Distribution consensus of nonlinear stochastic multi-agent systems based on sliding-mode control with probability density function compensation

https://doi.org/10.1016/j.jfranklin.2020.07.011Get rights and content

Abstract

Strict consensus is difficult to be implemented due to the stochastic behavior of multi-agent systems (MASs), so a new concept, distribution consensus, is proposed here to keep the agents’ consensus in the stochastic sense, i.e., the output errors do not converge to a fixed value but follow a desired distribution function. The appropriate control protocol, with the output error probability density function (PDF) as the target, is designed based on the combination of sliding mode control and PDF compensation. Sliding mode control is the core part to ensure the whole system’s stability, and the PDF compensator is used to compensate the random variation and reduce the chattering effect, respectively. In order to realize the complete control in real time, the PDF compensator is modeling by a radial basis function (RBF) neural network and its optimal control law is calculated by the iterative training of RBF network weights. Finally, the effectiveness of the proposed method is verified by MASs simulations with three different communication topologies. The PDF compensator can greatly improve the consensus effect for the nonlinear stochastic MASs.

Introduction

Multi-agent system (MAS) is evolved from distributed artificial intelligence, in which the complex control target cannot be solved by an agent alone. Multi-agent systems (MASs) usually have the characteristics of autonomy, distribution and coordination, and have the ability of self-organization, learning and reasoning [1], [2]. Therefore, the control problem of MASs, especially the consensus problem, is a hot and challenging research in present.

For example, Cai studied the leader-following consensus problem for multiple uncertain Euler-Lagrange systems under some restrictive assumptions on the network topology and proposed a distributed adaptive control law [3]. Yu investigated the distributed finite-time consensus problem of second-order MASs in the presence of bounded disturbances [4]. Cao addressed the leader-follower consensus tracking problem for MASs with identical general linear dynamics and unknown external disturbances and proposed observer-based consensus algorithm [5]. Li studied the robust consensus problem of a network of general discrete-time linear agents coordinating through unreliable communication channels [6]. Tang proposed a partial differential equations model of wave by using a pilot-following strategy and designed a distributed internal controller composed of Kelvin Voigt damping term to drive the follower [7]. Li investigated the output consensus problem of heterogeneous stochastic nonlinear MASs with directed communication topologies [8]. Liu addressed the event-based leader-following consensus of a class of MASs with switching networks [9]. Huang studied the robust consensus problem of a class of linear MASs with time-varying topology by using a non-linear method [10]. Liu studied the scaling consensus of a class of heterogeneous linear first-order MASs and proposed an adaptive consensus algorithm with a pilot-following structure [11]. Li studied the mean square scale consensus of MASs with Markovian switching topology and communication noise [12]. Du designed distributed fixed-time observers and tracking controllers, studied the problem of distributed fixed-time consensus for a class of heterogeneous nonlinear leader-follower MASs [13]. Wu considered distributed bipartite tracking consensus problem for nonlinear MASs subject to logarithmic quantization [14]. Li constructed an adaptive event triggering protocol based on the correlation information among agents, which is used for the consensus of linear MASs on undirected graphs [15].

The MASs are often stochastic in practical applications, so the study of stochastic systems is very important. The mean and variance are usually used as the goal to achieve consensus in the cooperative control of linear Gaussian MAS. Wang focused on the consensus control problems of grouped agents with linear or linearized nominal dynamics in stochastic framework [16]. Chen proposed an efficient framework of distributed protocol to ensure the finite time consensus of the stochastic MASs under the topology of undirected communication [17]. Ren studied the mean square consensus of nonlinear MASs with state dependent noise disturbance [18]. These conventional consensus algorithms have good results to the linear stochastic system with Gaussian randomness. However, the randomness usually is non-Gaussian for the non-linear stochastic MAS, so it is very important to consider the consensus directly in the perspective of the probabilistic sense.

Among many methods of controlling the stochastic systems, probability density function (PDF) control is an effective strategy for non-Gaussian stochastic systems. The control objective of this method is the PDF of output variables, but not the output itself [19]. Huang and Yue proposed a new method to model and control the shape of the output PDF of dynamic stochastic systems with arbitrary bounded random inputs [20]. Michael G presented a regulatory controller synthesis technique to make the output converge to a preselected PDF for the closed-loop process. The proposed design technique is referred to as PDF-shaping [21]. Michael G also extended a regulatory controller synthesis technique that was previously introduced for first-order processes to the higher-order processes [22]. A predictive control algorithm for PDF was presented based on the rational square-root B-spline based PDF model in 2009 [23]. A switching linear controller is proposed as an alternative for PDF shaping in 2011 [24]. Zhu proposed an innovative design procedure to control a nonlinear system subject to Poisson white noise excitation to target a specified stationary PDF [25]. Wang proposed a novel strategy in which the controller parameters are optimally obtained through the improved particle swarm optimization algorithm [26].

Furthermore, there are many kinds of control methods for MASs. Sliding mode variable structure control is a special kind of non-linear control in nature which has the advantages of fast response, insensitive to parameter changes and disturbances, no on-line identification of the system, and simple physical implementation [27]. Wu discussed dissipative consensus problems for multi-agent networks [28]. Mu investigated a novel integral sliding mode control(SMC) strategy for the waypoint tracking control of a quadrotor in the presence of model uncertainties and external disturbances [29]. N. Rahimi investigated an adaptive observer-based consensus control strategy for general nonlinear MASs [30]. Wang designed discontinuous and continuous sliding mode protocols to achieve finite-time consensus in spite of the disturbances based on the disturbance observer [31]. Huang proposed a new adaptive SMC method based on radial basis function (RBF), which combines the advantages of adaptive, neural network and inaccurate system model information SMC [32]. Combining the methods mentioned above, a SMC method based on PDF shaped is proposed [33]. Compared with the traditional SMC method, it can reduce the chattering effect of sliding mode and achieve good control effect. However, this method has not been applied to MAS.

In this paper, a SMC method based on PDF compensation is used to control the nonlinear stochastic MASs. PDF compensator based on RBF neural network and SMC are combined to achieve the distribution consensus of error PDF graphics for each agent. SMC has many important advantages, such as good robustness, but the chattering effect is inevitable. In this paper, PDF compensator based on RBF neural network not only can reduce the chattering effect of sliding mode to a certain extent, but also can suppress the random variations to keep system output in a desired distribution. Unlike the traditional consensus, the distribution consensus which achieved in this paper is not based on the strict consensus of position and speed, but on the final PDF shape of the tracking error. The simulation results show that the method not only achieves the distributed consensus of MASs, but also achieves good control results.

The rest of this paper is organized as follows. Section 2 introduces the concepts required needed by this method and describes the distribution consensus problems to be solved. Section 3 describes the design of the control protocol, including the design of sliding mode controller and PDF compensator. Section 4 presents the stability analysis of the control strategy. Section 5 shows the experimental simulation and results analysis. Section 6 draws the conclusions.

Notation: The superscript represents the matrix transpose. R is the space of the real numbers. Let Rn × n be the real matrix with dimension n × n Euclidean space, Rn be the real matrix with dimension n Euclidean space. In ∈ Rn × n is the identity matrix.

Section snippets

Preliminaries

The control method aiming at distribution consensus combines SMC with PDF compensator which is constructed by RBF neural network. The PDF compensator can reduce the chattering effect of sliding mode very well, and reshape the error to the desired distribution.

SMC-based PDF control algorithm

The whole control system is composed of SMC and PDF control shown in Fig. 3. SMC strategy is used as the core controller that aims to control the system output into steady status. This is a key step since the following PDF estimation and compensation are valuable only for stationary random variable. The PDF part is used as the compensator to fine adjust the output error as the desired shape. uiPDF is construed as RBF neural network, whose weight is adjusted by minimizing the K-L divergence

Convergence analysis

Consider the following Lyapunov functionV(i)=E{i=1nSiT(k)Si2Ciω¯iSi(k)+Ci2ω¯i2}ΔV(i)=E{i=1n[Si(k+1)2Si(k)2]2Ciω¯i[Si(k+1)Si(k)]}=E{i=1n((h21)Si2(k)+Ci2ωi2(k)2hSi(k)αsat(k)+(CiBiuiPDF)2+2hSi(k)Ciωi(k)+2hSi(k)CiBiuiPDF2Ciωi(k)αsat(k)+α2sat(k)22αsat(k)CiBiuiPDF+2Ci2Biωi(k)uiPDF2Ciω¯iSi(k+1)+2Ciω¯iSi(k))}=E{i=1n((h21)Si2(k)+Ci2ωi2(k)2hSi(k)αsat(k)+(CiBiuiPDF)2+2hSi(k)Ciωi(k)+2hSi(k)CiBiuiPDF2Ciωi(k)αsat(k)+α2sat(k)22αsat(k)CiBiuiPDF+2Ci2Biωi(k)uiPDF2Ciω¯i(h1)Si(k)2Ci2ω¯iωi(k)2Ci

Case 1

Consider the nonlinear system modeling from a real robot arm [33]:[xi1(k+1)xi2(k+1)]=Ai[xi1(k)xi2(k)]+Biui(k)+ε(xi(k))+[0ωi(k)][yi1(k)yi2(k)]=In[xi1(k)xi2(k)]+[di1(k)di2(k)]where i=1,...,5, Ai=[10.0101], i=1,...,3, Ai=[1.010.0101.01], i=4,5, Bi=[00.01], In is a unit matrix. ωi(k) is a β-distribution noise withωi(k)=0.01×β(0.2,1).εi(k)=[fi1(k)fi2(k)], where fij(k),j=1,2 represents unmodeled uncertainties and disturbances withfi1(k)=0.002sin(0.2πk)+0.005xi1(k)xi2(k)2andfi2(k)=(0.01sin(0.02πk)+0.01

Conclusion

Compared with the traditional MAS control problem, this paper introduces a new concept, distribution consensus. The control target is transformed to the desired error probability density function. Distribution consensus is different from traditional consensus because it is to keep the agent’s consensus in the stochastic sense. The output errors do not converge to a fixed value but follow a desired distribution function. The advantage of this distribution consensus is that it allows errors to

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This research is funded by the National Natural Science Foundation of China (61973023, 61573050) and Beijing Natural Science Foundation (4202052).

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