Adaptive output consensus for heterogeneous nonlinear multi-agent systems with multi-type input constraints under switching-directed topologies

Wei Zhang 1 · Zhaoxu Yu 1 · Shugang Li

Abstract This study concentrates on solving the output consensus problem for a class of heterogeneous uncertain nonstrict-feedback nonlinear multi-agent systems under switching-directed communication topologies, in which all followers are subjected to multi-type input constraints such as unknown asymmetric saturation, unknown dead-zone and their integration. A unif ied representation is presented to overcome the diffi culties originating from multi-agent input constraints. Moreover, the uncertain system functions in a non-lower triangular form and the interaction terms among agents are dealt with by exploiting the fuzzy logic systems and their special property. Furthermore, by introducing a nonlinear f ilter to alleviate the problem of“explosion of complexity” during the backstepping design, a distributed common adaptive control protocol is proposed to ensure that the synchronization errors converge to a small neighborhood of the origin despite the existence of multiple input constraints and arbitrary switching communication topologies. Both stability analysis and simulation results are conducted to show the eff ectiveness and performance of the proposed control methodology.

Keywords Nonlinear multi-agent system · Output consensus · Fuzzy logic system · Input constraint · Switching topology

1 Introduction

Multi-agent systems have received extensive attention in the past 2 decades due to their potential applications in various f ields, such as satellite formation, unmanned air vehicles, mobile robots, and so on [ 1- 4]. The leader-following consensus problems including state consensus and output consensus have been a fundamental issue for the research of multi-agent systems in [ 5- 19]. Especially, it is worth pointing out that the nonlinear dynamics in [ 5- 7] must be homogeneous. That is to say, all the agents must contain the identical dynamics. Actually, the agents often have diff erent dynamics, such that the multi-agent systems with heterogeneous dynamics are widely considered in [ 8- 16]. Fuzzy logic system (FLS) or neural networks have been used to deal with the heterogeneous dynamics in [ 10- 16]. Besides,all the subsystem functions of agent in a nonstrict-feedback form are related to the whole state variables; high-order nonstrict-feedback nonlinear multi-agent systems have a more representative form in comparison with the other nonlinear multi-agent systems, and there are many practical systems which can fall into this category. Thus, high-order nonstrict-feedback nonlinear multi-agent systems have been intensively studied in [ 17- 19]. However, the aforementioned results did not consider input constraints.

Note that actuators cannot respond to the command signal promptly and saturation is inevitable to restrict the magnitude of the command signal; dead-zone and saturation are two typical nonsmooth input nonlinearities in practical systems. Both dead-zone and saturation can severely degrade the control performance. Therefore, the investigations of dead-zone and saturation have been drawn a sustainable interest in the control community for a long time, and many techniques were presented to overcome the eff ects of such nonlinearities [ 20- 27]. Nevertheless, the results obtained in [ 20- 27] were only concerned with a single-agent system. For multi-agent nonlinear systems with input nonlinearities, some interesting control strategies were proposed in [ 28- 33], but the results in [ 28- 31] did consider deadzone or saturation individually. Actually, both dead-zone and saturation often happen in the same actuator, and it is quite diffi cult for the backstepping design to build the proper model representing the integration of dead-zone and saturation. Hence, few results on the consensus problem for heterogeneous uncertain nonlinear systems with integrating dead-zone and saturation have been published. In [ 33], the output consensus problem was solved for nonlinear multiagent systems subject to both linear dead-zone and asymmetric saturation, but the results are obtained based on a f ixed directed graph rather than the switching topologies.

In fact, the interaction topologies of practical multi-agent systems are often unreliable due to the constrained sensing region of sensors or the eff ect of obstacles [ 34]. In comparison with a plenty of research results on multi-agent systems with f ixed communication topology [ 5- 19, 28- 33], fewer works are paid attention to solving consensus problem under switching topologies. By constraining the time interval between the consecutive switching, i.e., dwell time, a number of multi-agent systems with switching topologies were studied in [ 35- 39] and some eff ective control protocols were presented for such systems. Besides, based on the fact that the graph is f ixed during some time intervals, several distributed adaptive consensus control schemes were proposed for nonlinear multi-agent systems under switching topologies in[ 40, 41]. However, the switching mechanism of communication topologies may be usually unknown or too complicated to be applied in the stability analysis and control design,such that it is necessary to investigate the case of arbitrary switching topologies. For a class of leader-following linear multi-agent systems under arbitrary switching topology, the output-feedback consensus tracking problem has been studied in [ 42] and suffi cient conditions are obtained to reach consensus in terms of linear matrix inequalities. In addition,a common Lyapunov function (CLF) is proposed to deal with the consensus problem of linear multi-agent system under switching-directed topologies in [ 43]. Unfortunately,still not enough attention has been paid to the problem of distributed adaptive control for uncertain nonlinear multiagent systems in the presence of multi-type input constraints and arbitrary switching topologies.

The preceding discussion motivates us to tackle the output consensus problem of heterogeneous uncertain nonlinear nonstrict-feedback multi-agent systems with multi-type input constraints under arbitrary switching communication topologies. In contrast to the existing literature, the main contributions of this paper are listed as follows:

1. It is the f irst time to address the adaptive output consensus problem of heterogeneous nonstrict-feedback nonlinear multi-agent systems with multi-type input constraints under arbitrary switching-directed graphs.Especially, the nonlinear multi-agent system with switching communication topologies is seen as switched nonlinear system, then an eff ective distributed adaptive control protocol is developed to attain the output consensus for such systems using the CLF method.

2. Unlike the results in [ 28- 31] and [ 33], where all the followers must be subject to the same input constraint, this paper considers the diff erent input constraints including not only asymmetric saturation or nonlinear dead-zone separately but also the integration of dead-zone and saturation for the followers. Furthermore, a unif ied model is proposed to cope with these input constraints.

3. Both the nonstrict-feedback structure and the interaction terms among agents are handled using the FLS and its property, such that the algebraic loop problem in designing a controller for such systems can be avoided.Besides, by replacing the conventional f irst-order f ilters (e.g., see in [ 10, 29, 41]) with some novel nonlinear f ilters, an improved dynamic surface control (DSC)method is proposed to alleviate the problem of “explosion of complexity” during the backstepping design.

2 Problem formulation and preliminaries

2.1 Graph theory

2.2 FLS and its approximation

Lemma 1Let f(Z)be a continuous function def ined on a compact set ΩZ.Then for any given positive constant ε,there exists a FLS WTΦ(Z)to satisfy[44]

2.3 Problem formulation

Suppose that there exists a multi-agent system with one leader labeled as 0 andNfollowers labeled as 1,2,…,N.The dynamics of theith (i∈V) follower can be described by

wheremi,l(ui) andmi,r(ui) are unknown smooth functions,andbi,l＜0 andbi,r＞0 are constants.

To facilitate the control design, some following assumptions on dead-zone are necessary.

Assumption 1 The output of dead-zone is unavailable [ 21].

Assumption 2 For smooth functionsmi,l(ui) andmi,r(ui) ,there exist unknown positive constantski,l0,ki,l1,ki,r0, andki,r1such that

whereui,Mandui,mrepresent the upper and lower bounds of the saturation nonlinearity, respectively. Similar to [ 27], both Gaussian error function and mean value theorem are used to represent the piecewise function ( 8). Then, fori∈I2,φi(ui)can be written as follows:

where

Remark 1As for the considered multi-agent nonlinear systems, some statements are given as follows:

(1) Noticed that the functionsfi,jandgi,jdepend on the whole state variablesxi,1,…,xi.ni; thus, system ( 3) is called as nonstrict-feedback form and can be used to represent various practical engineering systems, such as biochemical process, f light systems, manipulators,and so on.

(2) Though input nonlinearity integrating dead-zone and saturation has been addressed in [ 33], the dead-zone model is linear rather than nonlinear. Here, a new model ( 11) is proposed to describe the integration of nonlinear dead-zone and asymmetric saturation.

(3) The f ixed network topology can be taken as a special case of switching topologies when the index setQcontains only one element.

Def inition 1 For the nonlinear multi-agent systems ( 3) under the switching-directed topologies, the distributed consensus tracking errors between the followers and the leader are called to be cooperatively semiglobally uniformly ultimately bounded (CSGUUB) if for anyyi(t0)-yd(t0)∈Ωi,0withydbeing the output of leader agent andΩi,0being a given compact set, there exist constant∈＞0 and timeT(yi(t0)-yd(t0),∈) , such that ‖y-yd‖≤∈holds for?t≥t0+T, wherey=[y1,…,yN]Tandyd=[yd,…,yd]T[ 45].

The objective of this paper is to design the eff ective distributed adaptive consensus control protocolsui(i∈V)for the heterogeneous follower agents ( 3) under arbitrary switching-directed graphs, such that all the tracking errors are CSGUUB, i.e., there exist a constant∈＞0 and a timeTto satisfy |yi(t)-yd(t)|≤∈for allt≥T.

To attain the above control objective, some helpful assumptions and lemmas are introduced as follows.

Assumption 5 The leader’s outputydand its derivatives˙ydand¨ydare continuous and bounded. Moreover, they only can be available for theith agent satisfying 0∈Nki.

Remark 2There are some further statements on the above assumptions as follows: (1) As mentioned in [ 18], Assumption 3 means that the outputyiofith follower may track the leader’s outputydby developing the distributed control protocol appropriately fork,?k∈Q. Moreover, Assumption 3 also shows thatis true for anyi∈V andk∈Q;(2) Assumption 4 means that the constantg*iis not required to be known. Thus, no extra strict restriction is imposed on the system ( 3); (3) Assumptions 3- 5 are common in the existing literature, e.g. [ 18, 39- 41].

3 Main results

3.1 Distributed adaptive control design

In this section, FLS approximation and DSC approach are combined to develop a distributed adaptive control design procedure for the heterogeneous nonlinear multi-agent systems under the switching-directed topologies.

For anyi∈V andk∈Q, based on the directed graph Gk,the neighboring synchronization errorzi,1is f irst def ined as follows:

whereτi,j＞0 are time constants of the f ilters,ei,j∶=si,j-αi,jare boundary layer errors with the virtual control functionsαi,jbeing given below,κi,j＞0 are the constants to be determined, and＾Mi,jare the estimations of the unknown constantsMi,jto be given later. Let～Mi,j=＾Mi,j-Mi,jbe the estimation errors, and the updating laws of＾Mi,jare constructed as

whereγi,j＞0 andχi,j＞0 are the adjusted parameters.

For anyk∈Qand theith (i∈V) follower agent, the common adaptive control protocols are designed as follows:

wherepi,j＞0 andσi,j＞0 are the designed parameters forj=1,2,…,ni,＾θi,j(j=1,2,…,ni) are the adaptive parameters to be specif ied during the backstepping design. Furthermore, forj=1,2,…,ni, the adaptive laws of＾θi,jare given by

It is inspiring that the dynamics ( 20) of synchronization errorzi,jcan be taken as the switched nonlinear systems with the switching signal produced by the change of communication topologies according to [ 47]. Furthermore, the switching mechanisms of topologies in many cases are unknown or too complicated to be used in the consensus analysis and control design of each follower agent. Therefore, the CLF approach can be extended to solve the consensus control problem of multi-agent systems under arbitrary switching topologies [ 48].

First, to overcome the diffi culty from the system uncertainties of followers, the following unknown functioni,1,kis def ined and approximated by a FLS

With the help of ( 1), ( 21) and Lemma 4, one has

Some functions are lumped to be an unknown function,j,k,and a FLS is used to approximate it, i.e.,

Substituting ( 17), ( 19), ( 28) and ( 32) into ( 31), one has

StepniTo conduct the actual control design, the following unif ied description is exploited to represent the multitype input constraints

Accordingly, for anyi∈V , the dynamical system ofzi,nican be given by

Similar to ( 23) and ( 32), and substituting ( 19), ( 33)-( 36) and( 38) into ( 37) yields

3.2 Stability analysis

In this section, stability analysis for the whole closed-loop multi-agent systems is presented. First, according to the def inition ofei,jand ( 15), we have

Next, some continuous functionsBi,j,kare def ined as

For anyk∈Q,i∈V andj=1,2,…,ni-1 , there must exist positive constantsMi,jto satisfy |Bi,j,k|≤Mi,jon the compact setΩwhich will be given later. Therefore, it is easy to derive that

Theorem 1Consider the uncertain nonstrict-feedback nonlinear multi-agent systems(3)satisfying Assumptions1-5under switching-directed communication topologies,if the distributed adaptive control protocols(18)together with virtual control functions(17),nonlinear f ilters(15),and adaptive laws(16)and(19)are utilized. Then, for any initial condition satisfying V(0)≤R0and θ ＾i,j(0)＞0(i∈V,j=1,2,…,ni),the consensus tracking errors areCSGUUB and can remain in an enough small neighborhood of the origin by adjusting the parameters appropriately.

ProofApplying the def inition ofVand ( 46) gives

Fig. 1 Three communication topologies1-3

Fig. 2 The switching mechanism among 1-3

Remark 4Note that the repeated diff erentiations of virtual control functions must result in the problem of “explosion of complexity” during the traditional backstepping design;the conventional DSC (CDSC) method often introduce the f irst-order f ilter to solve the problem (e.g., see in [ 10, 21,29, 41]). However, the eff ects ofBi,j(·) in the dynamics of boundary layer errors are not compensated for the CDSC technique, which usually causes the degradation of synchronization performance. Motivated by the design of f ilter in [ 49], a novel nonlinear f ilter ( 15) and parameter adjusting laws ( 16) are included to compensate for the unknown boundsMi,jofBi,j(·).

4 Simulation example

To verify the eff ectiveness and applicability of the proposed control strategy, a set of one-link manipulators borrowed from [ 16] is considered. In the simulation, the communication topologies among one leader labeled as 0 and three followers labeled as 1-3 are shown in Fig. 1,and Fig. 2 exhibits the switching mechanism among three topologies1-3. The output of leader is chosen asyd=sin0.5t+sint, and the models of theith (i=1,2,3)follower agent are described by

whereqi,˙qi,¨qi, andμirepresent the link position, velocity,acceleration and the torque, respectively.videnotes the control input of theith follower. Moreover,μd,iandτd,idenote system uncertainties and model errors. Letxi,1=qi,xi,2=˙qi,andxi,3=μi, and take the multitype input constraints into consideration, then system ( 51) can be written as

Fig. 3 The output trajectories of three followers and one leader

Fig. 4 The input signals φ1(u1) and u1

The simulation results are shown in Figs. 3, 4, 5 and 6.Figure 3 depicts the output trajectories of three followers and one leader, which demonstrates the satisfactory consensus performance of the proposed distributed control protocol.Figures 4, 5 and 6 reveal the input signals of three followers satisfying diff erent constraints, respectively.

Fig. 5 The input signals φ2(u2) and u2

Fig. 6 The input signals φ3(u3) and u3

Remark 5In light of the preceding simulation results, it is worth pointing out that some small perturbations may occur with the switching of communication topologies, but the satisfactory tracking performance can recover quickly from the small perturbations using the common distributed adaptive control protocol presented in this paper.

5 Conclusions

This paper has studied the output consensus problem for a class of heterogeneous nonstrict-feedback multi-agent systems with multi-type input constraints under arbitrary switching-directed topologies. A distributed adaptive control strategy is developed to acquire the output consensus of such multi-agent systems. Meanwhile, the nonlinear f ilter is constructed to improve the CDSC method. Besides, the multitype input constraints including nonlinear dead-zone and asymmetric saturation are dealt with by proposing an unif ied method. Both stability analysis and simulation studies are provided to show the eff ectiveness of the proposed control protocol. In the future work, it is necessary to explore the distributed output-feedback control problem for heterogeneous multi-agent systems under the switching-directed communication topologies.

AcknowledgementsThis work was partially supported by the Chinese National Natural Science Foundation (No. 71871135), and the Fundamental Research Funds for the Central Universities (Nos.222201714055, 222201717006).