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

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.


Introduction
2][3][4][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. 6Olfati-Saber present a theoretical framework for design and analysis of distributed flocking algorithms to deal with the multiagent flocking problem. 7Guo et al. propose a novel technique to control the relative motion of multiple mobile agents as they stabilize to a desired configuration. 8Chen et al. propose four resilient state feedback based leader-follower tracking protocols. 9enerally speaking, the control methods of multiagent 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. 10,11However, for distributed control, the follower cannot get the state of the leader and needs to exchange information through the communication topology. 12,13Generally, 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.Wang 14 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. 13he switched stochastic nonlinear MASs control method is proposed. 15Zou 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. 21solve the distributed optimization problem of MASs subjected to exogenous disturbances.Guo and Kang 22 by constructing a twolayer 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. 23Through 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. 258][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.
1.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.2. 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.3. 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.

Graph theory
There is information interaction between multiple agents, and an undirected graph G = w, e, A ð Þis usually used to represent such information, in which w = n 1 , :::, n M f g .Where M is the number of agents.The set of edge is exhibited as e = n i , n j À Á È É 2 w3w, which expresses that there is information exchange between agent i and agent j.

Convex analysis
If a function f( Á ) : R n !R is convex we can have If a differentiable function f( Á ) : R n !R is strongly convex on R n we can have

Problem formulation
In this paper, we study the following high-order nonlinear multi-agent systems for agent i.
where r = 2, Á Á Á , n À 1, u i is the control input, y i is the system output and g i, l (x i, l , x i, 2 , Á Á Á , x i, r ) is an unknown nonlinear function defined on the system state vector.Define X i, l = (x i, 1 , x i, 2 , Á Á Á , x i, r ) T 2 R r as the system state vectors for agent i. Rewrite the system for agent i: where . For a given positive matrix Q T i = Q i , there exists a positive matrix P T i = P i satisfying 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 where x d, 1 and x d, 2 is the upper and lower bound of the trajectory of motion,

Define the global objective function as
Because the local objective function is an absolute convex function, the global objective function is also a strictly convex function.Define Therefore, we can design penalty term as follows Penalty function is defined as follows 36 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 Ã i, 1 for agent i is defined as Remark 1. From ( 12), we find that the value function has two parts.The first part ) is the global objective function.The second part x T 1 Lx 1 is the penalty term, which is used to reach consensus for all agents.Lemma 2. 37 For any x, y 2 R n , the following inequality relationship holds where l .1, m .1, n .0, and Lemma 3. 38 Let V : R n !R be a function satisfying and O 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 z such that 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.

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 where ð Þ 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 ð Þ.From this, the following assumptions can be derived Assumption 1.The unknown functions g i, l X i, l ð Þ, i = 1, Á Á Á , n can be expressed as where u i, l is the unknown constant vector, and u i, l X i, l ð Þ is Gaussian basis function vector.
By Assumption 1, we can obtain The observer model ( 15) can be converted into the following model Let e i = X i, n À b X i, n be state observation errors of system (4).According to equations ( 5) and ( 18), we have where . The vectors of optimal parameters are defined as where 14l4n, O i, l , and U i, l are compact regions for u i, l , X i, l , and b X i, l .Define errors of the optimal approximation e i, l and parameter estimation e u i, l as Assumption 2. The optimal approximation errors remain bounded, there exists positive constants e i0 , satisfying e i, l j j4e i0 .Remark 2. Neural network approximation has universal approximation 39 and assumption 2 is often designed in the design process of neural network controller. 40ssumption 3. The nonlinear function g i, l satisfies the Lipchitz property.There exists a set of known constants g i , the following relationship holds Where e i = e i, 1 , :::, e i, n ½ T , Dg i = Dg 1 , :::, Dg n ½ T .Constructing the Lyapunov function as: Then, we can obtain By Lemma 2 and Assumption 3, we obtain In a similar way, we have 1 2 e T i P T i P i e i + 1 2 Where l i, max (P i ) is the maximum eigenvalue of positive matrix P i .By equations ( 24)-( 26), we obtain Where 0 .
Then, we can obtain Where q 0 = P N i = 1 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 Ã i, l as input and v i, l as output to construct the following filter Where, m is the order of the multi-agent model and 24m4n.Define the error variable as follows: Where, s i, m is the error between multiple agents and the optimal trajectory, v i, m is the state variable of multiagent system, which is also the output of the filter.w i, m is the error between v i, m and x Ã i, m .xi, m is the estimation of x i, m .
By the equations ( 29) and (30), we have 570 Measurement and Control 57 (5)     Where l i, m is adjustable parameter and B i, m = À _ x Ã i, m .There exist constants M i, m .0, i = 1, . . ., N, such that jB i, m j4M i, m holds.
Step 1. First, we need find the extreme point of the penalty function ( 11) Where 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 By equations ( 11) and (32), we have Substitute the equation ( 7) into (33) to get Then according to (32) and (34), we have Let s 1 = ½s 1, 1 Á Á Á s N, 1 T .According to (35), we have Then, we construct the Lyapunov function Where s i, 1 is a adjustable parameter.According to (1), (18), and (30), we can obtain Take the derivative of V 1 and substitute the equation (37 where T , e 2 = ½e 1, 2 e 2, 2 Á Á Á e N, 2 T , According to Lemma 2, the following inequalities hold where g 1 = diag½g i, 1 , e 1 = ½e 1, 1 e 2, 1 Á Á Á e N, 1 T .Substituting (39)2(43) into (38), we have According to Lemma 1, we have Then, we can obtain and By equations ( 44), (46), and (47), the virtual controller x Ã i, 2 and adaptive law u i, 1 are designed as Where c i, 1 = 3 + 2 , r i, 1 is the adjustable parameters.Substituting (48) and ( 49) into (44), after (27) we can obtain Where q 1 = q 0 À N, Step 2. According to Theorem 1, design the error variable s i, 2 = xi, 2 À v i, 2 .By equations ( 15) and ( 16), we have Construct the Lyapunov function where s i, 2 is adjustable parameter.Substituting (51) into (52), we can obtain According to Lemma 2, we obtain 572 Measurement and Control 57( 5) Substituting (54)-( 56) into (53) can be written as According to Theorem 1, the virtual controller x Ã i, 3 and update laws u i, 2 are designed as where r i, 2 is adjustable parameter.Substituting equations (58), ( 59), (50), and ( 31) into (57), then we obtain According to Lemma 2, we have w i, 2 B i, 2 4 Then we can obtain where Step m.According to Theorem 1, define the m-order error variable s i, m = xi, m À v i, m .By equations ( 15) and ( 16), we have Construct the Lyapunov function where s i, m is adjustable parameter.Substituting (62) into (63), we can obtain According to Lemma 2, we obtain Substituting (65)2(67) into (64) can be written as According to Theorem 1, the m-order virtual controller x Ã i, m + 1 and update laws u i, m are designed as where r i, m is adjustable parameter.Substituting equations (69), (70), and ( 31) into (57), then we obtain According to Lemma 2, we have Then we can obtain Same as the equation (61), we can obtain where q m = q mÀ1 À 1 2 Step n.According to theorem 1, design the n-th error variable as follows Then, we have The Lyapunov function is constructed as where s i, n is designed parameter.Combining ( 73) and (76), we can obtain According to Lemma 2, the following inequalities hold Design the multi-agent system control law u i and update laws u i, n as follow where r i, n is designed parameter.substituting equations (78), ( 79), (80), and (81) into (77), then we can obtain 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 Substituting (36) into (90) yields, we have As can be seen from ( 91), when time approaches infinity, the error satisfies ∂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 closedloop 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 where i = 1, 2, 3, 4, 5 and the initial value selection for each multi-agent is as follows ð Þ as the upper bound of the bipartite containment trajectory.x d2 = 0:5 sin t ð Þ as the lower bound of the bipartite containment trajectory.Because there are negative adjacencies, Define x d3 = À sin t ð Þ and x d4 = À 0:5 sin t ð Þ as the negative bipartite containment trajectory.The unknown functions in system (92) are 2 À 0:1x 3, 2 g 4, 2 = 0:2x 4, 1 À 0:1x 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 The penalty function is shown below The condition of its optimal solution is 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  48), (49), (80), and (81), the design of the virtual control law, the adaptive weight update law and the control input are as follow Figure 1.Undirected topology among MASs.
The necessary parameters in equations ( 95), ( 96), (97), and (98) are selected as c i, 1 = 30, c i, 2 = 40, s i, 1 = s i, 2 = 10, r i, 1 = r i, 2 = 21, l i, 2 = 0:04.In the simulation, Figures 2-8 are the results of simulation.Figure 2 shows the trajectories of x d1 , x d2 , x d3 , x d4 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. 42As 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  .The value of the error between the system output and the optimal trajectory s i, 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.

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.

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.

Figure 7 .
Figure 7.The error of penalty function.