M} = {\bf{1}}_{N}$ [8].

Remark 1: In the research on multi-agent cooperative control, the leader-following consensus is a special situation of the containment problem with only one leader. Therefore, the robust tracking problem considered in [46] can be viewed as a special case of our problem. In [46], the virtual output error vectors were established for followers:

Additionally, virtual errors (5) can be regarded as a special containment error with a single leader. When the digraph ${\cal{G}}$ has a directed spanning tree, the leader-following consensus of [46] can be achieved if and only if the virtual errors converge to the origin asymptotically. However, this result is not valid when it is extended to a containment problem with more than one leader. Here, if the digraph ${\cal{G}}$ has a united spanning tree, the containment errors (3) converge to the origin asymptotically is only a sufficient condition for containment control.

C Problem Formulation

To solve the robust output containment problem of multi-agent systems (1) and (2) via the output regulation approach, some standard assumptions are necessary.

Assumption 1: Digraph ${\cal{G}}$ has a united spanning tree.

Assumption 2: $(A, \; B, \; C_{m})$ is stabilizable and detectable.

Assumption 3: S has no eigenvalue with a negative real part.

Assumption 4: $(S,Q_{m})$ is detectable.

Assumption 5: $\mathrm{rank}\left( \begin{matrix} A-\lambda_{i} I&B\\C&{\bf{0}}\end{matrix} \right) = n+p$, for all $\lambda_{i} \in \sigma(S)$.

These assumptions are loose for the output regulation problem, and similar entries can be found in [5], [6], [45], [46]. Here, if S has eigenvalues with a negative real part, the partial states of the leaders will converge to the origin asymptotically without any control command and have no influence on the final result. For a more interesting situation, consider Assumption 3. Assumption 5 is a transmission zero condition, which is a necessary condition to ensure that robust output regulation problem is solvable [43].

Definition 2 (Internal model): Given a square matrix S, the pair $({\cal{G}}_{1},{\cal{G}}_{2})$ incorporates a p-copy internal model of S if it admits

where Γ is any nonsingular matrix with compatible dimensions and $\Gamma_{i},\;i = 1,2,3$ are any constant matrices with compatible dimensions. The pair $(G_{1},G_{2})$ can be described as

where the pairs $(\beta_{i},\sigma_{i}),\;i = 1,\;2,\dots,p$ satisfy: 1) $\,\beta_{i}$ is a constant square matrix with its characteristic polynomial divisible by the minimum polynomial of S. 2) $\sigma_{i}$ is a column vector such that the pair $(\beta_{i},\sigma_{i})$ is controllable. Particularly, we say the pair $(G_{1},G_{2})$ incorporates a minimum p-copy internal model of S when $\mathrm{dim}(\Gamma_{i}) = 0,\;i = 1,2,3$ and $\Gamma = I$.

The following lemmas will be used to solve the robust containment problem.

Lemma 1: Under Assumption 1, all nonzero eigenvalues of Laplacian matrix L are eigenvalues of matrix $L_{1}$, and all have positive real parts. Furthermore, for the same eigenvalue, the eigenvector to $L_{1}$ is a subvector of the eigenvector to L.

Proof: See Appendix.■

Remark 2: There is a similar lemma for leader-following consensus (see Lemma 1 in [5]), but the proof here is more concise and clear and has given the connection between their eigenvectors.

Lemma 2 (See Lemma 1 in [48]): Given matrices $A\in \mathbb{R}^{n\times n}, B\in \mathbb{R}^{n\times m}$, $(A,B)$ is stabilizable, and the algebraic Riccati equation:

admits a unique solution $P = P^{{T}} > 0$. Then, $A-\nu BB^{{T}}P$ is Hurwitz for any $\nu\in \mathbb{C}$ with $\mathrm{Re}(\nu)\geq1$.

Lemma 3 (See Proposition A.2 in [43]): Given matrices $A\in \mathbb{R}^{n\times n},\ B\in \mathbb{R}^{m\times m}$ and $C\in \mathbb{R}^{m\times n}$, the Sylvester equation

provides a unique solution $X\in \mathbb{R}^{m\times n}$, if and only if $\sigma (A) \bigcap \sigma (B) = \emptyset$.

Lemma 4 (See Lemma 1.26 in [43]): Under Assumptions 2, 3, and 5, $(G_{1},G_{2})$ incorporates a minimum p-copy internal model of S,

then the pair $({\cal{A}},{\cal{B}})$ is stabilizable.

Lemma 5 (See Lemma 1.27 in [43]): If $(G_{1},G_{2})$ incorporates a minimum p-copy internal model of S and the Sylvester equation

provides a solution X, we have $\Pi = {\bf{0}}$.

Lemma 6 (See Chapter 7.2.1 in [49]): Given matrices $A\in\mathbb{R}^{n\times n}$ and $B\in\mathbb{R}^{n\times m}$, $(A,B)$ is stabilizable if and only if there exists a positive definite symmetric solution P to the Lyapunov inequality

It is well-known that the internal model principle can eliminate the steady-state error of the uncertain system by embedding a model of the exogenous signals [50]–[52]. It is also a key part of the solution to the cooperative robust output regulation problem.

The robust output regulation for a single plant has been provided in detail in [43] via the internal model principle, and it is theoretically proven that pure state feedback protocol cannot solve this problem. Subsequently, the robust leader-following consensus and containment problem of multi-agent systems were further considered in [6], [46], [47], [53] via the internal model principle and the compensator technique. In this paper, we continue to consider the robust output containment problem of linear multi-agent systems within the output regulation framework.

In practice, on the one hand, partial followers can not directly access information from the leader due to the lack of existence of an edge from the leaders to the followers in the communication network; on the other hand, the leaders’ state variate may not be available due to measurement difficulties, no actual physical meaning, etc. To address these problems, we design a distributed dynamic compensator for the followers based on the leaders’ measurable output:

where $\hat{w}_{i}\in \mathbb{R}^{q}$ is the state of the compensator and $\bar{L}$ is a constant matrix to be designed. Treat $\hat{y}_{i} = Q\hat{w}_{i},\ i\in {\cal{V}}_{{\cal{F}}}$ as the output term of the compensator (11), and rewrite them in a compact form:

where $\hat{w} \;=\; \mathrm{col}\;(\hat{w}_{1},\ldots,\;\hat{w}_{N}),\;\;\;\; \hat{y}_{{\cal{L}}} \;=\; \mathrm{col}\;(\hat{y}_{1},\ldots,\hat{y}_{N}),\;\;\; w =$ $\mathrm{col}(w_{N+1},\ldots, w_{N+M})$.

The following will show that the compensator (11) can be regarded as a state observer of the convex combination of the leaders.

Lemma 7: Under Assumption 4, for any initial states $w_{k}(0),\;k\in{\cal{V}}_{{\cal{L}}}$ and $\hat{w}_{i}(0),\;i\in{\cal{V}}_{{\cal{F}}}$, the states of compensator (11) can asymptotically converge to the convex hull spanned by the leaders’ states if and only if the digraph ${\cal{G}}$ has a united spanning tree.

Proof: See Appendix.■

Remark 3: Reference [12] has considered the state containment problem of linear multi-agent systems, in which a distributed dynamic compensator (13), based on the leaders’ states, was adopted to observe the convex combination of the leaders’ states. A similar compensator can be found in the research on containment problems [13], [40], [47] and leader-following consensus problems [5], [44]:

Despite playing the same role, the observer (11) differs from (13) in the following two aspects:

1) The distributed observer (11) is more realistic than (13) because each follower needs to know the leaders’ state information in (13) but only the measurable output information in (11). Generally, the system output information is easier to obtain than state information. Considering the same problem, [13] introduced a classical state observer for each leader to be suited for situations where the leaders’ states are not available. However, this approach increases the state dimension of the control protocol and makes the closed-loop system more complicated.

2) The distributed observer (11) contains the compensator (13) as a special case, where the measurable output matrix $Q_{m}$ is of full column rank. This scenario is equivalent to the state variate of leader systems (2) being available. We further make $\bar{L}$ satisfy $-\beta I_{q} = \bar{L}Q_{m}$, where β is a constant to be designed. Then, observer (11) is simplified to (13). At this time, let $\,\beta > {\mathrm{max}\{\mathrm{Re}(\lambda_{i}),\lambda_{i}\in\sigma(S)\}}/{\mathrm{min}\{\mathrm{Re}(\mu_{j}),\mu_{j}\in\sigma(L_{1})\}}$; then, matrix Ξ is Hurwitz. As a whole, the compensator (11) is a generalized case that includes the compensator (13) as a special case.

In this paper, we considered the robust output containment problem for linear heterogeneous multi-agent systems. First, for the case where the followers have identical nominal dynamics, the distributed dynamic state and output feedback protocols were introduced to drive uncertain followers into a convex hull spanned by the leaders. Subsequently, to consider a more realistic case where the followers may have nonidentical nominal dynamics and dimensions, with the aid of compensator (11), a new distributed dynamic output feedback protocol was established.

In the following part, two distributed dynamic control laws are introduced to solve the robust output containment problem using the internal model principle.

For convenience, the function ${\cal{G}}(\cdot)$ associated with the communication network ${\cal{G}}$ was defined as

where $\Phi_{i}\in\mathbb{R}^{n\times q},\;i\in{\cal{V}}_{{\cal{F}}}$ are any constant matrices.

The following forms of control protocols are constructed:

1) Distributed dynamic state feedback control law

where ${\textit{z}}_{i}\in \mathbb{R}^{n_{{\textit{z}}}}$ is the state of the internal model compensator, $(K_{1},K_{2})$ are the constant matrices to be designed, and $(G_{1},G_{2})$ incorporates a minimum p-copy internal model of S.

In practice, the state of followers is not always available, but the outputs are usually measurable.

2) Distributed dynamic output feedback control law^①

where $\hat{x}_{i}\in \mathbb{R}^{n}$, ${\textit{z}}_{i}\in \mathbb{R}^{n_{{\textit{z}}}}$ are the states of the compensators, $(K_{x},K_{\textit{z}},H)$ are the constant matrices to be designed, the pair $(G_{1},G_{2})$ incorporates a minimum p-copy internal model of S, and

Substituting the control law (15) or (16) into the follower system (1), and combining the leader system (2) and containment error (4) yields the closed-loop system:

where $x_{c}$ is the closed-loop state, $\bar{S} = I_{M}\otimes S$, and the details of $\bar{A}_{c}$, $\bar{B}_{c}$, $\bar{C}_{c}$, and $\bar{D}_{c}$ are provided in Theorems 1 and 2. Denote $\bar{A}_{c} = A_{c}$, $\bar{B}_{c} = B_{c}$, $\bar{C}_{c} = C_{c}$, and $\bar{D}_{c} = D_{c}$ for $\Delta = {\bf{0}}$. Thus, the robust output containment problem can be equivalently converted into a cooperative robust output regulation problem.

The problem of cooperative robust output regulation is: Substitute the control protocol (15) or (16) into linear multi-agent systems (1) and (2) under digraph ${\cal{G}}$, such that the closed-loop system (17) has the following properties:

1) When $\Delta = {\bf{0}}$, the origin of the system $\dot{x}_{c} = A_{c}x_{c}$ is exponentially stable, i.e., $A_{c}$ is Hurwitz.

2) For any initial states $x_{i}(0)$, $\hat{x}_{i}(0)$, ${\textit{z}}_{i}(0)$, $i\in{\cal{V}}_{{\cal{F}}}$, and $w_{k}(0)$, $k\in{\cal{V}}_{{\cal{L}}}$, there exists an open neighborhood W of $\Delta = {\bf{0}}$, for all $\Delta\in W$ such that:

III. Solution to Regulator Problem

In this section, two main results are presented under control laws (15) and (16).

A Distributed Dynamic State Feedback Control Law

Under control law (15), the closed-loop state is $x_{c} = \mathrm{col}(x,{\textit{z}})$, $x = \mathrm{col}(x_{1},\ldots,x_{N})$, ${\textit{z}} = \mathrm{col}({\textit{z}}_{1},\ldots,{\textit{z}}_{N})$. The dynamic matrices of closed-loop system (17) are as follows:

where $\bar{A} = \mathrm{diag}\{\bar{A}_{1},\ldots,\bar{A}_{N}\},$ $\bar{B} =$ diag$\{\bar{B}_{1},\ldots,\bar{B}_{N}\},$ $\bar{C} =$ diag$\{\bar{C}_{1},\ldots,\bar{C}_{N}\}$, $\Phi =$ diag$\{\Phi_{1},\ldots,\Phi_{N}\}$.

Theorem 1: Under Assumptions 1–3 and 5, $(G_{1},G_{2})$ incorporates a minimum p-copy internal model of S, and the robust output containment problem of multi-agent systems (1) and (2) can be solved by the distributed dynamic state feedback control law (15).

Proof: First, consider the first property of cooperative robust output regulation:

$A_{c}$, the nominal value of the uncertain system matrix ${\bar{A}_{c}}$ of (19), is as follows:

Let

Nonsingular transformation yields

Let $U_{1} =$ diag$\{U\otimes I_{n},U\otimes I_{n_{\textit{z}}}\}$. Nonsingular transformation yields:

Thus, $\tilde{A}_{c}$ is Hurwitz if and only if

are Hurwitz. Rewrite (24) as follows:

where ${\cal{A}} = \left( \begin{matrix} A & {\bf{0}} \\ G_{2}C & G_{1} \end{matrix} \right),\; {\cal{B}} = \left( \begin{matrix} B \\ {\bf{0}} \end{matrix} \right), \;{\cal{K}} = \left( \begin{matrix} K_{1} & K_{2} \end{matrix} \right)$. According to Assumptions 2, 3, and 5 and Lemma 4, the pair $({\cal{A}},{\cal{B}})$ is stabilizable. Furthermore, according to Lemma 6, there exists a ${\cal{P}} = {\cal{P}}^{{T}} > 0$ such that

Let

where $\, \beta\in\mathbb{R}$ satisfies $\,\beta\geq{1}/{\mathrm{min}\{\mathrm{Re}(\lambda_{i}),\lambda_{i}\in\sigma(L_{1})\}}$.

Let $\lambda_{i} = \sigma_{i}+\mathrm{j}w_{i}$ with $\sigma_{i},\ w_{i}\in\mathbb{R}$. Then

Therefore, $\tilde{A}_{ci},\;i\in{\cal{V}}_{{\cal{F}}}$ are Hurwitz; thus, $A_{c}$ is Hurwitz.

We now continue to analyze the second property of cooperative robust output regulation:

Based on the above analysis, there exists an open neighborhood W of $\Delta = {\bf{0}}$, for all $\Delta \in W$, such that $\bar{A}_{c}$ is Hurwitz. On the basis of the closed-loop system (17) and dynamic matrices (19), we establish a Sylvester equation:

Since $\bar{A}_{c}$ is Hurwitz, $\sigma(\bar{S})\bigcap \sigma(\bar{A}_{c}) = \emptyset$. By Lemma 5, (29) admits a unique solution

Substituting (30) into (29) yields:

From Lemma 5,

Let $\tilde{x}_{c} = x_{c}-X_{c}w$; taking the derivative of $\tilde{x}_{c}$ yields:

Since $\bar{A}_{c}$ is Hurwitz, $\mathop{\mathrm{lim}}\nolimits_{t \rightarrow \infty}\tilde{x}_{c} = 0$. For the containment errors:

Since $\mathop{\mathrm{lim}}\nolimits_{t \rightarrow \infty}\tilde{x}_{c} = {\bf{0}}$, and $((L_{1}\otimes I_{p})\bar{C}\quad {\bf{0}})$ is bounded, hence $\mathop{\mathrm{lim}}\nolimits_{t \rightarrow \infty}e = {\bf{0}}$.■

B Distributed Dynamic Output Feedback Control Law

Under control law (16), the closed-loop state of (17) is $x_{c} = \mathrm{col}(x,\hat{x},{\textit{z}})$, $x = \mathrm{col}(x_{1},\ldots,x_{N})$, $\hat{x} = \mathrm{col}(\hat{x}_{1},\ldots,\hat{x}_{N})$, ${\textit{z}}= \mathrm{col}({\textit{z}}_{1},\ldots,{\textit{z}}_{N})$. Denote $\Omega = I_{N}\otimes A+L_{1}\otimes(BK_{x}+HC_{m})$. The dynamic matrices are as follows:

where $\bar{C}_{m} =$ diag$\{\bar{C}_{m1},\ldots,\bar{C}_{mN}\}$.

Theorem 2: Under Assumptions 1–3 and 5, $(G_{1},G_{2})$ incorporates a minimum p-copy internal model of S; then, the cooperative robust output containment problem of multi-agent systems (1) and (2) can be solved by the distributed dynamic output feedback control law (16).

Proof: First, consider the first property of cooperative robust output regulation:

$A_{c}$, the nominal value of $\bar{A}_{c}$ within (34), is as follows:

Let

Denote $\Pi = I_{N}\otimes A+L_{1}\otimes HC_{m}$. Nonsingular transformation yields:

where $(K_{x},K_{\textit{z}}) = (K_{1},K_{2})$. $A_{c}$ is Hurwitz if and only if Π is Hurwitz. The pair $(A,C_{m})$ is detectable, so the pair $(A^{{T}},C_{m}^{{T}})$ is stabilizable. By Lemma 6, there exists a $P = P^{{T}} > 0$ such that the Lyapunov inequality

Let $H = -\varepsilon P^{-1}C_{m}^{{T}}$ with $\varepsilon\geq{1}/{\mathrm{min}\{\mathrm{Re}(\lambda_{i}),\lambda_{i}\in\sigma (L_{1})\}}$, so Π is Hurwitz. The convergence analysis of output containment error e is similar to that in Theorem 1 and is thus omitted here.■

Based on the above analysis, we have achieved robust output containment for linear heterogeneous multi-agent systems (1) and (2) using output regulation theory. Control laws (15) and (16) ensure that an agent accesses only local information from itself and its neighbors instead of global information, so they follow full information distributed strategies. For the stability analysis of the closed-loop system, the Lyapunov inequality method provided in [49] can make multiple subsystems reach a stable state simultaneously. However, a shortcoming remains; that is, the control protocols are associated with the relative state information between followers, so they can only cope with the case where followers have identical state dimensions. Moreover, our research framework is similar to that in [46]; the main differences include the following: 1) Different problems: The problem we consider is containment control with multiple leaders, while the problem in [46] is leader-following consensus with a single leader. 2) Different stability analysis methods for closed-loop system: Our method is based on a Lyapunov inequality, while [46] is based on an algebraic Riccati equation.

IV. Robust Containment Control of Heterogeneous Multi-Agent Systems

In this section, we further generalize the theoretical results to a more general case, where the followers have nonidentical state dimensions and the above strategies cannot be applied directly. Here, with the aid of the compensator (11), we introduce a new type of distributed dynamic output feedback control law by replacing the state-coupled item with the state variate of the compensator in the previous control law that lends itself to the avoidance of dependence on relative states, such that the new control protocol can cope with the case where the followers have nonidentical dynamic dimensions.

Consider N heterogeneous follower subsystems with the following dynamics:

where $x_{i}\in \mathbb{R}^{n_{i}}$, $y_{i}\in \mathbb{R}^{p}$, $y_{mi}\in \mathbb{R}^{p_{m_{i}}}$ and $u_{i}\in \mathbb{R}^{m_{i}}$ are the state, regulated output, measurable output and control input of the i-th follower, respectively. $\hat{A}_{i}$, $\hat{B}_{i}$, $\hat{C}_{i}$, and $\hat{C}_{mi}$ are constant matrices with compatible dimensions

where $A_{i}$, $B_{i}$, $C_{i}$, and $C_{mi}$ are the nominal values of these matrices, and $\hat{\Delta} A_{i}$, $\hat{\Delta} B_{i}$, $\hat{\Delta} C_{i}$, and $\hat{\Delta} C_{mi}$ are the uncertain parts. For convenience, denote

Two necessary assumptions must be added.

Assumption 6: $(A_{i},B_{i},C_{mi}),\;i\in{\cal{V}}_{{\cal{F}}}$ are stabilizable and detectable.

Assumption 7: $\mathrm{rank}\left( \begin{matrix} A_{j}-\lambda_{i} I&B_{j} \\C_{j} & {\bf{0}} \end{matrix} \right) = n_{j}+p,\ j\in {\cal{V}}_{{\cal{F}}}$, for all $\lambda_{i} \in \sigma(S)$.

Based on the state observer (12), establish the virtual error vectors for the followers:

Lemma 8: Under Assumptions 1 and 4, $\mathop{\mathrm{lim}}\nolimits_{t\rightarrow \infty}e_{i} = {\bf{0}},\;i\in {\cal{V}}_{{\cal{F}}}\Leftrightarrow\mathop{\mathrm{lim}}\nolimits_{t\rightarrow \infty}\hat{e}_{i} = {\bf{0}},\;i\in {\cal{V}}_{{\cal{F}}}$. $\bar{L}$ is given in Lemma 7.

Proof: Rewrite (40) in a compact form

where $\hat{e} = \mathrm{col}(\hat{e}_{1},\ldots,\hat{e}_{N})$.

■

At this point, our goal can be transformed from driving the containment error to zero to driving the virtual error to zero. A new distributed protocol of the following form is constructed:

Distributed dynamic output feedback control law^②

where ${\textit{z}}_{i}\in \mathbb{R}^{n_{zi}}$, $(G_{i1},G_{i2})$ incorporate a minimum p-copy internal model of S. $(K_{i1},K_{i2},H_{i})$ are the constant matrices to be designed.

Under control law (43), the closed-loop system can be described as

where the closed-loop state is $x_{c} = \mathrm{col}(x,\hat{x},{\textit{z}},\hat{w})$, $x = \mathrm{col}(x_{1},\;\ldots, x_{N}), \hat{x} = \mathrm{col}(\hat{x}_{1},\ldots,\hat{x}_{N}),\;{\textit{z}} = \mathrm{col}({\textit{z}}_{1},\ldots,{\textit{z}}_{N}), \hat{w} = \mathrm{col}(\hat{w}_{1},\ldots,\hat{w}_{N}).$ The dynamic matrices are as follows:

where $\hat{A} =$ diag$\{{\hat{A}_{1},\ldots,\hat{A}_{N}}\},$ $\hat{B} =$ diag$\{{\hat{B}_{1},\ldots,\hat{B}_{N}}\},$ $\hat{C} =$ diag$\{{\hat{C}_{1},\ldots,\hat{C}_{N}}\},$ $\hat{C}_{m} =$ diag$\{{\hat{C}_{m1},\ldots,\hat{C}_{mN}}\},$ $\check{A} =$ diag$\{A_{1},\ldots, A_{N}\},$ $\check{B} =$ diag$\{{B_{1},\ldots,B_{N}}\},$ $\check{C} =$ diag$\{{C_{1},\ldots,C_{N}}\},\;$ $\check{C}_{m} =$ diag$\{{C_{m1},\ldots,C_{mN}}\},\;$ $\hat{G}_{1} =$ diag$\{{G_{11},\ldots,G_{N1}}\},$ $\hat{G}_{2} =$ diag$\{G_{12},\ldots, G_{N2}\},$ $\hat{K}_{1} =$ diag$\{{K_{11},\ldots,K_{N1}}\},$ $\hat{K}_{2} =$ diag$\{{K_{12},\ldots,K_{N2}}\},$ $\hat{H} =$ diag$\{{H_{1},\ldots,H_{N}}\},$ $\hat{Q} = I_{N}\otimes Q$.

Theorem 3: Under Assumptions 1, 3, 4, 6, and 7, the pair $(G_{i1},G_{i2})$ incorporates a minimum p-copy internal model of S; thus, the robust output containment problem of linear heterogeneous multi-agent systems (39) and (2) can be solved by the distributed dynamic output feedback control law (43).

Proof: First, consider the first property of cooperative robust output regulation:

$A_{c}$, the nominal value of system matrix $\hat{A}_{c}$, is defined as follows:

Let

Nonsingular transformation yields:

Denote $\check{A}_{c1} = \left(\begin{matrix}\check{A}+\check{B}\hat{K}_{1}&\check{B}\hat{K}_{2}\\\hat{G}_{2}\check{C}&\hat{G}_{1}\end{matrix}\right)$, and rewrite it in the following form:

By Lemma 4, the pair $\left(\left(\begin{matrix}\check{A} & {\bf{0}} \\ \hat{G}_{2}\check{C} & \hat{G}_{1} \end{matrix}\right),\left(\begin{matrix} \check{B} \\ {\bf{0}} \end{matrix}\right)\right)$ is stabilizable, so there exist $(K_{i1},K_{i2}),\ i\in {\cal{V}}_{{\cal{F}}}$ such that $\check{A}_{c1}$ is Hurwitz. According to Assumption 6, there exists $H_{i}$ such that $\check{A}+\hat{H}\check{C}_{m}$ is Hurwitz, $i\in {\cal{V}}_{{\cal{F}}}$. Thus, $A_{c}$ is Hurwitz.

We now continue to analyze the second property of cooperative robust output regulation:

There exists an open neighbor W of $\hat{\Delta} = {\bf{0}}$, and $\hat{A}_{c}$ is Hurwitz for all $\hat{\Delta} \in W$. According to Lemma 3, the Sylvester equation

admits a unique solution

Substituting (51) into (50) yields:

According to the third term of (52) and Lemma 5,

From the fourth term of (52),

Let $\hat{x}_{c} = x_{c}-\hat{X}_{c}w$; taking the derivative of $\hat{x}_{c}$ yields:

Since $\hat{A}_{c}$ is Hurwitz, $\mathop{\mathrm{lim}}\nolimits_{t\rightarrow \infty}\hat{x}_{c} = {\bf{0}}$.

Furthermore,

Since $\mathop{\mathrm{lim}}\nolimits_{t\rightarrow \infty}\hat{x}_{c} = {\bf{0}}$ and $\hat{C}_{c}$ is bounded, $\mathop{\mathrm{lim}}\nolimits_{t\rightarrow \infty}\hat{e} = {\bf{0}}$. According to Lemma 8, $\mathop{\mathrm{lim}}\nolimits_{t\rightarrow \infty}e = {\bf{0}}$.■

Remark 4: In this section, we have achieved cooperative robust output containment control for linear heterogeneous multi-agent systems with followers having nonidentical dynamic dimensions. The core is the design of the distributed dynamic compensator (11), which can provide the followers with an estimate of the convex combination of the leaders’ states. By constructing the virtual error vectors, the containment problem was further transformed into an equivalent tracking problem between the follower system (39) and the compensator system (12). In this way, it helps the control protocol (43) avoid relying on the relative states of followers. Furthermore, under Assumption 6, we can choose an appropriate $(K_{i1},K_{i2},H_{i})$, $i\in{\cal{V}}_{{\cal{F}}}$ to stabilize the nominal closed-loop system, such that the tracking problem can be easily solved via the output regulation method. Finally, the objective of containment control was achieved by driving the virtual errors to converge to zero. However, compared to protocol (16), although, the relative states of followers are not needed for protocol (43), a shortcoming may remain; that is, the dynamic compensator (11) must to be added, which increases the dynamic dimensions of the control protocol and makes the closed-loop system complicated. This scenario may increase the difficulty of practical applications.

V. Example Simulation

In this section, two numerical simulation examples are adopted to illustrate the validity of the results of Theorems 2 and 3.

A Example 1

Consider a team of agents consisting of three followers $(i = 1,2,3)$ and three leaders $(k = 4,5,6)$. The information communication network is depicted in Fig 1. The dynamic matrices of the leaders are as follows:

Figure  1.  The topology network.

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The nominal values of the dynamic matrices of the followers are as follows:

The uncertain parts

Assume $Q_{m} = Q$, $\bar{C}_{mi} = \bar{C}_{i},\ i = 1,2,3$.

The pair $(G_{1},G_{2})$ are as follows:

By solving the Riccati equation and Lyapunov inequality:

Then, obtain the feedback matrices:

Let $\Phi_{i} = \left(\begin{matrix} 1 & 0 \\ 0 & 1 \\ 0 & 0 \end{matrix}\right),\;i = 1,2,3$.

In the simulation results, $y_{i1}$ and $y_{i2}$ are the components of the agent outputs $y_{i},\ i = 1,2,3,4,5,6$. Their output trajectories are depicted in Fig 2, where the convex hull spanned by the leaders is displayed at several time instants. The followers asymptotically enter the leaders’ convex hull, which is consistent with our expectation. Subsequently, the trajectories of the followers and the leaders over time are depicted in Fig 3. It can be found that the followers output components $y_{i1},\ i = 1,2,3$ and $y_{i2},\ i = 1,2,3$ converge asymptotically into the convex hull spanned by the leaders output components $y_{k1},\ k = 4,5,6$ and $y_{k2},\ k = 4,5,6$, respectively. Finally, the containment error and the compensator state trajectories are depicted in Figs. 4 and 5, respectively. The containment error curves asymptotically converge to zero, and the compensator state curves asymptotically converge to the convex hull spanned by the leaders’ states. Overall, the above analysis is consistent with the results in Theorem 2.

Figure  2.  The output trajectories of the followers and the leaders for the closed loop multi-agent system in Example 1.

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Figure  3.  The output trajectories of the followers and the leaders over time for the closed loop multi-agent system in Example 1

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Figure  4.  The containment error of the followers over time for the closed loop multi-agent system in Example 1.

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Figure  5.  The state trajectories of the compensators and leaders over time.

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B Example 2

Consider a team of agents consisting of three heterogeneous followers $(i = 1,2,3)$ and three leaders $(k = 4,5,6)$. The topology graph, leader dynamics are the same as in Example 1. Matrix pairs $(G_{i1},G_{i2}) = (G_{1},G_{2}),\;i = 1,2,3$. The nominal value of the followers dynamics matrices are as follows:

The uncertain parts are as follows:

Assume $\hat{C}_{mi} = \hat{C}_{i},\ i = 1,2,3$.

Under Assumptions 6, 7 and Lemma 4, by choosing some stable closed-loop poles for (48), we can obtain the feedback matrices as follows:

From the results of the simulation, it is demonstrated that the followers enter the leaders convex hull as $t\rightarrow \infty$ in Figs. 6 and 7. The trajectories are consistent with the expected results. Subsequently, the containment error and virtual error that decay to zero asymptotically are presented in Figs. 8 and 9, respectively. It can be seen that even if the followers have nonidentical state dimensions, multi-agent systems can still achieve output containment control. Overall, the above analysis results show the correctness of Theorem 3.

Figure  6.  The output trajectories of the followers and the leaders for the closed loop multi-agent system in Example 2.

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Figure  7.  The output trajectories of the followers and the leaders over time for the closed loop multi-agent system in Example 2.

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Figure  8.  The containment error of the followers over time.

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Figure  9.  The virtual error of the followers over time.

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VI. Conclusions

In this paper, we have solved the robust output containment control problem for linear multi-agent systems with a directed fixed network ${\cal{G}}$. A novel distributed dynamic compensator with measurable leaders’ output was designed to estimate the convex combination of the leaders’ states in combination with the communication network. First, based on the state-coupled item, the internal model compensator and the follower dynamic compensator, the distributed dynamic state and output feedback laws were adopted to drive uncertain followers with identical state dimensions into the convex hull spanned by the leaders. Subsequently, based on the leader compensator, the robust containment problem was converted into an equivalent tracking problem. A new distributed dynamic output feedback control law was designed to drive uncertain followers into the leaders’ convex hull under the output regulation framework. Unlike most existing control approaches, our method addressed the dependence on the relative states; therefore, it is capable of coping with containment problems where the followers have nonidentical state dimensions.

Appendix

A The Proof of Lemma 1

Proof: First, let all the leaders be a single node, where the Laplacian matrix can be expressed as

The eigenvalues of $L'$ satisfy the equation det$(\lambda I-L_{1})\lambda = 0$, and by Lemma 3.3 in [54], we have that the eigenvalues of $L_{1}$ have positive real parts. The eigenvalues of L satisfy the equation det$(\lambda I-L_{1})\lambda^{M} = 0$, so 0 is the M-copy eigenvalues of L and all the nonzero eigenvalues of L have positive real parts.

Thus, $\mathrm{Re}(\lambda_{1})\geq \cdots\geq \mathrm{Re}(\lambda_{N}) > \lambda_{N+1} = \cdots = \lambda_{N+M} = 0$, where $\lambda_{i}\in \sigma(L),\ i\in {\cal{V}}_{{\cal{F}}}\bigcup {\cal{V}}_{{\cal{L}}}$. Assume $\alpha_{i}$ is the right eigenvector of the non-zero eigenvalue $\lambda_{i}$, then $L\alpha_{i} = \lambda_{i}\alpha_{i}$ hold. Let $\alpha_{i} = [\alpha_{ i1(1\times N)}^{T}\quad {\bf{0}}_{1\times M}^{{T}}]^T$, then $L_{1}\alpha_{i1(1\times N)} = \lambda_{i}\alpha_{i1(1\times N)},\ i\in {\cal{V}}_{\cal{F}}$.■

B The Proof of Lemma 7

Proof (If part): Let $\tilde{w} = \hat{w}-(-L_{1}^{-1}L_{2}\otimes I_{q})w$, and denote $\Xi = I_{N}\otimes S+L_{1}\otimes \bar{L}Q_{m}$; taking the derivative of $\tilde{w}$ yields:

By Lemma 1, $L_{1}$ is nonsingular and there exists a unitary matrix U such that $\Lambda = U^{*}L_{1}U$ is an upper-triangular matrix with $\lambda_{i},\;i\in {\cal{V}}_{{\cal{F}}}$ as the i-th diagonal item. Let

$\bar{\Xi}$ is Hurwitz if and only if $S+\lambda_{i}\bar{L}Q_{m},\ i\in {\cal{V}}_{{\cal{F}}}$ are Hurwitz. Let

where $\mu\in \mathbb{R}$ satisfies $\mu\geq{1}/{\mathrm{min}\{\mathrm{Re}(\lambda_{i}),\lambda_{i}\in\sigma(L_{1})\}}$, and P is the solution to $SP+PS^{{T}}-PQ^{{T}}_{m}Q_{m}P+I_{q} = 0$. Thus, $\mathrm{Re}\{\mu\lambda_{i}\}\geq 1$ for all $\lambda_{i}\in\sigma(L_{1})$. Since $(S,Q_{m})$ is detectable, $(S^{{T}},Q_{m}^{{T}})$ is stabilizable. According to Lemma 2, $(S+\lambda_{i}\bar{L}Q_{m})^{{T}} = S^{{T}}- \mu\lambda_{i}Q_{m}^{{T}}Q_{m}P,\;\lambda_{i}\in\sigma(L_{1})$ are Hurwitz. That is, $S+\lambda_{i}\bar{L}Q_{m}, \lambda_{i}\in \sigma(L_{1})$ are Hurwitz; thus, Ξ is Hurwitz. Therefore,

Proof (Only if part): If the digraph ${\cal{G}}$ has no united spanning tree, at least one follower cannot receive information from the leaders through the information exchange network. Assuming that the initial states of these followers are all zero, they will always remain zero. Therefore, it is impossible to achieve control for any initial states $w_{k}(0),\;k\in{\cal{V}}_{{\cal{L}}}$.■