Elsevier

Automatica

Volume 82, August 2017, Pages 218-225
Automatica

Brief paper
Cooperative control of multiple stochastic high-order nonlinear systems

https://doi.org/10.1016/j.automatica.2017.04.052Get rights and content

Abstract

Distributed cooperative control of multiple stochastic high-order nonlinear systems has not been addressed in literature. This paper presents an approach to design of distributed cooperative controllers for multiple stochastic high-order nonlinear systems under directed leader–followers type network topology via the so-called distributed integrator backstepping method. By using the algebraic graph theory and stochastic analysis method, it is shown that the output tracking errors between the followers and the leader can be tuned arbitrarily small while all the states of the closed-loop system remain bounded in probability. Finally, the effectiveness of the proposed control approach is illustrated on a stochastic underactuated mechanical system.

Introduction

This paper considers distributed cooperative control of a network composed of N stochastic high-order nonlinear systems as followers and one leader. The followers’ dynamics are described as follows: dxij=(xi,j+1pij+fij(x̄ij))dt+gij(x̄ij)dω,j=1,,ni1,dxi,ni=(uipi,ni+fi,ni(x̄i,ni))dt+gi,ni(x̄i,ni)dω,yi=xi1, where x̄ij=(xi1,,xij)TRj, uiR, yiR are the state, input, output of the ith follower, respectively, i=1,,N. pijRodd1={qR:q1  and  q  is a ratio of odd integers} is called the “high-order” of system (1). ω is an m-dimensional independent standard Wiener process defined on the complete probability space (Ω,F,Ft,P) with a filtration Ft satisfying the typical conditions. Specifically, Ft is increasing and right continuous while F0 contains all P-null sets. The functions fij(x̄ij):RjR, gij(x̄ij):RjR1×m, i=1,,N, j=1,,ni, are known C1 functions. The leader’s output is defined as y0(t)R.

For the single agent case of system (1) (i.e.  N=1), there are fruitful results based on basic stability theory of stochastic control systems (Khasminskii, 2012, Kushner, 1967). For example, when p1j=1 for all j=1,,ni, Deng and Krstić (1999), Deng, Krstić, and Williams (2001), Krstić and Deng (1998) and Pan and Basar (1999) respectively introduce two kinds of Lyapunov functions for controller design: quadratic Lyapunov functions multiplied by different weighting functions and quartic Lyapunov functions. Subsequently, these design techniques are further developed by Liu et al., 2008, Liu et al., 2007 and Wu, Xie, and Zhang (2007). When p1j>1, j=1,,ni, Li, Jing, and Zhang (2011) and Xie and Tian (2007) address the state-feedback stabilization problem and Li, Xie, and Zhang (2011) study the output-feedback controller design by using homogeneous theory.

For linear multi-agent systems with stochastic noise (i.e.  N>1), Li, Li, Xie, and Zhang (2016), Li and Zhang (2009) and Ma, Li, and Zhang (2010) solve the cooperative control problems with various systems structure. When it turns to the cooperative control of stochastic nonlinear multi-agent system (1), there are only a few results with pij=1 for all i=1,,N, j=1,,ni in (1). Specifically, Song, Cao, and Yu (2010) present a pinning control and achieve leader-following consensus for multi-agent systems described by nonlinear second-order dynamics. Meng, Lin, and Ren (2013) study the distributed robust cooperative tracking problem for multiple non-identical second-order nonlinear systems with bounded external disturbances. Zhang and Lewis (2012) investigate the cooperative tracking control problem of higher-order nonlinear systems with Brunovsky form. Li and Zhang (2014) focus on the stochastic multi-agent systems with inherent nonlinear drift and diffusion terms. However, when some pij’s satisfy pij>1 and stochastic noise exists in system (1), to the best of authors’ knowledge, there is no result reported in open literature.

The following stochastic underactuated mechanical system gives a practical example of such stochastic high-order nonlinear system (1).

Example 1

The stochastic underactuated mechanical system (Xie & Duan, 2010), see Fig. 1, is described by: θ̈=glsinθ+ksm2l(xlsinθ)3cosθ,ẍ=km1xksm1(xlsinθ)3+vm1. The following coordinate changes x1=θ,x2=ẋ1,x3=(xlsinθ)(ksm2l)2cosθ3,x4=ksm2l3ẋ3,u=kscosx1m13m2l3v, transform the mechanical system (1) to dx1=x2dt,dx2=x33dt+f2(x1,x2)dt,dx3=x4dt,dx4=udt+f4(x1,x2,x3,x4)dt+g4(x1,x2,x3,x4)dω,y=x1, where f2(x1,x2), f4(x1,x2,x3,x4) and g4(x1,x2,x3,x4) are C1 functions whose concrete form can be found in Xie and Duan (2010). Obviously, (3) is a special form of the stochastic high-order nonlinear system (1). We will use this example for simulation in Section  4.

In this paper, the cooperative control problem of system (1) is investigated under a directed graph topology. By using the algebra graph theory and stochastic analysis method, distributed controllers are designed to ensure that the tracking error converges to an arbitrarily small neighborhood of zero. The main challenges of this work are twofold:

  • (1)

    This paper is the first result on cooperative control of system (1). Different from the traditional results for a single stochastic high-order nonlinear system (Li, Jing et al., 2011, Li, Xie et al., 2011, Xie and Tian, 2007), distributed cooperative control for system (1) needs to consider the interactions among agents, which makes the controller design and stability analysis of the closed-loop systems much more difficult. New design tools and stability analysis technique should be developed.

  • (2)

    Since some pij’s are allowed to satisfy pij>1 in system (1), the system considered in this paper is more general than those in Meng et al. (2013), Song et al. (2010) and Zhang and Lewis (2012). Due to the effect of pij’s, the Jacobian linearizations are neither controllable nor feedback linearizable, thus the existing design tools are hardly applicable and some new design technique should be developed.

The remainder of this paper is organized as follows. Section  2 is on preliminaries. Section  3 presents controllers design and stability analysis. Section  4 applies the theoretical results to cooperative control of multiple stochastic mechanical systems. Section  5 includes some concluding remarks.

Section snippets

Graph theory

For a given vector or matrix X, XT denotes its transpose. Tr{X} denotes its trace when X is square, and |X| is the Euclidean norm of a vector X. Let G=(V,E,A) be the followers’ weighted digraph of order N with the set of nodes V={1,2,,N}, set of arcs EV×V, and a weighted adjacency matrix A=(aij)N×N with nonnegative elements. The set of neighbors of vertex i is denoted by Ni={jV: j can directly send information to agent i, ij}. aij>0 if node j is a neighbor of node i and aij=0 otherwise.

Distributed controllers design and stability analysis

In this section, we focus on the distributed controllers design and stability analysis for system (1).

To proceed further, we need the following assumptions.

Assumption 1

p11=p21==pN1=1.

Assumption 2

The leader is globally reachable in Ḡ.

Assumption 3

The leader’s output y0(t)R and ẏ0(t) are bounded, and the bound is only available to a subgroup of followers whose neighbors contain the leader.

Remark 1

It can be observed from (3) that stochastic underactuated mechanical systems is a special class of stochastic high-order nonlinear systems.

A simulation example

Example 2

Consider cooperative control of a network composed of three followers and one leader. The followers’ dynamics are described by (3).

The communication topology of the concerned network is described in Fig. 2.

The goal is to design distributed controllers such that the outputs of followers can track the leader’s output y0(t)=0.5sint.

Define the output tracking errors as: ei=yiy0,i=1,2,3.

Initial conditions are randomly set as x11(0)=3.2, x12(0)=1, x13(0)=0.2, x14(0)=1, x21(0)=0.1, x22(0)=0.1, x23(0

Concluding remarks

This paper studies the cooperative control problem of multiple stochastic high-order nonlinear systems under directed network topology. The designed distributed controllers can ensure that the output tracking errors between the followers and the leader can be tuned arbitrarily small while all the states of the closed-loop system remain bounded in probability.

It is interesting to see whether or not these results can be extended to cooperative control of multiple stochastic nonlinear systems with

Wuquan Li received the Ph.D. degree in College of Information Science and Engineering, Northeastern University, China, in 2011. From 2012 to 2014, he carried out his postdoctoral research with Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, China. Since January 2011, he has been with School of Mathematics and Statistics Science, Ludong University, where he is currently an associate professor. He is a Young Taishan Scholar. His research

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    Wuquan Li received the Ph.D. degree in College of Information Science and Engineering, Northeastern University, China, in 2011. From 2012 to 2014, he carried out his postdoctoral research with Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, China. Since January 2011, he has been with School of Mathematics and Statistics Science, Ludong University, where he is currently an associate professor. He is a Young Taishan Scholar. His research interests include stochastic nonlinear systems control and cooperative control of multi-agent systems.

    Lu Liu received her Ph.D. degree in 2008 in the Department of Mechanical and Automation Engineering, Chinese University of Hong Kong, Hong Kong. From 2009 to 2012, she was an Assistant Professor in The University of Tokyo, Japan, and then a Lecturer in The University of Nottingham, United Kingdom. She is currently an Assistant Professor in the Department of Mechanical and Biomedical Engineering, City University of Hong Kong, Hong Kong. Her research interests are primarily in networked dynamical systems, control theory and applications, and biomedical devices. She received the Best Paper Award (Guan Zhao-Zhi Award) in the 27th Chinese Control Conference in 2008.

    Gang Feng received the B.Eng and M.Eng. degrees in Automatic Control from Nanjing Aeronautical Institute, China in 1982 and in 1984 respectively, and the Ph.D. degree in Electrical Engineering from the University of Melbourne, Australia in 1992. He has been with City University of Hong Kong since 2000 where he is now Chair Professor of Mechatronic Engineering. He was lecturer/senior lecturer at School of Electrical Engineering, University of New South Wales, Australia, 1992–1999. He was awarded an Alexander von Humboldt Fellowship in 1997, and the IEEE Transactions on Fuzzy Systems Outstanding Paper Award in 2007, and Changjiang chair professorship from Education Ministry of China in 2009. His current research interests include multi-agent systems and control, intelligent systems and control, and networked systems and control. Prof. Feng is an IEEE Fellow, an associate editor of IEEE Trans. Fuzzy Systems and Journal of Systems Science and Complexity, and was an associate editor of IEEE Trans. Automatic Control, IEEE Trans. Systems, Man & Cybernetics, Part C, Mechatronics, and Journal of Control Theory and Applications.

    This work was supported by National Natural Science Foundation of China under Grant (No. 61573179), the Young Taishan Scholars Program of Shandong Province of China under Grant (No. tsqn20161043), and grants from the Research Grants Council of the Hong Kong Special Administrative Region of China (Nos. CityU/11203714 and CityU/11213415). The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Xiaobo Tan under the direction of Editor Miroslav Krstic.

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