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Abstract

Many multi-agent systems (MASs) can be regarded as hybrid systems that contain continuous variables and discrete events exhibiting both continuous and discrete behavior. An MAS can accomplish complex tasks through communication, coordination, and cooperation among different agents. The complex, adaptive and dynamic characteristics of MASs can affect their stability that is critical for MAS performance. In order to analyze the stability of MASs, we propose a stability analysis method based on invariant sets and Lyapunov’s stability theory. As a typical MAS, an unmanned ground vehicle formation is used to evaluate the proposed method. We design discrete modes and control polices for the MAS composed of unmanned ground vehicles to guarantee that the agents can cooperate with each other to reliably achieve a final assignment. Meanwhile, the stability analysis is given according to the definition of MAS stability. Simulation results illustrate the feasibility and effectiveness of the proposed method.

Keywords

Multi-agent system hybrid system stability Lyapunov function unmanned ground vehicles

Introduction

Multi-agent systems (MASs) are dynamic systems where agents with autonomous behavior communicate with each other through communication networks to achieve cooperative goals. Many systems in different fields, such as biology, computer, and social networks, can be considered as MASs.^1–3 Due to the universal existence and wide application of such a kind of systems, the research on MASs has attracted much attention. Over the past two decades, according to different classification methods, a lot of meaningful results have been obtained on cooperative control for various types of MASs, such as linear or nonlinear MASs, first-order or higher-order MASs, continuous-time or discrete-time MASs.^4–13

To accomplish a complex task, the dynamic model of agents in an MAS may be different, or each agent may be affected by the mixing law, leading to hybrid system. The former hybrid systems composed of agents with different dynamic structures is also called heterogeneous MASs. There are many interesting research results for such systems. Zheng and Wang^14,15 investigated consensus and finite-time consensus problems of heterogeneous MASs consisting of first-order and second-order integrators. Zuo et al.¹⁶ considered an output containment control problem of general linear heterogeneous MASs with multiple unknown leaders. For a class of nonlinear heterogeneous MASs, the event-triggered cooperative global robust output regulation problem was studied.¹⁷ The above mentioned studies investigate the relevant control problems of heterogeneous MASs by designing appropriate controllers and using matrix theory, Lyapunov theory, adaptive theory, etc. Different from the hybrid MASs above, switching systems, as typical hybrid systems, have attracted more and more attention. Zheng and Wang¹⁸ focused on the consensus problem for a class of switched MASs, where a system can be switched arbitrarily on continuous-time and discrete-time subsystems. Zhu et al.¹⁹ further studied the containment control problem for such a switched MAS.

However, in the literatures mentioned above, the authors mainly study whether MASs or hybrid systems can achieve the final cooperative goal under designed controllers from the perspective of control. In practice, stability has always been an important criterion for determining the reliability of a system. The complex, adaptive, and dynamic characteristics of MASs can affect their system stability. Therefore, it is necessary to analyze the stability of the system under different environments. There are some different definitions and analytical methods of stability for hybrid systems. Lyapunov theory is a basic method. For hybrid systems composed of continuous-time and discrete-event autonomous individuals through some logic or decision-making processes, an agent is usually modeled by differential or difference equations, and the decision-making or logic is described by Petri nets or finite automata.^20,21 Such hybrid systems have wide applications in various industrial, electronic, and military fields.²² At present, some important results on the stability of such hybrid systems have been achieved.^23–27 Ye et al.²³ gave the definition of Lyapunov stability by invariant sets and sufficient conditions for uniform stability, asymptotic stability, and instability are obtained. Branicky²⁵ reported a multiple Lyapunov functions method for switching hybrid systems. Decarlo et al.²⁶ summarized the main results of Lyapunov stability for finite dimensional hybrid systems, and discussed stronger and more specialized results of switching linear systems (stable and unstable). In addition, there are some other significant results on the stability for hybrid systems.^28–31

From a single agent to an MAS with a hybrid structure, its dynamic characteristics become more complex that makes its stability analysis more difficult. Therefore, how to find a effective Lyapunov function that satisfies system stability is the main work in the stability study of hybrid systems. It is also a difficult problem in stability theory study, which has not been well solved so far. Therefore, based on the application value of the MASs with hybrid characteristics and the discussion above, this work mainly addresses the stability of the hybrid MASs composed of continuous variables and discrete events and give some application in practical problems.

In this work, an agent in a hybrid MAS is subject to the differential equations, but its state is always continuous and different continuous variables are switched according to discrete event mechanisms. First, with the help of the reachability, activity and fairness of MASs induced by the theory of discrete-event systems, we give the stability analysis method of a hybrid MAS based on the stability definition of discrete-event systems. Second, as a typical MAS, an unmanned ground vehicle formation is used in this paper to evaluate the reported method. The discrete modes and control polices for the MAS are proposed to guarantee that the agents can cooperate with each other to reliably achieve a final assignment. The stability analysis is also given according to the definition of MAS stability. Obviously, UGV movements and formation changes will affect the dynamic stability of UGVs, so the stability analysis for such a system is challenging.

The remainder of this work is organized as follows. Section II gives a brief discussion on an agent and MASs, and reviews agent stability. Section III proposes a stability analysis method of MASs. Section IV uses an example of UGVs to illustrate the proposed method. Section VI concludes the work.

Preliminaries

Agent and MASs

An agent is an entity with autonomous behavior in a complex environment, which can determine or control its next behavior according to internal state and external events. It may contain both continuous variables and discrete events, called hybrid system, which is a typical switched hybrid system.³² MASs composed of multiple agents interacting with each other can be regarded as discrete-event systems using aggregation, where their continuous dynamic behavior is integrated into discrete events.²⁴

Since an MAS consists of multiple autonomous agents communicating with each other through networks, such a system has complex dynamic characteristics. From the perspective of discrete-event systems, we should analyze and verify security, reachability, liveness, and boundness of systems. We can use Petri nets to model and analyze MASs. At the same time, to verify whether a system satisfies its specifications, model checking techniques can also be employed. From the perspective of continuous systems, we should discuss the stability of systems. A well-designed agent and MASs must be stable, robust, and adaptive systems, thus ensuring that the whole systems are safe and reliable. In this work, we focus on the stability analysis of multi-agent systems.

Stability analysis of agent

The basic research method for stability analysis of hybrid systems is the Lyapunov theory. The Lyapunov’s direct method analyzes stability of systems from the energy point of view. If the energy stored in a system is gradually decreasing, the system is stable; On the other hand, if the system constantly absorbs energy from an environment, and its energy is gradually increasing, it is unstable. How to design and obtain a suitable Lyapunov function is the key to analyze stability of hybrid systems.

A single agent can be seen as a switched hybrid system where continuous variables are switched according to discrete events. We regard discrete states as modes of an agent, where the states that continuously evolve in each mode are described by differential equations. When a state reaches a certain threshold, a corresponding discrete event is triggered, thus causing mode switching.

An autonomous agent can be defined as

{\begin{matrix} \overset{\cdot}{x} = f (x (t), q (t)) \\ q (t) = s (x (t), q (t^{-})) \\ y (t) = h (x (t), q (t)) \end{matrix}

(1)

where $x \in R^{n}$ is a continuous state and $q \in Q \subset {1, \dots, N}$ is a discrete mode. $f (\cdot, q)$ is the globally Lipschitz continuous representing the continuous dynamics of the agent. s denotes the finite dynamics of the agent, h is a hybrid state space $R^{n} \times Q$ , and $q (t^{-})$ is the left limit of $q (t)$ at time t. $y (t)$ denotes the system output at time t. We call $(x (t), q (t))$ the trajectory of the agent at time t.

Define an equilibrium $x_{e}$ for each mode in an agent and suppose that the agent switches into mode $q_{i}$ at $t_{0}$ . For $\forall ϵ > 0$ , $\exists δ (ϵ, t_{0}) > 0$ , when

∥ x (t_{0}) - x_{e} ∥ \leq δ (ϵ, t_{0}),

(2)

$x (t)$ starting from any initial state $x (t_{0})$ satisfies

∥ x (t) - x_{e} ∥ \leq ϵ,

(3)

then $x_{e}$ in $q_{i}$ is stable in the sense of Lyapunov.

Therefore, if internal control strategies of an agent make the equilibrium $x_{e}$ of $q_{i}$ stable, and the number of mode switching in $q_{i}$ is limited in a limited time, then the agent is stable.

We can analyze agent stability using the multiple Lyapunov function method.²⁵ Suppose that the initial state $(x_{0}, q_{i})$ and the evolution of discrete modes described by a switching sequence is given as

Δ_{(x_{0}, q_{0})} = (i_{0}, t_{0}), (i_{1}, t_{1}), \dots, (i_{n}, t_{n}), \dots

(4)

where $i_{k} \in Q, k \in N = {0, 1, 2, \dots}$ . When $t_{k} < t \leq t_{k + 1}$ , an agent evolves according to $\overset{\cdot}{x} = f (x (t), i_{k})$ . $Δ_{(x_{0}, q_{0})} | i$ represents the switching sequence of mode i. A strictly increasing time sequence $T = {t_{0}, t_{1}, \dots, t_{n}, \dots}$ is denoted as $I (T) = ⋃_{j \in N} [t_{2 j}, t_{2 j + 1}]$ , and $I (Δ | i)$ represents the time sequence set of mode i. The even sequence of T is denoted as $ϵ (T) = {t_{0}, t_{2}, \dots}$ .

Definition 1 (Lyapunov-like function²³) Suppose a strictly increasing time sequence T and equilibrium $\bar{x} \in Ω_{i} \subset R^{n}$ in mode i. For the function f of T and trajectory $x (\cdot)$ , if a function V satisfies the following conditions:

for $\bar{x} \neq x \in Ω_{i}$ , $V_{i} (\bar{x}) = 0, V_{i} (x) > 0$ ;

$\forall t \in I (T)$ , ${\overset{\cdot}{V}}_{i} (x (t)) \leq 0$ ;

then it is the Lyapunov-like function.

The multi-Lyapunov function can be used to study whether switching sequences are stable where equilibria are different in discrete modes. Meanwhile, mental states of an agent are described by using the Belief-Desire-Intention model that provides knowledge-based planning methods to generate switching sequences, driving the agent to reach equilibria of corresponding goals.

Stability analysis of multi-agent systems

MASs can be regarded as discrete events dynamic systems at a macro level, which can be modeled by automata. The stability of MASs means that multiple stable agents can achieve goals through negotiation and collaboration. In this section, we propose stability analysis methods for MASs.

Suppose that an MAS M is described by automata, $M = {Q, ε, f_{e}, g, E_{v}}$ , where

$Q = {q_{1}, \dots, q_{k}}$ is a finite set of states;

$ε = {e_{1}, \dots e_{l}}$ is a finite set of events;

$f_{e} : Q \to Q$ is a state switch function. For $e \in ε$ , $q_{k + 1} = f_{ek} (q_{k})$ represents that state $q_{k}$ is switched to $q_{k + 1}$ at time k as event $e_{k}$ occurs;

$g : Q \to 2^{ε} - ϕ$ is an enabled function, which represents that events only occur in state $q \in Q$ , $e_{k} \in g (q_{k})$ ;

$E_{v}$ represents all valid event sequences that have corresponding state sequences.

A brief introduction of the notation used in the analysis is given as follows. $E_{v} (Q_{k})$ is a set of all possible valid event sequences from $Q_{k} \subset Q$ and $E_{v} (Q_{k}) = ⋃_{q \in Q_{k}} E_{v} (q)$ . $E_{a}$ is an allowed event sequence set that makes an agent optimally achieves goals.³³ $E_{a} \subset E_{v}$ and $E_{a} (q_{0})$ represent the allowed event sequence set whose initial state is $q_{0} \in Q$ . For time k, $E_{k} = {e_{0}, e_{1}, \dots, e_{k - 1}}$ and $E_{0} = ϕ$ . An infinite event sequence is denoted as $E = e_{k}, e_{k + 1}, \dots$ . The conjunction of E and $E_{k}$ is denoted as $E_{k} E \in E_{v} (q_{0})$ . $Q (q_{0}, E_{k}, k)$ represents the state reached from the initial state $q_{0} \in Q$ at time k, which is called “motion” of the agent. It can also be expressed as $f_{E_{k}} (q_{0}) = f_{e_{k}} (f_{E_{k - 1}} (q_{0}))$ , where $E_{k} = E_{k - 1} e_{k}$ .

Definition 2 (Invariant Set³⁴) Suppose that a set $Q_{m} \subset Q$ . If $q_{0} \in Q_{m}$ , $E_{k}$ satisfies $E_{k} E \in E_{v} (q_{0})$ and $f_{E_{k}} (q_{0}) \in Q_{m}$ holds, then $Q_{m}$ is an invariant set of M.

It can be seen from the above definition that the states in an invariant set remain in it after switching. Consequently, we can construct a state set as an invariant set by setting control policies.

Definition 3 (Stability and Asymptotic Stability of Invariant Sets³⁴) If $\forall ϵ > 0$ , $\exists δ > 0$ , for all $E_{k}$ , $ρ (f_{E_{k}} (q_{0}), Q_{m}) < ϵ$ holds when $ρ (q_{0}, Q_{m}) < δ$ , then the invariant $Q_{m} \subset Q$ is Lyapunov stable, where $E_{k} E \in E_{a} (q_{0})$ and $E_{a} \subset E_{v}$ . For $\forall E \in E_{a} (q_{0})$ , $ρ (f_{E_{k}} (q_{0}), Q_{m}) \to 0$ when $k \to \infty$ , then the invariant set $Q_{m}$ is asymptotically stable.

According to the above definitions, the sufficient and necessary conditions for the stability and asymptotic stability of invariant sets in the sense of Lyapunov can be obtained.³⁴ For MASs, once a group of agents reaches goals and accomplishes tasks, mode switchings do not occur. Therefore, we can make agents reach a goal set by designing corresponding control policies, and regard the goal set as an invariant set. Denote an MAS initial state set $Q_{0}$ , metric $ρ$ , Lyapunov function $V (Q)$ , and goal set $Q_{m}$ . $ρ (Q_{0}, Q_{m})$ represents the metric between $Q_{0}$ and $Q_{m}$ , and $ρ (f_{E_{k}} (Q_{0}), Q_{m})$ represents the distance between the state set switched by $E_{k}$ and the goal set $Q_{m}$ . According to Definition 3, we can study the asymptotic stability of a group of agents to decide whether MASs are stable. Consequently, the stability criteria of MASs can be derived as follows.

Proposition 1 (MAS Stability) Consider an MAS. If the following conditions hold:

Each agent is stable;

There exist a set of allowed events $E_{a} \subseteq E_{v}$ , a metric $ρ$ and a positive constant $δ > 0$ . When $ρ (Q_{0}, Q_{m}) < δ$ , for $\forall E \in E_{a} (Q_{0})$ , $ρ (f_{E_{k}} (Q_{0}), Q_{m}) \to 0$ as $k \to \infty$ ,

then the MAS can achieve asymptotically stable.

We can find a Lyapunov-like function to ensure the stability of each agent, thus making condition (1) hold. Condition (2) indicates that at least one finite mode sequence can make the metric of different states tend to be zero. It becomes zero when goals are completed. $δ$ can be regarded as the upper bound of $ρ (Q_{0}, Q_{m})$ . It may represent the worst case for an initial configuration of MASs.³³ For example, the initial state is farthest from the goal set.

Example 1. Let us consider two agents random walk in Figure 1. Two agents $A_{1}$ and $A_{2}$ walk from $O_{1}$ and $O_{2}$ to the target points $P_{1}$ and $P_{2}$ , respectively. They must reach the target points simultaneously with the shortest path. Suppose that the two agents can find the shortest path without any conflicts between them. They can move only one cell per step. There are four directions to choose from: East (E), South (S), West (W), and North (N).

Figure 1.

Two agents random walk.

The state of an agent at time k is denoted as $Q_{k}$ , and the i th agent state is $Q_{i} (k)$ . The states are corresponding to the positions of agents in Figure 1. The position of the i th agent at time k is denoted as $(x_{i} (k), y_{i} (k))$ . We take $E_{a} (1) = {E^{01} E^{12} E^{23} E^{34} E^{45} E^{56} E^{67}}$ and $E_{a} (2) = {E^{0' 1'} E^{1' 2'} E^{2' 3'} E^{3' 4'} E^{4' 5'} E^{5' 6'} E^{6' 7'}}$ as allowed event sets of two agents $A_{1}$ and $A_{2}$ , respectively. $E^{ij}$ represents discrete state switching activated by events. The goal set of agent i is $Q_{im} = {({\bar{x}}_{i}, {\bar{y}}_{i}) : ({\bar{x}}_{i}, {\bar{y}}_{i}) \in P_{i}}$ that is an invariant set. Once an agent reaches its target position, its state switching will no longer occur.

According to the MASs stability criterion, we take the Manhattan distance of each agent and target sets as a metric, namely

ρ (Q, Q_{im}) = \sum_{i = 1}^{2} {| x_{i} - {\bar{x}}_{i} | + | y_{i} - {\bar{y}}_{i} |}

(5)

The simulation result in Figure 2 shows that the MAS converges in finite steps.

Figure 2.

Two agents random walk: the metric function $ρ (Q, Q_{im})$ .

Case study

Unmanned ground vehicles are typical MASs, where a UGV can be designed to work as a single agent to carry out a task and UGVs collaboratively perform complex tasks. UGV movements and formation changes will affect the dynamic stability of UGVs, which ensures the safety and reliability of UGVs. Therefore, the stability analysis of UGVs plays an important role for their performance.

UGV MASs

In this section, we consider a UGV formation that conducts reconnaissance, surveillance, and attack missions. This formation consists of five UGVs. Each UGV is regarded as an agent. Therefore, the UGVs are regarded as MASs with collaborative capabilities. Suppose that a UGV agent with some sensors can apperceive an environment and avoid obstacles. As soon as any one of the agents in the UGV formation detects targets, it will transmit the information to the other agents, so as to conduct coordinated attacks.

Each UGV agent is a switched hybrid system that includes several modes and discrete events, where each mode is described by a continuous time variable system and discrete events drive mode switching.

UGV agent state analysis

A continuous state of each UGV agent is $X = (x, y, α)$ , where $(x, y)$ is its coordinate and $α$ is its course angle. The dynamic equation of a UGV agent can be given as

{\begin{matrix} \overset{\cdot}{x} = v \cos α (t) \\ \overset{\cdot}{y} = v \sin α (t) \\ \overset{\cdot}{α} (t) = ω (t) \end{matrix}

(6)

where v and $ω$ are its velocity and angular velocity, respectively. $U = [v ω]^{T}$ is used to represent the control signals of a UGV agent. Namely, the system can be simply written as $\overset{\cdot}{X} = f (X, U)$ . Here, we suppose that a UGV agent consists of five discrete modes $Q = [q_{1}, q_{2}, \dots, q_{5}]$ and corresponding control policies $U (i)$ , where $K_{ij} (i = 1, 2, \dots, 5, j = 1, 2)$ is proportional control gain and $ρ$ is the distance between a UGV agent and a target, as shown in Table 1. Obviously, the model is accords with system (1).

Table 1.

The discrete modes and control policies of the UGV agent.

Discretemode $q_{i}$	Label	Control policy U
$q_{1}$	SearchTarget	$v = K_{11}, ω = K_{12}$
$q_{2}$	ApproachTarget	$v = K_{21} ρ_{1}, ω = 0$
$q_{3}$	LockOnAttack	$v = K_{31} ρ_{1} c os (α_{p} - α_{i})$ , $ω = K_{32} (α_{p} - α_{i})$
$q_{4}$	JoinPursuit	$v = K_{41} ρ_{2} c os (α_{1} - α_{2})$ , $ω = K_{42} (α_{1} - α_{2})$
$q_{5}$	ReturnToBase	$v = K_{51} ρ_{3}$ , $ω = K_{52} (α_{i} - α_{b})$

A UGV agent first initializes and then goes into mode $q_{1}$ (SearchTarget) where it can search targets with the velocity $K_{11}$ and angular velocity $K_{12}$ . When it detects targets, it goes into mode $q_{3}$ (ApproachTarget) and approaches targets, then locks and attacks them in $q_{4}$ (LockOnAttack). After the UGV agent accomplishes tasks, it goes into mode $q_{5}$ (ReturnToBase) mode and returns to its base. In mode $q_{4}$ (JoinPursuit), any one of the UGV agents apperceives targets and then disseminates the information to the other agents, so as to perform coordinated attacks.

The states in the above several discrete modes are described by time dynamic equations that describe state evolution of UGV agents. The discrete events in a UGV agent are as follows: $e_{1}$ : search target; $e_{2}$ : joint pursuit; $e_{3}$ : approach target; $e_{4}$ : target lock and attack; $e_{5}$ : return to base.

UGV agent stability analysis

Suppose that $x_{1} = x, x_{2} = y, x_{3} = α, u_{1} = v, u_{2} = ω$ in the continuous dynamic equations of a UGV agent and a control policy $U_{q} = [u_{1}$ $u_{2}]^{T}$ in each mode. Then

{\begin{matrix} {\overset{\cdot}{x}}_{1} = u_{1} \cos x_{3} \\ {\overset{\cdot}{x}}_{2} = u_{1} \sin x_{3} \\ {\overset{\cdot}{x}}_{3} = u_{2}, \end{matrix}

(7)

namely, $\overset{\cdot}{z} = f (z, U)$ . For $\overset{\cdot}{z} = f_{q} (z, U_{q})$ in each mode, we should design a control policy $U_{q}$ to make it satisfy the stability in the sense of Lyapunov. The goal of a UGV agent is to destroy targets and return to its base. Therefore, we set the positions of targets as the equilibria in modes $q_{2} ~ q_{4}$ , and set the base position as the equilibrium in mode $q_{5}$ .

The artificial potential field method is used to select Lyapunov functions.^35,36 It creates a potential field $P$ for a UGV agent. The potential field is composed of two parts: part one is to construct a gravitational potential field $P_{att}$ at a target position that increases along with the increase of the distance between a UGV agent and a target. The direction of $P_{att}$ points to the target and the gravitational force $F_{att}$ of $P_{att}$ attracts the UGV agent to move toward the target. Part two is to construct a repulsion potential field $P_{rep}$ around obstacles that decreases along with the increase of distance between the UGV agent and obstacles. The direction of $P_{rep}$ points away from obstacles and the repulsion force $F_{rep}$ of $P_{rep}$ keeps the UGV agent away from obstacles. $F_{att}$ and $F_{rep}$ make the UGV agent move toward the target.

The most common gravitational force function $F_{att}$ is

F_{att} (l) = 0.5 ξ ρ^{m} (l, l_{goal}),

(8)

where $ξ$ is a scaling factor. l and $l_{goal}$ are positions of a UGV agent and a target, respectively. $ρ (l, l_{goal})$ is the distance between the UGV agent and the target. m can be set as 1 or 2. If $m = 1$ , the shape of the attractive potential field is like a cone; if $m = 2$ , it is like a parabola.

The most common repulsion potential field function is revised to make a UGV agent reach targets with the minimum repulsive force. Consequently the total potential field force is the minimum. $P_{rep}$ is represented as

P_{rep} (l) = {\begin{matrix} 0.5 η (\frac{1}{ρ (l, l_{obs})} - \frac{1}{ρ_{0}}) ρ^{n} (l, l_{goal}), \\ if ρ (l, l_{obs}) \leq ρ_{0}, \\ 0, if ρ (l, l_{obs}) > ρ_{0}, \end{matrix}

(9)

where $ρ (l, l_{obs})$ is the distance between a UGV agent and an obstacle, $ρ (l, l_{goal})$ is the distance between a UGV agent and a target, n is tuning parameter, and $ρ_{0}$ is influenced distance of an obstacle. In general, $ρ_{0} \leq m in (d_{1}, d_{2})$ , where $d_{1}$ is a half of the minimum distance between each obstacle and $d_{2}$ is the minimum distance from a target to each obstacle.

According to the artificial potential field method, a Lyapunov function can be given as

V (x) = {\begin{matrix} 0.5 [(x_{1} - x_{t})^{2} + {(x_{2} - y_{t})}^{2} \\ + {(x_{3} - \arctan (\frac{x_{2} - y_{t}}{x_{1} - x_{t}}))}^{2}] \\ + 0.5 [\frac{1}{{[(x_{1} - x_{obs})^{2} + {(x_{2} - y_{obs})}^{2}]}^{0.5}} - \frac{1}{ρ_{0}}] \\ \times [(x_{3} - \arctan (\frac{x_{2} - y_{t}}{x_{1} - x_{t}} {))}^{2} \\ + {(x_{1} - x_{t})}^{2} + {(x_{2} - y_{t})}^{2}], \\ if [(x_{1} - x_{obs})^{2} + {(x_{2} - y_{obs})}^{2}]^{0.5} \leq ρ_{0}; \\ 0.5 [(x_{1} - x_{t})^{2} + {(x_{2} - y_{t})}^{2} \\ + {(x_{3} - \arctan (\frac{x_{2} - y_{t}}{x_{1} - x_{t}}))}^{2}], \\ if [(x_{1} - x_{obs})^{2} + {(x_{2} - y_{obs})}^{2}]^{0.5} \geq ρ_{0} . \end{matrix}

(10)

When a UGV agent is at the equilibrium $X (t) = (x_{t}, y_{t}, \arctan (\frac{x_{2} - y_{t}}{x_{1} - x_{t}}))$ , $V (X (t)) = 0$ . Otherwise, $V (X (t)) > 0$ .

To make a UGV agent stable, we should set control policies $u_{1}$ and $u_{2}$ in each mode to make $\overset{\cdot}{V} (x) \leq 0$ . Suppose that there are no obstacles in the battlefield environment, then

\begin{matrix} \overset{\cdot}{V} (x) = (x_{1} - x_{t}) {\overset{\cdot}{x}}_{1} + (x_{2} - y_{t}) {\overset{\cdot}{x}}_{2} + (x_{3} - \arctan \frac{x_{2} - y_{t}}{x_{1} - x_{t}}) \\ \times [{\overset{\cdot}{x}}_{3} - \frac{(x_{1} - x_{t}) {\overset{\cdot}{x}}_{2} - (x_{2} - y_{t}) {\overset{\cdot}{x}}_{1}}{{(x_{1} - x_{t})}^{2} + {(x_{2} - y_{t})}^{2}}] . \end{matrix}

(11)

For mode 2 (ApproachTarget), the control policy $U_{2} = [K_{21} ρ_{1}, 0]^{T}$ , where $ρ_{1}$ is the distance between a UGV agent and its target. To make

\begin{matrix} \overset{\cdot}{V} (x) = K_{21} [(x_{1} - x_{t}) (ρ \cos x_{3} - \frac{x_{3} - \arctan \frac{x_{2} - y_{t}}{x_{1} - x_{t}}}{ρ} \sin x_{3}) \\ + (x_{2} - y_{t}) (ρ \sin x_{3} + \frac{x_{3} - \arctan \frac{x_{2} - y_{t}}{x_{1} - x_{t}}}{ρ} \cos x_{3})] \\ \leq 0, \end{matrix}

(12)

we assume that

\begin{matrix} Δ = (x_{1} - x_{t}) (ρ \cos x_{3} - \frac{x_{3} - \arctan \frac{x_{2} - y_{t}}{x_{1} - x_{t}}}{ρ} \sin x_{3}) \\ + (x_{2} - y_{t}) (ρ \sin x_{3} + \frac{x_{3} - \arctan \frac{x_{2} - y_{t}}{x_{1} - x_{t}}}{ρ} \cos x_{3}) . \end{matrix}

(13)

Obviously, if $Δ < 0$ , $K_{21} > 0$ and if $Δ > 0$ , $K_{21} < 0$ .

In the same way, we can obtain the policies of modes 3, 4, and 5, and then set a proportional control gain $K_{ij}$ to make UGV agent stable.

Simulation and analysis

We first conduct a simulation where there are no obstacles in an environment. Suppose that two UGV agents are farthest from a target when the simulation is initialized, and they return their base after performing their goals. The simulation results are shown in Figure 3, where the triangles represent the initial positions of the UGV agents and target, the circles represent the positions where the UGV agents destroy the target, and the dotted lines represent the trajectories that the UGV agents return to the base. The simulation results show that the proposed model is effective and UGV agents can achieve their missions.

Figure 3.

The trajectories of UGV agents and the target with no obstacles in the environment.

Furthermore, we consider a complex situation. The initial configuration is shown in Figure 4, where UGV agents are farthest from a target and there are some obstacles in the environment. The goal of a UGV agent is to destroy target and return to its base. The simulation results are shown in Figures 5 to 7. Since the UGV Agent formation is stable under the worst initial conditions, we can be sure that the formation is still stable under other initial conditions. Figures 5 and 6 represent the trajectories of UGV agents attacking the target in X and Y coordinates, respectively. It can be seen from Figures 5 and 6, the UGV agents first discover the target around 5 s, and then the two UGV agents jointly pursue the target. In the pursuit process, the trajectory of UGV Agent and Target Agent is basically the same. Figure 7 shows the trajectories of UGV agents returning to the base.

Figure 4.

A complex scenario with obstacles.

Figure 5.

The trajectories of UGV agents and the target UGV Agent in X coordinate.

Figure 6.

The trajectories of UGV agents and the target UGV Agent in Y coordinate.

Figure 7.

The trajectories of UGV agents returning to the base.

The allowed event set of the two UGV agents is $Ea = [{e_{1}, e_{3}, e_{4}, e_{5}}, {e_{1}, e_{2}, e_{3}, e_{4}, e_{5}}]$ and the goal set $Q_{m} = (x_{m}, y_{m})$ is an invariant set. We regard the distance between a UGV agent and the target as a metric, which can be given as

ρ (Q, Q_{m}) = \sum_{i = 1}^{2} [(x_{i} - x_{m})^{2} + (y_{i} - y_{m})^{2}]^{0.5} .

(14)

The metric simulation result is shown in Figure 8, which shows that the UGV agent formation is convergent in limited steps. According to the proposition of MAS stability, the UGV agent formation is stable.

Figure 8.

The distance between UGV agents and the target.

Conclusion

This paper studies the stability of a switched hybrid MAS and provides a stability analysis method for the hybrid MAS based on invariant sets and Lyapunov’s stability theory. Then, for the UGVs formation, the stability of such a system is discussed in detail based on the given definition of MASs stability. Lyapunov functions are designed with the help of the artificial potential field method. The simulation results show that the proposed method is effective for the hybrid MASs. In the future, we will further consider the stability and group consensus problem for hybrid multi-agent systems under more general networks, such as the presence of attacks, disturbances, or communication delays.^9,37–39

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (Nos. 61873277 and 71571190), the Key Research and Development Program of Shaanxi Province (No. 2019GY-056) and the Special Scientific Research Plan Project of Shaanxi Provincial Department of Education (No. 21JK0488).

ORCID iDs

Zhenhua Yu

Xiaobo Li

Haitham A Mahmoud

References

Reynolds

. Flocks, herds, and schools: a distributed behavioral model. Comput Graph (ACM) 1987; 21: 25–34.

Cao

Anderson

Morse

. Sensor network localization with imprecise distances. Syst Control Lett 2006; 55: 887–893.

Yang

El-Meligy

, et al. On multiplexity-aware influence spread in social networks. IEEE Access 2020; 8: 106705–106713.

Liu

Jiang

. Distributed output-feedback control of nonlinear multi-agent Systems. IEEE Trans Automat Contr 2013; 58: 2912–2917.

Chen

Sun

. Distributed optimal control for multi-agent systems with obstacle avoidance. Neurocomputing 2016; 173: 2014–2021.

Jiang

Wang

. Consensus seeking of high-order dynamic multi-agent systems with fixed and switching topologies. Int J Control 2010; 83: 404–420.

Shi

Shao

Zheng

. Containment control of asynchronous discrete-time general linear multiagent systems with arbitrary network topology. IEEE Trans Cybern 2020; 50: 2546–2556.

Zhao

Hua

Guan

. A hybrid event-triggered approach to consensus of multi-agent systems with disturbances. IEEE Trans Control Netw Syst 2020; 7: 1259–1271.

Shao

Zheng

Shi

, et al. Bipartite tracking consensus of generic linear agents with discrete-time dynamics over cooperation-competition networks. IEEE Trans Cybern. Epub ahead of print 10 January 2020. DOI: 10.1109/TCYB.2019.2957415.

10.

, et al. Group consensus via pinning control for a class of heterogeneous multi-agent systems with input constraints. Inf Sci 2021; 542: 247–262.

11.

Shen

Wang

, et al. Consensus of upper-triangular multiagent systems with sampled and delayed measurements via output feedback. IEEE Trans Syst Man Cybern Syst 2020; 50: 600–607.

12.

Huang

. Two consensus problems for discrete-time multi-agent systems with switching network topology. Automatica 2012; 48: 1988–1997.

13.

Mustafa

Modares

Moghadam

. Resilient synchronization of distributed multi-agent systems under attacks. Automatica 2020; 115: 108869.

14.

Zheng

Wang

. Distributed consensus of heterogeneous multi-agent systems with fixed and switching topologies. Int J Control 2012; 85: 1967–1976.

15.

Zheng

Wang

. Finite-time consensus of heterogeneous multi-agent systems with and without velocity measurements. Syst Control Lett 2012; 61: 871–878.

16.

Zuo

Song

Lewis

, et al. Adaptive output containment control of heterogeneous multi-agent systems with unknown leaders. Automatica 2018; 92: 235–239.

17.

Liu

Huang

. Cooperative global robust output regulation for a class of nonlinear multi-agent systems by distributed event-triggered control. Automatica 2018; 93: 138–148.

18.

Zheng

Wang

. Consensus of switched multi-agent systems. IEEE Trans Circuits Syst II Express Briefs 2016; 63: 314–318.

19.

Zhu

Zheng

Wang

. Containment control of switched multi-agent systems. Int J Control 2015; 88: 2570–2577.

20.

Zan

Guo

, et al. A Pareto-based genetic algorithm for multi-objective scheduling of automated manufacturing systems. Adv Mech Eng 2020; 12: 1–15.

21.

Zhou

Nasr

, et al. Homomorphic encryption of supervisory control systems using automata. IEEE Access 2020; 8: 147185–147198.

22.

Antsaklis

. A brief introduction to the theory and applications of hybrid systems. Proc IEEE Spec Issue Hybrid Syst Theory Appl 2000; 88: 879–887.

23.

Michel

Hou

. Stability theory for hybrid dynamical systems. IEEE Trans Automat Contr 1998; 43: 461–474.

24.

Passino

Michel

Antsaklis

. Lyapunov stability of a class of discrete event systems. IEEE Trans Automat Contr 1994; 39: 269–279.

25.

Branicky

. Multiple Lyapunov functions and other analysis tools for switched and hybrid systems. IEEE Trans Automat Contr 1998; 43: 475–482.

26.

Decarlo

Branicky

Pettersson

, et al. Perspectives and results on the stability and stabilizability of hybrid systems. Proc IEEE 2000; 88: 1069–1082.

27.

Teel

Forni

Zaccarian

. Lyapunov-based sufficient conditions for exponential stability in hybrid systems. IEEE Trans Automat Contr 2013; 58: 1591–1596.

28.

Sanfelice

Teel

. Asymptotic stability in hybrid systems via nested Matrosov functions. IEEE Trans Automat Contr 2009; 54: 1569–1574.

29.

Xia

Boukas

Shi

, et al. Stability and stabilization of continuous-time singular hybrid systems. Automatica 2009; 45: 1504–1509.

30.

Goebel

Teel

. Preasymptotic stability and homogeneous approximations of hybrid dynamical systems. SIAM Rev 2010; 52: 87–109.

31.

Sanfelice

. Results on finite time stability for a class of hybrid systems. In: Proceedings of American control conference, Boston, MA, USA, 6–8 July 2016, pp.4264–4268. New York, NY: IEEE.

32.

Lemmon

Markovsky

. Supervisory hybrid systems. IEEE Control Syst 1999; 19: 42–55.

33.

Fregene

Kennedy

Wang

. Toward a systems- and control-oriented agent framework. IEEE Trans Syst Man Cybern Soc 2005; 35: 999–1012.

34.

Passino

Burgess

. Stability analysis of discrete event systems. New York, NY: Wiley, 1998.

35.

Adeli

Tabrizi

Hajipour

, et al. Path planning for mobile robots using iterative artificial potential field method. Int J Comput Sci Issues 2011; 8: 28–32.

36.

Chen

Luo

Mei

, et al. Uav path planning using artificial potential field method updated by optimal control theory. Int J Syst Sci 2016; 47: 1407–1420.

37.

Zhao

Hua

Guan

. Distributed event-triggered consensus of multi-agent systems with communication delays: a hybrid system approach. IEEE Trans Cybern 2020; 50: 3169–3181.

38.

Zhao

Hua

. A hybrid dynamic event-triggered approach to consensus of multi-agent systems with external disturbances. IEEE Trans Automat Contr 2021; 66: 3213–3220.

39.

Shen

. Out feedback

H_{\infty}

control of networked control systems based on two channel event-triggered mechanisms. Kybernetika 2021; 57: 118–140.

Stability analysis method and application of multi-agent systems from the perspective of hybrid systems

Abstract

Keywords

Introduction

Preliminaries

Agent and MASs

Stability analysis of agent

Stability analysis of multi-agent systems

Case study

UGV MASs

UGV agent state analysis

UGV agent stability analysis

Simulation and analysis

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iDs

References