Recently, the cooperative controls of multi-agent systems have drawn a large amount of attention in many fields, such as spacecraft formation flying,^[1] coordination of wireless sensor network,^[2] and synchronization of autonomous robots.^[3] In particular, leaderless consensus,^[4–7] leader-following tracking,^[8–12] and distributed containment with multiple leaders^[13–15] have received most attention in the field of cooperative control.

Unlike the distributed containment control problem where the cooperative containment control algorithm is designed for followers such that all the followers converge into the convex hull spanned by multiple leaders,^[14] the cooperatively surrounding control (CSC) problem is considered as an inverse of the distributed containment control problem where the CSC algorithm is designed for agents such that all the agents move cooperatively to surround the target or multiple targets.^[16] With this background, the CSC is quite suitable for many practical space applications such as orbital debris removal, on-orbit refueling, reconnaissance, and maintenance.^[17,18] However, the relationship between the agents and the target in the CSC problem is usually non-cooperative because the target could not initiatively send any state information to the agents.^[19] The non-cooperativeness poses challenges for the agents’ controller design in the CSC problem. Therefore, it is of great significance to conduct further research in the CSC problem. However, there are few results about this issue to date. Under a decentralized estimation-and-control framework, in Ref. [20] the CSC problem for first-order integrator linear systems was studied on the assumption that the followers were placed within a circle and a leader can communicate with at least one follower initially. In Ref. [21] the CSC problem was studied for first-order integrator linear systems under an undirected graph, where a CSC law is designed for agents such that all agents could encircle the dynamic targets by calculating the average position of the targets. There are three limitations in Ref. [21]: (i) the number of targets must be the same as the number of the agents; (ii) the working modes between agents and targets are collaborative because the targets’ velocity information is available to the agents, and each agent can always be connected to a target; (iii) all the agents finally form a formation of the same shape but larger than that of the targets. In Ref. [22] the results of Ref. [21] were extended to the second-order integrator linear system. Based on cyclic pursuit strategies, a special case of CSC problem was also studied in Refs. [23] and [24]. However, in the above Refs. [20]–[24] the practical constraints in systems were not considered, such as the uncertain dynamics and input saturation, the difficulty of real-time and bidirectional information interaction among agents, and the communication burden of relative velocity information.

Uncertain dynamics exists in many practical systems for the disturbance and the unmodeled dynamics.^[25] By comparing known parameters and unknown parameters, the synchronization control of two different chaotic systems was investigated in Ref. [26]. To deal with uncertain dynamics in the system, the backstepping design method was used in Ref. [27] to realize the distributed tracking control of nonlinear stochastic multi-agent systems under a directed graph. In Ref. [28] a distributed consensus tracking control problem under an undirected graph was investigated, in which the function approximation using neural networks was employed to compensate for the unknown dynamics induced from controller design procedure. With the limitation to controllers and actuators, input constraints, such as magnitude, rate and bandwidth limitations, cannot be ignored in practice.^[29,30] With the consideration of input saturation, the robust stabilization of state delayed discrete-time Takagi–Sugeno fuzzy system is studied in Ref. [31] by using the anti-windup fuzzy design method. In Ref. [32] the control of an unmanned air vehicle was studied, where a command filter was used to handle the intermediate state and the surface with magnitude, rate and bandwidth limitations. However, the reports in which the uncertain dynamics and input constraints are considered simultaneously in the CSC problem are few until now.

The objective of this paper is to study the CSC problem for a multi-agent system modeled by Euler–Lagrange (EL) equations under a directed graph. As many physical systems are inherently nonlinear in practice, it is significant to describe the model of agents by nonlinear equation. The EL system is a special case of the second-order nonlinear system which can be used to represent a large class of physical mechanical systems, such as underwater vehicles,^[33] robotic manipulators,^[34] and flying spacecrafts.^[35] The relationship between target and agents is non-collaborative, in which the position of the stationary target is only available to a portion of agents. With the consideration of uncertain dynamics in the EL system, by incorporating neural-networks (NN) technique into backstepping control design framework, a backstepping CSC algorithm is first investigated such that all agents move cooperatively to surround the stationary target. Then, by decoupling the design of the controllers for the backstepping iterations and derivative output of the command filter, we further develop a command filtered backstepping CSC algorithm to cope with the constraints on control input and the absence of neighbors’ velocity information. Numerical examples of eight satellites surrounding one space target illustrate the effectiveness of the obtained theoretical results.

To sum up, compared with those of other references studying the CSC problem, the main advantages of this paper are: i) the agents’ dynamics used in this paper are for nonlinear EL systems; ii) the uncertain dynamics and input constraints in systems are considered; iii) the numbers of targets and surrounding formation are not constrained; iv) the communication topology between agents is a directed graph without using the relative velocity information.

Some basic preliminaries about the EL systems, graph theory and problem statement will be introduced in this section.

In this section, two CSC algorithms are designed for agents to surround a static target. Firstly, we consider the EL systems in the presence of uncertain dynamics. Then, the constraints on control input and the absence of the relative velocity information are further considered.

3.1. Neural-network backstepping CSC for EL systems with uncertain dynamics

Construct the following auxiliary variable for the i-th agent

where δ_ij is the desired relative distance between agent i and j, and β is a positive constant. Then, we design the backstepping CSC based on NN. Backstepping-based method is an effective controller design tool for the nonlinear system.^[36] Following the backstepping control design procedure, we define the error variables Z_1i, Z_2i ∈ R^p as

where α_1i is a virtual variable which will be designed later.

Differentiating both sides of Eq. (3) yields

We choose the virtual control α_1i as

where K_1i is a symmetric positive-definite matrix. Construct the first Lyapunov function V_1i as

According to Eq. (6), we can obtain

Applying Eq. (4) to Eq. (1), we can obtain

We construct the second Lyapunov function

By using Property 1 and Eq. (9), we have

We give the following control law for the i-th agent

where K_2i is a symmetric positive definite matrix. Then, substituting Eq. (12) into Eq. (11), we obtain

From Eq. (13), we can easily conclude that the system under control law (12) is asymptotically stable. However, the system parameters M_i, C_i, g_i, and disturbance ω_i are all used in control law (12), which are difficult to obtain in practice because of uncertain dynamics. To solve this problem, artificial neural network function approximation techniques, neural-networks (NN) are used to approximate the unknown nonlinear parametric uncertainties , which can be represented as

where W_i is the ideal constant approximation weight matrix, θ_i(·) is the suitable basis set of functions, and ε_i is the approximation error for the i-th agent which is bounded by ∥ε_i∥ ≤ ε_Mi where ε_Mi is a positive constant. In practice, the nonlinear function cannot be approximated exactly by NN. Thus, the estimate of f_i (q_i,q̇_i,α̇_1i,α_1i) can be expressed as

where Ŵ_i is the estimate of W_i.

Thus, to deal with the uncertain dynamics in the system, we propose the following CSC law:

where γ and k₁ are positive constants.

Theorem 1 Suppose that Assumptions 1 and 2 hold, considering an arbitrary agent in this system, say, i-th agent, in the presence of the uncertain dynamics, using Eq. (16) for τ_i in Eq. (1), we can conclude that the static target will be surrounded by all agents with the desired formation or relative distance δ_i0.

Proof Construct the third Lyapunov function as

where W̃_i = W_i − Ŵ_i. Taking the derivative of V_3i using Eq. (11) gives

Substituting Eqs. (14), (16), and (17) into Eq. (19), we have

Because is a scalar, we can obtain . Meanwhile,

By choosing large enough k₁ such that k₁ ≥ ∥ω_Mi∥ + ∥ε_Mi∥, we can obtain

Because K_1i and K_2i are symmetrically positively definite, we can obtain that V̇_3i ≤ 0 and V_3i (t) ≤ V_3i (0), ∀t ≥ 0. Therefore, Z_1i (t), Z_2i (t), and W̃_i (t) are all bounded, which means that Z_1i (t) ∈ 𝕃_∞ and Z_2i (t) ∈ 𝕃_∞. From Eq. (4) and Property 2, we know that q_i, q̇_i and α_1i, Ż_1i, Ż_2i, and are all bounded, which leads to Ż_1i ∈ 𝕃_∞ and Ż_2i ∈ 𝕃_∞. Integrating both sides of inequality (20), we have

Thus, we can obtain Ż_1i ∈ 𝕃₂ and Ż_2i ∈ 𝕃₂. By using Barbalat’s lemma, we have

Note that equations (3) and (4) can be written as

and

Sustituting Eqs. (25) and (26) into Eqs. (23) and (24), we can obtain

According to the results in Ref. [37], we conclude that if the directed graph contains a spanning tree and the desired distance δ_ij is feasible, there is constant vector δ_i0,i = 1,…,N such that δ_ij = δ_i0 − δ_j0. Define vector , , and . For ∀i ∈ M, equation (27) holds true. Thus, equation (27) can be written as

By Lemma 1, we can obtain the following conclusion:

which leads to

This completes the proof of Theorem 1.

3.2. Command-filtered backstepping CSC for EL systems with input constraints

In Subsection 3.1, we design a CSC algorithm for a multiple EL system with uncertain dynamics. However, the input constraints in the system are not considered, which cannot be ignored in the presence of limitation of actuator in practice. Inspired by Refs. [32] and [38], the command filter (CF) can be used to solve the problem of input constraints in the system. As a byproduct in this study, the absence of relative velocity information in directed graph is also handled by the differential output of CF.

The structure of CF is shown in Fig. 1. When input signal is , CF can filter the input signal to produce a magnitude, rate and bandwidth-limited signal x_c and its derivation ẋ_c, which means that the derivative ẋ_c is computed without any differentiation but with integration.

Fig. 1.

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	Fig. 1. Structure of a command filter.

The transfer function from the variable to the output x_c can be derived as

From Ref. [32], we have the lemma about the CF as follows.

Lemma 2^[32] The CF can filter the input signal to produce a magnitude, rate and bandwidth-limited signal x_c and its derivative ẋ_c, which means

where ε = 1/ω_n and O(ε) denotes the infinitesimal of higher order about ε. Inspired by the above two advantages of CF, we will design a CSC algorithm by using two command filters to cope with the constraints on control input and the absence of the relative velocity information in directed graph.

Firstly, define the following compensated error vector:

where ψ_1i and ψ_2i are auxiliary variables, which will be designed later. Constructing the Lyapunov function as

The derivative of Eq. (36) follows

Because

Design the adaptive law as

where τ_oi is the input of the first CF which will be designed later, τ_i is the output of the first CF which is magnitude, rate and bandwidth-limited and will be applied to the actual system.

Applying Eqs. (38)–(41) to Eq. (30), we can obtain

Then, we design the following CSC algorithm:

Substituting Eq. (43) into Eq. (42), we can obtain

By using k₁ ≥ ∥ω_Mi∥ + ∥ε_Mi∥ ≥ ∥ω_i∥ + ∥ε_i∥, we can conclude

Like the statements in Subsection 3.1, we can obtain

From Eqs. (47) and (48), we can easily obtain the conclusion of asymptotical stability of the system with input constraints. However, the adaptive law (41) uses the system inertia matrix M_i, which is difficult to accurately obtain because of uncertain dynamics. At the same time, the relative velocity information is also used in α̇_1i in Eq. (15), which requires the system to be equipped with a velocity sensor. To tackle the first problem, we assume that M_i = M_oi + ΔM_i where M_oi is the nominal value and ΔM_i is the bounded uncertain part. To solve the second problem, we fully use the differential output of the CF. Thus, we propose the adjusted CSC algorithm

where α_1i is the input of the second CF, is the output of the second CF which converges to , and is approximated by NN which can be represented as

Like that in Subsection 3.1, ε_oi is bounded and the estimate of can be written as .

Theorem 2 Suppose that Assumptions 1 and 2 hold, considering an arbitrary agent in the system, say, the i-th agent, in the presence of the uncertain dynamics and input constraints, using Eq. (50) for Eq. (1), we can obtain that the static target will be surrounded by all agents with the desired formation or relative distance δ_i0.

Proof Construct the Lyapunov function as

Applying Eq. (50) to Eq. (39), we can obtain

By similar statements, we have

By using Property 2, we can obtain

where is bounded as it is related to CFs. Let , we can obtain that

Like the statements in Subsection 3.1, we can conclude

Finally, it comes to the conclusion of asymptotical stability of the system, which completes the proof of Theorem 2.

Remark 2 From the proof of Theorem 2, we can obtain lim_{t → ∞} Z̃_1i (t) = 0 and lim_{t → ∞} Z̃_2i (t) = 0 for the system with uncertain dynamics and input constraints. However, we cannot directly obtain lim_{t → ∞} Z_1i (t) = 0 and lim_{t → ∞} Z_2i (t) = 0 because ψ_1i and ψ_2i are not zero. According to Ref. [32], the control inputs are usually so aggressive during the initial stage that the magnitude, rate, or bandwidth limits will come into effect. As a result, the CF cannot completely track the inputs. With the ψ_1i and ψ_2i compensating for the tracking errors of the CF input and output, the limits will eventually be ineffective. Finally, the ψ_1i and ψ_2i variables converge to zero, and Z_1i and Z_2i converges to Z̃_1i and Z̃_2i, respectively.

Remark 3 The above designed CSC algorithms are mainly used to surround the single target. However, it is easy to extend these two CSC algorithms to surround multiple targets, which depends on the planning of the desired formation configuration δ_i0. In other words, as long as the desired relative distance δ_i0 between the agent i and the average position of all targets is longer than the convex hull spanned by all targets, the designed two CSC algorithms will still be effective.

In this section, we present a numerical simulation to verify the effectiveness of the proposed CSC algorithms in this study. With the consideration of a group of eight satellites (numbered as 1 to 8) surrounding a stationary target (numbered as 0), the equations of relative orbit motion in the local-vertical, local-horizontal (LVLH) rotating frame with a chief satellite can be described as^[39]

We assume that the relative orbit reference of the CSC system follows a near-circular orbit with the initial orbit elements [ae i Δ ω f] = [7136.0 0.00160° 10° 30° 0°] where a is the semi-major axis(km) of the reference orbit, e is the eccentricity, i is the inclination, Δ is the longitude of the ascending node, ω is the argument of periapsis, and f is the true anomaly. For each satellite i(∀i ∈ {1,…,8}), m_i = 35 kg and τ_doi = [− 1.025 sin (t), 6.248 cos (t),1]^T × 10^{− 4} N. The directed communication graph among eight satellites and the static target is shown in Fig. 2.

Fig. 2.

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	Fig. 2. The communication topology among the satellites and the target.

For each satellite (∀i = 1,…,8), the target (i = 0) has at least one directed path to each satellite (for example, to agent 6, there is a directed path: 0 → 4 → 5 → 1 → 2 → 6). Therefore, this communication topology satisfies Assumption 2. The Laplacian matrix of the agents is composed of

The activation function of NN for the i-th satellite used in this simulation can be denoted as follows:

To compare and illustrate the effectiveness and the advantages of the proposed CSC algorithms, we simulate the following three CSC algorithms: nominal CSC (NCSC) law (12), neural-networks CSC (NNCSC) law (16), and command filtered CSC (CFCSC) law (50) with four examples and the results are given below.

Example 1 NCSC for system with uncertain dynamics The NCSC law can be described as

Uncertain dynamics: .

Controller parameters: K_1i = 4I₃, K_2i = 4I₃, β = 0.01.

Simulation results are shown in Fig. 3 and 4.

Fig. 3.

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	Fig. 3. Trajectories of agents using NCSC.

Fig. 4.

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	Fig. 4. Control forces of agents using NCSC.

Example 2 NNCSC for system with uncertain dynamics

Uncertain dynamics: m_i, C_i, g_i, and τ_doi are all unknown. Controller parameters are taken as follows: K_1i = 4I₃, K_2i = 4I₃, β = 0.01, γ = 1, and k = 0.1.

Simulation results can be found in Fig. 5 and 6.

Fig. 5.

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	Fig. 5. Trajectories of agents using NNCSC.

Fig. 6.

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	Fig. 6. Control forces of agents using NNCSC.

Example 3 NNCSC for system with uncertain dynamics and input constraints

Uncertain dynamics: m_i, C_i, g_i, and τ_doi are all unknown.

Input constraints: control saturation with upper limitation of 30 N.

Controller parameters: K_1i = 4I₃, K_2i = 4I₃, β = 0.01, γ = 1, and k = 0.1.

Simulation results are presented in Fig. 7 and 8.

Fig. 7.

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	Fig. 7. Trajectories of agents using NNCSC.

Fig. 8.

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	Fig. 8. Control forces of agents using NNCSC.

Example 4 CFCSC for system with uncertain dynamics and input constraints

Uncertain dynamics: m_i, C_i, g_i, and τ_doi are all unknown.

Input constraints: control saturation with upper limitation of 10 N.

Controller parameters: K_1i = 0.002I₃, c, K_4i = 0.002I₃, β = 0.01, γ = 1, k = 50, , and .

Simulation results are displayed in Fig. 9, 10, and 11.

Fig. 9.

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	Fig. 9. Trajectories of agents using CFCSC.

Fig. 10.

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	Fig. 10. Control inputs of CF of agents using CFCSC.

Fig. 11.

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	Fig. 11. Control outputs of CF of agents using CFCSC.

We can see that the eight satellites can surround the stationary target in the above four simulation examples. The comparisons of performances among these four CSC algorithms are given below.

Comparing Example 1 with Example 2, we can obtain that the convergence time of NCSC is ten times shorter than that of NNCSC, however, the control force amplitude of the NCSC is four times larger than that of NNCSC which is difficult to realize for the limitation to actuators. This demonstrates the effectiveness of NN in solving the system uncertain dynamics.

From the comparison between Example 2 and Example 3, we conclude that the input constraints will extend the convergence time of the system. From Example 3 of the simulation, we find that the saturation upper limitation of 30 N is nearly the maximum tolerance of the stability in system, however, this upper limitation may still be larger than the actual system output. This comparison shows the influences of the input constraints on the system stability and convergence time.

From Examples 3 and 4, we can obtain that with the absence of relative velocity information in directed graph and the constant gain k₂ larger than k₁, the amplitude of CFCSC is twice as large as that of NNCSC, however, by using command filter with a saturation upper limitation of 10 N, the stability of CSC system can still be guaranteed. This comparison demonstrates the utility of solving input constraints by command filter.

All of the above simulation analyses illustrate the effectiveness of the proposed two CSC algorithms and show the respective advantages of these four CSC laws.

Cooperatively surrounding controls for multiple nonlinear Euler–Lagrange systems with uncertain dynamics and input constraints are studied. By fully using the relative formation distance between agents, the cooperatively surrounding control problem is solved in a different and improved way. By incorporating neural-networks into backstepping control design framework, the uncertain dynamics in the system is handled. Then, a cooperatively surrounding control algorithm is designed for the agents such that all agents can move collectively to surround the stationary target. With the consideration of the limited control force output and the difficulty in measuring relative velocity, a command filtered backstepping cooperatively surrounding control algorithm is further developed. Future work will focus on the dynamic target and the collision avoidance.