Distributed Adaptive Output Consensus for High-Order Multiagent Systems with Input Saturation and Uncertain Nonlinear Dynamics

-is paper deals with the leader-following output consensus problem for a class of high-order affine nonlinear strict-feedback multiagent systems with unknown control gains and input saturation under a general directed graph. Nussbaum gain function technique is used to handle the unknown control gains, and the uncertain nonlinear dynamics of each agent is approximated by radial basis function neural networks. Distributed adaptive controllers are designed via the backstepping technique as well as the dynamic surface control approach. It is proved that the closed-loop multiagent systems are semiglobally uniformly ultimately bounded, and the output consensus error can converge to a small region around the origin. Finally, the theoretical results are supported by a numerical simulation.

Consensus problems of MAS with unknown control gains, including unknown amplitudes of control gains or unknown signs of control gains (the so-called control directions), are gaining researchers' increasing attention in recent years. It is necessary and beneficial to study this topic since the control gains are often unknown in many practical control systems, such as the autopilot design of time-varying ships and unmanned sailboat heading control. ere are many results of consensus control for linearly parameterized MAS, while external disturbances were not considered. In [26], high-order MAS with nonlinear dynamics and partially unknown nonidentical time-varying control directions, as well as bounded external disturbances, was considered, where the communication digraph was assumed to be strongly connected. Note that most of the existing works including those mentioned above did not take input saturation into account, which is often encountered in practical applications due to physical limitations of actuators and may cause the instability or damage the control systems' performance [27][28][29][30][31].
Motivated by the limitation of the existing literatures, this paper studies the leader-following output consensus problem for a class of high-order nonlinear MAS subject to input saturation and bounded external disturbances. Moreover, the control gains are assumed to be time varying and unknown for the controller design. is issue has not been mentioned in any existing references to the best of the authors knowledge. Based on the backstepping technique combined with the dynamic surface control (DSC) method, and under the assumption that the communication digraph has a spanning tree, a distributed adaptive neural controller is proposed with the aid of the well-known Nussbaum gain function. It is proved that the closed-loop MAS is semiglobally uniformly ultimately bounded (SUUB), and the output consensus error can converge to a small region around the origin.
Compared with the existing results, the primary contributions of this paper can be summarized as follows. (1) First, the high-order MAS model discussed in this paper is a class of affine nonlinear systems in the strict-feedback form with external disturbances, which is more general than most existing systems regarding output consensus control with unknown control gains [26,32,33], where the system dynamics was described with the Brunovsky form or the input saturation was not considered. Hence, the consensus schemes in these references could not be applied. (2) Second, unlike some existing results where the signs of control gains were assumed to be known in advance and the derivative of each control gain function was assumed to be bounded [34] or the unknown nonlinear dynamics satisfied global Lipschitz conditions [35,36], the control gains (including their amplitudes and signs) and the nonlinear dynamics in this paper are completely unknown. e rest of this paper is organized as follows. In Section 2, some preliminaries are introduced and the problem of this paper is formulated. en, a backstepping-based control algorithm is proposed in Section 3, and the closedloop stability is proved in Section 4. In Section 5, a simulation example is given to verify the proposed control algorithm. Finally, some remarks of this paper are concluded in Section 6.

Preliminaries and Problem Statement
In this paper, a class of high-order affine nonlinear MAS with N followers is considered. e dynamics of each follower is described as follows: and y i ∈ R are the state vectors, actual controllers, and outputs of the MAS, respectively, i � 1, 2, . . . , N, d i,j (t) are the unknown bounded time-varying external disturbances, the unknown continuous function h i,j (·) represents the uncertain nonlinear dynamics, ℓ i,j (·) represents a continuous and unknown time-varying control gain function, and σ(·): R ⟶ R is a standard saturation function defined as where v * > 0 is a saturation level constant. e output of the leader is y 0 (t), which satisfies the following assumption.
Remark 1. e dynamics of the uncertain MAS (1) is affine nonlinear in the strict-feedback form with external disturbances, which cannot be converted to the Brunovsky form studied in most existing literatures [26,32,33] due to the unknown control gains. Hence, the MAS is more general, and the controller design is more challenging.

Remark 2.
e strict-feedback nonlinear MAS (1) may have great potential for practical applications since it can describe many dynamical behaviors, such as robotic systems, flight systems, and biochemical process [37].
In order to use the backstepping technique, the discontinuous saturation nonlinearity σ(·) is replaced by a smooth function [38]: x 0 e − t 2 dt is a Gaussian error function, which is a real-valued and continuous dif- e purpose of this paper is to present an adaptive neural output consensus controller for MAS (1) such that the semiglobal uniform ultimate boundedness for all the signals in the closed-loop system is ensured, and each follower's output y i (i � 1, 2, . . . , N) synchronically tracks the leader's output y 0 . Moreover, the output consensus error can converge to a small region around the origin. To this end, the following assumptions and preliminaries are required.
Lemma 1 (see [40]). Let Γ(t) and κ i (t) be smooth functions defined on [0, T) with Γ(t) ≥ 0, ∀t ∈ [0, T), and N(·) be an even smooth Nussbaum gain function. If the following inequality holds: where A represents some suitable constant, B > 0 is a positive constant, and the value of the time-varying parameter ℓ i (·) is located in the unknown intervals A digraph contains a spanning tree provided that there exists a directed path from one node (called root node) to every other node in the graph. A leader adjacency matrix R � diag(r 1 , r 2 , . . . , r N ) is used to demonstrate the communication between the leader and the follower, where r i > 0 represents that the leader is a neighbor of the i-th follower, and r i � 0 otherwise.

Assumption 4.
e communication digraph G among the N followers contains a spanning tree, and the leader is a neighbor of the root node.
Since the uncertain nonlinear dynamics of MAS (1) is unknown and continuous, many function approximators can be used, such as neural networks, spline functions, polynomials, and fuzzy systems. Gaussian radial basis function neural network (RBFNN) approach is easy to implement due to its small number of control parameters; therefore, it is widely used in nonlinear function approximation. In this paper, we employ an RBFNN to approximate the unknown function Ψ(Z): R m ⟶ R on a prescribed compact set Ω ⊂ R m , i.e., where c i and w i are the center and the width of the Gaussian function, respectively, i � 1, 2, . . . , s, and ϖ(Z) denotes the approximation error. It is known that given any positive constant ε, if s is large enough, there exist suitable vectors ϱ * and ϑ(Z) such that |ϖ(Z)| ≤ ε.

Consensus Controller Design
e distributed adaptive output consensus controller design procedure consists of n i steps using the backstepping technique. To begin with, the following error surfaces for the j-th follower are defined: where c i,j is the filtered virtual controller, which is acquired through a first-order filter with the virtual controller c i,j as the input. e boundary layer error between c i,j and c i,j is defined as Step 1. e time derivative of e i,1 in equation (5) is obtained as Define 2 , _ y 0 ] T . roughout this paper, let x be the estimates of x * , and the corresponding estimation error be x � x − x * . en, the unknown smooth function Ψ i,1 (ζ i,1 ) can be approximated by an RBFNN as en, we have e virtual controllers c i,2 and adaptation laws κ i,1 and ϱ i,1 are designed as follows: Mathematical Problems in Engineering where Λ i,1 is a positive-definite matrix, α i,1 > 0 and β i,1 > 0 are the design parameters. Adjusting the value of α i,1 properly will make the error e i,1 converge to a small neighborhood of the origin. β i,1 ϱ i,1 is a modification term with β i,1 being a small positive constant, such that ϱ i,1 will not drift to very large values.
Under Assumption 4, we have r i + d i > 0 ∀i ∈ V. Hence, we can construct the Lyapunov function candidate 2 , and then we have Using Young's inequality [41], the following inequalities hold: Substituting (14)- (16) into (13), one has In order to avoid differentiating c i,2 , the so-call explosion of the complexity problem is inherent in the backstepping technique, and let c i,2 pass through a first-order filter such that a filtered virtual controller c i,2 is acquired: where τ i,2 > 0 is a small time constant. A Lyapunov function candidate Γ i,1 (t) is given as Using (18), we have _ 1 is a continuous function. Since for any B 0 > 0 and Q > 0, the sets 2 . Similar analysis can be found in [42]. Hence, we have By taking the time derivative of Γ i,1 , one has where then we have

4
Mathematical Problems in Engineering Step j (2 ≤ j ≤ n i − 1). e derivative of the error surface e i,j � x i,j − c i,j+1 in (6) is obtained as can be approximated by an RBFNN as e virtual controllers c i,j+1 , j � 2, . . . , n i − 1, and adaptation laws κ i,j and ϱ i,j are designed as follows: where Λ i,j is a positive-definite matrix, α i,j > 0 and β i,j > 0 are the design parameters. Applying the Lyapunov function Υ i,j � (1/2)e 2 i,j and noting that c i,j+1 � c i,j+1 + c i,j+1 , we have where ξ i,j � ϖ i,j + d i,j . Using Young's inequality, we have Substituting the above equations into (29) gives Let c i,j+1 pass through a first-order filter with a small time constant τ i,j+1 > 0 to acquire c i,j+1 , j � 2, . . . , n i − 1: From (32), we have where B i,j+1 � (zc i,j+1 /zκ i,j ) _ κ i,j + (zc i,j+1 /ze i,j ) _ e i,j + (zc i,j+1 / zϱ i,j ) _ ϱ i,j is a continuous and bounded function satisfying Applying the Lyapunov function, Similar to the previous step, we have where then we have Step n i . At this final step, the actual controller v i (t) is designed. e derivative of the error surface e i,n i � x i,n i − c i,n i is obtained as

Mathematical Problems in Engineering
Define . is unknown smooth function Ψ i,n i (ζ i,n i ) can be approximated by an RBFNN as e actual controllers v i and adaptation laws κ i,n i and ϱ i,n i are designed as follows: where Λ i,n i is a positive-definite matrix, α i,n i > 0 and β i,n i > 0 are the design parameters. Applying the Lyapunov function where ω i,n i � ℓ i,n i η + d i,n i + ε i,n i .

Numerical Simulation
To test the availability of the proposed control algorithm, consider the following uncertain MAS with 3 followers: where h 11 (x 11 ) � − 2x 11 + x 2 11 , ℓ 11 (x 11 ) � − [3 + 2 sin(x 2 11 )], h 12 (x 12 ) � x 11 x 2 12 , e leader's output is given by y 0 (t) � 0.4 sin(t). It is easy to verify that the conditions in Assumptions 1-3 are all satisfied. Figure 1 depicts the communication digraph of the leader and the 3 followers.
In simulation, RBFNNs ϱ i,1 Tϑ i,1 (ζ i,1 ) and ϱ i,2 Tϑ i,2 (ζ i,2 ), which contain 11 neurons with centers spaced evenly in [− 1, 1] and widths being equal to w 1 � 0.5 and w 2 � 5, respectively, are used to approximate the unknown continuous functions h i1 (x i1 ) and h i2 (x i2 ), i � 1, 2, 3. e initial conditions are chosen as (61) Figures 2-4 display the evolution of the output trajectories of the 3 followers and the leader, the corresponding output consensus errors, and the actual controllers v i , respectively. One can see that the outputs of the 3 followers can track the leader's output since the output consensus error converges to a small neighborhood of zero. Additionally, it can be seen that there exist large differences between v i and the saturation level constant v * , which indicates the significant saturation effect. e simulation results illustrate that the proposed control algorithm is effective.

Remark 3.
e reason that why there exist large fluctuations in the output trajectories of y i lies in the large fluctuations of the actual controllers v i (see Figure 4). Because the Nussbaum gain function N(κ i,2 ) � κ 2 i,2 cos((π/2)κ i,2 ) will be very  large (positively or negatively) if κ i,2 becomes very large, correspondingly the actual controllers v i will be very large according to equation (40). To reduce the values of κ i,2 , the design parameters α i,2 should be turned smaller. However, from the proof of eorem 1, we know that smaller α i,2 may lead to a smaller value of δ and thereby causes a larger output consensus error (see equation (59)). In a word, there is a tradeoff between smaller output consensus errors and smooth output trajectories. is is another problem worthy to explore, which may be our future work.

Conclusion
In this paper, an output consensus scheme for a class of highorder affine nonlinear MAS with unknown control gains and input saturation under a directed communication graph is proposed. Combining the adaptive RBFNN approximation and the DSC methodology, consensus control laws are developed to guarantee the semiglobal uniform ultimate boundedness for all signals in the closed-loop system. Moreover, the output consensus error can converge to a small region around the origin provided that the design parameters are appropriately selected. e effectiveness of the proposed control algorithm is revealed by a numerical simulation.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest. Mathematical Problems in Engineering 9