Robust Leader-Following Output Regulation of Uncertain Multi-Agent Systems With Time-Varying Delay

Ⅰ. INTRODUCTION

During the last decade, cooperative control of multi-agent systems has been extensively studied due to its ever-increasing applications in many fields such as formation control, sensor network, flocking, synchronization of coupled oscillators and robot position synchronization [1]-[6]. Leader-following consensus problem as one of the essential problems in cooperative control has also received considerable attention in recent years. For example, Meng et al. [7] provided some analysis results for both leaderless and leader-following consensus algorithms for first-order and second-order multi-agent systems in presence of communication and input delays. An LMI approach for designing a leader-following consensus protocol was also introduced for general linear multi-agent systems in presence of time-delay by Ding et al. [8] in which the leader has the same dynamics as the followers. Furthermore, for time-varying delayed first-order multi-agent systems with a static leader, leader-following consensus analysis was studied in [9] and some sufficient conditions were obtained to guarantee the convergence of the agents to the leader. A leader-following consensus problem for second-order multi-agent systems with communication time-varying delays was also investigated in [10]. For systems with large delay sequences, leader-following consensus problem of discrete-time multi-agent systems has been addressed in [11]. Moreover, Cao et al. [12] studied the distributed containment control problem of mobile autonomous agents with multiple stationary or dynamic leaders under both fixed and switching directed network topologies. For uncertain multi-agent systems, Liu and Huang [13] addressed a leader-following consensus problem for a class of uncertain higher-order nonlinear multi-agent systems in presence of external disturbances. Recently, a leader-following consensus problem has been addressed in [14] for a class of uncertain multi-agent systems with actuator faults.

In many practical leader-following multi-agent systems, the control objective is to follow a combination of the leader states as an output. This is analogous to the case that the follower agents have a virtual leader which means that there is an exo-system acting as a leader. This problem is called leader-following output regulation that has attracted attention in recent years [15]-[18]. Therefore, the problem of leader-following output consensus can be viewed as the special case of cooperative output regulation problem. In [15], an algorithm for designing a distributed control law is provided for a robust cooperative control of multi-agent system with distributed output regulation. Yan and Huang [16] addressed the analysis of a cooperative output regulation for linear time-delay multi-agent systems. Moreover, a cooperative output regulation analysis of heterogeneous multi-agent systems is carried out in [17]. To the best of our knowledge, very few works have focused on output regulation problem of multi-agent systems with time-delay. Recently, a descriptor approach to robust leader-following output regulation design of multi-agent systems was given in [19] in the presence of both uncertainty and constant transmission delay. The presented method in this study works for both stable and unstable agents. Furthermore, Yan and Huang [18] addressed the cooperative robust output regulation problem for a discrete-time linear time-delay multi agent systems with stable agents. Since, the presented output regulation methods in the literature for time-delay multi-agent systems cannot handle the time-varying delays, we are motivated to investigate a robust leader-following output regulation for uncertain general linear multi-agent systems with stationary and ramp-type dynamic leaders as well as both stable and unstable agents in presence of time-varying delay. The contributions of this paper are summarized as follows:

1) A new regulation protocol is proposed in this paper that works for both stationary and ramp-type dynamic leaders.

2) Applying this protocol, a time-delay closed-loop system of retarded type is obtained that is simpler to deal with than the closed-loop system of neutral type presented in [19].

3) Using direct Lyapunov-Krasovskii method, robust analysis and design conditions for the leader-following output regulation of uncertain multi-agent systems with transmission time-varying delay are given in terms of linear matrix inequalities. The derived LMI conditions have smaller dimensions than the ones obtained in [19].

4) As mentioned, in this paper, a time-varying transmission delay is assumed in leader-following multi-agent system whereas the studies in [18] and [19] are based on a constant time-delay. Furthermore, our method can deal with both stable and unstable agents.

This paper is organized as follows. Problem formulation and preliminary results as well as a new regulation protocol are given in Section Ⅱ. In Section Ⅲ, The robust stability analysis conditions are provided in terms of certain linear matrix inequalities. Moreover, the design conditions are given in Section Ⅳ. Some illustrative examples are provided in Section Ⅴ to show the effectiveness of the proposed methods. Finally, the concluding remarks are given in Section Ⅵ.

Notations : $\mathbb{R}$ denotes the real number field. $\mathbb{R}$ $^{m\times m}$ denotes the set of all $m\times m$ dimensional real matrices. $I_{m}$ denotes m×m dimensional identity matrices and $\otimes$ denotes the Kronecker product. For a given real vector or matrix $X$ , $X^{T}$ denotes its transpose and $\left\| X\right\|$ denotes its Euclidean norm. For a square nonsingular matrix $X$ , $X^{-1}$ denotes its inverse matrix. Throughout the paper, the notations $\Delta A$ and $\Delta B$ are used for $\Delta A(t)$ and $\Delta B(t)$ in some of the equations for the sake of brevity.

Ⅱ. PRELIMINARIES AND PROBLEM STATEMENT

A Graph Theory

Let $G$ = ( $\mathcal {V}, \mathcal {E}, \mathcal {A}$ ) be a digraph with the set of vertices $\mathcal {V}$ = $\{1, 2, \ldots, N\}$ and the set of edges $\mathcal {E}$ $\subseteq$ $\mathcal {V}\times\mathcal {V}$ . In $G$ , the $i$ th vertex represents the $i$ th agent. The adjacency matrix $\mathcal {A}=[a_{ij}]\in\mathbb {R}^{N\times N}$ represents the communication topology among the agents in which $a_{ii} =0$ , $a_{ij} >0$ if the $j$ th agent is a neighbor of the $i$ th agent, otherwise $a_{ij} =0$ . The degree matrix of graph $G$ is denoted by $\mathcal {D}$ =diag $\{d_{1}, d_{2}, \ldots, d_{N}\}$ , in which $d_{j}=\sum_{j=1}^{N}a_{ij}$ . Correspondingly, the Laplacian matrix of the directed graph $G$ is defined as $L_{s}$ = $\mathcal {D}-\mathcal {A}$ . In a directed edge, the information flows from the agent $i$ to the agent $j$ and is represented as a pair ( $i$ , $j$ ) $\in\mathcal {E}$ . A sequence of directed edges is called directed path in which the edges all are in the same direction. A digraph is called connected if there is a directed path between every pair of vertices.

Assume $\bar{G}$ is a graph with a leader node 0 and N follower nodes. A diagonal matrix $M = {\rm{diag}}\left\{ {{m_1}, {m_2}, \ldots , {m_N}} \right\}$ is the leader adjacency matrix with $m_{i} \ge 0$ . If the ith agent has access to the information of the leader, then $m_{i} >0$ ; otherwise, $m_{i}$ = 0. The node 0 is globally reachable if there exists a path in $\bar{G}$ from every node i in G to node 0.

B Multi-Agent System Structure and Formulation

Consider a group of N uncertain $n$ th order agents represented by the following linear differential equation for each agent:

where $x_{i} (t)\in$ $\mathbb{R} ^{n}$ , $u_{i} (t)\in$ $\mathbb{R} ^{m}$ , $y_{i} (t)\in$ $\mathbb{R}^{r}$ are respectively the agent i's state, input and output, respectively, which can only use the local information from its neighbor agents. Moreover, the constant matrices $A\in \mathbb{R}^{n\times n}$ , $B\in\mathbb{R}^{n\times m}$ and $C\in\mathbb{R}^{r\times n}$ represent the identical nominal dynamics and $\Delta A(t)$ and $\Delta B(t)$ are real matrix functions representing time-varying parameter uncertainties. These uncertainties are the result of model linearization and unmodeled dynamics. These admissible uncertainties are assumed to be of the form

where $F_{a} (t)$ and $F_{b} (t)$ are unknown real time-varying matrices with Lebesgue measurable elements satisfying

and $D_{a} , E_{a} , D_{b} , E_{b}$ are real known constant matrices which represent how the uncertain parameters in $F_{a} (t)$ and $F_{b} (t)$ enter the nominal matrices $A$ and $B$ .

In this paper we consider the design of an appropriate feedback control $u_{i}(t)$ , $i=1, \ldots, N$ to render all the follower agents' output to asymptotically follow the leader. This is stated by the following definition.

Definition 1: For any initial condition $y_{i}(0)$ and the leader $y_{0}(t)$ , the leader-following output regulation is achieved if there exists a local feedback control $u_{i}(t)$ for each agent such that the closed-loop system satisfies

The control objective of the leader-following output regulation of a multi-agent system is to follow the output of a real or virtual leader. This output plays the role of a set-point for the followers. Moreover, in many practical problems, the leader has its own feedback independent of the followers in order to obtain the appropriate set-point in its output for the followers. Therefore, we find that in a general case, the model of the followers and the leader can be non-identical. On the other hand, in practice, many of the desired outputs of the agents can be approximated by stationary or ramp signals. Hence, we focus on designing leader-following output regulation of the uncertain multi-agent systems for stationary and ramp-type dynamic leaders in this paper. These leaders are labeled as $i$ =0 and considered by the following representations:

where $y_{0} (t)\in \mathbb{R}^{r}$ is the output of the leader and $a_{0} , \, \, r_{0} \in \mathbb{R}$ . The leaders introduced in (4) and (5) refer to stationary and ramp-type dynamic leaders respectively that should be followed by all the follower agents. The network topology of the multi-agent system is shown in Fig. 1. In this topology, it is assumed that $N'$ number of the agents ( $0<N'\le N$ ) have access to the leader. These agents are named as "reference followers". Moreover, we assume that the reference followers have a limited computational capability and no computation or processing is carried out in the rest of the followers. Together with the reference followers, the leader can also perform required computations in the network of the multi-agent system. Let $\tau _{r} (t)$ be the data transmission delay from the leader to the reference followers and $\tau _{s} (t)$ be the delay of transmitting data from the reference followers to the leader. It is supposed that the data transmission delay between the follower agents is negligible.

Figure  1.  Network topology of the directed multi-agent system.

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Now, we state the following assumptions that are needed throughout the paper.

Assumption 1: The graph $\bar{G}$ is fixed and directed.

Assumption 2: The pair ( $A, B$ ) is stabilizable and the pair ( $A, C$ ) is detectable.

Assumption 3: The digraph $\bar{G}$ is strongly connected.

C Leader-Following Regulation Protocol

To deal with the leaders (4) and (5) in the network of multi-agent system presented in Fig. 1, we propose a new consensus protocol for system (1) in presence of transmission delay shown in a general form that works for both stationary and dynamic leaders (4) and (5) as

in which

with $\tau (t)=\tau _{r} (t)+\tau _{s} (t)$ , $0\le \tau (t)<{\dot{\tau} }$ , $0\le \, \, |\dot{\tau} (t)|\, <\lambda$ , $0\le \tau _{r} (t)<\bar{\tau }_{r}$ and $0\le \tau _{s} (t)<\bar{\tau }_{s}$ , where $\bar{\tau }=\bar{\tau }_{r} +\bar{\tau }_{s}$ .

As seen in (6), the proposed regulation protocol is composed of two parts: consensus $(u_{iC} (t))$ and tracking $(u_{iT} (t))$ . Obviously, the reference followers receive both consensus and tracking parts as ${m}_{i}$ $\neq$ 0 whereas the rest of the followers merely receive the consensus part $(u_{iC} (t))$ of the protocol (6). Furthermore, we assume that the tracking part of the regulation protocol is computed in the reference followers. This is why apart from the induced transmission delay $\tau _{r} (t)$ in the leader signal, no other transmission delay is induced in the tracking part of the protocol (6). We further assume that $u_{iC} (t)$ is computed in the leader and is distributed among the followers through the reference followers. Thus, both delays $\tau _{r} (t)$ and $\tau _{s} (t)$ are induced in the consensus part of the protocol (6). Furthermore, $K, \, K_{I} , \, K_{II} , \, K'_{i} , \, K'_{Ii} , \, K'_{IIi} \in$ $\mathbb{R}$ $^{m\times r}$ are feedback matrices to be designed and ${m}_{i}$ is a scalar that is 1 if the agent i is the neighbor of the leader and 0 otherwise. As seen in (6), different controller gains $K'_{i} , \, K'_{Ii} , \, K'_{IIi}$ are considered for different agents. This strategy leads to more flexibility in design and less conservatism in the result. Applying the consensus protocol (6) to the system (1), the closed-loop system equation for the agent i is represented as

with $A_{\Delta}= A+\Delta A$ and $B_{\Delta}= B+\Delta B$ . Moreover, $y_{0} (t)=r_{0}$ and $y_{0} (t)=r_{0} t$ are respectively considered for the leaders (4) and (5). Then, the network dynamics of the multi-agent system is represented as

$L_s$ is the Laplacian matrix, $x(t)=\left[\begin{array}{c} {x_{1} (t)} \\ {\vdots } \\ {x_{N} (t)} \end{array}\right]$ , $K'={\text {diag}}\{K'_{11} , \ldots , K'_{1N}\}$ , $K'_{I} ={\text {diag}}\{K'_{21} , \ldots, K_{2N}\}$ , $K'_{II} ={\text {diag}}\{K'_{31} , \ldots , K'_{3N}\}, M={\text {diag}}\{m_{1} , \ldots , m_{N}\}.$

Remark 1: It is well-known that a Proportional-Integral (PⅠ) or Proportional-Integral-Double integral (PⅡ $^2$ ) controller designed for a stabilization problem can be further used for tracking problem of the same closed-loop system. Therefore, the leader-following output regulation of the system (8) is guaranteed as long as the stability of the following system is satisfied.

Remark 2: The simple and double integrator statements in (6) play important roles in achieving a general condition for the robust leader-following output regulation of the uncertain multi-agent systems with general linear dynamics in presence of each of the leaders (4) and (5). In other words, these integrators provide output tracking of a closed-loop multi-agent system for stationary or ramp-type dynamic leader that can prevalently occur in practice.

Now, we consider (9) and define new variables $\eta (t)=\int _{0}^{t}\xi (\alpha )d\alpha$ and $\xi (t)=\int _{0}^{t}x(\alpha )d\alpha$ . Then, the state space equation of the multi-agent system is represented as follows:

or

in which $\bar{x}(t)=\left[\begin{array}{c} {\eta (t)} \\ {\xi (t)} \\ {x(t)} \end{array}\right]$ , $x(t)=\left[\begin{array}{c} {x_{1} (t)} \\ {\vdots } \\ {x_{N} (t)} \end{array}\right]$

where

Now, we present the following lemmas which will be used in the main results of the paper.

Lemma 1 [20]: Let $H$ , $L$ and $F(t)$ be real matrices of appropriate dimensions with $F(t)$ being a matrix function. Then, for any $\sigma>$ 0 and $F^{T} (t)F(t)\le I,$ we have

Lemma 2 [21]: If $W>0$ , there exist $W^{-1}$ . Thus,

For the sake of brevity, the proofs of these two lemmas are omitted.

Ⅲ. ROBUST STABILITY OF LEADER-FOLLOWING OUTPUT REGULATION

In this section, we concentrate on deriving a robust stability analysis criterion for leader-following output regulation of uncertain time-delay multi-agent system (10). Defining

in (10), we will have

A Lyapunov-Krasovskii functional for system (14) has the form

where

in which $\bar{P}$ , $Z$ , $Q$ are real symmetric positive definite matrices. Now, we are in a position to state the following theorem which gives a sufficient condition for robust stability analysis of leader-following output regulation for the uncertain multi-agent system (10).

Theorem 1: Under Assumptions 1-3, for given time-delay $\tau (t)$ with $0\le \tau (t)<\bar{\tau }$ and $0\le \, \, |\dot{\tau }(t)|\, <\lambda$ , the system (14) is asymptotically stable if there exist scalars $\sigma _{1, 2} , i=1, 2, 3$ , positive definite symmetric matrices $\bar{P}$ , $X$ , $Q, Z\in \mathbb{R}^{3nN\times 3nN}$ and matrix $Y\in\mathbb{R}^{3nN\times 3nN}$ satisfying the following LMIs:

in which

with

Proof: Using Lyapunov-Krasovskii functional in (15) and (16), the delay-dependent sufficient conditions for the stability of the delayed multi-agent system (14) are presented. For sake of brevity, the detailed proof of this theorem is given in Appendix.

Remark 3: Theorem 1 provides robust stability analysis conditions for uncertain multi-agent system (14). Feasibility of the set of LMI conditions (17) and (18) guarantees the robust leader-following output regulation of the uncertain time-delay multi-agent system (14) with known state-space matrices. The significant advantage of this theorem is to give a set of stability analysis LMI conditions that are valid for both leaders (4) and (5).

Remark 4: It is worthwhile mentioning that the proposed leader-following output regulation protocol can be further used for low frequency sinusoidal leaders or the leaders that can be approximated by a combination of a number of sinusoidal signals. The tracking quality for high frequency sinusoidal leaders can be improved by incorporating a design criterion for widening the bandwidth of the closed-loop system.

Remark 5: In non-time-delay case, all the rows and columns containing $\bar{\tau}$ are omitted from the LMI conditions (17) and (18). Therefore, some less complicated LMI conditions are provided for this case. These delay free LMI conditions are also obtained by considering the state-space equation of the closed-loop multi-agent system with $\tau (t)=0$ }and the Lyapunov function as $V(t)=\bar{x}^{T} (t)\bar{P}\bar{x}(t)$ . All the proof procedure is given in the appendix.

Ⅳ. DESIGN OF ROBUST LEADER-FOLLOWING OUTPUT REGULATION CONTROL

In this section, using the analysis conditions provided in the previous section as well as the preliminaries given in the Section Ⅱ, design conditions for the proposed leader-following output regulation control are given. These conditions are presented by the following theorem.

Theorem 2: Consider the multi-agent system (1) and the control law (6) with time-varying transmission delay $\tau (t)$ that $0\le \tau (t)<\bar{\tau }$ and $0\le |\mathop{\tau }(t)|\, <\lambda$ . Suppose Assumptions 1-3 are satisfied. The leader-following output regulation is asymptotically achieved for the leaders (4) and (5) if there exist scalars $\varepsilon , \, \, \sigma _{1, 2} , \, \, \alpha _{i} \, >0, \, \, i=1, \, 2, \, 3$ , positive definite symmetric matrices $\bar{L}$ , $T_{1}$ , $T_{2}$ $R\in\mathbb{R}^{3nN\times 3nN}$ , matrix $N\in\mathbb{R}^{3nN\times 3nN}$ , matrices $K'$ , $K'_{I}$ , $K'_{II} \in\mathbb{R}^{mN\times rN}$ and scalars $K$ , $K_{I}$ , $K_{II}$ satisfying the following matrix inequalities:

in which

and $\bar{L}=EL$ where

Proof: See Appendix.

Remark 6: As we mentioned earlier, Theorem 1 gives a general set of matrix inequality conditions for stability analysis and design of leader-following output regulation of multi-agent system (1) with regulation protocol (8). Solving the presented conditions in (19) and (20), we obtain the appropriate solution for the regulation protocol (8) that works for each of the leaders (4) and (5). In a special case that the stationary leader is merely considered, one can set $K_{II} =K_{IIi} =0$ and derive a set of simpler stability analysis and design conditions than the general form. This special case will lead to the regulation protocol (21) that is a sub-case of the regulation protocol (7).

Then, the following corollary is obtained.

Corollary 1: Under Assumptions 1-3, the leader-following output regulation is asymptotically achieved for the multi-agent system (1) with $0\le \tau (t)<\bar{\tau }$ , $0\le \, \, |{\dot{\tau} }(t)|\, <\lambda$ and the control law (21) in presence of stationary leaders, if there exist scalars $\varepsilon$ , $\sigma_{1}$ , $\sigma_{2}>$ 0, positive definite symmetric matrices $\bar{L}$ , $T_{1}$ , $T_{2}$ , $R\in\mathbb{R}^{2nN\times 2nN}$ , matrix $N\in\mathbb{R}^{2nN\times 2nN}$ and matrices $K$ , $K_{I}$ , $K'$ , $K'_{I} \in\mathbb{R}^{m\times r}$ satisfying the following matrix inequalities:

in which

and $\bar{L}=EL$ where

Proof: The proof is omitted since it can be directly established using Theorem 2.

Remark 7: As seen in the leader-following conditions presented in Theorem 2 and Corollary 1, the matrix inequalities (19) and (22) are bilinear (BMI). Although BMIs are categorized in NP-hard problems, there exist practically effective algorithms for BMI solutions [22], [23]. Furthermore, the PENBMI solver in MATLAB environment can be used to solve BMIs [24]. It is noteworthy that for the first-order systems as a special case, the obtained BMI condition in (19) and (22) turn into LMI that are much easier to deal with as compared to BMIs. The details are given as follows.

A Special Case: First-Order Systems

Consider the following first-order system:

where $a$ , $\Delta a, b, \Delta b$ and $c$ are scalars. Replacing $A = a, ~B = b, \Delta A(t)=\Delta a(t)$ and $\Delta B(t)=\Delta b(t)$ , the BMI terms in the matrix inequalities (22) and (23) obtained in Theorem 2 can be written as $(M\otimes bc)K'_{II} \hat{L}$ , $(M\otimes bc)K'_{I} \hat{L}$ , $(M\otimes bc)K'\hat{L}$ , $L_{s} \otimes (bcK_{II} L_{1} )$ , $L_{s} \otimes (bcK_{I} L_{1} )$ , $L_{s} \otimes (bcKL_{1} )$ . Defining $V_{1} =K_{II} L_{1}$ , $V_{2} =K_{I} L_{1}$ , $V_{3} =KL_{1}$ , $W_{1} =K'_{II} \hat{L}$ , $W_{2} =K'_{I} \hat{L}$ , $W_{3} =K'\hat{L}$ , the matrix inequalities (22) and (23) turn into LMI. Moreover, the controller gains are given by $K=V_{3} L_{1}^{-1}$ , $K_{I} =V_{2} L_{1}^{-1}$ , $K_{II} =V_{1} L_{1}^{-1}$ , $K'=W_{3} \hat{L}^{-1}$ , $K'_{I} =W_{2} \hat{L}^{-1}$ , $K'_{II} =W_{1} \hat{L}^{-1}$ for $i=1, \, \, 2, \, \, 3$ . A similar procedure can be used for the results provided in Corollary 1 in order to obtain LMI design conditions for the leader-following output regulation problem for the stationary leaders. Thus, it is omitted here.

Ⅴ. SIMULATION

In this section, two examples are given to illustrate the effectiveness of the theoretical results.

A Example 1

In this example, we consider two multi-agent systems Ⅰ and Ⅱ with six uncertain first-order agents as follows:

The agents in system Ⅰ are stable in which $a$ = -5, $b$ = 5, $c$ = 1, $D_a$ = 0.15, $D_{b}$ = 0.1, $E_{a}$ = 1, $E_{b}$ = 1 with a time-delay bound $\bar{\tau }$ = 180 ms and $\bar{\tau }_{r}$ = 20 ms. For system Ⅱ, the agents are unstable with $a$ = 1.1, $b$ = 6, $c$ = 1, $D_{a}$ = 0.1, $D_{b}$ = 0.05, $E_{a}$ = 1, $E_{b}$ = 1, with a time-delay bound $\bar{\tau }$ = 70 ms and $\bar{\tau }_{r}$ = 20 ms. In both systems Ⅰ and Ⅱ, we have $F_{a}^{T} (t)F_{a} (t)<1$ , $F_{b}^{T} (t)F_{b} (t)<1$ . The communication topology is represented by the Laplacian matrix $L_{s}$ = $[l_{ij}]$ for $i$ , $j$ =1, 2, $\dots$ , 6 where $l_{ii}$ = 1 for $i$ = 1, 2, $\dots$ , 6 and $l_{12}$ = $l_{23}$ = $l_{34}$ = $l_{45}$ = $l_{56}$ = $l_{61}$ = 1, otherwise $l_{ij}$ = 0 for $i$ = 1, 2, $\dots$ , 6. Furthermore, the leader adjacency matrix $M$ is given as $M={\rm diag} \left\{1, 0, 1, 0, 1, 0\right\}$ . We set $\varepsilon$ = 0.2, $\alpha_{1}$ = $\alpha_{3}$ = 1, $\alpha_{2}$ = 0.25, $\sigma_{1}$ = $\sigma_{2}$ = 1. Using Theorem 2 and the results presented in Section Ⅳ-A. as well as LMI Toolbox in MATLAB, the designed controllers are obtained for the multi-agent system Ⅰ as

and the multi-agent system Ⅱ as

Two set points ramp and sinusoidal are applied to the agent 1 and 2 as the state of the leader. Figs. 2-5 display the simulation results of the multi-agent systems Ⅰ and Ⅱ. The initial conditions for the agents 1, 2, ${\ldots}$ , 6 are considered as 0.5, 2, -1, 1, -0.8, -2 and 3, 2, -1, 2.5, -1.5, -2 for systems Ⅰ and Ⅱ, respectively.

Figure  2.  Controlled outputs of the follower agents (solid) and output of the leader (dashed) in system Ⅰ.

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Figure  3.  Controlled outputs of the follower agents (solid) and output of the leader (dashed) in system Ⅰ.

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Figure  4.  Controlled outputs of the follower agents (solid) and output of the leader (dashed) in system Ⅱ.

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Figure  5.  Controlled outputs of the follower agents (solid) and output of the leader (dashed) in system Ⅱ.

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As seen in Figs. 2 and 3, a leader-following output regulation is achieved for multi-agent system Ⅰ and the agents' outputs follow a ramp path as well as sinusoidal path using the regulation protocol (6). Moreover, the unstable multi-agent system Ⅱ has been stabilized and the leader-following output regulation is obtained as seen in Figs. 4 and 5. These simulation results show that the proposed method can be used for the leader-following output regulation problems with hyperbolic leaders as well. Furthermore, as expected, the presented method provides leader-following output regulation for multi-agent systems in which the followers do not have necessarily the same dynamics as their leader.

B Example 2

In this example, we address the problem of the robust stability analysis of the leader-following output regulation for multi-agent systems investigated in Section Ⅲ. To this aim, we consider equation (1) with the following state-space matrices of an unstable nominal multi-agent system:

The communication topology and the leader adjacency matrices are represented as $L_{s} =\left[\begin{array}{ccc} {1}&{-1}&{0} \\ {0}&{1}&{-1} \\ {-1}&{-1}&{2} \end{array}\right]$ and $M={\rm diag}\, \left\{1, \, 0, \, 1\right\}$ , respectively. Furthermore, considering the dynamic leader (5) and the regulation protocol (6), the controller gains are assumed as follows:

Now, Theorem 1 can be applied for the stability analysis of the closed-loop system in presence of delay. For this purpose, we use the LMI Toolbox in MATLAB and solve the LMI conditions (17) and (18) with $\bar{\tau }=$ 20 ms and $\bar{\tau }_{r} =$ 10 ms. Simulation results show that these LMI conditions are feasible. It means that the regulation protocol (6) with the controller gains (27) stabilizes the closed-loop multi-agent system for the leader (5) in the presence of $\bar{\tau }=$ 20 ms. We set a step leader $r$ = 2 for the first output and a ramp-type leader with a slope of 1 for the second output. Furthermore, we assume that the initial conditions of the agents' state variables are 0.1, 0.2, 0 for the agents 1-3 respectively. The leader-following output regulation of the closed-loop multi-agent system is achieved as shown in Figs. 6 and 7.

Figure  6.  First controlled outputs of the follower agents (solid) and output of the leader (dashed).

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Figure  7.  Second controlled outputs of the follower agents (solid) and output of the leader (dashed).

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Now, we perform some simulations in order to examine the robust stability of the closed-loop system in the presence of system parameter variations. Since the stability of the closed-loop system is affected by the uncertainty in time-delay, gain and poles of the open-loop system, we investigate the performance of the closed-loop system in the presence of these uncertainties. To this aim, we set $\Delta A$ = 0, $\Delta B$ = 0 and increase $\bar{\tau }$ while repeating simulation. The simulation results show that the LMI conditions (17) and (18) are feasible for the maximum time-delay $\bar{\tau }_{\max }$ = 77 ms. This means that the stability of the closed-loop multi-agent system is guaranteed for all delays $\tau (t)\le$ 77 ms with $\Delta A$ = 0, $\Delta B$ = 0. Now, we set $\Delta A =$ 0 and define $\Delta B =$ $\alpha B$ . Then, solving the LMI conditions (17) and (18) in presence of the variations of $\alpha$ and $\bar{\tau }$ , the feasibility region for these LMI conditions is obtained as shown in Fig. 8. Therefore, the colored area in Fig. 8 shows the guaranteed stability region for the closed-loop multi-agent system for the variations of $\bar{\tau }$ and $\Delta B$ with $\Delta A$ = 0. Since the variations of $\Delta B$ directly affects the gain of the closed-loop multi-agent system, it is reasonable to observe that the closed-loop multi-agent system is stable for $\Delta B ~\mathit{\boldsymbol{\ge}}$ -0.83 $B$ where $\Delta A=0, \bar{\tau }=0$ . To see the stability region of the closed-loop multi-agent system in presence of the uncertainty on the poles of the system (26) that are at $\lambda_{1, 2}$ = 0.5 $\mathit{\boldsymbol{\pm}}$ 3.1225 $i$ , we set $\Delta B = 0$ . Then, varying the real part of the poles of the system (26) gives the range $\Delta Re(\lambda_{i})\le$ 5.05, $i$ = 1, 2 for the feasibility of the LMI conditions (17) and (18). As seen in Fig. 9, increasing the maximum time-delay $\bar{\tau}$ leads to decreasing the upper limit of the real part uncertainty of the system poles.

Figure  8.  Stability region in presence of $\bar{\tau }$ and $\Delta B$ variations when $\Delta A=0$ .

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Figure  9.  Stability region in presence of $\bar{\tau }$ and $\Delta A$ variations due to real part variations of $\lambda _{1, 2}$ when $\Delta B=0$ .

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Ⅵ. CONCLUSION

The problem of leader-following output regulation analysis and design of uncertain general linear multi-agent systems with transmission delay has been presented in this paper. The proposed method can be used for both stable and unstable follower agents under a directed graph. Many of the leader outputs can be approximated by the stationary and ramp signals. Therefore, two stationary and ramp-type dynamic leaders have been investigated in this paper that cover adequate variety of the leaders. To this aim, we proposed a new regulation protocol for the closed-loop system. The analysis conditions have been presented in terms of certain LMIs in which the provided results for design purposes are bilinear. It was shown that for first-order systems as special case, the presented results are turned into LMI. Finally, we presented two analysis and design examples for the leader-following output regulation of time-delay uncertain multi-agent systems. We showed that our proposed method effectively meets the quality requirements of the leader-following output regulation for both stationary and dynamic leaders. Moreover, we showed that this method can be further used in presence of low-frequency sinusoidal leaders. To achieve superior leader-following output regulation for high-frequency sinusoidal leaders, the bandwidth of the closed-loop system should be widened. To this aim, an appropriate performance objective is required to be added to the stability criterion that can be considered in the future works.

APPENDIX

Proof of Theorem 1: Differentiating $V_{1}$ in (16) with respect to $t$ and using the integral inequality in [25], results into

with

where

and

Also, the time-derivative of ${V}_{2}$ and ${{V}}_{3}$ can be represented as follow

Since $\dot{V}=\dot{V}_{1} +\dot{V}_{2} +\dot{V}_{3}$ , the inequalities (28)-(31) results a bound of $\dot{V}$ as follows

in which

If the condition $\Theta<$ 0 holds, the negative definiteness of $\dot{V}(t)$ is guaranteed and asymptotic stability of the system (10) is established. Using Schur complement on $\Theta <$ 0 the following matrix inequalities are obtained

in which

where

Partitioning the nominal and uncertain parts in (33) and considering the definitions in (2), we have

where

Since the conditions in (3) are satisfied, it can be easily shown that

Therefore, using Lemma 1, the inequality (34) is written as

Applying Schur complement and considering (29), the matrix inequalities (17) and (18) are obtained.

Proof of Theorem 2: Pre and post multiplying the matrix inequality (17) by diag $\{\bar{P}^{-1} , \, \bar{P}^{-1} , \, Z^{-1} , \, Z^{-1} , \, I, \, I, \, I, \, I\}$ as well as defining $\bar{L}=\bar{P}^{-1}$ , $T_{1} =\bar{L}X\bar{L}$ , $N=\bar{L}Y\bar{L}$ , $T_{2} =\bar{L}Q\bar{L}$ , $R=Z^{-1}$

in which $\bar{L}=EL$

the matrix inequality (22) is obtained. Furthermore, Pre and post multiplying the matrix inequality (29) by diag $[\bar{P}^{-1} , \, \bar{P}^{-1}]$ and using Lemma 2, the matrix inequality (23) is also obtained.