Leader-Following Consensus of High-Order Perturbed Multi-agent Systems under Multiple Time-Varying Delays

TL;DR

This paper develops a novel distributed control method for high-order perturbed multi-agent systems with multiple time-varying delays, ensuring stability and consensus through Lyapunov-Krasovskii functionals and linear matrix inequalities.

Contribution

It introduces an optimal control scheme combining sliding mode control and linear consensus for delayed multi-agent systems, with delay estimation and leader synchronization.

Findings

Ensures stability and convergence of multi-agent systems with delays.

Provides delay bounds via linear matrix inequalities.

Achieves leader-following consensus with optimized control gains.

Abstract

Solving an output consensus problem in multi-agent systems is often hindered by multiple time-variant delays. To address such fundamental problems over time, we present a new optimal time-variant distributed control for linearly perturbed multi-agent systems by involving an integral sliding mode controller and a linear consensus scheme with constant wights under directed topology. Lyapunov-Krasovskii functionals along with linear matrix inequalities are jointly employed to demonstrate the associated closed-loop stability and convergence features. Maximum delays for the communicating networks are also estimated by linear matrix inequalities. Synchronizing a network of linear time-variant systems to the associated leader dynamics is additionally taken into account by developing an optimization algorithm to find the constant control gains.

Equations

$$u(\Vert x\Vert )\le V(t,x(t))\le v(\Vert x\Vert ),t\in \mathbb{R},x\in {\mathbb{R}}^n,$$

$$V(t,x(t))\le -w(\Vert x\Vert )$$

$$V(t+\theta ,x(t+\theta ))\le V(t,x(t))$$

$$R(x)<0,Q(x)-S(x)R(x{)}^{-1}S(x{)}^{\prime }<0,$$

$$\left[\begin{array}{cc}Q(x)& S(x)\\ S(x{)}^{\prime }& R(x)\end{array}\right]<0$$

$$|2a(t{)}^{\prime }b(t)|$$

$$-2a^{\prime }{\int }_{t-h(t)}^t\dot{b}(s)ds\le \bar{h}a(t{)}^{\prime }{\psi }^{-1}a+{\int }_{t-h(t)}^t\dot{b}(s)\psi \dot{b}(s)ds$$

$$M_1+\tau M_2<0$$

$$M_1+\xi M_2<0\forall \text{leavevmode}\xi \text{leavevmode}(0,{\xi }_m]$$

$${\xi }_m=\frac{\Vert M_1\Vert }{\Vert M_2\Vert }$$

$${\dot{x}}_i(t)$$

$$A=\left[\begin{array}{ccccc}0& 1& 0& \dots & 0\\ 0& 0& 1& \ddots & 0\\ \dots & \dots & \dots & \ddots & ⋮\\ 0& 0& 0& \ddots & 1\\ -a_1& -a_2& \dots & \dots & -a_n\end{array}\right]B=\left[\begin{array}{c}0\\ ⋮\\ 0\\ b\end{array}\right]$$

$$|{\omega }_i(t)|$$

$${\dot{x}}_0$$

$$u_i(t)=$$

$$\displaystyle-\rho\operatorname{sign}\big{(}s_{i}(t)\big{)}$$

$$s_i(t)=1_n^{\prime }{x}_i(t)+z_i(t)$$

$$\displaystyle\dot{z}_{i}(t)=\mathbb{1}_{n}^{\prime}\Big{(}Ax_{i}(t)+B\sum_{j=0}^{N}\alpha_{ij}k_{ij}\big{(}x_{i}(t-\tau_{ij}(t))-x_{j}(t-\tau_{ij}(t))\big{)}\Big{)}$$

$$z_i(0)=-1_n^{\prime }{x}_i(0)$$

$${\tau }_{i0}(t)\in$$

$${\tau }_{ij}(t)\in$$

$$|{\dot{\tau }}_{i0}(t)|$$

$$|{\dot{\tau }}_{ij}(t)|$$

$$\left[\begin{array}{ccc}{M}_1+3{\tau }^{\star }{A}_0^{\prime }H{A}_0& {\sum }_{l=1}^N{\tau }^{\star }P{\hat{A}}_l& {\sum }_{p=1}^m{\tau }^{\star }P{\tilde{A}}_p\\ \ast & -{\sum }_{l=1}^N{\tau }^{\star }{R}_l& 0_{Nn\times Nn}\\ \ast & \ast & -{\sum }_{p=1}^m{\tau }^{\star }{\bar{R}}_p\end{array}\right]<0$$

$$3\sum _{l=1}^N{\tau }^{\star }{\hat{A}}_l^{\prime }H\sum _{l=1}^N{\hat{A}}_l+{\bar{M}}_1<0$$

$$3\sum _{p=1}^m{\tau }^{\star }{\tilde{A}}_p^{\prime }H\sum _{p=1}^m{\tilde{A}}_p+{\bar{M}}_1<0$$

$$H$$

$$M_1$$

$${\bar{M}}_1$$

$${\tilde{M}}_1$$

$$A_0$$

$${\hat{A}}_l=\mathrm{diag}\left([A_{(1,1)},\dots ,A_{(N,N)}]\right)\in {\mathbb{R}}^{Nn\times Nn}$$

$$A_{(i,i)}=\{\begin{array}{ll}{A}_{i0}& i=l,\text{leavevmode}{\tau }_l(\cdot )={\tau }_{i0}\\ 0_{n\times n}& i\ne l,\text{leavevmode}{\tau }_l(\cdot )\ne {\tau }_{i0}.\end{array}$$

$${\tilde{A}}_{p(r,q)}=\{\begin{array}{ll}{A}_{ij}& \text{mbox}withi\ne j\text{mbox}if\text{leavevmode}{\sigma }_p(\cdot )={\tau }_{ij}(\cdot ),r=q=i\\ -A_{ij}& \text{mbox}with\text{leavevmode}i\ne j\text{mbox}if\text{leavevmode}{\sigma }_p(\cdot )={\tau }_{ij}(\cdot ),r=i,q=j\\ 0& \text{mbox}otherwise.\end{array}$$

Full text

I Introduction

MuLTI-AGENT Systems (MASs) are extensively networked to describe very large numbers in natural and engineered environments, ranging from biological, social and communication systems [1, 2, 3] to transport, power grids and robotic swarms [4, 5, 6]. The problem with their consensus, however, remains an attractive challenge in controlling multiple interacting agents within the system. Consensus can thereof be regarded as a control objective in which all the agents in a network converge to (or, agree upon) a common value. This is achieved through a given control strategy, referred to usually as a consensus algorithm. Most of the early works on this topic [7, 8, 9] were based on the leaderless consensus in multi-agent systems, whereas all participating agents had to agree with certain coordinate values as problems. Similar studies were then shifted toward a leader-follower approach, wherein a network of follower-agents has to be regulated by the leader coordinates [10, 11, 12, 13].

Inclusion of time-variant delays in any multi-agent model has become a challenge due to the limitation of axonal signal transmission and switching speeds. Being inevitable in many physical systems, a time delay, as an integral part of the convergence in consensus protocols, is therefore to be studied [14]. The stability of a single-agent system with time delays are realized [15, 16, 17], with emphasis on the effect of a delay in exchanging the information in linear stochastic models [15]. The connection between the dynamics of a single oscillator with delayed feedback and a feedforward ring of identical oscillators was also highlighted [16]. Moreover, the real-time data was recorded by a delay scheduled impulsive controller [17]. In this context, a multi-agent model with input and constant communication delays can then be analysed in frequency-domain to underline the fact that the consensus is indeed independent of the communication delays [18]. A distributed and robust $H\infty$ rotating consensus control within a single time delay was then reported [19]. In addition, the problems of synchronization and nonmonotonic transitions in oscillator communities with distributed constant delays were investigated [20]. As for synchronizing a heterogeneous network, multiple constant time delays were similarly involved [21]. Consensus problems of MASs within second-order continuous time were as well investigated by considering single time delays and jointly-connected topologies [22].

The cases for continuous-time and discrete-time first-order MASs with uniform input constant delays could marginalize the maximum consensus delay [23], whereas general second-order models with single constant delay conditioned by an integrator consensus problem [24]. Instantaneous states of the leader itself and the delayed states of its followers were also focused upon [25]. In result, a distributed control algorithm with multiple constant delays for these second-order models was proposed [26]. More to this, the delayed state synchronization of homogeneous discrete-time MASs in the presence of unknown non-uniform communication delays was studied [27]. An observer-based triggering control problem was then based for leader-following consensus of MASs with time-varying delays [28]. Another consensus problem of discrete-time linear MASs was then controlled by a distributed prediction within directed switching topologies and constant delays [29]. Nonlinear agents, however, were synchronized in directed networks within single delays [30].

It is worth to mention that most of these consensus problems are bounded to only a single delay, particularly in those of systems where the agent dynamics, either single or double, integrates with specific consensus protocols. We herein introduce a new linear distributed control in MASs within multiple time-varying delays. This is achieved by combining an integral sliding mode controller with tunable constant weights in well-networked perturbed linear MASs. Compared to that of the existing models, the performance of the proposed scheme is thoroughly analyzed by combining both the Lyapunov- Krasovskii theorem and the Linear Matrix Inequality (LMI) approach. An upper bound for maximum tolerable input delays through LMIs is also provided. Furthermore, the net gain of the proposed control is tuned and optimized.

As follows, preliminaries in the section II present the basic concepts of algebraic graph theory. Section III introduces the problem statement. Our main results are then discussed in the Sections IV and V. Section VI details the numerical models, and finally, the conclusions are collected in the section VII.

II Preliminaries

The set of natural, real, and strictly positive real numbers are denoted by $\mathbb{N}$, $\mathbb{R}$ and $\mathbb{R}^{>0}$, respectively. For $d\in\mathbb{N}$ and a column vector of $x\in\mathbb{R}^{d}$, let $x^{\prime}$ be its transpose and $\|x\|_{2}$ the corresponding 2-norm. Then, define $\mathcal{G}=\left(\mathcal{V},\mathcal{E}\right)$ as a directed graph (digraph), where $\mathcal{V}=\{1,\ldots,N\}$ is a set of nodes (i.e. agents) and $\mathcal{E}\subseteq\{\operatorname{\mathcal{V}}\times\mathcal{V}\}$ is the set of edges. $\mathcal{A}=[\alpha_{ij}]\in\mathbb{R}^{N\times N}$ is the adjacency matrix of $\operatorname{\mathcal{G}}$, with non-zero weight if $i$ communicates with $j$ $(i;j)\subseteq\operatorname{\mathcal{E}}$, $\alpha_{ij}=0$ otherwise. Let ${\mathcal{N}}_{i}=\{j\in\operatorname{\mathcal{V}}:(i,j)\in\operatorname{\mathcal{E}}\}$ be the set of neighbors around agent $i$, showing those of agents that share an edge with agent $i$. Next, let agent “0” be an additional virtual object in the augmented graph $\operatorname{\mathcal{G}}_{N+1}$, while the remainder “0” is considered as the virtual leader for the proposed protocol assuming to be globally reachable in $\operatorname{\mathcal{G}}_{N+1}$.

In the remainder of the paper the following useful properties are exploited:

Theorem 1**.**

Suppose that $f:\mathbb{R}\times C\in({\small\begin{bmatrix}-h&0\\ \end{bmatrix}},\mathbb{R}^{n})\longrightarrow\mathbb{R}^{n}$ in $\dot{x}(t)=f(t,x_{t})$ maps $\mathbb{R}\times$ (bounded sets of $C$) into bounded sets of $\mathbb{R}^{n}$ and also that $u,v,w:$ ${\mathbb{R}}_{>0}\longrightarrow{\mathbb{R}}_{>0}$ are continuous non-decreasing functions, $u(s)$ and $v(s)$ are positive for $s>0$, $u(0)=v(0)=0$ and $v$ is always increasing. If there exists a continuously differentiable function $V:\mathbb{R}\times\mathbb{R}^{n}\longrightarrow\mathbb{R}$ such that

[TABLE]

and the derivative of $V(t,x(t))$ along the solution, $x(t)$, of $\dot{x}(t)=f(t,x_{t})$ satisfies

[TABLE]

whenever

[TABLE]

for $\theta\in{\small\begin{bmatrix}-h&0\\ \end{bmatrix}}$ where $h>0$ is the delay, then the trivial solution of $\dot{x}(t)=f(t,x_{t})$ is uniformly stable. $\blacksquare$

Lemma 1**.**

The Schur complement lemma converts a class of convex nonlinear inequalities that appears regularly in control problems to an LMI. The convex nonlinear inequalities are

[TABLE]

where $Q(x)=Q(x)^{\prime}$, $R(x)=R(x)^{\prime}$, and $S(x)$ depend affinely on $x$. The Schur complement lemma converts this set of convex nonlinear inequalities into the equivalent LMI

[TABLE]

$\blacksquare$**

Lemma 2**.**

By letting $a(t),b(t)\in\mathbb{R}^{n}$ and $\psi\in\mathbb{R}^{n\times n}_{>0}$, the following inequalities are in force

[TABLE]

Moreover, let $h\in\mathbb{R}$ and $\bar{h}$ be the maximum value assumed by a time delay, then

[TABLE]

$\blacksquare$**

Theorem 2**.**

Let $M_{1}\in\mathbb{R}^{n\times n}$ be a negative symmetric matrix and $M_{2}\in\mathbb{R}^{n\times n}$ be a positive definite matrix, and then let $\tau\in\mathbb{R}_{>0}$ be a positive constant. If the following relation is in force:

[TABLE]

then it yields

[TABLE]

where

[TABLE]

$\blacksquare$**

III Problem Statement

We consider a continues-time MAS with topology represented by a directed graph $\mathcal{G}=\left(\mathcal{V},\mathcal{E}\right)$ where the agents have the dynamics described by

[TABLE]

where $x_{i}\in\mathbb{R}^{n}$ represents the state of the $i$-th agent and $u_{i}(t)\in\mathbb{R}$ is a control protocol which needs to be designed. The partial function of $\omega_{i}(t)\in\mathbb{R}$, denotes unknown exogenous perturbations. And also $A\in\mathbb{R}^{n\times n}$ and $B\in\mathbb{R}^{n}$ have the following expression:

[TABLE]

with $b>0$ and $a_{1},a_{2},\ldots,a_{n}$ are positive constants.

Assumption 1**.**

We assume that for the system (8), the unknown perturbations are bounded:

[TABLE]

where $\Gamma_{i}$ is a positive known constant. $\blacksquare$

Our objective is to design a continues-time distributed consensus protocol $u_{i}(t)$ for agents as in Eq. (8) to enable each agent to track the time-varying virtual leader despite delayed communications among agents, wherein the reference dynamic is described as

[TABLE]

IV Proposed Distributed Consensus Protocol

To synchronize the agent’s states to the state’s virtual leader, we present the following local interaction protocol of

[TABLE]

where $k_{ij}^{\prime}\in\mathbb{R}^{n}$ and $\rho\in\mathbb{R}$ are the constant tuning-parameter vectors and scalars, respectively. $\tau_{ij}(t)$ shows the time-varying delay between the communicating agents. We also assume that $\tau_{ij}(t)=\tau_{ji}(t)$. The switching function $s_{i}(t)$ in (12) set as follows:

[TABLE]

where

[TABLE]

To exploit a more compact notation, delays $\tau_{ij}(t)$ can be represented as elements of the following delay set: $\sigma_{p}(t)\in\left\{\tau_{ij}(t):i,j=1,\ldots,N,i\neq j\right\}$ for $p=1,\ldots,m$ with $m\leq N(N-1).$ Analogously, delays $\tau_{i0}$ are elements of the set: $\tau_{l}(t)\in\left\{\tau_{i0}(t):i=1,\ldots,N\right\}$ for $l=1,\ldots,N.$

The next assumption refers to [31]-[32] to solve the communication’s delays.

Assumption 2**.**

Let the known bounds to $\tau_{l}^{\star}$,$\sigma_{p}^{\star}$,$d_{l}$ and $\bar{d}_{p}\in\mathbb{R}_{>0}$ exist and be known in advance such that

[TABLE]

$\blacksquare$**

Theorem 3**.**

Consider the multi-agent system dynamics (8) operating over a communication network whose topology can be described by a directed connected graph $\operatorname{\mathcal{G}}$. Let Assumption 2 be satisfied and there exits $\rho>\varGamma$. Then let (12) be local control protocol for the $i$-th agent communication and assume the state of the agent “0” be globally reachable over $\operatorname{\mathcal{G}}_{N+1}$. Given an upper bound of time-delay function $\tau^{\star}=max\left\{\tau_{l}^{\star},\sigma_{p}^{\star}\right\}>0$, If there exist symmetric positive definite matrices $P,Q_{l},\bar{Q}_{p},R_{l}$ and $\bar{R}_{p}\in\mathbb{R}^{Nn\times Nn}$, such that the following LMIs are feasible

[TABLE]

[TABLE]

[TABLE]

being

[TABLE]

[TABLE]

[TABLE]

with diagonal blocks as

[TABLE]

and matrix $\tilde{A}_{p}=[\tilde{A}_{p(r,q)}]\in\mathbb{R}^{Nn\times Nn}$ within the following entries

[TABLE]

with

[TABLE]

where $r,q=\left\{1,2,\ldots,N\right\}$, and

[TABLE]

Then, the delayed MAS (8) under constant gains $k_{ij}$, achieves synchronization, i.e.

[TABLE]

$\blacksquare$**

Remark 1**.**

In accordance with Theorem 2, it results that agents (8) under the control protocol (12) perform synchronization on the leader’s states in accordance with (30). From (77)-(85) it further results that an estimation of the maximum admissible delay tolerated by MASs (8)-(12), where it can be lower estimated as follows

[TABLE]

$\blacksquare$**

Remark 2**.**

Due to the discontinuous nature of (12), high-frequency chattering on the agents state variables arises in practical implementation. To relax this phenomenon, several useful methods were proposed in [33],[34]. We thereby reformulate (12) as:

[TABLE]

by which a relatively high gain filtering is retained to validate an equivalent control, and subsequently estimate the perturbation in the system by chattering alleviation method. Note that if the time constant $\widetilde{T}$ of the filter is small enough such that the filter preserves the slow component of an equivalent control, and under the realistic assumption that the spectrum of the perturbation $\omega_{i}(t)$ does not overlap with the high-frequency components of the switching control, then:

[TABLE]

so that the unwanted chattering effect is mitigated, while the accuracy of the original discontinuous system is better preserved compared to more of the conventional saturation-based chattering alleviation methods. $\blacksquare$

V Solving optimization over LMI

In this section, we explain how to solve the LMIs feasibility problem (16)-(18) so that we can find the robust values of the constant gains $k_{ij}$ that guarantee the output consensus of the MAS in (8) to the leader dynamics in (11). This problem can be converted into the following LMI optimization

[TABLE]

[TABLE]

In the proposed optimization, we obtain the largest upper bound of the delay by solving (33)-(34) within the variables $k_{ij}$. The process of using the optimization algorithm to solve the LMI is shown in Fig. 1 and summarizes in Algorithm 1.

According to this optimization algorithm, we find the maximum input delays and the best tuning gains $k_{ij}$ such that the system (8) remains consensusable under the proposed protocol (12).

VI Verification of Results

To test the performance of the proposed protocol (12), we consider here a generic MAS composed by 4 agents plus a leader. The $i$-th agent dynamics are defined as

[TABLE]

and the leader dynamic is considered as

[TABLE]

According to Routh–Hurwitz stability, $\phi_{i}$ in (60) is negative if the following condition satisfies

[TABLE]

From (61) and (35), (37) can be recast as

[TABLE]

The communication topology $\mathcal{G}$ is chosen in accordance to the following adjacency matrix

[TABLE]

Numerical calculations were carried out in MATLAB. Tunable gains of the protocol according to the optimization algorithm are obtained as $k_{10}=\begin{bmatrix}0.01&0.011&3.87\end{bmatrix},\leavevmode\nobreak\ k_{13}=\begin{bmatrix}0.001&0.822&0.188\end{bmatrix},\leavevmode\nobreak\ k_{24}=\begin{bmatrix}0.01&0.01&0.143\end{bmatrix},\leavevmode\nobreak\ k_{34}=\begin{bmatrix}0.01&0.01&0.01\end{bmatrix},\leavevmode\nobreak\ k_{41}=\begin{bmatrix}0.01&0.01&0.01\end{bmatrix},\leavevmode\nobreak\ k_{42}=\begin{bmatrix}0.80&0.11&1.61\end{bmatrix}$. The initial states equal to $x_{0}=\begin{bmatrix}-12&9&4\end{bmatrix},\leavevmode\nobreak\ x_{1}=\begin{bmatrix}-1&13&-8\end{bmatrix},\leavevmode\nobreak\ x_{2}=\begin{bmatrix}-4&8&5\end{bmatrix},\leavevmode\nobreak\ x_{3}=\begin{bmatrix}4&-13&5\end{bmatrix}$, and $x_{4}=\begin{bmatrix}13&-12&0\end{bmatrix}$, as well as the time derivatives of the communication delays between agents $\tau_{ij}$, that are modeled as random variables with an uniform discrete distribution in the range of $|\dot{\tau}_{ij}|\leq d_{l}=\bar{d}_{p}=1$, so that conditions $\begin{bmatrix}0&\tau^{\star}\end{bmatrix}$ are enforced by means of limiters wherein $\tau^{\star}=0.8s$ obtained by solving Algorithm 1. By randomly selecting the disturbances, defined as biased sinusoidal signals, we have:

[TABLE]

within random coefficients of $a_{i}\in[-3,3]$, $b_{i}\in[1,6]$ and $f_{i}\in[1,3]$. The upper bounding constant of the disturbance, $\Gamma$, is obtained from (40) as $\Gamma=9$.

We should note that the linear part of (12) is initially tested to acknowledge that the leader’s states are being successfully tracked by the follower’s state variables when the disturbances $\omega_{i}(t)$ set to zero and $\tau^{\star}=0.8s$. Time tracking of the follower’s state variables compared to the leader is shown in Fig. 2, supporting the fact that the linear part of the proposed distributed control (12) is consistent with each of agents tracking the leader’s behavior when the disturbances $\omega_{i}(t)$ set to zero and $\tau^{\star}=0.8s$. Note that the performance of this algorithm is undermined when $\tau^{\star}$ is chosen greater than the optimal value obtained from Algorithm 1 ($\tau^{\star}=0.8s$). As shown in Fig. 3, the follower’s state variables cannot provide a consensus on the desired values when the disturbances $\omega_{i}(t)$ are present.

Figs. 4(a-c) show the agent’s states of the distributed control (12) upon time-variant delays in a perturbed environment as defined in (40). As expected, the proposed optimization algorithm rejects the perturbations on the agent’s dynamics and enables the agent’s systems to track the leader’s states along their communication delays and unknown perturbations. The signal control inputs are therefore not smooth (see Fig. 4(d)).

To fix chattering problem, the smoothed protocol (32) is implemented with the time constant value of $\widetilde{T}=0.01s$ and its performance is verified in during the perturbations.

The results obtained from Fig. 4(a-c) and Fig. 4(e) justify that the proposed distributed control (32) correctly synchronizes the agent’s states to the reference value while fixing the chattering problem along multiple time-variant delays in association with agents and unknown perturbations.

VII Conclusions

This paper investigates a leader-following consensus of MASs in time-variant domains. A new distributed and robust consensus control is jointly programmed under Lyapunov-Krasovskii functional equations and LMIs to show that, with sufficient conditions, the leader-following consensus problem can be solved within multiple time-varying communication delays. The LMI criterion is thus involved to estimate of the upper bound delay in support of the consensus convergence in the system. Furthermore, the optimal constant control gains are mapped to subsequently guarantee the consensus of the MAS to the leader dynamics. These analytical protocols are to broaden their applicability from communication networks to engineered biological systems, where it could lead to the development of new automatic controls in scalable artificial and natural intelligence.

** Proof of Theorem 1****.**

See [35] for details. $\blacksquare$

Proof of Lemma 1**.**

See [36] for details. $\blacksquare$

Proof of Lemma 2**.**

See [37] for details. $\blacksquare$

** Proof of Theorem 2****.**

Let $X\in\mathbb{R}^{n}$ and let $M\in\mathbb{R}^{n\times n}$ , then

[TABLE]

According to (41), we can write

[TABLE]

By denoting ${\lambda_{max}}^{M_{1}}<0$ and ${\lambda_{max}}^{M_{2}}>0$ be the maximum eigenvalues of $M_{1}$ and $M_{2}$, (42) can be recast as follows

[TABLE]

it follows that

[TABLE]

Since $M_{1}<0,M_{2}>0$ and $\|M\|={\lambda_{max}}^{M}$, we can thus rewrite (44) as follows

[TABLE]

This concludes the proof. $\blacksquare$

** Proof of Theorem 3****.**

By substituting (12) into the MAS dynamics (8), we obtain:

[TABLE]

and then, by computing the time derivative of (13) along with the trajectories of (14), we reach to:

[TABLE]

Let us now select the following Lyapunov function:

[TABLE]

so that the time derivative of $\bar{V}(t)$ correspondingly takes the form:

[TABLE]

Then by referring to Assumption 1, we manipulate (49) as:

[TABLE]

so that by reaching to (50), $V(t)=0\leavevmode\nobreak\ \forall\leavevmode\nobreak\ t\geq 0$ is concluded. Consequently, the condition $s_{i}=\dot{s}_{i}=0$ is invariant since the initial instant of time $t=0$. Hence, by letting $\dot{s}_{i}=0$, the following function is supported:

[TABLE]

Therefore, by substituting (51) into (46), we get:

[TABLE]

Let us define errors between the $i$-th and $j$-th agent’s states with respect to the leader as

[TABLE]

after algebraic manipulations, one derives from (11) and (52) that

[TABLE]

Then, by taking (27)-(28) into account to, we reaches to:

[TABLE]

To describe the multi-agent dynamics we define the error state vector as

[TABLE]

Regarding (23)-(26), the multi-agent closed loop dynamics can be written as

[TABLE]

We now present a model transformation. Using the Leibniz Newton formula, it holds:

[TABLE]

Thus, the multi-agent closed loop dynamics (57) can be transformed to

[TABLE]

where $A,A_{i0}$ and $A_{ij}$ are respectively defined in (9),(27) and (28). Therefore, to show that $F$ is negative definite, it suffices to prove that $F_{(i,i)}$, for $i=1,\ldots,N$, is a negative. By considering $k_{ij}>0$ and according to (28) , the term $\sum_{j=1}^{N}{A}_{ij}$ is negative semi-definite. Consequently, blocks $F_{(i,i)}$ in (62) are negative definite if the following matrix $\phi_{i}=A+A_{i0}$ are negative definite for $i=1,\ldots,N$.

[TABLE]

with

[TABLE]

It is worth mentioning that $F$ is a strictly diagonally dominant block matrix, that its generic block element $F_{(i,i)}\in\mathbb{R}^{n\times n}$ is defined on the main diagonal as

[TABLE]

Let us construct the following Lyapunov-Krasovskii functional

[TABLE]

with

[TABLE]

where, in accordance with the statement of Theorem 3, $P,Q_{l},\bar{Q}_{p},\bar{R}_{p}$ and $R_{l}\in\mathbb{R}^{Nn\times Nn}$ are symmetric positive definite matrices. Now, the time derivative of ${V}_{1}(e(t))$ in (64) along the trajectories of the system in (59) are given by

[TABLE]

According to (7) in Lemma 2, (69) can be rewritten as

[TABLE]

From (65) and (66), by differentiating ${V}_{2}(e(t))$,${V}_{3}(e(t))$ and exploiting the bound on delays according to Assumption 2, we get

[TABLE]

[TABLE]

By taking the time derivative of ${V}_{4}(e(t))$ and ${V}_{5}(e(t))$, it yields to

[TABLE]

Considering the upper bound of time-delays $\tau^{\star}$, we can write

[TABLE]

Next, by summing (70)-(76) and from (19)-(22), one derives

[TABLE]

Let us expand the term $\tau^{\star}\dot{e}(t)^{\prime}H\dot{e}(t)$ according to (57) as follows

[TABLE]

From (6) (see Lemma 2), and assuming

[TABLE]

the following results are then obtained

[TABLE]

Therefore, according to (80)-(82), (78) can be recast as

[TABLE]

Now, by substituting (83) into (77), one derives

[TABLE]

being

[TABLE]

with

[TABLE]

Therefore, to have $\dot{V}(e(t))\leq 0$, $H_{1},H_{2}$ and $H_{3}$ in (84) are to be negative definite. It should be noted that $H_{1}$ is a non-linear inequality due to the presence of terms $R_{l}^{-1}$ and $\bar{R}_{p}^{-1}$. Therefore, performing the Schur complement on $H_{1}$ (see Lemma 1), (84) can be rewritten as in (16)-(18), which consists of LMIs. Their solutions can be easily found by using standard numerical solvers based on the the interior point method. Hence, if (16)-(18) are satisfied, then $\dot{V}(e(t))\leq 0$ in (84) and it results ${V}(e(t))$ converges to zero and thus condition (30) is in force. This concludes the proof. $\blacksquare$

Acknowledgment

We thank Alessandro Pilloni for the useful discussions on developing the optimization algorithm.