1. Introduction
The control of uncertain nonlinear systems has been researched for several decades, such that so many remarkable results have been obtained on this topic [
1,
2,
3,
4,
5,
6,
7,
8,
9]. However, most of them are for SISO or MIMO systems, and their methods or techniques cannot be directly applying to multi-agent systems, as the information of each agent or subsystems is only available for part of others. According to the topology of information transformation graph, the graph can be divided into undirected and directed graphs. Generally, the consensus control of a MAS with the directed graph is more difficult than the undirected case, since the methods for the directed case are always applicable for the undirected case, but not vice versa.
Recently, some significant progress has been made in the control of a MAS [
10,
11,
12]. For a linear MAS with undirect graphs, fully distributed adaptive consensus controller is present in [
10]. Adaptive asymptotically consensus for an uncertain MAS is achieved in [
11], and adaptive asymptotically consensus is achieved in [
12] for an uncertain MAS, and so on. However, their methods are only applicable for a MAS with an undirected graph and are in vain for a MAS with a directed graph. For a MAS with a directed graph and constant control coefficients, adaptive consensus for a MAS with system nonlinearities satisfying match conditions is researched in [
13] to solve the problem of actuator faults; a fully distributed adaptive consensus control is studied for a MAS with unknown control directions in [
14] by using a Nussbaum gain technique; actuator faults in a MAS are considered in [
15] with integral chain dynamics; and prescribed performance consensus control for uncertain MAS is investigated in [
16]. Though much progress has been made [
17,
18,
19,
20], it should be noted that there are still some nonnegligible problems to be solved. Firstly, the existing methods require the control coefficients to be constants, or even known, for a MAS with a directed graph. The main difficulty is that the Laplace matrix for a directed graph is asymmetric and thus the selections of control parameters must always resort to adaptive methods, which falls into trouble when the control coefficients are time-varying and unknown. Secondly, to the best of our knowledge, there is no global consensus control method for a MAS with a directed graph and the systems functions thereof completely unknown, except for [
21], wherein the unknown system nonlinearities required to satisfy the Lipschitz conditions and control coefficients are one. Universal approximators such as neural networks (NN) or fuzzy logic systems (FLS) have been attempted to solve the consensus control problem of a MAS with completely unknown system nonlinearities [
22,
23,
24], however, it is well known that these methods are semi-global in the sense that their stabilities depend on the initial conditions of systems and the careful selection of controller parameters. Therefore, NN or FLS-based approaches cannot guarantee the global consensus of the MAS, though they are very favorable to solve the problem of MAS with unknown nonlinearities.
As for the global control of systems with completely unknown nonlinearities, a pioneering work is [
25], wherein a low-complexity controller is presented that cannot only achieve global convergence of all the system signals, but which can also guarantee the prescribed performance of tracking error and state errors. In view of the low complexity and strong robustness of this method, much research has been carried on this method for solving different nonlinear control problems [
26,
27,
28,
29,
30]. By introducing a novel barrier function, a fault-tolerant controller is designed for a class of unknown nonlinear systems in [
26]. With consideration to the constraints of system states, a barrier function-based adaptive control method is proposed in [
27]. Addressing systems with unknown control direction and system dynamics, a Nussbaum function-based low-complexity control scheme is designed in [
28]. As regards asymptotic tracking control for systems with unknown nonlinearities, an universal global low-complexity controller is proposed in [
31]. Nevertheless, it is worth mentioning that the global control of a MAS with unknown nonlinearities is still an unsolved problem, since these methods are based on the condition that the desired output for systems are known, but this knowledge cannot be obtained for some agents of a MAS. Moreover, considering the control coefficients of each agent are time-varying functions, these traditional methods will fall into trouble when solving for the consensus control of a MAS with unknown dynamics.
Motived by the above discussion, we investigate the fully distributed control of a MAS with a directed graph, time-varying control coefficients and completely unknown system nonlinearities. The main contributions of this paper are summarized as follows.
- (1)
To address the time-varying control coefficients of a MAS, a two-order filter is firstly designed for each agent to produce estimates of the signals from the leader, so that an asymmetric Laplace matrix for a directed graph will not be used to design the controller for each agent of the MAS, by which the difficulty of control design is solved.
- (2)
To address the completely unknown system nonlinearities in MAS, barrier functions are used to propose a fully distributed controller by combining novel filters; barrier functions are well-suited to dealing with the effects of unknown system nonlinearities, such that global results are achieved, for the first time, in a MAS with completely unknown system nonlinearities in this paper.
- (3)
To guarantee the prescribed tracking performance by the proposed controller, such that the consensus of the controlled MAS is rigorously proved and all the closed signals are globally bounded.
2. Problem Statement and Preliminaries
Consider a class of uncertain MAS as follows
where
,
,
, are the states, the control input and the output of the
th subsystem, respectively. The system nonlinearities
are unknown continuous functions with respect to
.
represent the system uncertainties and external disturbances.
The desired trajectory for the outputs of the subsystems is bounded and only known by part of the subsystems, with being bounded and unknown to all subsystems.
Suppose that the information transmission condition among the group of subsystems can be represented by a directed graph , where denotes the set of indexes corresponding to each subsystem. The edge indicates that subsystem could obtain information from subsystem , but not necessarily vice versa. In this case, subsystem is called a neighbor of subsystem , and vice versa. Denoting the set of neighbors for subsystem as . Self-edging is not allowed, thus and . The connectivity matrix of is defined as if and if . An in-degree matrix is introduced, such that with being the th row sum of . Then, the Laplacian matrix of is defined as . Defining , where means the is accessible directly by subsystem , and otherwise, we have . Throughout this paper, the following notations are used. is the Euclidean norm of a vector. Letting and be two vectors, then define the vector operator as . Letting be a matrix, then denotes the minimum eigenvalue of .
Assumption 1. The directed graphcontains a spanning tree, and the desired trajectoryis accessible to at least one subsystem, i.e.,.
Assumption 2. There exist unknown local Lipschitz functions such that, for Assumption 3. The unknown control coefficients is strictly positive or negative. Without a loss of generality, it is assumed to be strictly positive, namely, for Lemma 1. (Ref. [17]) Based on Assumption 1, the matrixis nonsingular. Definingthen for and is definitely positive. Remark 1. In contrast to the methods in [13,14,15,16] for a MAS with a directed graph, the control coefficients, , are time-varying and unknown continuous functions in this paper, which makes the control design much more difficult, since the matrixin (4) is always unknown and required to be estimated adaptively while the unknown control coefficientsmakeinestimable. To cope with this problem, a novel two-order filter will be given for each agent (shown later). Remark 2. The system nonlinearities, and, are completely unknown functions so that there is little knowledge with which to construct the controller. To deal with this problem, neural networks and fuzzy logic systems have been used to approximate the unknown functions caused by the system nonlinearitiesand in [22,23,24], however, only semi-global results can be obtained by use of these approximators. To construct a distributed controller for a MAS with these unknown system nonlinearities with global consensus is a challenging problem, which is solved by the skillfull cooperation of novel two-order filters and barrier functions in the following. 4. Stability Analysis
In this section, we will give the main results with the designed fully distributed controller and present the stability analysis. The main results of this article are as follows.
Theorem 1. Consider the closed-loop system consisting of uncertain agents as (1) satisfying Assumptions 1–3, the intermediate control signals (21) and the distributed controller (22). Then, we have the following properties:
- (1)
All the signals in the closed-loop system are globally bounded
- (2)
Prespecified tracking performance can be guaranteed, namely,, for.
- (3)
The output of each agent ultimately satisfies.
Proof (of Theorem 1). From (18), (19) and (21), we have
It can be observed from (23) that
is continuous function of
,
and
, where
and
are bounded time-varying functions. Thus,
can be rewritten as the form of continuous function of
and
. Similar analysis can be made for
. Therefore, we obtain
where
, and
are some continuous functions. Defining
and in view of (25) and (26), we obtain
Let us define the open set:
It is easily observed that
. Additionally,
are continuous with respect to all its variables, owing to the fact that
are all continuous differentiable functions. Therefore, it follows from [
32] that the conditions on
ensure the existence and uniqueness of a maximal solution
on the time interval
, namely,
for
, which implies
for
and
.
In the following, we will prove that by seeking a contradiction. Suppose that ; then the related analysis is performed as follows, and all of what follows is based on .
Step 1: Consider the following positive definite functions
Let
. It follows from (21), (24), (25) and (30) that the time derivative of
is
where
and
. Note that
,
and
are bounded on
because (23) and (29), respectively. Utilizing the fact that
are bounded and employing the extreme value theorem, owing to the continuity of
,
and
, we arrive at
where
,
, and
are some unknown positive constants.
Then, substituting (32) and (33) into (31) yields
From (34), it follows that
is negative when
and subsequently that
which implies
As a result, the control signal
is bounded. Moreover, invoking (24), we also can conclude the boundedness of
. Therefore, the time derivative of
is
where
Noting (36) and using the same analysis as (33), it also easy to conclude the boundedness of , and hence .
Step
: Consider the following positive definite functions
Let
. In a similar fashion to that in the former step, by noting Assumption 1, it follows from (21), (24), (26) and (39) that the time derivative of
is
where
and
. Noting that
are bounded on
because the boundedness of
,
and
are bounded on
in view of (29). Utilizing the fact that
are bounded and employing the extreme value theorem owing to the continuity of
,
and
, we arrive at
with
,
and
being some unknown positive constants.
Then, substituting (41) and (42) into (40) yields
From (43), it follows that
is negative when
and subsequently that
which implies
As a result, the control signal
is bounded. Moreover, we also can conclude the boundedness of
by noting (24). Finally, the time derivative of
is
where
Noting (45) and using the same analysis as (42), it also easy to conclude the boundedness of and hence .
Step
: Consider the following Lyapunov functions
Let
. Similar as the former steps, we can have
where
and
are some unknown positive constants. It follows from (49) that
is negative when
and subsequently that
which implies
As a result, the control signal
is bounded. Moreover, we also can conclude the boundedness of
. Notice that (36), (45) and (51) imply that
, where the set
is nonempty and compact, defined as
Owing to (36), (45) and (51) it is straightforward to verify that
. Therefore, assuming
dictates the existence of a time instant
, such that
, which is a clear contradiction. Therefore,
. Hence, all closed-loop signals remain bounded and moreover
. Furthermore, from (36) we conclude that
Then, for all
. In view of Lemma 2 and (52), we have
This completes the proof.
5. Simulation Study
Two examples will be given to demonstrate the effectiveness of the proposed distributed adaptive controller in this section, as follows.
Example 1. Consider the following multi-agent systemswith the system functions chosen as follows: ,, , , , , , , , ,,, , , , . The communication topology for these subsystems are depicted inFigure 1.
The desired trajectory for the outputs of each subsystem is
. The initial conditions for each subsystems are set as:
,
,
,
and
. Then, the intermediate control signals are designed and the distributed controllers are designed as follows
where their control parameters and functions are selected as:
,
,
,
,
,
,
and
,
for
. For the filters, the parameters are chosen as:
and
. Then, the simulation results are reported as
Figure 2,
Figure 3 and
Figure 4.
It can be observed from
Figure 2,
Figure 3 and
Figure 4 that under the designed distributed controllers, the outputs of the subsystems track the desired trajectory very quick, and the tracking performance is satisfactory.
Example 2. Consider the consensus for four high-maneuver fighters, with communication topologies as in Figure 5 and their flight control systems as follows [33].withwhere are the roll angle, attack angle, sideslip angle, roll angular velocity, pitching angular velocity, yaw angular velocity and pitch angle of fighter, respectively. .
are the left and right elevators, left and right ailerons, front and rear flaps, and rudder, respectively. Detailed explanations for the parameters and variables of this model can be found in [26]. Suppose that they are all flying at an altitude of 40,000 feet, at a speed of 0.6 Mach. The desired output for these fighters is. The signal is only available for fighter 1. According to Theorem 1, we design the distributed flight controller as follows
with
where
,
and
are the signals produced by filter (6) with
being the filter inputs, respectively.
and
for
, and
represents the pseudo-inverse for
.
For the purposes of comparison, we use the control method of [
17] under the same conditions. Following [
17], the controller for the distributed flight controller is designed as follows
where the variables and controller parameters are the same as in our proposed methods. The simulation results are then reported in
Figure 6,
Figure 7,
Figure 8,
Figure 9 and
Figure 10. In
Figure 6, the dotted curves denote the outputs of fighters under the control of the method in [
17], while the solid curves denote the outputs of fighters under the control of method in this paper. It can be seen from
Figure 6 that our control performance is better than [
17] since the outputs of ours track the desired value more accurately.
Figure 7,
Figure 8,
Figure 9 and
Figure 10 show the actions of actuators of four fighters under our method.
Figure 11 show the controller performance of our method and that from [
17]. In
Figure 11, the blue curves denote the control efforts
of the fighters with our method, while the red curves denote the control efforts
of Fighters in the method from [
17], where
and
are defined as
It can be seen from
Figure 11 that, initially, the control efforts of our method are greater than those in [
17], and finally, there is little difference in effort between these methods, which means that the control performance of our method is better under similar control efforts.
It can be seen from these results that the consensus between the four fighters is achieved and the tracking performance is very good, while fairly good control performance is achieved.