#### Abstract

This 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.

#### 1. Introduction

Consensus control of multiagent systems (MAS) has drawn considerable attention in the past two decades due to its broad applications in multiple ground-moving robots [1], unmanned aerial vehicles (UAVs) [2], unmanned surface vessels [3], sensor networks [4], smart grids [5], and synchronization and flocking models [6–8], for instance. The investigation on this topic has been carried out from different perspectives, to mention a few, such as single-integrator or double-integrator MAS [9, 10], general linear MAS [11–13], nonlinear MAS [14–16], fractional-order MAS [17, 18], and high-order MAS [19, 20].

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. There are many results of consensus control for MAS with unknown control gains, for instance, [21–26]. In [21], Nussbaum function-based adaptive control was developed to handle the consensus problem for a first-order MAS provided that the communication digraph was balanced and weakly connected. By using a Nussbaum-like switching function, the authors in [22] also addressed the first-order consensus problem under a strongly connected communication digraph. In [23], using a novel Nussbaum gain function, consensus for a second-order linearly parameterized MAS was realized under the assumption that the control directions were the same and the communication graph was undirected and connected. Some progress has been made with respect to high-order MAS. In [24], adaptive backstepping-based controllers were designed to guarantee the consensus for high-order MAS under an undirected and connected communication graph. In [25], the authors investigated high-order 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 [27–31].

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. This 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.

The rest of this paper is organized as follows. In Section 2, some preliminaries are introduced and the problem of this paper is formulated. Then, a backstepping-based control algorithm is proposed in Section 3, and the closed-loop 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.

#### 2. Preliminaries and Problem Statement

In this paper, a class of high-order affine nonlinear MAS with followers is considered. The dynamics of each follower is described as follows:where , , and are the state vectors, actual controllers, and outputs of the MAS, respectively, , are the unknown bounded time-varying external disturbances, the unknown continuous function represents the uncertain nonlinear dynamics, represents a continuous and unknown time-varying control gain function, and is a standard saturation function defined as , where is a saturation level constant. The output of the leader is , which satisfies the following assumption.

*Assumption 1. *, , and are bounded for .

*Remark 1. *The 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. *The 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 is replaced by a smooth function [38]: , where and is a Gaussian error function, which is a real-valued and continuous differentiable function. Defining , , and then we have .

The 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 synchronically tracks the leader’s output . Moreover, the output consensus error can converge to a small region around the origin. To this end, the following assumptions and preliminaries are required.

*Assumption 2. *There exist constants , , , and , such that , , and , respectively, and .

*Assumption 3. *There exist positive constants and such that the unknown control gains satisfy .

*Definition 1 (see [39]). *A continuous function is called a Nussbaum gain function if it satisfiesIn this paper, a Nussbaum gain function is chosen.

Lemma 1 (see [40]). *Let and be smooth functions defined on with , , and be an even smooth Nussbaum gain function. If the following inequality holds:where represents some suitable constant, is a positive constant, and the value of the time-varying parameter is located in the unknown intervals with , then , , and are bounded on .**The communication topology in MAS (1) with followers is expressed via a digraph , where followers are denoted by a node set , is an edge set, and is an adjacency matrix. if the -th follower can receive information from the -th follower, otherwise . Self-loop is not considered, i.e., . is defined as an in-degree matrix with being the in-degree of the -th follower. The Laplacian matrix of is denoted by . 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 is used to demonstrate the communication between the leader and the follower, where represents that the leader is a neighbor of the -th follower, and otherwise.*

*Assumption 4. *The communication digraph among the 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 on a prescribed compact set , i.e.,where is the input vector, is the ideal weight vector with being the number of neurons, is the Gaussian basis function vector with , where and are the center and the width of the Gaussian function, respectively, , and denotes the approximation error. It is known that given any positive constant , if is large enough, there exist suitable vectors and such that .

#### 3. Consensus Controller Design

The distributed adaptive output consensus controller design procedure consists of steps using the backstepping technique. To begin with, the following error surfaces for the -th follower are defined:where is the filtered virtual controller, which is acquired through a first-order filter with the virtual controller as the input. The boundary layer error between and is defined as .

*Step 1. *The time derivative of in equation (5) is obtained asDefine , . Throughout this paper, let be the estimates of , and the corresponding estimation error be .

Then, the unknown smooth function can be approximated by an RBFNN asThen, we haveThe virtual controllers and adaptation laws and are designed as follows:where is a positive-definite matrix, and are the design parameters. Adjusting the value of properly will make the error converge to a small neighborhood of the origin. is a modification term with being a small positive constant, such that will not drift to very large values.

Under Assumption 4, we have . Hence, we can construct the Lyapunov function candidate . Note that , and then we havewhere . Using Young’s inequality [41], the following inequalities hold:Substituting (14)–(16) into (13), one hasIn order to avoid differentiating , the so-call explosion of the complexity problem is inherent in the backstepping technique, and let pass through a first-order filter such that a filtered virtual controller is acquired:where is a small time constant.

A Lyapunov function candidate is given asUsing (18), we have , where is a continuous function. Since for any and , the sets and , , are compact in and , respectively, and is also compact in . Therefore, has a maximum on , i.e., . Similar analysis can be found in [42]. Hence, we haveBy taking the time derivative of , one haswhere . Note thatthen we haveStep (). The derivative of the error surface in (6) is obtained asDefine , . This unknown smooth function can be approximated by an RBFNN asThe virtual controllers , , and adaptation laws and are designed as follows:where is a positive-definite matrix, and are the design parameters.

Applying the Lyapunov function and noting that , we havewhere . Using Young’s inequality, we haveSubstituting the above equations into (29) givesLet pass through a first-order filter with a small time constant to acquire , :From (32), we havewhere is a continuous and bounded function satisfying .

Applying the Lyapunov function,Similar to the previous step, we havewhere . Similar to (22), one hasthen we haveStep . At this final step, the actual controller is designed. The derivative of the error surface is obtained asDefine , . This unknown smooth function can be approximated by an RBFNN asThe actual controllers and adaptation laws and are designed as follows:where is a positive-definite matrix, and are the design parameters.

Applying the Lyapunov function ; similar to the aforementioned step, we havewhere .

#### 4. Stability Analysis

Theorem 1. *Suppose that the initial conditions are bounded and the design parameters fulfill the following inequalities (44), then the output consensus problem for MAS (1) can be addressed by the controllers (10), (26), and (40) combining with adaptation laws (11), (12), (9), (28), (41), and (42), as well as the filters (18) and (32) under Assumptions 1–4. Moreover, all signals in the closed-loop MAS are SUUB, and the output consensus error can converge to a small region around the origin provided that the design parameters are appropriately selected:*

*Proof. *Select the following constants and asFrom (23), one hasMultiplying (46) by and then integrating it with respect to over , one hasSince , we havewhere . Then, according to Lemma 1, we can conclude that , , , , and are all SUUB on . Let be the upper bound of , i.e., .

For , select the following constants and asFrom (37), we haveSimilar to (48), one haswhere . Then, according to Lemma 1, one can conclude that , , , , and are all SUUB. Let be the upper bound of , i.e., .

Finally, applying the Lyapunov function,Denotethen substitute (23), (37), and (43) into the time derivative of (52), and based on (44), we haveMultiplying the above inequality by and then integrating it with respect to over , we haveBy Lemma 1, it can be concluded that , , , , and are all SUUB; i.e., all the signals in the closed-loop system are SUUB. Besides, from (52) and (55), one hasSince the matrix is nonsingular under Assumption 4, denote its minimum singular value as , which is a positive constant [10]. Define and with and . Then, we haveHence, the output consensus error can converge to a small region around the origin provided that the design parameters , , and are appropriately selected.

#### 5. Numerical Simulation

To test the availability of the proposed control algorithm, consider the following uncertain MAS with 3 followers:where , , , , , ,