Formation deployment control of multi-agent systems modeled with PDE

1.   Introduction

In recent years, as the development of complex systems theory and its wide application in the fields of transportation systems, UAVs, satellites, vehicles and control engineering, the cooperative control of MASs has been extensively researched ^[1,2,3]. Consensus control and formation control are the primary issues studied in cooperative control. Consensus control emphasizes multiple agents converging to the same target state ^[4,5,6,7], while formation control is the design of a cooperative control scheme to move the agents to multiple target states, which in turn form the desired formation. Hence, formation control technology is widely applied to radar, navigation, aerial photography, military exercises, and other fields ^[8,9].

In formation studies, the majority of MASs are modeled by ordinary differential equations (ODEs) based on graph theoretic Laplace consistent action ^{[10,11,12,13,14]}. However, when the number of agents approaches infinity, it is difficult to achieve the control goal using ODEs modeling. In addition, since the Laplace operator in PDEs has the same consistent action as the Laplace consistent control law, more and more scholars have applied the continuum model based on PDEs to describe MASs with interactions in recent years ^{[15,16,17,18]}. In ^[19], the dynamics of a MAS described by a Burgers-type PDE was considered and an extended Luenberger state observer was designed. It is also combined with motion planning and stability feedback control to enable the MASs to perform the desired formation in finite time. In ^[20], the MAS was modeled by the reaction-advection-diffusion equation, and the control laws of the agents were designed along with the PDE-backstepping method, which stabilizes the agents in the desired formation under a fixed communication topology. In ^[21], a modeling, analysis and control design approach for large MASs deployed in three-dimensional space was presented. Based on two reaction-convection-diffusion two-dimensional PDEs, the issue of stabilizing PDEs by boundary control on a disk was innovatively solved using a new explicit backstepping kernel containing a Poisson kernel. In ^[22], the MASs in finite time were deployed into planar formation profiles along predefined spatio-temporal paths with use of formal power series, appropriate summation and generalized sequence transformations.

The commonly adopted control methods for formation control problems of the MASs are the leader-follower method ^[23], the behavior-based method ^[24], and the virtual leader method ^[25]. In this paper, the leader-follower method will be used, in which followers only need to know the relative position information of itself and its neighbors, and leaders hold the global information and drive followers to the position of the desired formation in combination with the controller, so that the MASs stabilizes in the desired formation ^[26]. There are two main types of desired formations, static and time-varying formations. Static formation refers to the formation of agents in a fixed formation under the action of a control law ^[27]. In particular, time-varying formation means that the agents form a specified formation and move along a certain trajectory ^[28]. The static formation deployment issue under the grid-grid two-dimensional cylindrical surface topology in polar coordinates was studied in ^[29]. In addition, the formation of MASs under arbitrarily large boundary input delays was innovatively investigated in ^[30]. The leader agents were affected by input delays caused by communication, measurement, and actuator constraints, so the authors designed boundary control laws to compensate for them and to drive all agents into the desired formation. In ^[31], the time-varying formation control for MASs was discussed. The PDE-backstepping technique was applied to design the boundary controller and observer, which transform it into a leader-driven model using only local information, thus converging the MASs to the desired time-varying formation. In ^[32], a large-scale MASs formation tracking control based on wave equations was studied. Employing the leader-following model, a control algorithm was developed and the robustness of the control law to velocity measurement disturbances was discussed.

The main idea of dealing with the formation control problem for large-scale MASs modeled by PEDs is that ^[33]: 1) The discrete agents are modeled as a continuum and the MASs dynamic equation are established based on PDEs; 2) The control is designed based on PDEs; 3) The PDE system and controller are discretized to realize the revivification from PDEs to the agents themselves. Theoretically, provided the number of agents is large enough, then this approach is independent of the number of agents and the communication structure.

In this paper, the wave equation with spatial-varying coefficient is adopted to describe the MASs and the formation control problem of MASs in polar coordinates is discussed. A boundary controller is proposed by combining the PDE-backstepping technique, Lyapunov stability theory and state feedback control. The Lyapunov direct method is employed to prove the stability of the closed-loop system. In the closed-loop system, the first leader arrives directly at the desired position, the followers only need to know the position information of themselves and their neighbors, and the second leader drives the MASs to stabilize in the desired formation under the action of this control law. The main contributions of this paper are as follows: 1) The PDEs framework considered in this paper is a second-order hyperbolic equation with spatial-varying coefficient. The parameters of the reaction term are a space-dependent function that describe more general and extensive formations of MASs. 2) A collaborative control law for the second leader at the boundary is designed by selecting the appropriate target system using the PDE-backstepping technique. 3) The existence uniqueness of the kernel function is analyzed using the variable substitution method and the iterative method, which further shows that the closed-loop system satisfies the well-posedness.

2.   System description

Notations: ${\left(\Delta \right)_s}$ and ${\left(\Delta \right)_{ss}}$ mean the first and second order derivatives of $\left(\Delta \right)$ in terms of the variable $s$, i.e., ${\left(\Delta \right)_s} = {{\partial \left(\Delta \right)} \mathord{\left/ {\vphantom {{\partial \left(\Delta \right)} {\partial s}}} \right. } {\partial s}}$, ${\left(\Delta \right)_{ss}} = {{{\partial ^2}\left(\Delta \right)} \mathord{\left/ {\vphantom {{{\partial ^2}\left(\Delta \right)} {\partial {s^2}}}} \right. } {\partial {s^2}}}$. Similarly, for the variable $t$, ${\left(\Delta \right)_t} = {{\partial \left(\Delta \right)} \mathord{\left/ {\vphantom {{\partial \left(\Delta \right)} {\partial t}}} \right. } {\partial t}}$, ${\left(\Delta \right)_{tt}} = {{{\partial ^2}\left(\Delta \right)} \mathord{\left/ {\vphantom {{{\partial ^2}\left(\Delta \right)} {\partial {t^2}}}} \right. } {\partial {t^2}}}$.

A chain communication structure with $M$ nodes is considered, which is shown in Figure 1. The label $k$ denotes the node, which is the position of the $k$th agent. The agents at the ends of the chain structure are leader agents, labeled 1 and M, and the others are follower agents, labeled $k$, $\left({k = 2, ..., M - 1} \right)$. From ^[20], it is clear that the Laplace operator in PDEs has the same consensus action as the Laplace consensus control laws in ODEs. Therefore, the number of agents can be represented as continuous variables and view the cluster of agents as a continuum. When $M \to \infty$, The discrete chain topology approximates $s = \left[{{G_1}, {G_2}} \right]$, i.e., $k \to {s_k} \to s$. Let $z\left({s, t} \right) = x + iy$ denote the position coordinates of the agent $s$.

In this paper, a leader-follower MAS is studied based on wave equation with spatial-varying coefficient, whose dynamics are described as follows

where $\left({s, t} \right) \in \left[{{G_1}, {G_2}} \right] \times {\mathbb{R}^ + }$, with the boundary conditions

and the initial conditions

In the above equations, $s$ and $t$ are the independent spatial and time variables, respectively. $z\left({s, t} \right)$ represents the position of agents, and ${z_0}\left(s \right) \in {H^1}\left({{G_1}, {G_2}} \right), {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {z_1}\left(s \right) \in {L^2}\left({{G_1}, {G_2}} \right)$. $\Delta z = \frac{{{{\left({{z_s}\left({s, t} \right) \cdot s} \right)}_s}}}{s} = {z_{ss}} + \frac{{{z_s}}}{s}$ is the Laplace operator, $\beta \left(s \right)$ is a spatial-varying bounded complex function, and $\beta \left(s\right)z\left(s, t\right)$ serves to describe the different formations. ${Q_1}\left(t \right), {\kern 1pt} {\kern 1pt} {\kern 1pt} {Q_2}\left(t \right) \in C\left[{0, \infty } \right]$ are complex functions, which designate the dynamic trajectories that agents should follow. Moreover, $s = {G_1}$ is the position of the first leader. During the formation changing process, it arrives directly at the desired position and sends information to its neighbors unidirectionally. At the same time, the control force $\Gamma \left(t \right)$ on the second leader $s = {G_2}$ is applied to communicate bilaterally with its neighbors and drive the MAS to stabilize in the desired formation.

Similarly to (1), the desired formation of the MAS is described as

with the boundary conditions

and the initial conditions

where $z_0^d\left(s \right) \in {H^1}\left({{G_1}, {G_2}} \right), {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} z_1^d\left(s \right) \in {L^2}\left({{G_1}, {G_2}} \right)$.

Definition 1. Introducing the space $H_L^1\left({{G_1}, {G_2}} \right): = \left\{ {y \in {H^1}\left({{G_1}, {G_2}} \right); y\left({{G_1}} \right) = 0} \right\}$, $\mathcal{H} = \;{H^1}\left({{G_1}, {G_2}} \right) \times {L^2}\left({{G_1}, {G_2}} \right)$ and the domain $\Omega = \left\{ {\left({s, \alpha } \right)} \right.\left. { \in {R^2}:{G_1} \leqslant \alpha \leqslant s \leqslant {G_2}} \right\}$. The ${L^2}$-norm and ${H^1}$-norm are defined by ${\left\| z \right\|_{{L^2}}} = {\left({\int_{{G_1}}^{{G_2}} {z{{\left({s, t} \right)}^2}sds} } \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. } 2}}}$, ${\left\| z \right\|_{{H^1}}} = {\left({\int_{{G_1}}^{{G_2}} {z{{\left({s, t} \right)}^2}sds} + \int_{{G_1}}^{{G_2}} {{z_s}{{\left({s, t} \right)}^2}sds} } \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. } 2}}}$.

It is important to note that for positive function $\beta \left(s \right)$, the MAS (1)–(3) may be open-loop unstable. In this case, the agents will not spontaneously move to the desired position. Therefore, it is essential to design a leader-driven state feedback boundary controller.

3.   Boundary controller design and stability analysis

Firstly, the error variable of the system is defined as $q\left({s, t} \right) = z\left({s, t} \right) - {z^d}\left({s, t} \right)$. Then the following error system is obtained

The primary method applied for the control design in this paper is the PDE-backstepping method. The main idea is to first design a Volterra transformation that enables to transform the q-system (7) into a suitable stable target system (g-system). Then substituting the g-system into the Volterra transformation, the PDEs with respect to kernel functions are obtained. To proceed, using the variable substitution method and the iterative method, the existence and the uniqueness of the solution of the PDEs is proved, and then derive the boundary control law. Finally, by the exponential stability of the g-system and the invertibility of the Volterra transform, the stability of the closed-loop system q-system is guaranteed. The clearer control design process chart is presented in Figures 2 and 3, respectively.

Define the following Volterra integral transformation

where $F\left({s, \alpha } \right)$ is a kernel function defined on $\Omega$. The kernel function $F\left({s, \alpha } \right)$ will be suitably selected later to realize converting the unstable q-system (7) into the following target system (g-system) by using the defined Volterra integral transformation (8).

where $c\left(s \right)$ satisfies $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{c} \leqslant {\left\| {c\left(s \right)} \right\|_{{L^2}}} \leqslant \bar c$ for any $s \in \left[{{G_1}, {G_2}} \right]$. $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{c}, {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \bar c$ are positive constants ^[34]. With simple computation, it is easy to obtain the boundary control

In the remainder of this section, the PDEs satisfied by the kernel $F\left({s, \alpha } \right)$ are derived, and the existence and uniqueness of the kernel $F\left({s, \alpha } \right)$ are analyzed.

3.1.   PDE for the kernel

To start with, we compute ${g_s}\left({s, t} \right)$, ${g_{ss}}\left({s, t} \right)$ and ${g_{tt}}\left({s, t} \right)$ respectively based on the transformation (8). Substituting (8) into (9), one gets

where $\frac{{dF\left({s, s} \right)}}{{ds}} = {F_s}\left({s, s} \right) + {F_\alpha }\left({s, s} \right)$. Calculating the second time derivative of $g\left({s, t} \right)$, one yields

Considering $\Delta g - c\left(s \right)g\left({s, t} \right) = {g_{ss}}\left({s, t} \right) + \frac{{{g_s}}}{s}\left({s, t} \right) - c\left(s \right)g\left({s, t} \right)$, and combining (8), (9), (11), (12) with (13), one obtains

Hence, the following Eqs (15)–(18) hold

To simplify the expression, define $(·)\left(s, \alpha \right) = (·)$ and $a\left(s \right) = \beta \left(s \right) + c\left(s \right)$. Then, from (15)–(18), the following equations are derived

Next, the well-posedness of (19) will be proved by using the variable substitution method and the iterative method, and then the boundary control law is obtained.

3.2.   Existence and uniqueness of kernel function

3.2.1.   The existence of kernel function

To prove the existence of the kernel function $F\left({s, \alpha } \right)$, a PDE equivalent to the system (19) is derived. Firstly, the functional transformation $F\left({s, \alpha } \right) = \sqrt {\frac{\alpha }{s}} w\left({s, \alpha } \right)$ is introduced, and the variables replacement are defined

which is equivalent to

Further, define function $\Psi = \Psi \left({\gamma, \varphi } \right)$ relevant to variables $\gamma$ and $\varphi$ as

Meanwhile, let $\lambda \left(\gamma \right) = - \frac{1}{2}\int_{{G_1}}^{{\gamma \mathord{\left/ {\vphantom {\gamma 2}} \right. } 2}} {a\left(\tau \right)d\tau }$. Next, based on $F\left({s, \alpha } \right) = \sqrt {\frac{\alpha }{s}} w\left({s, \alpha } \right)$, the first and second order derivatives of the variables $s$ and $\alpha$ are obtained, respectively.

In addition, the derivative relationship between $w$ and $\Psi$ can be calculated from (22)

Substituting (24) into (23), and combining (21) with (22), the left-hand side of (18) can be rewritten as

Furthermore, one obtains

where ${b_1}\left({\gamma, \varphi } \right) = \beta \left({\frac{{\gamma - \varphi }}{2}} \right) + c\left({\frac{{\gamma + \varphi }}{2}} \right)$. Thereby, in light of (25) and (26), (19) with respect to the kernel $F\left({s, \alpha } \right)$ is converted to

when $\gamma = \gamma, {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi = 0$, one has

and when $\gamma = \varphi + 2{G_1}, {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi = 0$, one has

Synthesis of (27)–(29), the PDEs equivalent to (19) is given

where $b\left({\gamma, \varphi } \right) = \frac{{4\gamma \varphi }}{{{{\left({{\gamma ^2} - {\varphi ^2}} \right)}^2}}} + {b_1}\left({\gamma, \varphi } \right)$. Next, integrating (30), first from 0 to $\varphi$ for variable $\varphi$, and then from $\varphi + 2{G_1}$ to $\gamma$ for variable $\gamma$, then $\Psi \left({\gamma, \varphi } \right)$ is obtained

Combining the above analysis, it is concluded that (31) has a solution. Further, according to the variable and function substitution (20)–(22), one yields that (31) is equivalent to (19) satisfied by the kernel function. Thus, the PDEs (19) satisfied by the kernel function has a solution. In the next step, we will focus on the uniqueness of the PDEs (19).

3.2.2.   The uniqueness of kernel function

In the following, the iterative method is applied to prove that the PDEs (19) has a unique solution. Firstly, the function ${\Psi ^0}$ is defined as

and set the following recursive formula, for $n \in \mathbb{N}$:

Then, one obtains

Next, let $N = 2{G_2}{\left\| {\lambda '} \right\|_{{L^\infty }}}$, we assume that the following equation holds for some $n \in \mathbb{N}$:

Given the recursive formula (33), ${\Psi ^{n + 1}}$ satisfies the following relationship

Therefore, it can be shown that (35) holds for a given constant $P$, and $P = \frac{{\left({{G_2} - {G_1}} \right){{\left\| b \right\|}_{{L^\infty }}}}}{2}$. Then, combining (36) with it, one obtains the solution of (31) as

which is a continuous function. From (31), it follows that if $b$ is a continuous function, then $\Psi \in {\mathbb{C}^2}$. Therefore, the following conclusion is drawn.

Theorem 1. Let ${\kern 1pt} \beta \left(s \right) \in {\mathbb{C}^0}\left({\left[{{G_1}, {G_2}} \right]} \right)$ and $c\left(s \right) \in {\mathbb{C}^0}\left({\left[{{G_1}, {G_2}} \right]} \right)$. Then (19) has a unique solution $F \in {\mathbb{C}^2}\left(\Omega \right)$.

By making $s = {G_2}$ in (8), and combining the boundary conditions in (7) and (9), the control law $\Phi \left(t \right)$ of the q-system as is derived as

and the control law $\Gamma \left(t \right)$ of the MAS (1)–(3) is

This completes the formation control design for the MAS. Next we will pay attention to the stability of MAS under the action of the formation control. Before that, it is necessary to analyze the equivalence of the original system q-system and the target system g-system. Then, it further shows that the stability of the original system q-system is ensured by proving that the target system g-system is stable.

3.3.   Conversion Between q-system and g-system

Define the following map ^[35]

where ${y_1}, {y_2}$ are respectively defined as

The defined map is linear and continuous, thus there exists a constant ${K_1} > 0$, such that

The map $\Xi$ maps solutions $\left({q\left(t \right), {q_t}\left(t \right)} \right)$ of (44) to solutions $\left({y\left(t \right), {y_t}\left(t \right)} \right): = \Xi \left({q\left(t \right), {q_t}\left(t \right)} \right)$ of (45).

The map ${\Xi ^{ - 1}}$ that convert q-system into g-system is reversible, and the inverse transformation is

Similar to the integral transformation (8), the kernel functions $J\left({s, \alpha } \right)$ in (46) can be computed. The existence of $J\left({s, \alpha } \right)$ can be shown by a method similar to that in Section 3.2. Therefore, the inverse map exists

Furthermore, there exists a constant ${K_2} > 0$ such that

Consequently, the norm of the q-system is equivalent to the norm of the g-system. Therefore, it is only to prove the stability of the g-system (9), which can guarantee the stability of the q-system (7).

4.   Stability analysis

This section will discuss the stability of g-system (9). The candidate Lyapunov function is chosen as

where

with $0 < \delta < 1$ being a constant. To prove the stability of g-system, the following lemma is firstly presented.

Lemma 1. The upper and lower bounds of the candidate Lyapunov function $\Pi \left(t \right)$ are ${\eta _1}{\Pi _1}\left(t \right)$ and ${\eta _2}{\Pi _1}\left(t \right)$, respectively. This means that

where ${\eta _1}$ and ${\eta _2}$ are two positive constants, whose definition will be given later.

Proof. Applying Young's inequality to $\left| {{\Pi _2}\left(t \right)} \right|$, one gets

where ${\xi _1} > 0$ is a constant. Let ${C_0} = \max \left\{ {\delta {\xi _1}, {\delta \mathord{\left/ {\vphantom {\delta {{\xi _1}}}} \right. } {{\xi _1}}}} \right\}$, and one gets $\left| {{\Pi _2}\left(t \right)} \right| \leqslant {C_0}{\Pi _1}\left(t \right)$. Therefore, it is satisfied that ${C_0} < 1$ by choosing $\delta$ small enough, and then

Define ${\eta _1} = 1 - {C_0}, {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\eta _2} = 1 + {C_0}$, then ${\eta _1}{\Pi _1}\left(t \right) \leqslant \Pi \left(t \right) \leqslant {\eta _2}{\Pi _1}\left(t \right)$ holds.

This completes the proof of Lemma 1. Following, the time derivatives for ${\Pi _1}\left(t \right)$ and ${\Pi _2}\left(t \right)$ are caculated, respectively.

Firstly, taking the derivative of ${\Pi _1}\left(t \right)$ along time, one has

To proceed, integrating the first term by parts and substituting the target system (9) into the second term of the above equation, one gets

Then, it is simplified to obtain that

By utilizing Young's inequality yields

Taking the time derivative of ${\Pi _2}\left(t \right)$ and substituting (9) into ${\dot \Pi _2}\left(t \right)$, one has

Performing integration by parts on the second term, one obtains

Then, simplifying it, one gets

The time derivative of (49) yields $\dot \Pi \left(t \right) = {\dot \Pi _1}\left(t \right) + {\dot \Pi _2}\left(t \right)$. Combining (58) with (61), one obtains

Based on the assumptions of $c\left(s \right)$ and $d\left(s \right)$, it holds that

where ${\eta _3} = \min \left\{ {2\delta, {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 2\bar c\left({1 + \delta } \right) - 2, {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 2\left({\bar c - 1 - \delta } \right)} \right\} > 0$ is a constant.

Finally, according to (52) and (63), $\dot \Pi \left(t \right)$ is further expressed as

where $\eta = {{{\eta _3}} \mathord{\left/ {\vphantom {{{\eta _3}} {{\eta _2}}}} \right. } {{\eta _2}}} > 0$.

Through the above analysis, the following lemma for the g-system is obtained.

Lemma 2. There exist constants ${\varepsilon _1} > 0$ and ${K_3} > 0$, such that for any initial condition $\left({{g_0}, {g_1}} \right) \in H_L^1 \times {L^2}$, the g-system (9) with the boundary compatibility ${g_0}\left({{G_1}} \right) = 0$, ${g_1}\left({{G_1}} \right) = 0$, ${g_0}\left({{G_2}} \right) = 0$ and ${g_1}\left({{G_2}} \right) = 0$ is exponentially stable and its solution satisfies

Proof. Multiplying ${e^{\eta t}}$ on both sides of (64), one has

The above equation is equivalent to

Then, integrating (67) with respect to $t$ from 0 to $t$, one gets

Thus, (65) holds.

This completes the proof of Lemma 2.

Based on equivalence between the q-system and the g-system in Section 3.3, it is concluded that the q-system is also exponentially stable. Detailed conclusion is stated as follows.

Theorem 2. There exists a unique solution of the closed-loop system (7) with the control (38) and the initial condition $\left({{q_0}, {q_1}} \right) \in H_L^1 \times {L^2}$ satisfying the compatibility condition

Moreover, there exists positive constants ${\varepsilon _2}$ and ${K_4}$, such that the q-system (7) is exponentially stable and its solution satisfies

Together with the above analysis, it is obtained that the MASs (1)–(3) are stable under the action of the boundary control law (39).

Remark 1. The differences and advantages of using PDEs to model multi-agent systems versus ODEs modeling focus on two main points. Firstly, if the ODEs are used for modeling, each agent corresponds to one ODE. As the number of agents increases, the number of ODEs also increases. Therefore, when MASs have a large number of nodes, it is difficult to add control inputs to each agent. While PDEs can model large-scale MASs based on a continuum model perspective. Secondly, the dynamics of the agent actually has spatio-temporal distribution characteristics. PDEs can not only describe the temporal characteristics, but also simulate the spatial characteristics. Thus, the PDEs-based approach has the advantage of large-scale evolution of MASs compared with the ODEs model.

Remark 2. It is worth noting that the modeling of MASs using the PDE framework must focus on the well-posedness problem of the closed-loop system (i.e., the existence of uniqueness of the system solution). In this paper, the considered MASs satisfies the well-posedness as a prerequisite for its control design. Compared with the ODE model, the PDE model involves spatial and temporal variables and is more difficult to analyze for well-posedness. In this paper, the problem of the well-posedness of MASs is properly handled and solved by using the variable substitution method and the iterative method, and the existence of uniqueness of the closed-loop system solution is guaranteed.

5.   Simulation

To verify the effectiveness of the proposed distributed control law, $31$ agents are simulated on a chain topology. We consider the space step as $\Delta s = {{\left({{G_2} - {G_1}} \right)} \mathord{\left/ {\vphantom {{\left({{G_2} - {G_1}} \right)} {\left({M - 1} \right)}}} \right. } {\left({M - 1} \right)}}$. Thus, each agent on the chain topology can be represented as ${s_k} = {G_1} + \left({k - 1} \right)\Delta s$, $k = 1, 2, \ldots, M$. The positions of the first leader, second leader, and followers are denoted as ${z_1}{\kern 1pt}$, ${\kern 1pt} {z_M}$ and ${z_k}, {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} k = 2, \ldots, M - 1$, respectively. Next, the PDE model (1) is discretized using the finite difference approximation method, which yields the control laws of leaders and followers.

The control law of the first leader is

The control laws for followers are

Using Simpson's numerical integration rule, the second leader's control law is obtained as

where ${a_k}$ is the Simpson coefficients, and according to Simpson's rule of numerical integration, ${a_k}$ is

in which $M$ must be chosen as an odd number according to Simpson's rule.

The topology parameters are selected as: ${G_1} = 0.2$ and ${G_2} = 1$. The spatial-varying functions are chosen as $\beta \left(s \right) = 0.154 + 0.075\cos \left({5.14s} \right) - 0.0317i$ and $c\left(s \right) = - 0.042 - 0.075\cos \left({5.1s} \right)$$- 0.03i$ The finite difference method is applied to discretize the hyperbolic PDE (21) satisfied by the kernel $F\left({s, \alpha } \right)$, which leads to the kernel gain $F\left({{G_2}, \alpha } \right)$ of the controller, as shown in Figure 4. Then, under the distributed control laws (71)–(73), the evolution of the agents from the initial formation (circular) to the desired formation (V-shaped) is simulated. The initial position is the circle ${z_0}\left(s \right) = 3\cos \left({{{2\pi s} \mathord{\left/ {\vphantom {{2\pi s} {\left({{G_2} - 0.18} \right)}}} \right. } {\left({{G_2} - 0.18} \right)}}} \right) + 3i\sin \left({{{2\pi s} \mathord{\left/ {\vphantom {{2\pi s} {\left({{G_2} - 0.18} \right)}}} \right. } {\left({{G_2} - 0.18} \right)}}} \right)$ and the desired position is V-shaped ${z^d}\left(s \right) = \left({ - 7 - 3.67i} \right){J_2}\left({{{\left({17 - 6i} \right)}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. } 2}}}s} \right) + \left({5 - 11i} \right){Y_0}\left({{{\left({17 - 6i} \right)}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. } 2}}}s} \right) + 3i$. The first leader reaches the desired position directly. The second leader communicates with its neighbors in both directions under the action of the actuator, and thus gradually reaches the desired position with the followers. The transient evolution process is shown in Figure 5. The above analysis and the simulation results show that the proposed control scheme is effective.

6.   Conclusions

In this paper, a distributed cooperative deployment scheme was constructed using the Volterra integral transformation, which is used for two-dimensional spatial deployment control of the MAS on a one-dimensional chain topology. The communication graph of agents is determined by the discretization of the PDE space derivatives, which leads to a distributed protocol in which each agent only needs to exchange information with its neighbors. The well-posedness of the kernel function in the control scheme is proved using the iterative and inductive methods, so that the feasibility of the control protocol was guaranteed. Then, the stability of the closed-loop system is illustrated by a rigorous mathematical analysis combined with Lyapunov stability theory. Finally, the effectiveness of the proposed control protocol is verified by simulation. Moreover, considering finite-time control, event-triggered control and bipartite cooperative control of MASs based on the PDE continuum model will be significant and interesting. All these issues will be our future work.

Acknowledgments

This work is supported in part by the National Natural Science Foundation of China under Grants 61973147, 62273171 and 62025303, and in part by the LiaoNing Revitalization Talents Program under Grant XLYC2007177, in part by the Natural Science Foundation of Liaoning Province under Grant 2021-MS-318.

Conflict of interest

The authors declare that there is no conflict of interest.