Sage Journals: Discover world-class research

Abstract

This paper studies the distributed convex optimization of bipartite containment control problem for a class of higher order nonlinear multi-agent systems with uncertain states. For the optimization problem, the penalty function is constructed by summing the local objective function of each agent and combining the penalty term formed by the adjacency matrix. For the unknown nonlinear function and unpredictable states in the system, this paper construct radial basis function Neural-networks and state observer for approaching, respectively. In order to avoid “explosion of complexity,” under the framework of Lyapunov function theory, we propose the dynamic surface control (DSC) technology and design the distributed adaptive backstepping neural network controller to ensure all the signals remain semi-global uniformly ultimately bounded in the closed-loop system and all agents can converge to the convex hull containing each boundary trajectory as well as its opposite trajectory different in sign. Simulation results confirm the feasibility of the proposed control method.

Keywords

Distributed optimization bipartite containment observer multi-agent systems neural network adaptive backstepping control

Introduction

Over the recent decades, multi-agent systems (MASs) have been widely concerned by scholars in the field of control because of their importance in practical applications.^1–5 Many control problems for multi-agent systems are presented. Such as neural network and disturbance observer are used to deal with the influence of the input dead zone and the external disturbance on multi-agent formation respectively.⁶ Olfati-Saber present a theoretical framework for design and analysis of distributed flocking algorithms to deal with the multi-agent flocking problem.⁷ Guo et al. propose a novel technique to control the relative motion of multiple mobile agents as they stabilize to a desired configuration.⁸ Chen et al. propose four resilient state feedback based leader–follower tracking protocols.⁹

Generally speaking, the control methods of multi-agent systems can be divided into two categories. One is decentralized control, the other is distributed control. With the decentralized control method, the follower can get the state of the leader.^10,11 However, for distributed control, the follower cannot get the state of the leader and needs to exchange information through the communication topology.^12,13 Generally, distributed control is more widely used and currently it is the main multi-agent control method. Ren et al.¹² study the finite-time positiveness and distributed control problem for a class of Lipschitz nonlinear multi-agent systems. Wang¹⁴ propose a distributed consensus algorithm to deal with the leaderless consensus control problem for higher-order nonlinear MASs with completely unknown non-identical control directions. The leader-follower issue of MASs considering network transmission delay is solved by an observer-based distributed control triggered by adaptive event-triggered.¹³ The switched stochastic nonlinear MASs control method is proposed.¹⁵ Zou et al.¹⁶ focus on the mean square practical leader-following consensus of nonlinear MASs with noises and unmodeled dynamics.

The bipartite containment problem of MASs will make agents converge to a convex hull, it’s not necessarily the optimal convergence route. Therefore, an optimal problem is introduced to create a distributed optimal controller, so that all agents converge to the optimal solution. In the existing bipartite containment control papers, the main focus is that the model and does not take into account output optimization. Through the study of Zhang et al.,¹⁷ the distributed bipartite containment control problem for high-order nonlinear MASs with time-varying powers is solved. A bipartite containment fuzzy controller for nonlinear MASs with unknown external interference and quantized inputs is designed by Li et al.¹⁸ Wu et al.¹⁹ proposed a fixed-time adaptive fuzzy quantization controller in order to ensure that the nonlinear MASs with unknown external disturbances and unknown Bouc–Wen hysteresis is controlled by bipartite containment. Similarly, controllers designed for distributed optimization problems in MASs do not take into account the bipartite containment problem. Kang et al.²⁰ proposed a backstepping controller for distributed optimization of high order nonlinear MASs with strict feedback. Guo et al.²¹ solve the distributed optimization problem of MASs subjected to exogenous disturbances. Guo and Kang²² by constructing a two-layer control framework, the optimal trajectory is obtained by adaptive control technology, and then MASs are tracked to the optimal trajectory by state integral feedback control (SIFC). Yu Zhiyong proposes some distributed methods to solve the MASs optimization problem with equality constraints.²³ Through the study of Wang et al.,²⁴ the distributed convex optimization problem of multi-agent systems with nonlinear terms interfered by random noise is solved. The bipartite containment of fractional order system is analyzed and a controller with good control performance is proposed by Chen and Yuan.²⁵ The control analysis of consistency optimization problem is carried out by Yang et al.²⁶

In the actual model, there are many nonlinear uncertain functions, which will seriously affect the operation of the agent, so scholars use neural network(NN) and fuzzy logic systems to approximate unknown nonlinear functions.^27–29 Guo et al.³⁰ use the command-filtered backstepping control method, approximates the unknown nonlinear function by NNs, so as to solve MASs bipartite containment control problem. An adaptive NNs output feedback controller is designed for MASs with time delays and unmodeled dynamics by Li et al.³¹

Based on the aforementioned research, this paper proposes an adaptive backstepping neural network dynamic surface controller to solve the distributed optimization problem for MASs with unknown nonlinear functions and bipartite containment problem. Every agent in the MASs need to solve their local objective function optimally. Compared with the previous research work, the main contributions of the method in this paper are as follows.

In this paper, a controller is proposed for the optimization of MASs with bipartite containment control. A penalty function is designed to make each agent gradually approach the optimal solution of the global objective function while being contained in the convex hull. Unlike the study of Guo and Zhang,³² where the controller was designed only for consensus control problem, but the bipartite containment problem in the control process is analyzed in this paper. Compared with the study of Zhang et al.,¹⁷ the optimization problem of minimizing the sum of squares of distance difference between the agent and the upper and lower bounds is considered when designing the controller.

Compared with the study of Liu et al.,³³ this paper uses backstepping control method to extend the optimal bipartite containment control method to higher-order MASs, and solve the explosion of complexity problem caused by high order system by using filter.

Compared with the study of Liu et al.,³⁴ this paper solves the problem of unmeasurable state of high order system by introducing observer, and approximates the nonlinear term of high order system by RBF neural network.

The rest of this paper is as follows. Section 2 introduces the theoretical knowledge of distributed optimization and the model of MASs. In section 3, an observer model is established to estimate the state variables of the system, and then the controller and the adaptive weight update law are designed according to the backstepping method. Section 4 proved the effectiveness of the controller by simulation. In section 5, the work of the full text is summarized.

Prerequisites

Graph theory

There is information interaction between multiple agents, and an undirected graph $G = (w, ε, \bar{A})$ is usually used to represent such information, in which $w = {n_{1}, . . ., n_{M}}$ . Where $M$ is the number of agents. The set of edge is exhibited as $ε = {(n_{i}, n_{j})} \in w \times w$ , which expresses that there is information exchange between agent $i$ and agent $j$ . The $\bar{A} = {a_{ij}} \in R^{N \times N}$ in the undirected graph is the Adjacency matrix. $a_{ij}$ of $\bar{A}$ is represented as if $(n_{i}, n_{j}) \notin ε$ , $a_{ij} = 0$ ; if not, $a_{ij} \neq 0$ . Then we assume that $N_{i} = {j | (n_{i}, n_{j}) \in ε}$ means the neighbor set of agents $i$ and $D = diag (d_{1}, . . ., d_{N})$ as the diagonal matrix, in which $d_{i} = \sum_{j \in n = N_{i}} a_{ij}$ . Define the Laplacian matrix of a signed undirected graph as $L = D - \bar{A}$ .

Lemma 1. Define a column vector $1_{M}$ with $M$ elements and all elements being one. The symmetric matrix $L \in R^{M \times M}$ is used to represent the Laplace matrix of the undirected graph $G$ . $1_{M}$ is eigenvector for eigenvalue 0 of Laplacian matrix.³⁵

Convex analysis

If a function $f (\cdot) : R^{n} \to R$ is convex we can have

\begin{matrix} f (ax + (1 - a) y) \leq af (x) + (1 - a) f (y) \\ \forall x, y \in R^{n}, 0 \leq α \leq 1 . \end{matrix}

(1)

If a differentiable function $f (\cdot) : R^{n} \to R$ is strongly convex on $R^{n}$ we can have

\begin{matrix} {(x - y)}^{T} (\nabla f (x) - \nabla f (y)) \geq ω ∥ x - y ∥^{2} \\ \forall x, y \in R^{n}, ω > 0 . \end{matrix}

(2)

if a function $f (\cdot) : R^{n} \to R$ is $Γ$ -Lipschitz $(Γ > 0)$ on $R^{n}$ we can have

∥ f (x) - f (y) ∥ \leq Γ ∥ x - y ∥, \forall x, y \in R^{n} .

(3)

Problem formulation

In this paper, we study the following high-order nonlinear multi-agent systems for agent $i$ .

{\begin{matrix} {\overset{\cdot}{x}}_{i, 1} (t) = x_{i, 2} + g_{i, 1} (x_{i, 1}) \\ {\overset{\cdot}{x}}_{i, r} (t) = x_{i, r + 1} + g_{i, r} (x_{i, 1}, x_{i, 2}, \dots, x_{i, r}) \\ {\overset{\cdot}{x}}_{i, n} (t) = u_{i} (t) + g_{i, n} (x_{i, 1}, x_{i, 2}, \dots, x_{i, n}) \\ y_{i} = x_{i, 1} \end{matrix}

(4)

where $r = 2, \dots, n - 1$ , $u_{i}$ is the control input, $y_{i}$ is the system output and $g_{i, l} (x_{i, l}, x_{i, 2}, \dots, x_{i, r})$ is an unknown nonlinear function defined on the system state vector. Define $X_{i, l} = (x_{i, 1}, x_{i, 2}, \dots, x_{i, r})^{T} \in R^{r}$ as the system state vectors for agent $i$ . Rewrite the system for agent $i$ :

\begin{matrix} {\overset{\cdot}{X}}_{i, n} = A_{i} X_{i, n} + K_{i} y_{i} + \sum_{l = 1}^{n} B_{i, l} [g_{i, l} (X_{i, l})] + B_{i, n} u_{i} (t) \\ y_{i} = X_{i, 1} \end{matrix}

(5)

where $A_{i} = [\begin{matrix} - k_{i, l} \\ ⋮ & I_{n - 1} \\ - k_{i, n} 0 & \dots & 0 \end{matrix}]$ , $K_{i} = [\begin{matrix} k_{i, 1} \\ ⋮ \\ k_{i, n} \end{matrix}]$ , $B_{i, l} = {[\begin{matrix} \underset{l}{\underset{︸}{0 \dots 1}} & \dots 0 \end{matrix}]}^{T}$ . For a given positive matrix $Q_{i}^{T} = Q_{i}$ , there exists a positive matrix $P_{i}^{T} = P_{i}$ satisfying

A_{i}^{T} P_{i} + P_{i} A_{i} = - 2 Q_{i} .

(6)

The distributed optimization problem

In this paper, we must not only solve the optimization problem of the global objective function, but also solve the optimization problem of the local objective function of agents $N$ . The local objective function of the $i th$ agent is defined as

\begin{matrix} f_{i} (x_{i, 1}) = a_{i, 1} {(x_{i, 1} - x_{d, 1})}^{2} + a_{i, 2} {(x_{i, 1} - x_{d, 2})}^{2} + c \\ = a_{i} x_{i, 1}^{2} + b_{i} x_{i, 1} + c_{i} \end{matrix}

(7)

where $x_{d, 1}$ and $x_{d, 2}$ is the upper and lower bound of the trajectory of motion, $a_{i} = a_{i, 1} + a_{i, 2} > 0$ , $b_{i} = - 2 a_{i, 1} x_{d, 1} - 2 a_{i, 2} x_{d, 2}$ , $c_{i} = a_{i, 1} x_{d, 1}^{2} + a_{i, 2} x_{d, 2}^{2} + c$ , $1 < i < N$ . Define the global objective function as

f (x_{i, 1}) = \sum_{i = 1}^{N} f_{i} (x_{i, 1})

(8)

Because the local objective function is an absolute convex function, the global objective function is also a strictly convex function. Define $x_{1} = {[x_{1, 1} x_{2, 1} \dots x_{N, 1}]}^{T}$ . According to Lemma 1, for some $α \in R$ , if $x_{1} = α \cdot 1_{N}$ we obtain

L x_{1} = 0 .

(9)

Therefore, we can design penalty term as follows

\begin{matrix} x_{1}^{T} L x_{1} = 0 . \end{matrix}

(10)

Penalty function is defined as follows³⁶

P (x_{1}) = \sum_{i = 1}^{N} f_{i} (x_{i, 1}) + x_{1}^{T} L x_{1} .

(11)

Because the global objective function is an absolute convex function, the penalty function is also a strictly convex function.

Let each multi-agent have its own local objective function, and finally make its objective function optimized. Thus obtaining the optimal trajectory $x_{i, 1}^{*}$ for agent $i$ is defined as

(x_{1, 1}^{*}, \dots, x_{N, 1}^{*}) = \underset{(x_{1, 1}, \dots, x_{N, 1})}{\arg min} P (x_{1}) .

(12)

Remark 1. From (12), we find that the value function has two parts. The first part $\sum_{i = 1}^{N} f_{i} (x_{i, 1})$ is the global objective function. The second part $x_{1}^{T} L x_{1}$ is the penalty term, which is used to reach consensus for all agents.

Lemma 2. ³⁷ For any $x, y \in R^{n}$ , the following inequality relationship holds

x^{T} y \leq \frac{n^{l}}{l} ∥ x ∥^{l} + \frac{1}{m n^{m}} ∥ y ∥^{m}

(13)

where $l > 1$ , $m > 1$ , $n > 0$ , and $(l - 1) (m - 1) = 1$ .

Lemma 3. ³⁸ Let $V : R^{n} \to R$ be a function satisfying $V (0) = 0$ and $0 \leq V_{1} (∥ x (t) ∥) \leq V (x (t)) \leq V_{2} (∥ x (t) ∥)$ , for any initial state $x (t_{0}) \in Ω$ , where $V_{1} (\cdot)$ and $V_{2} (\cdot)$ are class $K$ functions, and $Ω$ is a compact set. Then, all the signals of system (1) are semi-globally uniformly ultimately bounded (SGUUB) if there exist two positive constants $C$ and $ζ$ such that

\overset{\cdot}{V} (x (t)) \leq - CV (x (t)) + ζ .

(14)

Control objectives: This paper aims to design an neural network controller $u_{i}$ , so that all the signals remain semi-global uniformly ultimately bounded in the closed-loop system and enable all agents $N$ to converge to the convex hull containing each target trajectory as well as its opposite trajectory different in sign.

Main results

Observer design

The state variables of the system (4) in this article are agnostic, so we design an observer to estimate the system variables of Agent $i$ , Define the observer as follows

\begin{matrix} \overset{\cdot}{\hat{X}} = A_{i} {\hat{X}}_{i, n} + K_{i} y_{i} + \sum_{l = 1}^{n} B_{i, l} [{\hat{g}}_{i, l} ({\hat{X}}_{i, l})] + B_{i, n} u_{i} (t) \\ {\hat{y}}_{i} = {\hat{X}}_{i, 1} \end{matrix}

(15)

where $A_{i} = [\begin{matrix} - k_{i, l} \\ I_{n - 1} \\ - k_{i, n} 0 & \dots & 0 \end{matrix}]$ , $K_{i} = [\begin{matrix} k_{i, 1} \\ ⋮ \\ k_{i, n} \end{matrix}]$ , $B_{i, l} = {[\begin{matrix} \underset{l}{\underset{︸}{0 \dots 1}} & \dots 0 \end{matrix}]}^{T}$ . ${\hat{X}}_{i, n} = {({\hat{x}}_{i, 1}, {\hat{x}}_{i, 2}, . . ., {\hat{x}}_{i, n})}^{T}$ are the estimated values of $X_{i, n} = {(x_{i, 1}, x_{i, 2}, . . ., x_{i, n})}^{T}$ .

Since $g_{i, l} (X_{i, l})$ in the system (5) is an unknown nonlinear function. we use the neural network method to approximate the unknown nonlinear function $g_{i, l} (X_{i, l})$ . From this, the following assumptions can be derived

Assumption 1. The unknown functions $g_{i, l} (X_{i, l})$ , $i = 1, \dots, n$ can be expressed as

g_{i, l} (X_{i, l} | θ_{i, l}) = θ_{i, l}^{T} φ_{i, l} (X_{i, l}), 1 \leq i \leq n

(16)

where $θ_{i, l}$ is the unknown constant vector, and $φ_{i, l} (X_{i, l})$ is Gaussian basis function vector.

By Assumption 1, we can obtain

{\hat{g}}_{i, l} ({\hat{X}}_{i, l} | θ_{i, l}) = θ_{i, l}^{T} φ_{i, l} ({\hat{X}}_{i, l}) .

(17)

The observer model (15) can be converted into the following model

\begin{matrix} \overset{\cdot}{\hat{X}} = A_{i} {\hat{X}}_{i, n} + K_{i} y_{i} + \sum_{l = 1}^{n} B_{i, l} [{\hat{g}}_{i, l} ({\hat{X}}_{i, l} | θ_{i, l})] + B_{i, n} u_{i} (t) \\ {\hat{y}}_{i} = {\hat{X}}_{i, 1} \end{matrix}

(18)

Let $e_{i} = X_{i, n} - {\hat{X}}_{i, n}$ be state observation errors of system (4). According to equations (5) and (18), we have

\begin{matrix} {\overset{\cdot}{e}}_{i} = A_{i} e_{i} + \sum_{l = 1}^{n} B_{i, l} [g_{i, l} ({\hat{X}}_{i, l}) - {\hat{g}}_{i, l} ({\hat{X}}_{i, l} | θ_{i, l}) + Δ g_{i, l}] \end{matrix}

(19)

where $Δ g_{i, l} = g_{i, l} (X_{i, l}) - g_{i, l} ({\hat{X}}_{i, l})$ .

The vectors of optimal parameters are defined as

θ_{i, l}^{*} = \arg min_{θ_{i, l} \in Ω_{i, l}} [{sup}_{{\hat{X}}_{i, l} \in U_{i, l}} | {\hat{g}}_{i, l} ({\hat{X}}_{i, l} | θ_{i, l}) - g_{i, l} ({\hat{X}}_{i, l}) |]

(20)

where $1 \leq l \leq n$ , $Ω_{i, l}$ , and $U_{i, l}$ are compact regions for $θ_{i, l}$ , $X_{i, l}$ , and ${\hat{X}}_{i, l}$ .

Define errors of the optimal approximation $ε_{i, l}$ and parameter estimation ${\tilde{θ}}_{i, l}$ as

\begin{matrix} ε_{i, l} = g_{i, l} ({\hat{X}}_{i, l}) - {\hat{g}}_{i, l} ({\hat{X}}_{i, l} | θ_{i, l}^{*}) \\ {\tilde{θ}}_{i, l} = θ_{i, l}^{*} - θ_{i, l}, l = 1, 2, . . ., n . \end{matrix}

(21)

Assumption 2. The optimal approximation errors remain bounded, there exists positive constants $ε_{i 0}$ , satisfying $| ε_{i, l} | \leq ε_{i 0}$ .

Remark 2. Neural network approximation has universal approximation³⁹ and assumption $2$ is often designed in the design process of neural network controller.⁴⁰

Assumption 3. The nonlinear function $g_{i, l}$ satisfies the Lipchitz property. There exists a set of known constants $γ_{i}$ , the following relationship holds

| g_{i, l} (X_{i, l}) - g_{i, l} ({\hat{X}}_{i, l}) | \leq γ_{i, l} ‖ X_{i, l} - {\hat{X}}_{i, l} ‖ .

(22)

By equations (19) and (21), we have

\begin{matrix} {\overset{\cdot}{e}}_{i} = A_{i} e_{i} + \sum_{l = 1}^{n} B_{i, l} [g_{i, l} ({\hat{X}}_{i, l}) - {\hat{g}}_{i, l} ({\hat{X}}_{i, l} | θ_{i, l}) + Δ g_{i, l}] \\ = A_{i} e_{i} + \sum_{l = 1}^{n} B_{i, l} [ε_{i, l} + Δ g_{i, l} + {\tilde{θ}}_{i, l}^{T} φ_{i, l} ({\hat{X}}_{i, l})] \\ = A_{i} e_{i} + Δ g_{i} + ε_{i} + \sum_{l = 1}^{n} B_{i, l} [{\tilde{θ}}_{i, l}^{T} φ_{i, l} ({\hat{X}}_{i, l})] \end{matrix}

(23)

Where $ε_{i} = {[ε_{i, 1}, . . ., ε_{i, n}]}^{T}$ , $Δ g_{i} = {[Δ g_{1}, . . ., Δ g_{n}]}^{T}$ .

Constructing the Lyapunov function as:

V_{0} = \sum_{i = 1}^{N} \frac{1}{2} e_{i}^{T} P_{i} e_{i} .

Then, we can obtain

\begin{matrix} {\overset{\cdot}{V}}_{0} \leq \sum_{i = 1}^{N} {\frac{1}{2} e_{i}^{T} (P_{i} A_{i}^{T} + A_{i} P_{i}) e_{i} + e_{i}^{T} P_{i} (ε_{i} + Δ g_{i}) \\ + \sum_{l = 1}^{n} e_{i}^{T} P_{i} B_{i, l} [{\tilde{θ}}_{i, l}^{T} φ_{i, l} ({\hat{X}}_{i, l})]} \\ \leq \sum_{i = 1}^{N} {- e_{i}^{T} Q_{i} e_{i} + e_{i}^{T} P_{i} (ε_{i} + Δ g_{i}) \\ + e_{i}^{T} P_{i} \sum_{l = 1}^{n} B_{i, l} {\tilde{θ}}_{i, l}^{T} φ_{i, l} ({\hat{X}}_{i, l})} . \end{matrix}

(24)

By Lemma 2 and Assumption 3, we obtain

\begin{matrix} e_{i}^{T} P_{i} (ε_{i} + Δ g_{i}) \\ \leq \frac{1}{2} ‖ e_{i} ‖^{2} + \frac{1}{2} ‖ P_{i} ε_{i} ‖^{2} + \frac{1}{2} ‖ e_{i} ‖^{2} \\ + \frac{1}{2} ‖ P_{i} ‖^{2} ‖ Δ g_{i} ‖^{2} \\ \leq ‖ e_{i} ‖^{2} + \frac{1}{2} ‖ P_{i} ε_{i} ‖^{2} + \frac{1}{2} ‖ P_{i} ‖^{2} \sum_{l = 1}^{n} {| Δ g_{i, l} |}^{2} \\ \leq ‖ e_{i} ‖^{2} + \frac{1}{2} ‖ e_{i} ‖^{2} ‖ P_{i} ‖^{2} \sum_{l = 1}^{n} γ_{i, l}^{2} + \frac{1}{2} ‖ P_{i} ε_{i} ‖^{2} \\ \leq ‖ e_{i} ‖^{2} (1 + \frac{1}{2} {‖ P_{i} ‖}^{2} \sum_{l = 1}^{n} {γ_{i, l}}^{2}) + \frac{1}{2} ‖ P_{i} ε_{i} ‖^{2} \end{matrix}

(25)

In a similar way, we have

\begin{matrix} e_{i}^{T} P_{i} \sum_{l = 1}^{n} B_{i, l} {\tilde{θ}}_{i, l}^{T} φ_{i, l} ({\hat{X}}_{i, l}) \\ \leq \frac{1}{2} e_{i}^{T} P_{i}^{T} P_{i} e_{i} + \frac{1}{2} \sum_{l = 1}^{n} {\tilde{θ}}_{i, l}^{T} φ_{i, l} ({\hat{X}}_{i, l}) φ_{i, l}^{T} ({\hat{X}}_{i, l}) {\tilde{θ}}_{i, l} \\ \leq \frac{1}{2} λ_{i, max}^{2} (P_{i}) {‖ e_{i} ‖}^{2} + \frac{1}{2} \sum_{l = 1}^{n} {\tilde{θ}}_{i, l}^{T} {\tilde{θ}}_{i, l} \end{matrix}

(26)

Where $λ_{i, \max} (P_{i})$ is the maximum eigenvalue of positive matrix $P_{i}$ . By equations (24)–(26), we obtain

{\overset{\cdot}{V}}_{0} \leq \sum_{i = 1}^{N} (- q_{i, 0} {‖ e_{i} ‖}^{2} + \frac{1}{2} {‖ P_{i} ε_{i} ‖}^{2} + \frac{1}{2} \sum_{l = 1}^{n} {\tilde{θ}}_{i, l}^{T} {\tilde{θ}}_{i, l})

(27)

Where $0 < φ_{i, l} (\cdot) φ_{i, l}^{T} (\cdot) \leq 1$ and $q_{i, 0} = λ_{i, min} (Q_{i}) - (1 + \frac{1}{2} {‖ P_{i} ‖}^{2} \sum_{l = 1}^{n} γ_{i, l}^{2} + \frac{1}{2} λ_{i, max}^{2} (P_{i}))$ .

Then, we can obtain

\begin{matrix} {\overset{\cdot}{V}}_{0} \leq - q_{0} {‖ e ‖}^{2} + \frac{1}{2} {‖ P ε ‖}^{2} + \sum_{i = 1}^{N} \sum_{l = 1}^{n} \frac{1}{2} {\tilde{θ}}_{i, l}^{T} {\tilde{θ}}_{i, l} \end{matrix}

(28)

Where $q_{0} = \sum_{i = 1}^{N} q_{i, 0}$ .

Controller design

For the MASs (4), design state observer (15), by designing a Neural network optimal backstepping controller (80), virtual control laws (48), (58), and (69), filter (29), together with the presented designs can ensure that all the signals remain semi-global uniformly ultimately bounded in the closed-loop system and enables all agents converge to the convex hull of the target trajectory.

Proof: In this section, we combine backstepping design, filter and Lyapunov method to design virtual control laws and control input.

This paper uses virtual controller $x_{i, l}^{*}$ as input and $v_{i, l}$ as output to construct the following filter

λ_{i, m} {\overset{\cdot}{v}}_{i, m} + v_{i, m} = x_{i, m}^{*}, v_{i, m} (0) = x_{i, m}^{*} (0)

(29)

Where, $m$ is the order of the multi-agent model and $2 \leq m \leq n$ . Define the error variable as follows:

\begin{matrix} s_{i, 1} = x_{i, 1} - x_{i, 1}^{*} \\ s_{i, m} = {\hat{x}}_{i, m} - v_{i, m} \\ w_{i, m} = v_{i, m} - x_{i, m}^{*} m = 2, \dots, n \end{matrix}

(30)

Where, $s_{i, m}$ is the error between multiple agents and the optimal trajectory, $v_{i, m}$ is the state variable of multi-agent system, which is also the output of the filter. $w_{i, m}$ is the error between $v_{i, m}$ and $x_{i, m}^{*}$ . ${\hat{x}}_{i, m}$ is the estimation of $x_{i, m}$ .

By the equations (29) and (30), we have

\begin{matrix} {\overset{\cdot}{w}}_{i, m} = {\overset{\cdot}{v}}_{i, m} - {\overset{\cdot}{x}}_{i, m}^{*} \\ = - \frac{v_{i, m} - x_{i, m}^{*}}{λ_{i, m}} - {\overset{\cdot}{x}}_{i, m}^{*} \\ = - \frac{w_{i, m}}{λ_{i, m}} + B_{i, m} \end{matrix}

(31)

Where $λ_{i, m}$ is adjustable parameter and $B_{i, m} = - {\overset{\cdot}{x}}_{i, m}^{*}$ . There exist constants $M_{i, m} > 0$ , $i = 1, \dots, N$ , such that $| B_{i, m} | \leq M_{i, m}$ holds.

Step 1. First, we need find the extreme point of the penalty function (11)

\frac{\partial P (x_{1})}{\partial x_{1}} = vec (\frac{\partial f_{i} (x_{i, 1} (t))}{\partial x_{i, 1}}) + L x_{1}

(32)

Where $vec (\frac{\partial f_{i} (x_{i, 1} (t))}{\partial x_{i, 1}})$ is a column vector. The point where the gradient is zero is the extreme point, because the function is absolutely convex, so the extreme point is the optimal solution of the distributed problem. So, we let

\frac{\partial P (x_{1}^{*})}{\partial x_{1}^{*}} = 0 .

By equations (11) and (32), we have

\frac{\partial f_{i} (x_{i, 1}^{*} (t))}{\partial x_{i, 1}^{*}} + \sum_{j \in N_{i}} a_{ij} (x_{i, 1}^{*} - x_{j, 1}^{*}) = 0 .

(33)

Substitute the equation (7) into (33) to get

\begin{matrix} 2 a_{i, 1} (x_{i, 1}^{*} - x_{d, 1}) + 2 a_{i, 2} (x_{i, 1}^{*} - x_{d, 2}) \\ + \sum_{j \in N_{i}} a_{ij} (x_{i, 1}^{*} - x_{j, 1}^{*}) = 0 \end{matrix}

(34)

Then according to (32) and (34), we have

\begin{matrix} \frac{\partial P (x_{1})}{\partial x_{i, 1}} = 2 a_{i, 1} (x_{i, 1} - x_{d, 1}) + 2 a_{i, 2} (x_{i, 1} - x_{d, 2}) \\ + \sum_{j \in N_{i}} a_{ij} (x_{i, 1} - x_{j, 1}) \\ = 2 a_{i, 1} (x_{i, 1} - x_{d, 1}) + 2 a_{i, 2} (x_{i, 1} - x_{d, 2}) \\ + \sum_{j \in N_{i}} a_{ij} (x_{i, 1} - x_{j, 1}) \\ - 2 a_{i, 1} (x_{i, 1}^{*} - x_{d, 1}) - 2 a_{i, 2} (x_{i, 1}^{*} - x_{d, 2}) \\ - \sum_{j \in N_{i}} a_{ij} (x_{i, 1}^{*} - x_{j, 1}^{*}) \\ = 2 a_{i, 1} s_{i, 1} + 2 a_{i, 2} s_{i, 1} + \sum_{j \in N_{i}} a_{ij} (s_{i, 1} - s_{j, 1}) \end{matrix}

(35)

Let $s_{1} = [s_{1, 1} \dots s_{N, 1}]^{T}$ . According to (35), we have

\frac{\partial P (x_{1})}{\partial x_{1}} = H s_{1}

Where $H = A + L$ , $A = diag {2 a_{i}}$ and $a_{i} = a_{i, 1} + a_{i, 2}$ .

Then, we construct the Lyapunov function

\begin{matrix} V_{1} = & V_{0} + \frac{1}{2} s_{1}^{T} H s_{1} + \sum_{i = 1}^{N} \frac{1}{σ_{i, 1}} {\tilde{θ}}_{i, 1}^{T} {\tilde{θ}}_{i, 1} \end{matrix}

(36)

Where $σ_{i, 1}$ is a adjustable parameter. According to (1), (18), and (30), we can obtain

\begin{matrix} {\overset{\cdot}{s}}_{i, 1} = {\hat{x}}_{i, 2} + θ_{i, 1}^{T} φ_{i, 1} + {\tilde{θ}}_{i, 1}^{T} φ_{i, 1} + Δ g_{i, 1} + ε_{i, 1} + e_{i, 2} . \end{matrix}

(37)

Take the derivative of $V_{1}$ and substitute the equation (37) into ${\overset{\cdot}{V}}_{1}$ , we have

\begin{matrix} {\overset{\cdot}{V}}_{1} = {\overset{\cdot}{V}}_{0} + s_{1}^{T} H {\overset{\cdot}{s}}_{1} + \sum_{i = 1}^{N} \frac{1}{σ_{i, 1}} {\tilde{θ}}_{i, 1}^{T} {\tilde{θ}}_{i, 1} \\ = {\overset{\cdot}{V}}_{0} + s_{1}^{T} H ({\hat{x}}_{2} + vec (θ_{i, 1}^{T} φ_{i, 1}) + vec ({\tilde{θ}}_{i, 1}^{T} φ_{i, 1}) \\ + Δ g_{1} + ε_{1} + e_{2}) + \sum_{i = 1}^{N} \frac{1}{σ_{i, 1}} {\tilde{θ}}_{i, 1}^{T} {\tilde{θ}}_{i, 1} \\ = {\overset{\cdot}{V}}_{0} + s_{1}^{T} H s_{2} + s_{1}^{T} H w_{2} \\ + s_{1}^{T} H (x_{2}^{*} + vec (θ_{i, 1}^{T} φ_{i, 1}) + vec ({\tilde{θ}}_{i, 1}^{T} φ_{i, 1})) \\ + s_{1}^{T} H Δ g_{1} + s_{1}^{T} H ε_{1} + s_{1}^{T} H e_{2} - \sum_{i = 1}^{N} \frac{1}{σ_{i, 1}} {\tilde{θ}}_{i, 1}^{T} {\overset{\cdot}{θ}}_{i, 1} \end{matrix}

(38)

where $s_{2} = [s_{1, 2} s_{2, 2} \dots s_{N, 2}]^{T}$ , $w_{2} = [w_{1, 2} w_{2, 2} \dots w_{N, 2}]^{T}$ , $x_{2}^{*} = [x_{1, 2}^{*} x_{2, 2}^{*} \dots x_{N, 2}^{*}]^{T}$ , $Δ g_{1} = [Δ g_{1, 1} Δ g_{2, 1} \dots Δ g_{N, 1}]^{T}$ , $ε_{1} = [ε_{1, 1} ε_{2, 1} \dots ε_{N, 1}]^{T}$ , $e_{2} = [e_{1, 2} e_{2, 2} \dots e_{N, 2}]^{T}$ , According to Lemma 2, the following inequalities hold

s_{1}^{T} H s_{2} \leq \frac{1}{2} s_{1}^{T} H H^{T} s_{1} + \frac{1}{2} s_{2}^{T} s_{2}

(39)

s_{1}^{T} H w_{2} \leq \frac{1}{2} s_{1}^{T} H H^{T} s_{1} + \frac{1}{2} w_{2}^{T} w_{2}

(40)

s_{1}^{T} H Δ g_{1} \leq \frac{1}{2} s_{1}^{T} H γ_{1} γ_{1}^{T} H^{T} s_{1} + \frac{1}{2} e_{1}^{T} e_{1}

(41)

s_{1}^{T} H ε_{1} \leq \frac{1}{2} s_{1}^{T} H H^{T} s_{1} + \frac{1}{2} ε_{1}^{T} ε_{1}

(42)

s_{1}^{T} H e_{2} \leq \frac{1}{2} s_{1}^{T} H H^{T} s_{1} + \frac{1}{2} e_{2}^{T} e_{2}

(43)

where $γ_{1} = diag [γ_{i, 1}]$ , $e_{1} = [e_{1, 1} e_{2, 1} \dots e_{N, 1}]^{T}$ . Substituting (39)−(43) into (38), we have

\begin{matrix} {\overset{\cdot}{V}}_{1} \leq {\overset{\cdot}{V}}_{0} + s_{1}^{T} H (x_{2}^{*} + vec (θ_{i, 1}^{T} φ_{i, 1}) + vec ({\tilde{θ}}_{i, 1}^{T} φ_{i, 1})) \\ + \frac{1}{2} s_{1}^{T} H H^{T} s_{1} + \frac{1}{2} w_{2}^{T} w_{2} \\ + \frac{1}{2} s_{1}^{T} H H^{T} s_{1} \\ + \frac{1}{2} s_{2}^{T} s_{2} + \frac{1}{2} s_{1}^{T} H γ_{1} γ_{1}^{T} H^{T} s_{1} + \frac{1}{2} e_{1}^{T} e_{1} \\ + \frac{1}{2} s_{1}^{T} H H^{T} s_{1} + \frac{1}{2} ε_{1}^{qT} ε_{1}^{q} \\ + \frac{1}{2} s_{1}^{T} H H^{T} s_{1} + \frac{1}{2} e_{2}^{T} e_{2} - \sum_{i = 1}^{N} \frac{1}{σ_{i, 1}} {\tilde{θ}}_{i, 1}^{T} {\overset{\cdot}{θ}}_{i, 1} . \end{matrix}

(44)

According to Lemma 1, we have

\begin{matrix} s_{1}^{T} H = s_{1}^{T} A + s_{1}^{T} L \\ = [2 a_{1} s_{1, 1} \dots 2 a_{n} s_{n, 1}] \\ + [\sum_{j \in N_{i}} a_{1 j} (s_{1, 1} - s_{j, 1}) \dots \sum_{j \in N_{i}} a_{nj} (s_{n, 1} - s_{j, 1})] \\ = [2 a_{1} s_{1, 1} + \sum_{j \in N_{i}} a_{1 j} (s_{1, 1} - s_{j, 1}) \\ \dots 2 a_{n} s_{n, 1} + \sum_{j \in N_{i}} a_{nj} (s_{n, 1} - s_{j, 1})] \end{matrix}

(45)

Then, we can obtain

\begin{matrix} s_{1}^{T} H H^{T} s_{1} = {(2 a_{1} s_{1, 1} + \sum_{j \in N_{i}} a_{1 j} (s_{1, 1} - s_{j, 1}))}^{2} \\ + \dots + \\ {(2 a_{N} s_{N, 1} + \sum_{j \in N_{i}} a_{Nj} (s_{N, 1} - s_{j, 1}))}^{2} \\ = \sum_{i = 1}^{N} [2 a_{i, 1} (x_{i, 1} - x_{d, 1}) \\ + 2 a_{i, 2} (x_{i, 1} - x_{d, 2}) \\ {+ \sum_{j \in N_{i}} a_{ij} (x_{i, 1} - x_{j, 1})]}^{2} \end{matrix}

(46)

and

\begin{matrix} s_{1}^{T} H γ_{1} γ_{1}^{T} H^{T} s_{1} = \sum_{i = 1}^{N} γ_{i, 1}^{2} [2 a_{i, 1} (x_{i, 1} - x_{d, 1}) \\ + 2 a_{i, 2} (x_{i, 1} - x_{d, 2}) \\ {+ \sum_{j \in N_{i}} a_{ij} (x_{i, 1} - x_{j, 1})]}^{2} . \end{matrix}

(47)

By equations (44), (46), and (47), the virtual controller $x_{i, 2}^{*}$ and adaptive law $θ_{i, 1}$ are designed as

\begin{matrix} x_{i, 2}^{*} = - c_{i, 1} [2 a_{i, 1} (x_{i, 1} - x_{d, 1}) + 2 a_{i, 2} (x_{i, 1} - x_{d, 2}) \\ + \sum_{j \in N_{i}} a_{ij} (x_{i, 1} - x_{j, 1})] - θ_{i, 1}^{T} φ_{i, 1} ({\hat{X}}_{i, 1}) \end{matrix}

(48)

\begin{matrix} {\overset{\cdot}{θ}}_{i, 1} = σ_{i, 1} φ_{i, 1} ({\hat{X}}_{i, 1}) [2 a_{i, 1} (x_{i, 1} - x_{d, 1}) + 2 a_{i, 2} (x_{i, 1} - x_{d, 2}) \\ + \sum_{j \in N_{i}} a_{ij} (x_{i, 1} - x_{j, 1})] - ρ_{i, 1} θ_{i, 1} \end{matrix}

(49)

Where $c_{i, 1} = 3 + \frac{γ_{i, 1}^{2}}{2}$ , $ρ_{i, 1}$ is the adjustable parameters. Substituting (48) and (49) into (44), after (27) we can obtain

\begin{matrix} {\overset{\cdot}{V}}_{1} \leq - q_{0} ∥ e ∥^{2} + \frac{1}{2} ∥ P ε ∥^{2} + \sum_{i = 1}^{N} \sum_{l = 1}^{n} \frac{1}{2} {\tilde{θ}}_{i, l}^{T} {\tilde{θ}}_{i, l} + \frac{1}{2} e_{2}^{T} e_{2} \\ + \frac{1}{2} e_{1}^{T} e_{1} + \frac{1}{2} ε_{1}^{T} ε_{1} + \sum_{i = 1}^{N} \frac{ρ_{i, 1}}{σ_{i, 1}} {\tilde{θ}}_{i, 1}^{T} θ_{i, 1} + \frac{1}{2} s_{2}^{T} s_{2} + \frac{1}{2} w_{2}^{T} w_{2} \\ - s_{1}^{T} H H^{T} s_{1} \\ \leq - q_{1} ∥ e ∥^{2} + η_{1} + \sum_{i = 1}^{N} \sum_{l = 1}^{n} \frac{1}{2} {\tilde{θ}}_{i, l}^{T} {\tilde{θ}}_{i, l} \\ + \sum_{i = 1}^{N} \frac{ρ_{i, 1}}{σ_{i, 1}} {\tilde{θ}}_{i, 1}^{T} θ_{i, 1} + \sum_{i = 1}^{N} \frac{1}{2} s_{i, 2}^{2} + \sum_{i = 1}^{N} \frac{1}{2} w_{i, 2}^{2} \\ - s_{1}^{T} H H^{T} s_{1} \end{matrix}

(50)

Where $q_{1} = q_{0} - N$ , $η_{1} = \frac{1}{2} ∥ P ε ∥^{2} + \frac{1}{2} ε_{1}^{T} ε_{1}$ .

Step 2. According to Theorem 1, design the error variable $s_{i, 2} = {\hat{x}}_{i, 2} - v_{i, 2}$ . By equations (15) and (16), we have

{\overset{\cdot}{s}}_{i, 2} = {\hat{x}}_{i, 3} + k_{i, 2} e_{i, 1} + ε_{i, 2} + Δ g_{i, 2} + θ_{i, 2}^{T} φ_{i, 2} + {\tilde{θ}}_{i, 2}^{T} φ_{i, 2} - {\overset{\cdot}{v}}_{i, 2}

(51)

Construct the Lyapunov function

\begin{matrix} V_{2} = V_{1} + \frac{1}{2} \sum_{i = 1}^{N} {s_{i, 2}^{2} + \frac{1}{σ_{i, 2}} {\tilde{θ}}_{i, 2}^{T} {\tilde{θ}}_{i, 2} + w_{i, 2}^{2}} . \end{matrix}

(52)

where $σ_{i, 2}$ is adjustable parameter. Substituting (51) into (52), we can obtain

\begin{matrix} {\overset{\cdot}{V}}_{2} = {\overset{\cdot}{V}}_{1} + \sum_{i = 1}^{N} [s_{i, 2} (s_{i, 3} + w_{i, 3} + x_{i, 3}^{*} + k_{i, 2} e_{i, 1} + ε_{i, 2} + Δ g_{i, 2} \\ + θ_{i, 2}^{T} φ_{i, 2} + {\tilde{θ}}_{i, 2}^{T} φ_{i, 2} - {\overset{\cdot}{v}}_{i, 2}) + \frac{1}{σ_{i, 2}} {\tilde{θ}}_{i, 2}^{T} {\overset{\cdot}{\tilde{θ}}}_{i, 2} + w_{i, 2} {\overset{\cdot}{w}}_{i, 2}] . \end{matrix}

(53)

According to Lemma 2, we obtain

s_{i, 2} k_{i, 2} e_{i, 1} \leq \frac{1}{2} s_{i, 2}^{2} + \frac{1}{2} k_{i, 2}^{2} ∥ e_{i, 1} ∥^{2}

(54)

s_{i, 2} (s_{i, 3} + w_{i, 3}) \leq s_{i, 2}^{2} + \frac{1}{2} s_{i, 3}^{2} + \frac{1}{2} w_{i, 3}^{2}

(55)

s_{i, 2} ε_{i, 2} + s_{i, 2} Δ g_{i, 2} \leq s_{i, 2}^{2} + \frac{1}{2} ∥ ε_{i, 2} ∥^{2} + \frac{1}{2} γ_{i, 2}^{2} ∥ e_{i, 2} ∥^{2}

(56)

Substituting (54)–(56) into (53) can be written as

\begin{matrix} {\overset{\cdot}{V}}_{2} \leq {\overset{\cdot}{V}}_{1} + \sum_{i = 1}^{N} [s_{i, 2} (x_{i, 3}^{*} + θ_{i, 2}^{T} φ_{i, 2} + {\tilde{θ}}_{i, 2}^{T} φ_{i, 2} - {\overset{\cdot}{v}}_{i, 2}) \\ + \frac{5}{2} s_{i, 2}^{2} + \frac{1}{2} (s_{i, 3}^{2} + w_{i, 3}^{2}) + \frac{1}{2} k_{i, 2}^{2} ∥ e_{i, 1} ∥^{2} + \frac{1}{2} ∥ ε_{i, 2} ∥^{2} \\ + \frac{1}{2} γ_{i, 2}^{2} ∥ e_{i, 2} ∥^{2} - \frac{1}{σ_{i, 2}} {\tilde{θ}}_{i, 2}^{T} {\overset{\cdot}{θ}}_{i, 2} + w_{i, 2} {\overset{\cdot}{w}}_{i, 2}] . \end{matrix}

(57)

According to Theorem 1, the virtual controller $x_{i, 3}^{*}$ and update laws $θ_{i, 2}$ are designed as

x_{i, 3}^{*} = - (c_{i, 2} + 3) s_{i, 2} - θ_{i, 2}^{T} φ_{i, 2} ({\hat{X}}_{i, 2}) + \frac{x_{i, 2}^{*} - v_{i, 2}}{λ_{i, 2}}

(58)

{\overset{\cdot}{θ}}_{i, 2} = σ_{i, 2} φ_{i, 2} ({\hat{X}}_{i, 2}) s_{i, 2} - ρ_{i, 2} θ_{i, 2} .

(59)

where $ρ_{i, 2}$ is adjustable parameter. Substituting equations (58), (59), (50), and (31) into (57), then we obtain

\begin{matrix} {\overset{\cdot}{V}}_{2} \leq - q_{1} ∥ e ∥^{2} + η_{1} - s_{1}^{T} H H^{T} s_{1} + \sum_{i = 1}^{N} \sum_{l = 1}^{n} \frac{1}{2} {\tilde{θ}}_{i, l}^{T} {\tilde{θ}}_{i, l} \\ + \sum_{i = 1}^{N} \frac{ρ_{i, 1}}{σ_{i, 1}} {\tilde{θ}}_{i, 1}^{T} θ_{i, 1} + \sum_{i = 1}^{N} \frac{1}{2} s_{i, 2}^{2} + \sum_{i = 1}^{N} \frac{1}{2} w_{i, 2}^{2} \\ + \sum_{i = 1}^{N} [s_{i, 2} (- (c_{i, 2} + 3) s_{i, 2} - θ_{i, 2}^{T} φ_{i, 2} ({\hat{X}}_{i, 2}) \\ + \frac{x_{i, 2}^{*} - v_{i, 2}}{λ_{i, 2}} + θ_{i, 2}^{T} φ_{i, 2} + {\tilde{θ}}_{i, 2}^{T} φ_{i, 2} \\ - {\overset{\cdot}{v}}_{i, 2}) + \frac{5}{2} s_{i, 2}^{2} + \frac{1}{2} (s_{i, 3}^{2} + w_{i, 3}^{2}) + \frac{1}{2} k_{i, 2}^{2} ∥ e_{i, 1} ∥^{2} \\ + \frac{1}{2} ∥ ε_{i, 2} ∥^{2} + \frac{1}{2} γ_{i, 2}^{2} ∥ e_{i, 2} ∥^{2} \\ - \frac{1}{σ_{i, 2}} {\tilde{θ}}_{i, 2}^{T} (σ_{i, 2} φ_{i, 2} ({\hat{X}}_{i, 2}) s_{i, 2} \\ - ρ_{i, 2} θ_{i, 2}) + w_{i, 2} (- \frac{w_{i, 2}}{λ_{i, 2}} + B_{i, 2})] . \end{matrix}

(60)

According to Lemma 2, we have $w_{i, 2} B_{i, 2} \leq \frac{1}{2} w_{i, 2}^{2} + \frac{1}{2} M_{i, 2}^{2}$ . Then we can obtain

\begin{matrix} {\overset{\cdot}{V}}_{2} \leq - q_{2} ∥ e ∥^{2} + η_{2} + \sum_{i = 1}^{N} \sum_{l = 1}^{n} \frac{1}{2} {\tilde{θ}}_{i, l}^{T} {\tilde{θ}}_{i, l} + \sum_{i = 1}^{N} \frac{ρ_{i, 1}}{σ_{i, 1}} {\tilde{θ}}_{i, 1}^{T} θ_{i, 1} \\ + \sum_{i = 1}^{N} \frac{ρ_{i, 2}}{σ_{i, 2}} {\tilde{θ}}_{i, 2}^{T} θ_{i, 2} - \sum_{i = 1}^{N} c_{i, 2} s_{i, 2}^{2} - \sum_{i = 1}^{N} (\frac{1}{λ_{i, 2}} - 1) w_{i, 2}^{2} \\ + \sum_{i = 1}^{N} [\frac{1}{2} M_{i, 2}^{2} + \frac{1}{2} (s_{i, 3}^{2} + w_{i, 3}^{2})] - s_{1}^{T} H H^{T} s_{1} \end{matrix}

(61)

where

\begin{matrix} q_{2} = q_{1} - \frac{1}{2} \sum_{i = 1}^{N} (k_{i, 2}^{2} + γ_{i, 2}^{2}) \\ η_{2} = η_{1} + \frac{1}{2} \sum_{i = 1}^{N} ∥ ε_{i, 2} ∥^{2} . \end{matrix}

Step m. According to Theorem 1, define the m-order error variable $s_{i, m} = {\hat{x}}_{i, m} - v_{i, m}$ . By equations (15) and (16), we have

\begin{matrix} {\overset{\cdot}{s}}_{i, m} = {\hat{x}}_{i, m + 1} + k_{i, m} e_{i, 1} + ε_{i, m} + Δ g_{i, m} + θ_{i, m}^{T} φ_{i, m} \\ + {\tilde{θ}}_{i, m}^{T} φ_{i, m} - {\overset{\cdot}{v}}_{i, m} \end{matrix}

(62)

Construct the Lyapunov function

\begin{matrix} V_{m} = V_{m - 1} + \frac{1}{2} \sum_{i = 1}^{N} {s_{i, m}^{2} + \frac{1}{σ_{i, m}} {\tilde{θ}}_{i, m}^{T} {\tilde{θ}}_{i, m} + w_{i, m}^{2}} . \end{matrix}

(63)

where $σ_{i, m}$ is adjustable parameter. Substituting (62) into (63), we can obtain

\begin{matrix} {\overset{\cdot}{V}}_{m} = {\overset{\cdot}{V}}_{m - 1} + \sum_{i = 1}^{N} [s_{i, m} (s_{i, m + 1} + w_{i, m + 1} + x_{i, m + 1}^{*} \\ + k_{i, m} e_{i, 1} + ε_{i, m} + Δ g_{i, m} + θ_{i, m}^{T} φ_{i, m} + {\tilde{θ}}_{i, m}^{T} φ_{i, m} \\ - {\overset{\cdot}{v}}_{i, m}) + \frac{1}{σ_{i, m}} {\tilde{θ}}_{i, m}^{T} {\tilde{θ}}_{i, m} + w_{i, m} {\overset{\cdot}{w}}_{i, m}] . \end{matrix}

(64)

According to Lemma 2, we obtain

s_{i, m} k_{i, m} e_{i, 1} \leq \frac{1}{2} s_{i, m}^{2} + \frac{1}{2} k_{i, m}^{2} ∥ e_{i, 1} ∥^{2}

(65)

s_{i, m} (s_{i, m + 1} + w_{i, m + 1}) \leq s_{i, m}^{2} + \frac{1}{2} s_{i, m + 1}^{2} + \frac{1}{2} w_{i, m + 1}^{2}

(66)

s_{i, m} ε_{i, m} + s_{i, m} Δ g_{i, m} \leq s_{i, m}^{2} + \frac{1}{2} ∥ ε_{i, m} ∥^{2} + \frac{1}{2} γ_{i, m}^{2} ∥ e_{i, m} ∥^{2}

(67)

Substituting (65)−(67) into (64) can be written as

\begin{matrix} {\overset{\cdot}{V}}_{m} \leq {\overset{\cdot}{V}}_{m - 1} + \sum_{i = 1}^{N} [s_{i, m} (x_{i, m + 1}^{*} + θ_{i, m}^{T} φ_{i, m} + {\tilde{θ}}_{i, m}^{T} φ_{i, m} \\ - {\overset{\cdot}{v}}_{i, m}) + \frac{5}{2} s_{i, m}^{2} + \frac{1}{2} (s_{i, m + 1}^{2} + w_{i, m + 1}^{2}) \\ + \frac{1}{2} k_{i, m}^{2} ∥ e_{i, 1} ∥^{2} + \frac{1}{2} ∥ ε_{i, m} ∥^{2} \\ + \frac{1}{2} γ_{i, m}^{2} ∥ e_{i, m} ∥^{2} - \frac{1}{σ_{i, m}} {\tilde{θ}}_{i, m}^{T} {\overset{\cdot}{θ}}_{i, m} + w_{i, m} {\overset{\cdot}{w}}_{i, m}] . \end{matrix}

(68)

According to Theorem 1, the m-order virtual controller $x_{i, m + 1}^{*}$ and update laws $θ_{i, m}$ are designed as

x_{i, m + 1}^{*} = - (c_{i, m} + 3) s_{i, m} - θ_{i, m}^{T} φ_{i, m} ({\hat{X}}_{i, m}) + \frac{x_{i, m}^{*} - v_{i, m}}{λ_{i, m}}

(69)

{\overset{\cdot}{θ}}_{i, m} = σ_{i, m} φ_{i, m} ({\hat{X}}_{i, m}) s_{i, m} - ρ_{i, m} θ_{i, m} .

(70)

where $ρ_{i, m}$ is adjustable parameter. Substituting equations (69), (70), and (31) into (57), then we obtain

\begin{matrix} {\overset{\cdot}{V}}_{m} \leq {\overset{\cdot}{V}}_{m - 1} + \sum_{i = 1}^{N} [s_{i, m} (- (c_{i, m} + 3) s_{i, m} - θ_{i, m}^{T} φ_{i, m} ({\hat{X}}_{i, m}) \\ + \frac{x_{i, m}^{*} - v_{i, m}}{λ_{i, m}} + θ_{i, m}^{T} φ_{i, m} + {\tilde{θ}}_{i, m}^{T} φ_{i, m} - {\overset{\cdot}{v}}_{i, m}) \\ + \frac{5}{2} s_{i, m}^{2} + \frac{1}{2} (s_{i, m + 1}^{2} + w_{i, m + 1}^{2}) + \frac{1}{2} k_{i, m}^{2} ∥ e_{i, 1} ∥^{2} \\ + \frac{1}{2} ∥ ε_{i, m}^{q} ∥^{2} + \frac{1}{2} γ_{i, m}^{q 2} ∥ e_{i, m} ∥^{2} \\ - \frac{1}{σ_{i, m}} {\tilde{θ}}_{i, m}^{T} (σ_{i, m} φ_{i, m} ({\hat{X}}_{i, m}) s_{i, m} - ρ_{i, m} θ_{i, m}) \\ + w_{i, m} (- \frac{w_{i, m}}{λ_{i, m}} + B_{i, m})] . \end{matrix}

(71)

According to Lemma 2, we have $w_{i, m} B_{i, m} \leq \frac{1}{2} w_{i, m}^{2} + \frac{1}{2} M_{i, m}^{2}$ . Then we can obtain Same as the equation (61), we can obtain

\begin{matrix} {\overset{\cdot}{V}}_{m} \leq - q_{m} ∥ e ∥^{2} + η_{m} + \sum_{i = 1}^{N} \sum_{l = 1}^{n} \frac{1}{2} {\tilde{θ}}_{i, l}^{T} {\tilde{θ}}_{i, l} \\ + \sum_{i = 1}^{N} [\sum_{l = 1}^{m} \frac{ρ_{i, l}}{σ_{i, l}} {\tilde{θ}}_{i, l}^{T} θ_{i, l} - \sum_{l = 2}^{m} c_{i, l} s_{i, l}^{2} \\ - \sum_{l = 2}^{m} (\frac{1}{λ_{i, l}} - 1) w_{i, l}^{2} + \frac{1}{2} \sum_{l = 2}^{m} M_{i, m}^{2} \\ + \frac{1}{2} (s_{i, m + 1}^{2} + w_{i, m + 1}^{2})] - s_{1}^{T} H H^{T} s_{1} \end{matrix}

(72)

where

\begin{matrix} q_{m} = q_{m - 1} - \frac{1}{2} \sum_{i = 1}^{N} (k_{i, m}^{2} + γ_{i, m}^{2}) \\ η_{m} = η_{m - 1} + \frac{1}{2} \sum_{i = 1}^{N} ∥ ε_{i, m} ∥^{2} . \end{matrix}

Step n. According to theorem 1, design the n-th error variable as follows

\begin{matrix} s_{i, n} = {\hat{x}}_{i, n} - v_{i, n} \end{matrix}

(73)

w_{i, n} = v_{i, n} - x_{i, n}^{*}

(74)

Then, we have

\begin{matrix} {\overset{\cdot}{s}}_{i, n} = u_{i} + k_{i, n} e_{i, 1} + ε_{i, n} \\ + Δ g_{i, n} + θ_{i, n}^{T} φ_{i, n} + {\tilde{θ}}_{i, n}^{T} φ_{i, n} - {\overset{\cdot}{v}}_{i, n} \end{matrix}

(75)

The Lyapunov function is constructed as

\begin{matrix} V_{n} = V_{n - 1} + \frac{1}{2} \sum_{i = 1}^{N} {s_{i, n}^{2} + \frac{1}{σ_{i, n}} {\tilde{θ}}_{i, n}^{T} {\tilde{θ}}_{i, n} + w_{i, n}^{2}} \end{matrix}

(76)

where $σ_{i, n}$ is designed parameter.

Combining (73) and (76), we can obtain

\begin{matrix} {\overset{\cdot}{V}}_{n} = {\overset{\cdot}{V}}_{n - 1} + \sum_{i = 1}^{N} [s_{i, n} (u_{i} + k_{i, m} e_{i, 1} \\ + θ_{i, n}^{T} φ_{i, n} + {\tilde{θ}}_{i, n}^{T} φ_{i, n} + ε_{i, n} + Δ g_{i, n} - {\overset{\cdot}{v}}_{i, n}) \\ + \frac{1}{σ_{i, n}} {\tilde{θ}}_{i, n}^{T} {\tilde{θ}}_{i, n} + w_{i, n} {\overset{\cdot}{w}}_{i, n}] \end{matrix}

(77)

According to Lemma 2, the following inequalities hold

s_{i, n} k_{i, n} e_{i, 1} \leq \frac{1}{2} s_{i, n}^{2} + \frac{1}{2} k_{i, n}^{2} ∥ e_{i, 1} ∥^{2}

(78)

s_{i, n} ε_{i, n} + s_{i, n} Δ g_{i, n} \leq s_{i, n}^{2} + \frac{1}{2} ∥ ε_{i, n} ∥^{2} + \frac{1}{2} γ_{i, n}^{2} ∥ e_{i, n} ∥^{2}

(79)

Design the multi-agent system control law $u_{i}$ and update laws $θ_{i, n}$ as follow

u_{i} = - c_{i, n} s_{i, n} - 2 s_{i, n} - θ_{i, n}^{T} φ_{i, n} ({\hat{X}}_{i, n}) + \frac{x_{i, n}^{*} - v_{i, n}}{λ_{i, n}}

(80)

{\overset{\cdot}{θ}}_{i, n} = σ_{i, n} φ_{i, n} ({\hat{X}}_{i, n}) s_{i, n} - ρ_{i, n} θ_{i, n}

(81)

where $ρ_{i, n}$ is designed parameter. substituting equations (78), (79), (80), and (81) into (77), then we can obtain

\begin{array}{l} {\dot{V}}_{n} \leq - q_{n - 1} ∥ e ∥^{2} + η_{n - 1} + \sum_{i = 1}^{N} \sum_{l = 1}^{n} \frac{1}{2} {\tilde{θ}}_{i, l}^{T} {\tilde{θ}}_{i, l} \\ + \sum_{i = 1}^{N} [\sum_{l = 1}^{n - 1} \frac{ρ_{i, l}}{σ_{i, l}} {\tilde{θ}}_{i, l}^{T} θ_{i, l} - \sum_{l = 2}^{n - 1} c_{i, l} s_{i, l}^{2} \\ + \sum_{l = 2}^{n - 1} (\frac{1}{λ_{i, l}} - 1) w_{i, l}^{2} + \frac{1}{2} \sum_{l = 2}^{n - 1} M_{i, l}^{2} + \frac{1}{2} (s_{i, n}^{2} + w_{i, n}^{2})] \\ + \sum_{i = 1}^{N} [s_{i, n} (- c_{i, n} s_{i, n} - 2 s_{i, n} - θ_{i, n}^{T} φ_{i, n} ({\hat{X}}_{i, n}) \\ + \frac{x_{i, n}^{*} - v_{i, n}}{λ_{i, n}} + θ_{i, n}^{T} φ_{i, n} + {\tilde{θ}}_{i, n}^{T} φ_{i, n} - {\dot{v}}_{i, n}) \\ + \frac{3}{2} s_{i, n}^{2} + \frac{1}{2} k_{i, n}^{2} ∥ e_{i, 1} ∥^{2} + \frac{1}{2} ∥ ε_{i, n} ∥^{2} \\ + \frac{1}{2} γ_{i, n}^{2} ∥ e_{i, n} ∥^{2} - \frac{1}{σ_{i, n}} {\tilde{θ}}_{i, n}^{T} (σ_{i, n} φ_{i, n} ({\hat{X}}_{i, n}) s_{i, n} \\ - ρ_{i, n} θ_{i, n}) + w_{i, n} (- \frac{w_{i, n}}{λ_{i, n}} + B_{i, n})] - s_{1}^{T} H H^{T} s_{1} \end{array}

(82)

According to Lemma 2, we have $w_{i, n} B_{i, n} \leq \frac{1}{2} w_{i, n}^{2} + \frac{1}{2} M_{i, n}^{2}$ . Then we can obtain Same as the equation (72), we can obtain

\begin{matrix} {\overset{\cdot}{V}}_{n} \leq - q_{n} ∥ e ∥^{2} + η_{n} + \sum_{i = 1}^{N} \sum_{l = 1}^{n} \frac{1}{2} {\tilde{θ}}_{i, l}^{T} {\tilde{θ}}_{i, l} \\ + \sum_{i = 1}^{N} [\sum_{l = 1}^{n} \frac{ρ_{i, l}}{σ_{i, l}} {\tilde{θ}}_{i, l}^{T} θ_{i, l} - \sum_{l = 2}^{n} c_{i, l} s_{i, l}^{2} \\ - \sum_{l = 2}^{n} (\frac{1}{λ_{i, l}} - 1) w_{i, l}^{2} + \frac{1}{2} \sum_{l = 2}^{n} M_{i, l}^{2}] \\ - s_{1}^{T} H H^{T} s_{1} \end{matrix}

(83)

where

\begin{array}{l} q_{n} = q_{n - 1} - \frac{1}{2} \sum_{i = 1}^{N} (k_{i, n}^{2} + γ_{i, n}^{2}) \\ η_{n} = η_{n - 1} + \frac{1}{2} \sum_{i = 1}^{N} ∥ ε_{i, n} ∥^{2} . \end{array}

According to Lemma 2, we obtain

{\tilde{θ}}_{i, l}^{T} θ_{i, l} \leq - \frac{1}{2} {\tilde{θ}}_{i, l}^{T} {\tilde{θ}}_{i, l} + \frac{1}{2} θ_{i, l}^{* T} θ_{i, l}^{*} .

(84)

Then, we can obtain

\begin{matrix} {\overset{\cdot}{V}}_{n} \leq - q_{n} ∥ e ∥^{2} + η_{n} + \sum_{i = 1}^{N} \sum_{l = 1}^{n} \frac{1}{2} {\tilde{θ}}_{i, l}^{T} {\tilde{θ}}_{i, l} \\ + \sum_{i = 1}^{N} [- \sum_{l = 1}^{n} \frac{ρ_{i, l}}{2 σ_{i, l}} {\tilde{θ}}_{i, l}^{T} {\tilde{θ}}_{i, l} + \sum_{l = 1}^{n} \frac{ρ_{i, l}}{2 σ_{i, l}} θ_{i, l}^{* T} θ_{i, l}^{*} \\ - \sum_{l = 2}^{n} c_{i, l} s_{i, l}^{2} - \sum_{l = 2}^{n} (\frac{1}{λ_{i, l}} - 1) w_{i, l}^{2} + \frac{1}{2} \sum_{l = 2}^{n} M_{i, l}^{2}] \\ - s_{1}^{T} H H^{T} s_{1} \end{matrix}

(85)

Define

\begin{matrix} ζ & = η_{n} + \sum_{i = 1}^{N} \sum_{l = 1}^{n} \frac{ρ_{i, l}}{2 σ_{i, l}} θ_{i, l}^{* T} θ_{i, l}^{*} + \frac{1}{2} \sum_{i = 1}^{N} \sum_{l = 2}^{n} M_{i, l}^{2} . \end{matrix}

(86)

Then, equation (85) can be written as

\begin{matrix} {\overset{\cdot}{V}}_{n} \leq - q_{n} ∥ e ∥^{2} - s_{1}^{T} H H^{T} s_{1} + \sum_{i = 1}^{N} [- \sum_{l = 2}^{n} c_{i, l} s_{i, l}^{2} \\ - \sum_{l = 1}^{n} (\frac{ρ_{i, l}}{2 σ_{i, l}} - \frac{1}{2}) {\tilde{θ}}_{i, l}^{T} {\tilde{θ}}_{i, l} - \sum_{l = 2}^{n} (\frac{1}{λ_{i, l}} - 1) w_{i, l}^{2}] + ζ \end{matrix}

(87)

where $c_{i, l} > 0$ , $(\frac{ρ_{i, l}}{2 σ_{i, l}} - \frac{1}{2}) > 0$ , $(\frac{1}{λ_{i, l}} - 1) > 0, (l = 2, \dots, n)$ .

Define

\begin{matrix} C = \min {2 \frac{q_{n}}{λ_{\min} (P)}, 2 c_{i, l}, 2 (\frac{ρ_{i, l}}{2 σ_{i, l}} - \frac{1}{2}), 2 λ_{max} (H), \\ 2 (\frac{1}{λ_{i, l}} - 1)} . \end{matrix}

(88)

Then, equation (87) becomes

\overset{\cdot}{V} (x (t)) \leq - CV (x (t)) + ζ .

(89)

According to Lemma 3, we know that the output variables of each multi-agent remain SGUUB in the entire multi-agent nonlinear closed-loop system. And ensure that the sum of the local objective functions of each multi-agent is minimal, other words, each multi-agent converges to the optimal containing position. The margin of error is shown below.

By solving the inequality (89), we have

0 \leq V (t) \leq (V (0) - \frac{ζ}{C}) e^{- Ct} + \frac{ζ}{C}

(90)

Substituting (36) into (90) yields, we have

| \frac{\partial P (x_{1})}{\partial x_{1}} | \leq \sqrt{2 (V (0) - \frac{ζ}{C}) e^{- Ct} + \frac{2 ζ}{C}}, \forall t > 0

(91)

As can be seen from (91), when time approaches infinity, the error satisfies $| \frac{\partial P (x_{1})}{\partial x_{1}} | \leq \sqrt{\frac{2 ζ}{C}}$ . So $\frac{\partial P (x_{1})}{\partial x_{1}}$ is uniformly ultimately bounded. Also, it can be concluded that the rest of signals such as $s_{i, 2}$ , $s_{i, n}$ in the closed-loop system are uniformly ultimately bounded.

Simulations

In this section, we will use simulation to verify the control effect of the control method. The system model used in this section is as follows,⁴¹

{\begin{matrix} {\overset{\cdot}{x}}_{i, 1} = x_{i, 2} + g_{i, 1} (X_{i, 1}) \\ {\overset{\cdot}{x}}_{i, 2} = u_{i} + g_{i, 2} (X_{i, 2}) \\ y_{i} = x_{i, 1} \end{matrix}

(92)

where $i = 1, 2, 3, 4, 5$ and the initial value selection for each multi-agent is as follows $x_{1} (0) = [0.1, 0.1]$ , $x_{2} (0) = [0.15, 0.15]$ , $x_{3} (0) = [0.2, 0.2]$ , $x_{4} (0) = [0.25, 0.25]$ , $x_{5} (0) = [- 0.25, - 0.25]$ . Define $x_{d 1} = \sin (t)$ as the upper bound of the bipartite containment trajectory. $x_{d 2} = 0.5 \sin (t)$ as the lower bound of the bipartite containment trajectory. Because there are negative adjacencies, Define $x_{d 3} = - \sin (t)$ and $x_{d 4} = - 0.5 \sin (t)$ as the negative bipartite containment trajectory. The unknown functions in system (92) are

\begin{matrix} g_{1, 1} = g_{2, 1} = g_{3, 1} = g_{4, 1} = g_{5, 1} = 0 \\ g_{1, 2} = 0.2 x_{1, 1} - 0.1 x_{1, 1}^{3} - 0.1 x_{1, 2} \\ g_{2, 2} = 0.2 x_{2, 1} - 0.1 x_{2, 1}^{3} + 0.1 {(x_{2, 1}^{2} + x_{2, 2}^{2})}^{\frac{1}{2}} - 0.1 x_{2, 2} \\ g_{3, 2} = 0.2 x_{3, 1} - 0.1 x_{3, 1}^{3} + 0.2 \sin (t) {(x_{3, 1}^{2} + 2 x_{3, 2}^{2})}^{\frac{1}{2}} \\ - 0.1 x_{3, 2} \\ g_{4, 2} = 0.2 x_{4, 1} - 0.1 x_{4, 1}^{3} + 0.2 \sin (t) {(2 x_{4, 1}^{2} + 2 x_{4, 2}^{2})}^{\frac{1}{2}} \\ - 0.1 x_{4, 2} \\ g_{5, 2} = 0.2 x_{5, 1} - 0.1 x_{5, 1}^{3} + 0.2 \sin (t) {(x_{5, 1}^{2} + x_{5, 2}^{2})}^{\frac{1}{2}} \\ - 0.1 x_{5, 2} \end{matrix}

We assumption multi-agent system to exchange information in the manner shown in Figure 1. The local objective function of multi-agent system is as follows

\begin{matrix} f_{1} (x_{1, 1}) = 0.2 x_{1, 1}^{2} - 0.2 x_{1, 1} x_{d 1} - 0.1 x_{1, 1} x_{d 2} \\ + 0.1 x_{d 1}^{2} + 0.1 x_{d 2}^{2} + 1 \\ f_{2} (x_{2, 1}) = 0.4 x_{2, 1}^{2} - 0.4 x_{2, 1} x_{d 1} - 0.2 x_{2, 1} x_{d 2} \\ + 0.2 x_{d 1}^{2} + 0.2 x_{d 2}^{2} + 2 \\ f_{3} (x_{3, 1}) = 0.6 x_{3, 1}^{2} - 0.6 x_{3, 1} x_{d 1} - 0.3 x_{3, 1} x_{d 2} \\ + 0.3 x_{d 1}^{2} + 0.3 x_{d 2}^{2} + 3 \\ f_{1} (x_{4, 1}) = 0.8 x_{4, 1}^{2} - 0.8 x_{4, 1} x_{d 1} - 0.4 x_{4, 1} x_{d 2} \\ + 0.4 x_{d 1}^{2} + 0.4 x_{d 2}^{2} + 4 \\ f_{1} (x_{5, 1}) = x_{5, 1}^{2} - x_{5, 1} x_{d 3} - 0.5 x_{5, 1} x_{d 4} \\ + 0.5 x_{d 3}^{2} + 0.5 x_{d 4}^{2} + 5 \end{matrix}

Figure 1.

Undirected topology among MASs.

The penalty function is shown below

P (x_{1}) = \sum_{i = 1}^{5} f_{i} (x_{i, 1}) + x_{1}^{T} L x_{1}

(93)

The condition of its optimal solution is

\frac{\partial P (x_{1}^{*})}{\partial x_{1}^{*}} = 0

(94)

where $x_{1}^{*} = {[x_{1, 1}^{*}, x_{2, 1}^{*}, \dots, x_{5, 1}^{*}]}^{T}$ . In this paper, the control input, the parameters update laws and the virtual control law are established using the state variables obtained from the observer. The parameters of the observer are selected as $k_{1, 1} = k_{2, 1} = k_{3, 1} = k_{4, 1} = k_{5, 1} = 100$ , $k_{1, 2} = k_{2, 2} = k_{3, 2} = k_{4, 2} = k_{5, 2} = 500$ . The initial states of the observer are ${\hat{x}}_{1} = [0.1, 0.1]$ , ${\hat{x}}_{2} = [0.15, 0.15]$ , ${\hat{x}}_{3} = [0.2, 0.2]$ , ${\hat{x}}_{4} = [0.25, 0.25]$ , ${\hat{x}}_{5} = [- 0.25, - 0.25]$ .

According to Theorem 1 and equations (48), (49), (80), and (81), the design of the virtual control law, the adaptive weight update law and the control input are as follow

\begin{matrix} x_{i, 2}^{*} = - c_{i, 1} [2 a_{i 1} (x_{i, 1} - x_{d 1}) + 2 a_{i 2} (x_{i, 1} - x_{d 2}) \\ + \sum_{j \in N_{i}} a_{ij} (x_{i, 1} - x_{j, 1})] - θ_{i, 1}^{T} φ_{i, 1} ({\hat{X}}_{i, 1}) \end{matrix}

(95)

\begin{matrix} {\overset{\cdot}{θ}}_{i, 1} = σ_{i, 1} φ_{i, 1} ({\hat{X}}_{i, 1}) [2 a_{i 1} (x_{i, 1} - x_{d 1}) + 2 a_{i 2} (x_{i, 1} - x_{d 2}) \\ + \sum_{j \in N_{i}} a_{ij} (x_{i, 1} - x_{j, 1})] - ρ_{i, 1} θ_{i, 1} ({\hat{X}}_{i, 1}) \end{matrix}

(96)

u_{i} = - c_{i, 2} s_{i, 2} - 2 s_{i, 2} - θ_{i, 2}^{T} φ_{i, 2} ({\hat{X}}_{i, 2}) + \frac{x_{i, 2}^{*} - v_{i, 2}}{λ_{i, 2}}

(97)

{\overset{\cdot}{θ}}_{i, 2} = σ_{i, 2} φ_{i, 2} ({\hat{X}}_{i, 2}) s_{i, 2} - ρ_{i, 2} θ_{i, 2}

(98)

The necessary parameters in equations (95), (96), (97), and (98) are selected as $c_{i, 1} = 30, c_{i, 2} = 40, σ_{i, 1} = σ_{i, 2} = 10, ρ_{i, 1} = ρ_{i, 2} = 21, λ_{i, 2} = 0.04$ .

In the simulation, Figures 2 –8 are the results of simulation. Figure 2 shows the trajectories of $x_{d 1}, x_{d 2}, x_{d 3}, x_{d 4}$ and $x_{i, 1}$ . It can be seen from the Figure 2 that all the agents can converge to the convex hull containing each trajectory. Figure 3 is a comparison control effect diagram designed by the method in the research paper of Yuan and Chen.⁴² As can be seen from Figure 3, the control method in the reference does not consider the optimization problem, resulting in that the controller trajectory in the convex hull cannot guarantee the minimum sum of the squares of the distance to the boundary. Figure 4 shows the error between the observer output and the system state, we can find that the approximation effect of the observer is very well. Figure 5 shows the trajectories of $u_{i, 1}$ . The size of the initial state is proportional to the control input. Figure 6 shows the trajectories of the error $s_{i, 1}$ , the tracking error can converge very quickly to near zero, it shows excellent tracking performance. And Figure 7 compares the penalty function with the optimal solution, the error converges very rapidly to near zero. Figure 8 shows that the error value estimated by the neural network is about $0.2$ . From simulation results, the method presented in this paper has good control performance for multi-agent systems.

Figure 2.

The trajectories of $x_{d}$ and $x_{i, 1} (i = 1 \dots 5)$ .

Figure 3.

Bipartite containment control without considering distributed optimization.

Figure 4.

The values of $x_{i, 1} (i = 1 \dots 5)$ estimation errors.

Figure 5.

The values of $u_{i, 1} (i = 1 \dots 5)$ .

Figure 6.

The value of the error between the system output and the optimal trajectory $s_{i, 1} (i = 1 \dots 5)$ .

Figure 7.

The error of penalty function.

Figure 8.

Approximation error of nonlinear function $g_{i 1} (i = 1, 2, \dots, 5)$ .

Conclusions

In this paper, the optimal bipartite containment problem for multi-agent systems with unknown nonlinear functions is studied. the penalty function is constructed by combining the bipartite containment definition. Moreover, We define a local objective function for each agent to ensure agents can track the target accurately. We use DSC technology to construct an adaptive inversion controller to avoid “explosion of complexity.” We construct Lyapunov functions to guarantee the stability of systems. The result of simulation show that the control method can control agents to converge to the optimal solution quickly under the condition of obtaining the optimal solution of the optimization problem, so that agents can satisfy the optimal solution and achieve the bipartite containment objective. On the basis of this paper, distributed optimization problems will continue to be studied in the future.

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) received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Lihui Hao

Jiaxin Yuan

References

Uwano

. A cooperative learning method for multi-agent system with different input resolutions. In: 2021 4th International symposium on agents, multi-agent systems and robotics (ISAMSR), 2021, pp.84–90. New York, NY: IEEE.

Abdulghafor

Turaev

Zeki

, et al. The convergence consensus of multi-agent systems controlled via doubly stochastic quadratic operators. In: 2015 International symposium on agents, multi-agent systems and robotics (ISAMSR), 2015, pp.59–64. New York, NY: IEEE.

Yang

Tong

. Fuzzy adaptive distributed event-triggered consensus control of uncertain nonlinear multiagent systems. IEEE Trans Syst Man Cybern Syst 2019; 49(9): 1777–1786.

Che

Deng

, et al. Observer-based event-triggered containment control for mass under dos attacks. IEEE Trans Cybern 2022; 52(12): 13156–13167.

Hua

You

, et al. Leader-following consensus control for uncertain feedforward stochastic nonlinear multiagent systems. IEEE Trans Neural Netw Learn Syst 2023; 34: 1049–1057.

Zhao

Wang

Zhang

, et al. Neural network based boundary control of a vibrating string system with input deadzone. Neurocomputing 2018; 275: 1021–1027.

Olfati-Saber

. Flocking for multi-agent dynamic systems: Algorithms and theory. IEEE Trans Automat Contr 2006; 51(3): 401–420.

Guo

Zavlanos

Dimarogonas

. Controlling the relative agent motion in multi-agent formation stabilization. IEEE Trans Automat Contr 2014; 59(3): 820–826.

Chen

Lewis

Xie

, et al. Resilient adaptive and h_∞ controls of multi-agent systems under sensor and actuator faults. Automatica 2019; 102: 19–26.

10.

Lindemann

Dimarogonas

. Decentralized control barrier functions for coupled multi-agent systems under signal temporal logic tasks. In 2019 18th European Control Conference (ECC), 2019, pp.89–94. New York, NY: IEEE.

11.

Ribeiro

Silvestre

. Decentralized control for multi-agent missions based on flocking rules. In: CONTROLO 2020: Proceedings of the 14th APCA international conference on automatic control and soft computing, Bragança, Portugal, 1–3 July 2020, pp. 445–454. Springer.

12.

Ren

Nie

. Finite-time positiveness and distributed control of lipschitz nonlinear multi-agent systems. J Franklin Inst 2019; 356(15): 8080–8092.

13.

Zhao

, et al. Observer-based adaptive sampled-data event-triggered distributed control for multi-agent systems. IEEE Trans Circuits Syst II Express Briefs 2020; 67(1): 97–101.

14.

Wang

. Distributed control of higher-order nonlinear multi-agent systems with unknown non-identical control directions under general directed graphs. Automatica 2019; 110: 108559.

15.

Zou

Shi

Xiang

, et al. Consensus tracking control of switched stochastic nonlinear multiagent systems via event-triggered strategy. IEEE Trans Neural Netw Learn Syst 2020; 31(3): 1036–1045.

16.

Zou

Xiang

Ahn

. Mean square leader–following consensus of second-order nonlinear multiagent systems with noises and unmodeled dynamics. IEEE Trans Syst Man Cybern Syst 2019; 49(12): 2478–2486.

17.

Zhang

Liu

Hua

. Distributed bipartite containment control of high-order nonlinear multi-agent systems with time-varying powers. IEEE Trans Circuits Syst I Regul Pap 2023; 70: 1371–1380.

18.

Pan

. Disturbance observer-based fuzzy adaptive containment control of nonlinear multi-agent systems with input quantization. Int J Fuzzy Syst 2022; 24(1): 574–586.

19.

Chen

, et al. Observer-based fixed-time adaptive fuzzy bipartite containment control for multiagent systems with unknown hysteresis. IEEE Trans Fuzzy Syst 2022; 30(5): 1302–1312.

20.

Kang

Guo

Yang

. Distributed optimization of high-order nonlinear systems: Saving computation and communication via prefiltering. IEEE Trans Circuits Syst II Express Briefs 2022; 69(3): 1144–1148.

21.

Guo

Kang

, et al. Distributed model reference adaptive optimization of disturbed multiagent systems with intermittent communications. IEEE Trans Cybern 2022; 52(6): 5464–5473.

22.

Guo

Kang

. Distributed optimization of multiagent systems against unmatched disturbances: A hierarchical integral control framework. IEEE Trans Syst Man Cybern Syst 2022; 52(6): 3556–3567.

23.

Jiang

, et al. Distributed fixed-time optimization for multi-agent systems over a directed network. Nonlinear Dyn 2021; 103: 775–789.

24.

Wang

, et al. Distributed convex optimization for nonlinear multi-agent systems disturbed by a second-order stationary process over a digraph. Sci China Inf Sci 2022; 65(3): 132201.

25.

Chen

Yuan

. Command-filtered adaptive containment control of fractional-order multi-agent systems via event-triggered mechanism. Trans Inst Meas Contr 2023; 45: 1646–1660.

26.

Yang

Zhao

Yuan

, et al. Distributed optimization for fractional-order multi-agent systems based on adaptive backstepping dynamic surface control technology. Fractal fract 2022; 6(11): 642.

27.

Chen

Liu

. Finite-time fuzzy adaptive consensus for heterogeneous nonlinear multi-agent systems. IEEE Trans Netw Sci Eng 2020; 7(4): 3057–3066.

28.

Zhang

Cui

. Impulsive consensus of nonlinear fuzzy multi-agent systems under dos attack. Nonlinear Anal Hybrid Syst 2022; 44: 101155.

29.

Wang

Dong

. Event-triggered adaptive consensus for fuzzy output-constrained multi-agent systems with observers. J Franklin Inst 2020; 357(1): 82–105.

30.

Guo

Liang

, et al. Command-filter-based fixed-time bipartite containment control for a class of stochastic multiagent systems. IEEE Trans Syst Man Cybern Syst 2022; 52(6): 3519–3529.

31.

Park

Hua

, et al. Distributed adaptive output feedback containment control for time-delay nonlinear multiagent systems. Automatica 2021; 127: 109545.

32.

Guo

Zhang

. Lyapunov redesign-based optimal consensus control for multi-agent systems with uncertain dynamics. IEEE Trans Circuits Syst II Express Briefs 2022; 69(6): 2902–2906.

33.

Liu

Zhan

Han

, et al. Distributed adaptive finite-time bipartite containment control of linear multi-agent systems. IEEE Trans Circuits Syst II Express Briefs 2022; 69(11): 4354–4358.

34.

Liu

Zhan

, et al. Distributed adaptive bipartite containment control of linear multi-agent systems with structurally balanced graph. Int J Control Autom Syst 2022; 20(11): 3476–3486.

35.

Duan

. Cooperative control of multi-agent systems: a consensus region approach. Boca Raton, FL: CRC Press, 2017.

36.

Wang

. Finite-time distributed approximate optimization algorithms of higher order multiagent systems via penalty-function-based method. IEEE Trans Syst Man Cybern Syst 2022; 52(10): 6174–6182.

37.

Liu

Zhu

Zhao

, et al. Adaptive fuzzy backstepping control for nonstrict feedback nonlinear systems with time-varying state constraints and backlash-like hysteresis. Inf Sci 2021; 574: 606–624.

38.

Zhao

Wang

Zhang

, et al. Adaptive neural backstepping control design for a class of nonsmooth nonlinear systems. IEEE Trans Syst Man Cybern Syst 2019; 49(9): 1820–1831.

39.

Park

Sandberg

. Universal approximation using radial-basis-function networks. Neural Comput 1991; 3(2): 246–257.

40.

Chen

. Adaptive neural output feedback control of uncertain nonlinear systems with unknown hysteresis using disturbance observer. IRE Trans Ind Electron 2015; 62(12): 7706–7716.

41.

Chen

Yuan

Yang

. Event-triggered adaptive neural network backstepping sliding mode control of fractional-order multi-agent systems with input delay. J Vib Control 2022; 28(23-24): 3740–3766.

42.

Yuan

Chen

. Observer-based adaptive neural network dynamic surface bipartite containment control for switched fractional order multi-agent systems. Int J Adapt Control Signal Process 2022; 36(7): 1619–1646.

Observer-based distributed convex optimization of bipartite containment control for higher order nonlinear uncertain multi-agent systems

Abstract

Keywords

Introduction

Prerequisites

Graph theory

Convex analysis

Problem formulation

The distributed optimization problem

Main results

Controller design

Simulations

Conclusions

Footnotes

Declaration of conflicting interests

Funding

ORCID iDs

References