Adaptive Fuzzy Containment Control for Uncertain Nonlinear Multiagent Systems

This paper considers the containment control problem for uncertain nonlinear multiagent systems under directed graphs. The followers are governed by nonlinear systems with unknown dynamics while the multiple leaders are neighbors of a subset of the followers. Fuzzy logic systems (FLSs) are used to identify the unknown dynamics and a distributed state feedback containment control protocol is proposed. This result is extended to the output feedback case, where observers are designed to estimate the unmeasurable states. Then, an output feedback containment control scheme is presented.The developed state feedback and output feedback containment controllers guarantee that the states of all followers converge to the convex hull spanned by the dynamic leaders. Based on Lyapunov stability theory, it is proved that the containment control errors are uniformly ultimately bounded (UUB). An example is provided to show the effectiveness of the proposed control method.


Introduction
Since the scales of practice control system became larger and larger, much attention has been paid to complex systems, such as interconnected systems and multiagent systems.An interconnected system means a system that consists of interacting subsystems.The main control objective of interconnected systems is to find some decentralized feedback laws for adapting the interconnections from the other subsystems, where no state information is transferred [1][2][3][4].Multiagent systems consist of some intelligent agents, which have the capability of reacting to the variety of environments automatically, such as robots, automatic vehicles, and sensors.The main control objective of multiagent systems is to establish distributed control laws based on the information of the agent and its neighbors to realize collective behavior [5][6][7][8].In the past decades, cooperative control problem of multiagent systems has attracted significant research interests, which mainly focuses on consensus [9][10][11][12][13][14], formation control [15,16], and containment control [17].
Containment control aims at guiding the states or outputs of the followers to converge to a convex hull formed by the multiple leaders using a distributed control protocol.
The problem has many applications, for example, securing a group of followers in the area spanned by the leaders so that they can be away from dangerous sources outside the area.Recently, distributed containment control problem has been investigated and numerous research results have been obtained [18][19][20][21][22][23][24][25][26][27].Containment control strategies were proposed for multiagent systems with single-integrator [17][18][19], double-integrator [20][21][22], or general linear dynamics [23].However, the reported methods can only deal with the containment control problem of linear multiagent systems.By now, there have been some results on containment control for nonlinear multiagent systems in [24][25][26][27].It should be noted that the proposed containment controllers required each agent satisfying Lagrangian dynamics with known nonlinearities [24,25], linearly parameterized nonlinearities [26], or unknown nonlinearities [27].Therefore, containment control problem for uncertain nonlinear multiagent systems needs to be further investigated.
Motivated by the above observations, in this paper, containment control problem for multiagent systems with more general nonlinear dynamics is studied.The nonlinear dynamics of each follower can be totally unknown.Using FLSs to identify the unknown nonlinear dynamics, distributed state 2 Mathematical Problems in Engineering feedback and output feedback containment control schemes are proposed to drive the states of all followers into the convex hull spanned by the leaders.It is proved that the containment control errors converge to a residual set.The rest of the paper is organized as follows.Section 2 formulates problem formulation.Section 3 provides the design of distributed state feedback containment controllers.Section 4 provides the design of distributed state feedback containment controllers.Section 5 gives an illustrative example to show the effectiveness of the proposed approaches.Section 6 concludes the paper.
Compared with the existing results on nonlinear multiagent systems, the main advantages of the proposed containment control scheme in this paper are listed as follows.
(1) In [14], the consensus scheme was proposed for general nonlinear multiagent systems, which drive all followers to track the states of the leader.In this paper, we develop a containment control method for the nonlinear multiagent systems to drive all followers to converge to a convex hull formed by the multiple leaders.It should be noted that consensus and containment control are two different problems in the cooperative control of multiagent systems.
(3) In [24][25][26][27], containment control scheme was developed for nonlinear multiagent systems with Lagrangian dynamics, where the nonlinearities were assumed to be known, linearly parameterized, or unknown.In this paper, we design a state feedback containment control scheme for more general nonlinear multiagent systems with unknown dynamics.Besides, considering that some states in the systems are unmeasurable in practice, an output feedback containment control scheme is proposed.
Notations.Throughout this paper,  + is a set of positive real numbers. × is a set of  ×  real matrices.  is an identity matrix with the dimension of .‖ ⋅ ‖ is the Euclidean norm of a vector.‖ ⋅ ‖  is the Frobenius norm of a matrix.tr(⋅) is the trace of a matrix.(⋅) and (⋅) are the maximum and minimum singular values of a matrix, respectively.diag(  ) is a diagonal matrix with   being the th diagonal element.⊗ is the Kronecker product.

Problem Formulation
Consider a class of nonlinear multiagent systems consisting of  followers and  leaders.The dynamics of follower  are described by where If   =   , for all , , the graph  is undirected; otherwise the graph  is directed.A directed graph has a spanning tree if there is a root node, such that there is a directed path from the root node to every other node in the graph.The Laplacian matrix  = [  ] ∈  (+)×(+) is defined as Then, the Laplacian matrix  =  − Λ, where  = diag(  ) is the degree matrix with   = ∑  =1   ( = 1, . . ., ).An agent is called a follower if the agent has at least one neighbor.An agent is called a leader if the agent has no neighbor.Without loss of generality, we assume that the agents indexed by 1, . . .,  are followers, whereas the agents indexed by  + 1, . . .,  +  are leaders.Then, the Laplacian matrix  can be partitioned as where  1 ∈  × and  2 ∈  × .
Assumption 1.For each follower, there exists at least one leader that has a directed path to that follower.
In this paper, we adopt the singleton fuzzifier, product inference, and the center-defuzzifier to deduce the following fuzzy rules [28][29][30].
: IF  1 is  1  , and . . .and   is    , THEN  is   ( = 1, . . ., ), where  = [ 1 , . . .,   ] ∈   and  ∈  are the input and output of the fuzzy system, respectively.   ( = 1, . . ., ) and   are fuzzy sets in .The fuzzy inference engine performs a mapping from fuzzy sets in   to a fuzzy set in  based on the IF-THEN rules in the fuzzy rule base and the compositional rule of inference.The fuzzifier maps a crisp point x into a fuzzy set   in .The defuzzifier maps a fuzzy set in R to a crisp point in R. Since the strategy of singleton fuzzification, center-average defuzzification, and product inference is used, the output of the fuzzy system can be formulated as where   is the point at which fuzzy membership function    (  ) achieves its maximum value.It is assumed that () = [ 1 (), . . .,   ()]  , and  = [ 1 , . . .,   ]  .Then the fuzzy logic system (6) can be rewritten as It has been proved in [31] that if Gaussian functions are used as membership functions, the following lemma holds.Lemma 4. Let () be a continuous function defined on a compact set Ω.Then, for any constant  > 0, there exists an FLS such as where Ω is a compact region for . = [ 1 , . . .,   ]  is an adjustable vector.() = [ 1 (), . . .,   ()]  is a fuzzy basis function vector.Optimal parameter vector  * is defined as where   is the compact set of .Then where  is the minimum fuzzy approximation error with an unknown bound.

The Design of Distributed State Feedback Containment Controllers
where  1 and  2 are designed as follows: where  ∈  + is a coupling gain. ∈  × is a controller gain with  = −   1 , and  1 is positive definite satisfying the following Riccati inequality: where  1 is positive definite.By Lemma 4, the multipleinput multiple-output unknown dynamics   (  ) can be approximated by FLSs as [24,25]   (  ) =  *    (  ) +   .
Then,  2 are designed as where where Substituting ( 16) into the derivative of (17), we have
Under Assumption 1, the communication graph is directed and has a spanning tree.Select the containment controllers (11), (12), and (15) with the coupling gain  satisfying where   are the eigenvalues of  1 .  are updated by where    > 0,  > 0.Then, all the signals in the closedloop multiagent systems are UUB, and the containment control errors satisfy where Proof.Consider the Lyapunov function candidate where   = diag(   ).Substituting  = −   1 and ( 18) into the derivative of ( 22), we have It follows from (20) that By Assumption 1 and Lemma 2, all the eigenvalues of  1 have positive real parts.Thus, there exists a unitary matrix  ∈  × such that    −1 1  = diag( −1  ),  = 1, . . ., .Let  = ( ⊗   ), where  = [  1 , . . .,    ]  .Then, it follows from ( 24) that Substituting ( 13) and ( 19) into ( 25), one has Rewrite (27) in the following matrix form: where Noting the fact that ( 1 ) > 0 and  > 0, it follows that Σ 1 is positive definite.Then Let Then From ( 30) and ( 32), we have where Since lim  → ∞ √ 1 () = /, we obtain that all signals in the closed-loop multiagent systems are UUB.Then Then, it follows from (17) that Then, we get (21) with  1 = ‖ℎ 1 ‖ 1 /( −1 1 )(Σ 1 )√ 1 ( 1 ).It means that the states of the followers converge to the convex hull formed by those of the leaders with the containment errors being UUB.The containment control problem is solved.Remark 6.In [24][25][26][27], the distributed containment control approaches were proposed for nonlinear Lagrangian systems.However, the previous approaches cannot be applied to the nonlinear multiagent systems (1), (2).Therefore, it is significant to investigate the distributed containment control problem for more general nonlinear multiagent systems in the presence of unknown dynamics.

The Design of Distributed Output Feedback Containment Controllers
The method proposed in Section 3 required the states of the followers being measurable.However, in practice, some states in the systems are unmeasurable.In this section, the output feedback containment controllers will be designed.
We assume here that the states of the leaders are measurable and   () = 0.

Output Feedback Containment Controller Design.
Design distributed observers to estimate the unmeasurable states.Let x be the estimations of   .Similar to [32][33][34], the observers are designed in the following form: where   ∈  + is a coupling gain.ỹ =   − ŷ . ∈  × is an observer gain with  =  −1 2   , and  2 is positive definite satisfying the following linear matrix inequality (LMI): where  is an adjustable parameter to guarantee the existence of  2 . =   −  2  and  2 is positive definite.
Based on the developed observers, the output feedback containment controllers are designed in (11) with Then are updated by where    > 0,  > 0, and  > 0.Then, all the signals in the closed-loop systems are UUB, and the containment control errors satisfy where  2 ∈  + .
Proof.Consider the Lyapunov function candidate Substituting  =  −1 2   and (41) into the derivative of (47), we have Let Rewrite (53) in the following form: Using a similar analysis process to Section 3.2, it follows that x and θ are UUB and the bound of x is given by where

Consider another Lyapunov function candidate
Substituting  = −   1 and (43) into the derivative of (57), we have Then Then, the containment control problem is solved.
Remark 8.In [14], consensus scheme was developed for nonlinear multiagent systems (1); that is, the proposed method can guarantee all states of the followers synchronize to that of a single leader.In this paper, containment control approach is designed to guarantee all states of the followers stay in a dynamic convex hull formed by multiple leaders.

Simulation
In this section, a simulation example is provided to show the effectiveness of the proposed distributed output feedback containment controllers.Consider a network of harmonic oscillators described by (1), with [35]    Choose fuzzy membership functions as where  = 1, . . ., 5. The communication graph is described in Figure 1.Let  = 2/3 and solve (13) In simulation,   = 10,  = 2,   = 10, and  = 0.01.
The containment results and containment errors using the output feedback containment controllers of this paper are shown in Figures 2 and 3.It can be observed that the proposed containment scheme can realize that the states of followers converge to the convex hull formed by those of the leaders; that is, the states of all followers stay in the area formed by the leaders.Figure 4 shows the states of developed observer, from which we can see that the designed  observer can estimate unmeasurable states with the estimation errors in a small neighborhood of the origin.The profiles of the designed distributed output feedback containment controllers are shown in Figure 5.It can be observed that the designed containment controllers guarantee both the stability and good containment performance of the closedloop multiagent systems with unknown dynamics.Figure 6 shows the profiles of  12 , from which it can be observed that the unknown dynamics   (  ) can be compensated by  2 .

Conclusions
In this paper, the containment control problems were considered for uncertain nonlinear multiagent systems with measurable and unmeasurable states under directed graphs.Based on FLSs identifying the unknown dynamics of the followers, distributed state feedback containment controllers were designed first.Then, adaptive fuzzy observers were designed to estimate the unmeasurable states.Based on   the developed observers, distributed output feedback containment controllers were designed.Both of the developed containment controllers ensure that the states of the followers converge to the convex hull formed by those of the leaders with the containment control errors in a small residual set.Future research efforts will be devoted to the containment control problem of uncertain nonlinear multiagent systems with time-delay.

Figure 4 :
Figure 4: The estimation effect of observers.