#### Abstract

In this paper, a distributed asymptotic tracking control strategy is investigated by establishing filters and barrier function-based consensus control scheme to address the control of heterogenous power-chained multiagent systems (MASs) under a directed graph subject to the unknown input deadzone nonlinearities and unknown control coefficients. First, to generate estimation information from the leader, a two-order filter is exploited for every agent which solves the difficultly of the time-varying control coefficients in multiagent systems with a directed topology. Then, based on the two-order filters, prescribed performance method and barrier functions are utilized to establish the distributed tracking protocol to handle the power-chained deadzone input nonlinearities, such that the MAS can reach the global consensus while guaranteeing the prescribed tracking error performance. Using the Lyapunov stability theorem, the proof of the convergence is accomplished rigorously. Ultimately, the efficacy and advantage of the devised method are validated by two simulation examples.

#### 1. Introduction

The problem of distributed control for nonlinear MASs has been investigated for several decades, and a lot of significant achievements have been achieved [1–4]. Recently, various control methodologies, including the function approximation control techniques [5], adaptive control [6], and robust control [7], have been studied extensively for nonlinear MASs. In [8, 9], a consensus tracking protocol for uncertain pure feedback MASs is addressed. The distributed tracking problem based on observer is studied for nonlinear MASs with uncertain networks in [10]. In [11], on the strength of Nussbaum-type gain technique, both leaderless and leader-following consistent tracking control methodologies can make the errors converge be guaranteed asymptotically to zero for a second-order MAS with unknown control directions, without any information about the global communication graph. Under directed topologies or undirected topologies, a distributed adaptive control approach is applied to the leader-following networked Lagrangian systems with unknown control directions in [12]. However, the leader-following system assumes that there is only one leader in the system, and the communication graph among the followers is an undirected connected topology. Different from adaptive designs, [13] proposes a prescribed performance control method for an uncertain multiagent system. For resolving the consensus difficulty in MASs with unknown nonlinearities, neural networks or fuzzy logic systems are used as universal approximators in [14–16]. To settle the issue of unknown control directions, [17] presents a novel Nussbaum-type function. Although these above studies can obtain good command tracking performance, there is still a lack of research on fault-tolerant control design.

From a practical perspective, input deadzone nonlinearity is one of the key problems in nonlinear control system design, which results in the unaccomplished desired control efforts [18]. In the process of multiagent control design, if the control input constraints are not considered, the closed-loop system may be limited by performance and even lose stability. Therefore, the control investigations on restricted MASs has already attracted extensive attention in recent years [19]. On the basis of meeting preset trigger conditions, in [20], a distributed adaptive control approach is investigated for the uncertain MAS to deal with intermittent actuator faults. By introducing a predictor to obtain each error surface estimation, a distributed adaptive fuzzy consensus tracker design is used for MASs with input saturations [21]. In [22], the time-varying formation tracking control protocol for multileader MASs is studied in order to address the problem of actuator failure and saturation nonlinearity simultaneously. Whereas, the augmented plant of [22] is constructed complicatedly, and the first-order low-pass filter greatly increases the uncertainty of numerical simulation results. To compensate for both actuator bias fault and loss of actuator effectiveness fault, a fully distributed control scheme is constructed in [23]. In [24], a fully distributed protocol based on prescribed performance control is proposed to set predefined overshoot range, guarantee convergence rates, and limit neighbourhood synchronization steady-state errors for uncertain nonlinear MASs with deadzone inputs. Despite the workload of [24] is very sufficient and it is greatly improved compared with other aforementioned papers, the proposed strategy is still limited in the scope to the unity low power of MASs. As a matter of fact, there are a lot of practical heterogenous power-chained MASs, such as the coupled underactuated unstable 2-DOF machinery model (i.e., equation (38)) of [25] and the roll dynamic model of multiple axisymmetric skid to turn missiles as Example 2 of [26]. Thus, it is of necessity to do the research on MASs with heterogenous high powers. However, most control approaches applied to date have been limited in their boundary to the unity low power of MASs owing to the uncontrollable linearization of nonlinear systems caused by the high-power chains of control variables.

Contrary to these, the distributed consensus control for power-chained MASs with limited communication networks is seldom studied. Based on the controller of single nonlinear power-chained systems, aiming at the networked nonlinear power-chained MASs, [27] proposes a robust cooperative output tracking control protocol. Adding a power integrator technique and adaptive disturbance compensator are introduced to reduce the degradation of system performance for complex nonlinear MASs with input noise in [28]. By utilizing the distributed integrator backstepping approach, the boundness of the closed-loop system for multiple strict-feedback stochastic nonlinear systems under a directed leader-followers type communication graph is achieved in [29]. In [30], a neural adaptive tracking control scheme is employed to nonstrict feedback nonlinear MASs. A distributed low-complexity control strategy is proposed in [26] to guarantee consensus tracking performance of uncertain heterogenous power-chained MASs. We note that considerable progress has been made in the distributed control of MASs, but there are still some nonnegligible issues that need to be solved. First, for MASs with high powers, there are some restrictions on the heterogenous high powers. Specifically, the existing design methods usually require the high powers to be identical or to be known for establishing the control protocols. For example, in order to enable the control design for high power systems, [26] assumes the powers of the first order to be known (i.e., see Assumption 3 in [26]). Second, for MASs with a directed communication graph, the existing approaches do not consider the control coefficients or require that the control coefficients must be constants. The reason making it difficult is that the directed graph Laplace matrix is asymmetric. Considering the time-varying control coefficients of MASs are unknown, traditional adaptive control techniques [3, 20] used to address the time-varying coefficients will make the unknown matrix inestimable, where is defined by the Laplacian matrix and the communication directed matrix (i.e., see Lemma 1 in [5]). To circumvent the difficulties, the symmetric positive definite property of Laplacian matrix between followers is implicitly used in most theoretical analyses [24]. Whereas, this means that these methods are not applicable to the MAS with a directed topology. Third, there is no performance envelope scheme to predetermine the distributed consensus tracking error in the controller design of MASs with heterogeneous unknown uncertain powers. Moreover, it should be stressed out that the global control of heterogenous power-chained MASs with unknown input deadzone nonlinearities and unknown control coefficients is still an unsolved problem.

Inspired by the aforementioned discussion, the paper proposes the design of a distributed control for uncertain MASs with a directed graph. Compared to the relevant existing researches in the literature, the main contributions of this paper can be concluded as follows. (1)The investigated control scheme is able to relax the aforementioned four restrictions in the existing consensus researches [26–30] for power-chained MASs with input deadzone nonlinearities(2)To deal with the time-varying difficulty of multi-agent control coefficients, a second-order filter is exploited to make each agent generate the estimations of the leader signals, thus avoiding to use the asymmetric Laplacian matrix of the directed topology(3)To solve the unknown nonlinearities in MAS, barrier functions are applied to the prescribed tracking distributed protocol, which make the specified tracking performance guaranteed; thus, the convergence of the controlled system is proved, and all the closed signals are globally bounded

#### 2. Problem Formulation

##### 2.1. Problem Statement

In this subsection, the analysis on control system with nonlinear deadzone is given first. Then, the new mathematical model under nonlinear model with asymmetric deadzone is constructed.

In practical physical systems, certain abnormal events and emergencies may happen in time of operation, which will result in the failure of the actuator. Besides, as a result of the practical constraints on manufacturing and assembly process, the actual output of actuators often presents the characteristic of asymmetric deadzone nonlinearity. The actual control input nonlinear model can be modelled as follows. where is the control input, is the control output with nonlinear deadzone, and are unknown time-varying positive continuous functions denoting the left and right slopes of the deadzone, and are unknown parameters representing the break-points of deadzone nonlinearity, as shown in Figure 1.

According to (1), for the purpose of facilitating the control law design, the deadzone function can be modelled as

with

The uncertain high-order nonlinear MASs with -th order agents can be modelled by the following dynamic equations where are the states with initial conditions , , , , is the actual control input, is the output; is all unknown odd integers. The system nonlinearities and are unknown continuous functions of . Without loss of generality, the unknown control coefficients are supposed to be strictly positive.

The desired trajectory for the subsystem output is bounded and known only to some subsystems, and is bounded and unknown to all subsystems.

The goal of the paper is to design a distributed tracking protocol for the multiagent system (4) with heterogenous high powers under the directed graph, so that all the agents’ output tracking errors (i.e., , ) converge to the predesigned transient and steady state performance boundaries, and simultaneously, make sure all signals of the closed loop system bounded.

##### 2.2. Algebraic Graph Theory

Let the telecommunication network among agents be denoted by a directed graph , where represents the index set corresponding to each node with being the -th agent, represents the set of the edges, represents the adjacency matrix. denotes an edge of the graph , and indicates that there is an agent -to-agent communication. The element of corresponding to the edge represents the quality of communication between the agents and (i.e., ), otherwise, . is referred to as an undirected graph if and only if . Stipulate that the diagonal elements . Clearly, for a directed graph, is unsymmetrical. The in-degree matrix , with being the -th row sum of . Moreover, the Laplacian matrix for the directed digraph can be defined as . Besides, defining to denote the communication directed topology between the agents and the desired trajectory , where means that is directly accessible by agent , otherwise, . A sequence of edges in a graph of the form is called a path. The following notation is utilized over all the paper. Have and being two vectors, next, a vector operator is defined as .

*Assumption 1. *The directed graph includes a directed spanning tree, and the desired trajectory is accessible to at least one subsystem.

*Assumption 2. *There are unknown and continuous functions and nonnegative such that and .

Lemma 3 (see [1]). *For any real number , time-varying function in field of real number and any odd integer , the following inequality is always true
*

Lemma 4 (see [31]). *Let and be real variables. Then, for any integers , and any real number , the following inequality is always true
*

*Remark 5. *Assumption 1 and Assumption 2 are not violated conditions. Under a directed topology, the expected trajectory can only be accessible to a subset of followers (i.e., ) in Assumption 1. In Assumption 2, denotes the left and right slopes of the deadzone when and (i.e., when and , ). In engineering practice, the slope is usually bounded, because for any real physical system, the slope of the control input cannot be infinite, which means that there must be an upper bound of . Simultaneously, the unknown input nonlinearity function is bounded obviously due to the upper boundness of . Besides, it is usually supposed that (i.e., see Assumption 2 in [32]), but this assumption cannot be satisfied when there are unknown deadzone nonlinearities because the derivative between the breakpoints is zero [24], (i.e., , for ). In this work, when , (i.e., the aforementioned positive nature requirement on , will not restrict a zero value appearing in the dead-band as ). Therefore, Assumption 2 is realistic and reasonable.

*Remark 6. *In this paper, the time-varying system nonlinearities and are unknown continuous functions, such that little knowledge can be used to establish the control scheme. To handle the difficulty, neural networks and fuzzy logic [14–16] are utilized for approximating the unknown functions as a consequence of the system nonlinearities and . But only semiglobal results can be got due to the employment of the approximators. It is a challenge problem to establish a distributed protocol for the unknown nonlinear MASs with global consensus, which will be solved skillfully by cooperating two-order filters and barrier functions as follows.

#### 3. Main Results

In this section, first, we design a filter for each agent to generate estimate information from the leader. Subsequently, a distributed asymptotic tracking controller for an uncertain MAS with deadzone input nonlinearities and heterogenous high powers will be designed. Finally, we shall demonstrate that it results in the solution for the problem of predesigned performance for (4).

##### 3.1. Filter Design

To facilitate the distributed control design, a filter is utilized for each agent , where .

Denote where , . Then, design the filters as with where , , and are positive constant parameters selected as , , and , is the upper boundary of , is the upper boundary of , .

Theorem 7. *Consider a closed-loop system composed of filters (8) under Assumption 1 with control design (9). The asymptotic and consistent tracking of all outputs of the filter to can be realized (i.e., , ). In addition, and are bounded.*

*Proof. *Based on Assumption 1, the matrix is nonsingular. Same as [5], define , , , where for . It can be summarized from [5] that is positive definite.

Construct the following Lyapunov function
where with , , , , , and is a constant satisfying with . Denote , , and , . Then, the following equation can be obtained
Through (11) and (12), by denoting , the time derivative of is
It should be noticed that , , , , and , with representing the minimum eigenvalue of .

Combing these above inequality, (13) becomes
where , . It is easy to verify that and . Thus, it can be obtained from (14) that and accordingly, . The boundedness of and can be deduced on the basis of the boundedness of and . The proof is completed.

##### 3.2. Control Scheme

In this subsection, the distributed tracking control scheme is constructed cooperating with the filter (8). We define the error variables , , and as where is the intermediate control signal, which will be defined later.

Define and select an output performance function for where , and the constant design parameters , , and are chosen appropriately to satisfy and with any given initial condition .

Define the barrier functions as where and are the normalized errors. The -th virtual control signals , , and are designed as where is a positive control parameter. At this stage, the actual tracking control signal is designed as where is a positive control parameter.

*Remark 8. *It is worth pointing out that the output performance function is applied to limit the error within the imposed time-varying constraints. Given any initial condition, the constants , is selected satisfying , and represents the maximum allowable value of the steady-state , which can be assigned to an arbitrarily small value. Hence, the convergence of to a predefined set of arbitrary small residuals is realized. Furthermore, the rate of decline is influenced by and results in the lower bound of the required rate of convergence. Therefore, is restricted to a preassigned boundary by using the performance function.

*Remark 9. *Under transient and steady state responses, for the construction of (17), the proposed control scheme (19) and (20) merely made up of nonlinear transformation error surfaces and design constants can be designed to preallocate the tracking performance boundary.

##### 3.3. Stability and Performance Analysis

Theorem 10. *Consider system (4) obeying Assumption 1 and Assumption 2 controlled by the virtual control signals (19) and the proposed distributed controller (20), all the signals in the closed-loop system are globally bounded. Then, we have the following properties:
*(1)*The normalized error , , and satisfies , which can be guaranteed by prespecified tracking performance*(2)*The outputs of each agent eventually satisfy , where *

*Proof. *From the definition of the errors, the states can be rewritten as
Let , from the definition of , we can get that
By denoting , , the time derivative of can be given by
where .

The performance functions have been chosen to meet requirements as , which equals to where an open set with , . Additionally, the fact that from (17), the desired trajectory and the performance functions are bounded and continuously differentiable with respect to time. The intermediate control signals , , and the control law are smooth over the set . It is deduced that is bounded and piecewise continuous in and locally Lipschitz on over . According to Theorem 54 of [33], the conditions on ensure the existence and uniqueness of a maximal solution of (22)–(24) over the set , for , such that , or equivalently that for , the following formula holds
Since is well established.

Next, will be proved by looking for a contradiction. It is supposed that , thus, the following analysis is performed. And a systematic procedure for the proof of the aforementioned statements is given below for .

*Step 1. *From (17) and (28), for any , is bounded as
where is a constant such that . Moreover, the boundedness of results in the boundedness of the functions as
where is a positive nonlinear function.

By denoting , establish the first Lyapunov function candidate as

Substituting (18) and (22), the time derivative of is derived as follows

Using (30) and yields where and is a positive constant.

Applying Lemma 3 to the terms , using and , one has where and are both positive constants satisfying .

Combining (33) and (34), we obtain where .

Applying Lemma 4 to the terms , one has

Therefore, (35) becomes where , . From (37), for all , it follows that is negative when , where and subsequently that , which implies that the normalized error is bounded as for . According to (19), for all , the boundedness of results in the boundedness of . Additionally, from , we conclude that

Step : construct the Lyapunov function as

Differentiating along (40) with respect to time, using (26), yields

According to the boundedness of in the last step, using , , and (28), is bounded as , , and . Furthermore, the boundedness of results in the boundedness of . From (38), for all , is also bounded. Moreover, is bounded. Similar to (30) and (33), there is a nonlinear function , and we have where and is a positive constant.

Applying Lemma 3 to the terms , using , (42) becomes where , , , and are both constants satisfying .

Then, applying Lemma 4 to the terms yields

Thus, (43) becomes where , . From (45), we conclude that when , where and subsequently that for all . Additionally, for all by using (18), we get

From , (46) implies that

In addition, according to (19), the boundedness of results in the boundedness of , for all .

Step : construct the final Lyapunov function as

Differentiating along (48) with respect to time and invoking (18) and (24), one has

It can be deduced that for all , is bounded from (46). Based on the boundedness of in the step. Additionally, from (17), (28), it is obviously that is bounded as for all . From (19), is also bounded. Then, the boundedness of results in the boundedness of . Therefore, there is a constant satisfying , such that

Under Assumption 2 and Lemma 3, invoking (20) and the previous analysis yields where is an unknown positive constant satisfying , with an unknown positive continuous function . Invoking (51) and Assumption 2, we have where with an unknown positive constant satisfying , and are both positive constants satisfying .

Substituting (20) into the third term of (52) yields

Applying Lemma 4 to the term , we obtain

Thus, (53) becomes where and . Then, where is negative when , where , we obtain for all . Invoking (18), for , we get

From , (56) implies that

As a result, due to (20), the control signal is bounded from the boundedness of . Moreover, for all