# Distributed Average Tracking for Lipschitz-Type Nonlinear Dynamical Systems

In this paper, a distributed average tracking problem is studied for Lipschitz-type nonlinear dynamical systems. The objective is to design distributed average tracking algorithms for locally interactive agents to track the average of multiple reference signals. Here, in both the agents' and the reference signals' dynamics, there is a nonlinear term satisfying the Lipschitz-type condition. Three types of distributed average tracking algorithms are designed. First, based on state-dependent-gain designing approaches, a robust distributed average tracking algorithm is developed to solve distributed average tracking problems without requiring the same initial condition. Second, by using a gain adaption scheme, an adaptive distributed average tracking algorithm is proposed in this paper to remove the requirement that the Lipschitz constant is known for agents. Third, to reduce chattering and make the algorithms easier to implement, a continuous distributed average tracking algorithm based on a time-varying boundary layer is further designed as a continuous approximation of the previous discontinuous distributed average tracking algorithms.

## Authors

• 24 publications
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## 1 Introduction

In the past two decades, there have been lots of interests in the distributed cooperative control [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], and [13], for multi-agent systems due to its potential applications in formation flying, path planning and so forth. Distributed average tracking, as a generalization of consensus and cooperative tracking problems, has received increasing attentions and been applied in many different perspectives, such as distributed sensor networks [14], [15] and distributed coordination [16], [17]. For practical applications, distributed average tracking should be investigated for signals modeled by more and more complex dynamical systems.

The objective of distributed average tracking problems is to design a distributed algorithm for multi-agent systems to track the average of multiple reference signals. The motivation of this problem comes from the coordinated tracking for multiple camera systems. Spurred by the pioneering works in [18], and [19] on the distributed average tracking via linear algorithms, real applications of related results can be found in distributed sensor fusion [14], [15], and formation control [16]. In [20], distributed average tracking problems were investigated by considering the robustness to initial errors in algorithms. The above-mentioned results are important for scientific researchers to build up a general framework to investigate this topic. However, a common assumption in the above works is that the multiple reference signals are constants [19] or achieving to values [18]. In practical applications, reference signals may be produced by more general dynamics. For this reason, a class of nonlinear algorithms were designed in [21] to track multiple reference signals with bounded deviations. Then, based on non-smooth control approaches, a couple of distributed algorithms were proposed in [22] and [23] for agents to track arbitrary time-varying reference signals with bounded deviations and bounded second deviations, respectively. Using discontinuous algorithms, further, [24] studied the distributed average tracking problems for multiple signals generated by linear dynamics.

Motivated by the above mentioned observations, this paper is devoted to solving the distributed average tracking problem for Lipschitz-type nonlinear dynamical systems. Three DAT algorithms are proposed in this paper. First of all, based on relative states of neighboring agents, a class of distributed discontinuous DAT algorithms are proposed with robustness to initial conditions. Then, a novel class of distributed algorithms with adaptive coupling strengths are designed by utilizing an adaptive control technique. Different from [22], [23] and [24], the proposed algorithms are based on node adaptive lows. Further, a class of continuous algorithms are given to reduce chattering. Compared with the above existing results, the contributions of this paper are three-fold. First, main results of this paper extend the dynamics of the reference signals and agents from linear systems [22] and [23] to nonlinear systems, which can describe more complex dynamics. Second, by using adaptive control approaches, the requirements of all global information are removed, which greatly reduce the computational complexity for large-scale networks. Third, compared with existing results in [24], new continuous algorithms are redesigned via the boundary layer concept to reduce the chattering phenomenon. Continuous algorithms in this paper is more appropriate for real engineering applications.

Notations: Let and be sets of real numbers and real matrices, respectively.

represents the identity matrix of dimension

. Denote by

a column vector with all entries equal to one. The matrix inequality

means that is positive (semi-) definite. Denote by the Kronecker product of matrices and . For a vector , let denote the 2-norm of , , . For a set , represents the number of elements in .

## 2 Preliminaries

### 2.1 Graph Theory

An undirected (simple) graph is specified by a vertex set and an edge set whose elements characterize the incidence relations between distinct pairs of . The notation is used to denote that node is connected to node , or equivalently, . We make use of the incidence matrix, , for a graph with an arbitrary orientation, i.e., a graph whose edges have a head (a terminal node) and a tail (an initial node). The columns of are then indexed by the edge set, and the th row entry takes the value if it is the initial node of the corresponding edge, if it is the terminal node, and zero otherwise. The diagonal matrix of the graph contains the degree of each vertex on its diagonal. The adjacency matrix, , is the symmetric matrix with zero in the diagonal and one in the th position if node is adjacent to node . The graph Laplacian [25] of , , is a rank deficient positive semi-definite matrix.

An undirected path between node and node on undirected graph means a sequence of ordered undirected edges with the form . A graph is said to be connected if there exists a path between each pair of distinct nodes.

###### Assumption 1.

Graph is undirected and connected.

###### Lemma 1.

[25] Under Assumption 1

, zero is a simple eigenvalue of

with

as an eigenvector and all the other eigenvalues are positive. Moreover, the smallest nonzero eigenvalue

of L satisfies .

## 3 Main results

### 3.1 Robust distributed average tracking algorithms design

Consider a multi-agent system consisting of physical agents described by the following nonlinear dynamics

 ˙xi(t)=Axi(t)+Bf(xi,t)+Bui, (1)

where and both are constant matrices with compatible dimensions, and is the state and control input of the th agent, respectively, and is a nonlinear function. Suppose that there is a time-varying reference signal, , which generated by the following Lipschitz-type nonlinear dynamical systems:

 ˙ri(t)=Ari(t)+Bf(ri,t), (2)

where is the state of the th reference signal.

It is assumed that agent has access to , and agent can obtain the relative information from its neighbors denoted by .

###### Assumption 2.

is stabilizable.

The main objective of this paper is to design a class of distributed controller for physical agent in (1) to track the average of multiple reference signals generated by the general nonlinear dynamics (2), i.e.,

 limt→∞(xi(t)−1NN∑i=1ri(t))=0,

where each agent has only local interaction with its neighbors.

###### Assumption 3.

For , and , the nonlinear function satisfies a Lipschitz-type condition: , where and .

As it was mentioned, there are many applications that the physical agents should track a time varying trajectory, where each agent has an incomplete copy of this trajectory. While, the physical agents and reference trajectory might be described by more complicated dynamics rather than the linear dynamics in real applications. Therefore, we consider a more general group of physical agents, where the nonlinear function in their dynamics satisfies the Lipschitz-type condition.

Therefore, a distributed average tracking controller algorithm is designed as

 ui(t) = K1(pi(t)−ri(t))+K2~xi(t)+μϕih[K2~xi(t)] (3) +αϑiBh(∑j∈NiK1(pi(t)−pj(t))),

with a distributed average tracking filter algorithm is proposed as follows:

 pi(t) = si(t)+ri(t), ˙si(t) = Asi(t)+BK1(pi(t)−ri(t)) (4) +αϑiBh(∑j∈NiK1(pi(t)−pj(t))),

where , , are the states of the DAT algorithm, , and state-dependent time-varying parameters, , , and constant parameters, and control gain matrices, respectively, to be determined.

Then, using the controller (3) for (1), one gets the tracking error system

 ˙~xi(t) = (A+BK2)~xi(t)+B(f(xi,t)−f(ri,t)) (5) +μϕiBh[K2~xi(t)].

Following from (2) and (3.1), one gets

 ˙pi(t) = (A+BK1)pi(t)−BK1ri(t)+Bf(ri,t) (6) + αϑiBh(∑j∈NiK1(pi(t)−pj(t))).

Let , , , , , , and . In matrix form, one obtains the closed-loop system as follows:

 ˙~x(t) = (I⊗(A+BK2))~x(t)+(I⊗B)(F(x,t)−F(r,t)) (7) +μ(Φ⊗B)H[(I⊗K2)~x(t)],

with

 ˙p(t) = (I⊗(A+BK1))p(t)−(I⊗BK1)r(t) (8) + (I⊗B)F(r,t)+α(Θ⊗B)H((L⊗K1)p(t)),

where

 H((I⊗K2)~x(t))=⎛⎜ ⎜⎝h(K2~x1(t))⋮h(K2~xN(t))⎞⎟ ⎟⎠,

and

 H((L⊗K1)p(t))=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝h(∑j∈N1K2(p1(t)−pj(t)))⋮h(∑j∈NNK2(pN(t)−pj(t)))⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠.

Define . Then satisfies following properties: Firstly, it is easy to see that is a simple eigenvalue of with as the corresponding right eigenvector and is the other eigenvalue with multiplicity , i.e., . Secondly, since , one has . Finally, .

Define , where . Then, it follows that if and only if . Therefore, the consensus problem of (6) is solved if and only if asymptotically converges to zero. Hereafter, we refer to as the consensus error. By noting that , it is not difficult to obtain from (8) that the consensus error satisfies

 ˙ξ(t) = (M⊗(A+BK1))ξ(t)−(M⊗BK1)r(t) (9) + α(MΘ⊗B)H(L⊗K1)ξ(t)+(M⊗B)F(r,t).

Algorithm 1: Under Assumptions 1 and 2, for multiple reference signals in (2), the distributed average tracking algorithms (3.1) and (3) can be constructed as follows

1. Solve the following algebraic Ricatti equations (AREs):

 PiA+ATPi−PiBBTPi+Qi=0, (10)

with to obtain matrices , where . Then, choose .

2. Choose the parameters , and .

###### Theorem 1.

Under Assumptions 1-3, by using the distributed average tracking controller algorithm (3) with the distributed average tracking filter algorithm (3.1), the state in (1) will track the average of multiple reference signals , generated by the Lipschitz-type nonlinear dynamical systems (2) if the parameters , , , and the feedback gains are designed by Algorithm 1.

Proof: The proof contains three steps. First, it is proved that for the th agent, . Consider the Lyapunov function candidate

 V1(t)=ξT(L⊗P1)ξ. (11)

By the definition of , it is easy to see that . For the connected graph , it then follows from Lemma 1 that

 V1(t)≥λ2λmin(P1)∥ξ∥2, (12)

where is the smallest eigenvalue of the positive matrix . The time derivative of along (9) can be obtained as follows

 ˙V1 = ˙ξT(L⊗P1)ξ+ξT(L⊗P1)˙ξ (13) = ξT(M⊗(A+BK1)T)(L⊗P1)ξ +ξT(L⊗P1)(M⊗(A+BK1))ξ −2ξT(L⊗P1)(M⊗BK1)r(t) +2αξT(L⊗P1)(MΘ⊗B)H(L⊗K1)ξ(t) +2ξT(L⊗P1)(M⊗B)F(r,t).

Substituting into (13), it follows from the fact and Assumption 3 that

 ˙V1 = ξT[L⊗(ATP1+P1A)−2(L⊗P1BBTP1)]ξ (14) +2ξT(L⊗P1BBTP1)r(t) −2αξT(LΘ⊗PB)H[(L⊗BTP1)ξ] +2ξT(L⊗P1B)F(r,t) = ξT[L⊗(ATP1+P1A)−2(L⊗P1BBTP1)]ξ +2N∑i=1(∑j∈Ni[BTP1(ξi(t)−ξj(t))])TBTP1ri −2αN∑i=1ϑi(∑j∈Ni[BTP1(ξi(t)−ξj(t))])T h(∑j∈Ni[BTP1(ξi(t)−ξj(t))]) +2N∑i=1(∑j∈Ni[BTP1(ξi(t)−ξj(t))])T[f(ri,t)−f(0,t)] ≤ ξT[L⊗(ATP1+P1A)−2(L⊗P1BBTP1)]ξ +2∥∥∥N∑i=1(∑j∈Ni[BTP1(ξi(t)−ξj(t))])T∥∥∥∥BTP1ri∥ −2αN∑i=1ϑi∥∥∥∑j∈Ni[BTP1(ξi(t)−ξj(t))]∥∥∥ +2N∑i=1∥∥∥∑j∈Ni[BTP1(ξi(t)−ξj(t))]∥∥∥∥f(ri,t)−f(0,t)∥ ≤ ξT[L⊗(ATP1+P1A)−2(L⊗P1BBTP1)]ξ −2αN∑i=1ϑi∥∥∥∑j∈Ni[BTP1(ξi(t)−ξj(t))]∥∥∥ +2N∑i=1∥∥∥∑j∈Ni[BTP1(ξi(t)−ξj(t))]∥∥∥(γ+∥BTP∥1)∥ri∥ = ξT[L⊗(ATP1+P1A)−2(L⊗P1BBTP1)]ξ −2N∑i=1[(α−γ−∥BTP1∥)∥ri∥+αβ] ∥∥∥∑j∈Ni[BTP1(ξi(t)−ξj(t))]∥∥∥.

Since , one has

 ˙V1 ≤ ξT(L⊗(P1A+ATP1−2P1BBTP1))ξ (15) ≤ λ2ξT(I⊗(P1A+ATP1−2P1BBTP1))ξ.

It follows from (10) that . Therefore, we have

 ˙V1 < −η1V1, (16)

where . Thus, one has

 limt→∞ξi(t)=limt→∞(pi(t)−1NN∑k=1pk(t))=0.

Second, it is proved that . Let . It follows from (2) that

 ˙r∗(t)=Ar∗(t)+1NBN∑i=1f(ri(t),t). (17)

Let . It follows from (2) that

 ˙p∗(t) = (A+BK1)p∗(t)−BK1r∗(t)+1NBN∑i=1f(ri(t),t) (18) + αN∑i=1ϑih(∑j∈NiK1(pi(t)−pj(t))).

Denote , one has

 ˙ζ(t) = ˙p∗(t)−˙r∗(t) (19) = (A+BK1)p∗(t)−BK1r∗(t)−Ar∗(t) + αN∑i=1ϑih(∑j∈NiK1(pi(t)−pj(t))) = (A+BK1)ζ(t)+ω(t),

where . We then use input-to-state stability to analyze the system (19) by treating the term as the input and as the states. Since (10) with , one has is Hurwitz. Thus, the system (19) with zero input is exponentially stable and hence input-to-state stable. Since . One has . Thus, it follows that , which implies that . Therefore, one obtains .
Third, it is proofed that . Consider the candidate Lyapunov function

 V2=~xT(I⊗P2)~x, (20)

with . By taking the derivative of along (7), one gets

 ˙V2 = ~xT(I⊗((A+BK2)TP2+P2(A+BK2)))~x (21) +2~xT(I⊗P2B)(F(x,t)−F(r,t)) +2μ(Φ⊗P2B)H[(I⊗K2)~x(t)].

Using , one has

 ˙V2 = ~xT(I⊗(ATP2+P2A−2P2BBTP2))~x (22) +2~xT(I⊗P2B)(F(x,t)−F(r,t)) −2μ~xT(Φ⊗P2B)H[(I⊗BTP2)~x(t)] = ~xT(I⊗(ATP2+P2A−2P2BBTP2))~x +2N∑i=1(BTP2~xi(t))T(f(xi,t)−f(ri,t)) −2μN∑i=1ϕi(BTP2~xi(t))Th(BTP2~xi) ≤ ~xT(I⊗(ATP2+P2A−2P2BBTP2))~x +2N∑i=1∥BTP2~xi(t)∥∥(f(xi,t)−f(ri,t))∥ −2μN∑i=1ϕi∥BTP2~xi(t)∥ ≤ ~xT(I⊗(ATP2+P2A−2P2BBTP2))~x +2N∑i=1∥BTP2~xi(t)∥γ∥xi−ri∥ −2μN∑i=1(∥xi−ri∥+ν)∥BTP2~xi(t)∥ ≤ ~xT(I⊗(ATP2+P2A−2P2BBTP2))~x −2N∑i=1((μ−γ)∥xi−ri∥+μν)∥BTP2~xi(t)∥.

Since and , one has

 ˙V2 ≤ ~xT(I⊗(ATP2+P2A−2P2BBTP2))~x. (23)

Using , one has

 ˙V2 ≤ −η2V2. (24)

where . Thus, one has . Therefore, the distributed average tracking problem is solved. This completes the proof.

### 3.2 Adaptive distributed average tracking algorithms design

Note that, in above subsection, the proposed distributed average tracking algorithms (3) and (3.1) require that the parameters and satisfy the conditions and , which depend the Lipschitz constant . Since the is a global information, for a local agent, it becomes difficult to obtain . Therefore, to overcome the global information restriction, we design an adaptive distributed average tracking controller algorithm

 ui(t) = K1(pi(t)−ri(t))+K2~xi(t)+μi(t)ϕih[K2~xi(t)] (25) +αi(t)ϑiBh(∑j∈NiK1(pi(t)−pj(t))),

and an adaptive distributed average tracking filter algorithm

 pi(t) = si(t)+ri(t), ˙si(t) = Asi(t)+BK1(pi(t)−ri(t)) (26) +αi(t)ϑiBh(∑j∈NiK1(pi(t)−pj(t))),

with two time-varying parameters and satisfying the following adaptive update strategies:

 ˙μi(t)=κiϕi∥K2˜xi(t)∥, (27)

and

 ˙αi(t)=χiϑi∥∥∥∑j∈NiK1(ξi(t)−ξj(t))∥∥∥, (28)

respectively, where are adaptive parameters to be determined.