# Learning nonlinear dynamics in synchronization of knowledge-based leader-following networks

Knowledge-based leader-following synchronization problem of heterogeneous nonlinear multi-agent systems is challenging since the leader's dynamic information is unknown to all follower nodes. This paper proposes a learning-based fully distributed observer for a class of nonlinear leader systems, which can simultaneously learn the leader's dynamics and states. The class of leader dynamics considered here does not require a bounded Jacobian matrix. Based on this learning-based distributed observer, we further synthesize an adaptive distributed control law for solving the leader-following synchronization problem of multiple Euler-Lagrange systems subject to an uncertain nonlinear leader system. The results are illustrated by a simulation example.

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12/24/2020

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## I Introduction

Synchronization of networked multi-agent systems has found its applications in various scenarios, such as flocking (zhang2014route), coupled distributed estimation (su2012cooperative; olfati2012coupled), and formation of multiple robots (feng2021finite), and has been extensively studied for the past several decades. A central class of synchronization problems are the so-called leader-following synchronization problems (also called cooperative tracking problems (zhang2012adaptive)), which aim to drive each follower system to behave in the same motion or oscillate concurrently with the leader system (chen2014pattern; isidori2014robust; radenkovic2018distributed).

According to wang2006theoretical, leader-following synchronization problems can be divided into two categories: power-based leader-following synchronization problems and knowledge-based leader-following synchronization problems. For the former one, both followers and the leader node share the same system dynamics. For the latter one, the leader’s dynamic knowledge (i.e., parameters) is unaccessible to any followers and each follower adaptively learns the leader’s dynamic knowledge through neighborhood interaction. In this sense, the leader is also called an uncertain leader in the literature (wanghuang2018; baldi2020distributed).

However, in practical applications, dynamics of the leader are often described by a nonlinear system (wangslotine2004; wang2006theoretical; zhang2012adaptive), and the nonlinear leader system may contain some parameters unknown to any of the followers. It is noted that the techniques developed in wanghuang2018 and baldi2020distributed cannot be applied to nonlinear systems. Compared with its linear counterpart, the knowledge-based leader-following synchronization problem with a nonlinear leader is much challenging, and is worthy of further investigation.

Note that by considering the multi-agent system consisting only of the leader node and the learning-based observers, the distributed observer design problem is essentially the same with the knowledge-based leader-following synchronization problem (wang2006theoretical) and has the well-known frequency estimation problem as a special case (HsuOrtega1999TAC-globally). In the literature, synchronization of coupled nonlinear systems can be achieved under either of the following three common conditions: the Lipschitz condition (wangslotine2004; zhou2006adaptive; yu2009pinning), the quadratic condition (lu2004synchronization), and the contracting condition (wang2006theoretical). The relationship among these conditions was reported in delellis2010quad

. Essentially, the aforementioned works try to use a sufficiently large constant to stabilize the coupled nonlinear dynamic networks. For more general scenarios, when the nonlinear dynamics of the leader do not satisfy the above conditions, a practical approach for synchronization of networked nonlinear systems is to use neural networks to approximate the unknown nonlinearities

(zhang2012adaptive), which only guarantees boundedness of the synchronization error, without estimating the leader’s dynamics. Moreover, some parameters of the controller require the knowledge of the whole communication graph, such as its Laplacian matrix, which is a certain global information.

The rest of this paper is organized as follows: In Section II, we formulate our problem. In Section III, we establish a learning-based distributed observer for a nonlinear leader system whose parameters are unknown to any follower nodes. As an application of the learning-based distributed observer, we apply the learning-based distributed observer to synthesize an adaptive distributed control law for solving the leader-following synchronization problem of multiple Euler - Lagrange systems subject to an uncertain nonlinear leader system in Section IV. A simulation example is given in Section V and Section VI concludes the paper.

Notation: Let denote the Kronecker product of matrices. For , and

 blkdiag(X1,…,Xk)=⎡⎢ ⎢⎣X1⋱Xk⎤⎥ ⎥⎦.

For , unless indicated otherwise, denotes the -th component of , and denote the -norm and the -norm of , respectively.

## Ii Preliminaries and problem formulation

### ii.1 Graph theory

The communication/interaction network of a multi-agent system composed of followers and one leader is described by a graph with and being the node set and the edge set, respectively. Here node is associated with the leader and nodes , , are associated with the followers. For and , if and only if node can get the information of node for control purpose. Let denote the neighbor set of agent . Let denote the induced subgraph of with , which captures the interaction among follower nodes. Assume that contains a spanning tree with node being the root, and is an undirected graph. Let be the Laplacian matrix of the graph , and is obtained by deleting the last row and column of . Then, is a symmetric positive definite matrix (HW2015Constructing). More details of the graph theory can be found in godsil2013algebraic.

### ii.2 Problem formulation

Let the follower nodes be described by general nonlinear systems

 ˙xi= fi(xi,τi), yi= hi(xi,τi),i=1,…,N, (1)

where , and are the state, measurement output and control input of the th follower, and and are globally defined and sufficiently smooth functions vanishing at the origin.

The leader’s output signal , is generated by the following nonlinear system

 ˙v =p(v,ω), (2a) qN+1 =Ev, (2b)

where , is a globally defined and sufficiently smooth function vanishing at the origin,

is a constant vector consisting of the unknown parameters of the leader node,

is a known constant matrix. The uncertain nonlinear system (2) is assumed to satisfy

 p(v,ω)=ϕ(v)ω, (3)

where the regressor matrix is known. It is assumed that given any compact set , there exists a compact set such that, for any , for all . The system (2) encompasses a class of nonlinear systems that can generate stable limit cycles, such as the well-known Van der Pol system.

For , we consider the following class of distributed control laws

 τi= ki(xi,^vi,^ωi), (4a) ˙^vi= gi(^ωi,^vi,∑j∈¯Ni(^vj−^vi)), (4b) ˙^ωi= pi(^vi,∑j∈¯Ni(^vj−^vi)), (4c)

where , and for , and are the estimation of and , respectively; , and are sufficiently smooth functions vanishing at the origin.

Now we are ready to formulate the knowledge-based leader-following synchronization (KLFS) problem for a class of nonlinear multi-agent systems.

###### Problem 0 (KLFS Problem)

Consider the multi-agent system (II.2) and (2), and the corresponding communication graph . Given any compact set containing the origin, design a distributed control law in the form of (4), such that for any initial condition , , , and , is bounded for all , and

Note that for the special case that and are known to all followers, the control law (4) will reduce to the following form

 τi=ki(xi,v,ω). (5)

Then the KLFS problem will reduce to the traditional adaptive nonlinear output regulation problem, which has been well studied in huang2004nonlinear and isidori2014robust.

In practice, the leader’s dynamic knowledge (i.e., parameters) may be unknown to all followers, and only a few followers who have direct links to the leader can sense the leader’s state or output information. Then, in order to solve the KLFS problem, we need to design a learning-based distributed observer for each follower node to estimate the state and simultaneously learn the parameters of the leader node. The learning-based distributed observer, which is the main kernel in solving Problem 1, is defined as follows.

###### Definition 1 (Learning-based distributed observer)

A dynamic of the form (4b) and (4c) is called a learning-based distributed observer for the leader (2) if, given a digraph and any compact set containing the origin, there exit global defined and sufficiently smooth functions and , such that for any initial condition , and , is bounded for all , and

Further, under certain conditions, the observer (4c) can adaptively learn the actual values of the knowledge-based leader in the sense that

 limt→∞(^ωi(t)−ω)=0.

Before proceeding, we recall the definition of persistent excitation, which plays an important role in analyzing the convergence of the estimated parameters.

###### Definition 2

(Anderson) A bounded piecewise continuous function is said to be persistently exciting if there exist positive constants , , and such that,

 1T0∫t+T0tf(s)fT(s)ds≥ϵIn,    ∀t≥t0.

## Iii Learning-based distributed observer design

This section devotes to the design and analysis of the learning-based distributed observers. Then in Section IV, we will apply this observer to solve a leader-following synchronization problem of multiple Euler-Largrange systems.

### iii.1 Learning-based distributed observers design

Note that the parameter vector in the leader system (2) is constant, and thus bounded. To estimate both states and parameters of the leader node, we design the following learning-based distributed observer for node ,

 ˙^vi =p(^vi,^ωi)+^κiρi(zi)zi, (6a) ˙^ωi =μϕT(^vi)zi, (6b) ˙^κi =ρi(zi)zTizi, (6c) zi =∑j∈¯Ni(^vj−^vi),i=1,…,N, (6d)

where , is the estimation of , is the estimation of , is a dynamic gain to estimate some sufficiently large positive unknown bound induced by the matrix , uniform bound of state and unknown parameter ,

 p(^vi,^ωi)=ϕ(^vi)^ωi,

where is defined in (3), is a smooth positive function to be designed and is a positive scalar to be designed.

###### Remark 0

A salient feature of the observer (6) is its fully distributed nature, and only those follower nodes who have direct links from the leader need to get access to the leader’s state information, instead of its parameters . Closely related results can be found in wangslotine2004; wang2006theoretical and zhou2006adaptive. However, they use a fixed linear gain to stabilize the coupled nonlinear dynamic networks under assumptions that each coupled nonlinear agent must satisfy the global bounded Jacobian matrix assumption or the quadratic condition proposed in lu2004synchronization.

Moreover, the observer design (6) is fully distributed in the sense that the coupled global information induced by the communication graph and knowledge-based leader system has been completely removed by using an adaptive technique, i.e., equation (6c). Note that this technique is different from the one proposed in li2014designing, which is not applicable in our case of nonlinear uncertain leader dynamics.

###### Remark 0

The choice of depends on the structure of , where is unknown. For a class of polynomial differential systems (2), with being real polynomials of degree ,

 ∥p(v,ω)∥2≤∑2m0k=0ck(ω)∥v∥k

where is some unknown constant induced by the uncertain parameter . Then satisfies the following function

 ρi(zi)=ai∑2m0−2k=0∥zi∥k+bi,

where and are positive numbers. Details can be found in Lemma 4.

### iii.2 Convergence analysis

Define the estimation errors as , , and , respectively. Their time derivatives along the trajectories (2) and (6) are

 ˙~vi= ¯κiρi(zi)zi+ϕ(^vi)~ωi +~κiρi(zi)zi+(ϕ(^vi)−ϕ(v))ω, (7a) ˙~ωi= μϕT(^vi)zi, (7b) ˙~κi= ρi(zi)zTizi,i=1,…,N. (7c)

Define , , , , , , and . Then, we have

 z=−(H⊗Im)~v, (8)

where is defined in Section II.1. Equations (7a) and (7b) can be rewritten in the following compact form

 ˙~v= (¯κdρd⊗Im)z+ϕd(^v)~ω+(~κdρd⊗Im)z +[ϕd(^v)−IN⊗ϕ(v)](1N⊗ω), (9a) ˙~ω= μϕTd(^v)z. (9b)

Before analyzing the convergence of the proposed learning-based distributed observer, we first establish the following result.

###### Lemma 0

Consider systems (2) and (6).

1. There exit sufficiently smooth positive function and positive constants and such that

 N∑i=1∥p(^vi,ω)−p(v,ω)∥2≤β2MλMN∑i=1γi(zi)∥zi∥2. (10)
2. Moreover, if system (2) is a class of polynomial differential system with degree , then

 γi(zi)=∑2m0−2k=0∥zi∥k.

Proof: 1): Define

 fi(~vi,v,ω)=∥p(~vi+v,ω)−p(v,ω)∥, (11)

which is a continuously differentiable function satisfying , for . By (iv) of Lemma 11.1 in chen2015stabilization, there exists a smooth function such that

 fi(~vi(t),v(t),ω)≤mi(~vi(t),v(t),ω)∥~vi(t)∥,    ∀t≥0.

For the continuous function , by (i) of Lemma 11.1 in chen2015stabilization, there exist smooth functions and , such that

 mi(~vi(t),v(t),ω)≤αi(~vi(t))×βi(v(t),ω),    ∀t≥0.

Since is uniform bounded in , and is an unknown constant vector, there exists a positive constant such that , for all . Define

 gi(z)=αi(~vi(z))∥~vi(z)∥,

where we consider as a function of via the relationship

 ~v=−(H−1⊗Im)z.

Since the smooth function satisfies , by Lemma 7.8 in huang2004nonlinear, there exist smooth functions and for some positive constant , for such that

 ∑Ni=1f2i(~vi,v,ω)≤β2MλM∑Ni=1γi(zi)∥zi∥2.

2): For , are continuously differentiable functions and are a class of polynomial differential systems with the largest degree . It is noted that , for . Then, for any and , we have

 pl(^vi,ω)−pl(v,ω)= [∫10∂pl(x,ω)∂x∣∣x=v+ϑl~vidϑl]~vi.

Then, for , we will have

 ∥pl(^vi,ω)−pl(v,ω)∥≤∥∥∫10∂pl(x,ω)∂x∣∣x=v+ϑl~vidϑl∥∥∥~vi∥.

In addition, is in real polynomials form with the largest degree . Thus

 m∑l=1∥∥∫10∂pl(x,ω)∂x∣∣x=v+ϑl~vidϑl∥∥≤∑m0−1k=0βk(ω,v)∥~vi∥k,

where are some unknown positive smooth functions induced by the uncertain parameter and unknown state . Since is uniform bounded in , and is an unknown constant vector, there exists a positive constant such that , for all . Hence, we have

 ∥p(v,ω)−p(^vi,ω)∥2≤β2M∑2m0−2k=0∥~vi∥k+2.

It is noted that

 ∥~v∥=∥(H−1⊗Im)z∥≤¯h∥z∥,

where , and

 ∥x∥p≤∥x∥≤n12−1p∥x∥p

for any and , which further implies

 ∥x∥pp≤∥x∥p≤np2−1∥x∥pp.

Then, for any , we have

 ∑Ni=1∥~vi∥k≤ (∑Ni=1∥~vi∥2)k/2 ≤ ¯hk(∑Ni=1∥zi∥2)k/2 = ¯hk(∥z∥)k≤Nk2−1¯hk∑Ni=1∥zi∥k.

Thus

 N∑i=1∥p(v,ω)−p(^vi,ω)∥2≤ β2MN∑i=12m0∑k=2Nk−1¯hk+2∥zi∥k.

Then, by letting , we have

 ∑Ni=1∥p(v,ω)−p(^vi,ω)∥2≤β2MλM∑Ni=1γi(zi)∥zi∥2

for some positive constant .

###### Theorem 5

Consider systems (2) and (6). There exist some sufficiently smooth positive functions for all , such that, for all , and , , and exist and are bounded for all and satisfy

 limt→∞~vi(t)=0, (12) limt→∞˙~ωi(t)=0, (13) limt→∞ϕ(^vi(t))~ωi(t)=0. (14)

Moreover, if is persistently exciting, then

 limt→∞~ωi(t)=0. (15)

Proof: Consider the Lyapunov function candidate

 V=12[~vT(H⊗Im)~v+μ−1~ωT~ω+∑Ni=1~κTi~κi], (16)

where is defined in Section II.1 and is symmetric positive definite. Differentiating (16) along the trajectory of (7) gives

 ˙V= ~vT(H⊗Im)˙~v+μ−1~ωT˙~ω+∑Ni=1~κTi˙~κi = −zT(¯κdρd⊗Im)z−zT(~κdρd⊗Im)z −zT[ϕd(^v)−IN⊗ϕ(v)](1N⊗ω) −zTϕd(^v)~ω+μ−1~ωT˙~ω+∑Ni=1~κTi˙~κi = −∑Ni=1[¯κiρi(zi)∥zi∥2−zTi[ϕ(^vi)−ϕ(v)]ω] −∑Ni=1[~κiρi(zi)zTizi−~κTi˙~κi] = −N∑i=1[¯κiρi(zi)∥zi∥2−zTi[p(^vi,ω)−p(v,ω)]]. (17)

By using the procedure and the inequality (10) in Lemma 4, the equation in (III.2) implies if and only if there exits a such that

 ∑Ni=1[−¯κiρi(zi)1/21/20]≤λλMβ2M∑Ni=1[−γi(zi)001].

A sufficient condition to make the above inequality hold is

 [−¯κiρi(zi)+λλMβ2Mγi(zi)1/21/2−λβ2M]<0. (18)

Therefore, for any sufficiently smooth positive function with the form

 ρi(zi)=aiγi(zi)+bi≥λλMβ2Mγi(zi)/¯κi (19)

where and are any positive numbers, there exist and to make (18) hold and

 ˙V≤−ε∑Ni=1zTizi. (20)

for some . Then, is bounded, which means , and are bounded, for all , and exits and is finite. Since is bounded, is bounded from (8), and is bounded, for all . From (10) and the smoothness of , , we know that is bounded, for all . Again, using the smoothness of , , and are all bounded from (19) and the fact that is bounded, for all . From (9), is bounded because , , , , , and are all bounded for all , which further implies is bounded from (III.2), for all . By the Lyapunov-Like Lemma in r18, we have . From (20), we have

 −˙V≥εzTz≥0

which implies that . From (8) and (9b), we can further have (12) and (13), respectively.

To show (14), differentiating gives,

 ¨~v= (¯κdρd⊗Im)˙z+(¯κd˙ρd⊗Im)z+˙~ϕd(^v,v)(1N⊗ω) +˙ϕd(^v)~ω+ϕd(^v)˙~ω+(˙~κdρd⊗Im)z +(~κd˙ρd⊗Im)z+(~κdρd⊗Im)˙z. (21)

We have shown that and are smooth, and , , , , and are all bounded, for all . We can also show that is bounded from (7c), for all . Thus, is bounded from (III.2), for all . By the Barbalat’s lemma, we have , which further implies

 limt→∞[ϕ(^vi(t))~ωi(t)+(ϕ(^vi(t))−ϕ(v(t)))ω]=0.

Besides, for ,

 limt→∞

due to (12), which we have proved earlier. Then,

 limt→∞ϕ(^vi(t))~ωi(t)=0.

Thus, (14) holds. Eqs. (12) and (14) imply

 limt→∞ϕ(^vi(t)−~vi(t))~ωi(t)=limt→∞ϕ(v(t))~ωi(t)=0.

If is persistently exciting, by Lemma 2.4 of chen2015stabilization, we have (15) from (13).

###### Remark 0

As a result of Theorem 5, it is trivial to show that

 limt→∞(E^vi(t)−qN+1(t))= 0, (22a) limt→∞(E˙^vi(t)−˙qN+1(t))= 0,  i=1,…,N. (22b)
###### Remark 0

The learning-based distributed observer (6) features itself in two aspects. Unlike the nonlinear distributed observers proposed in many literatures, such as LJHuang2018auto and DongChen202021, the observer (6) is independent of the leader’s dynamic knowledge, i.e., parameters . Moreover, the observer (6

) does not depend on the eigenvalues of either the Laplacian matrix or the adjacency matrix of the communication graph, which is normally required in the literature of distributed observer design. Therefore, our proposed observer is fully distributed. In particular, the observer (

###### Example 1

This example shows that the well-known Van der Pol system (r18) satisfies the requirements of the leader node. Suppose the leader’s states are generated by a Van der Pol oscillator of the following form

 ˙v=p(v,ω)=[av2−bv1+c(1−v21)v2], (23)

where and . When , , and are positive constants, the Van der Pol system will have a stable limit cycle with periodic for any initial condition, which implies there exists such that Then, we have

 ∫t+Tt v2(s)ds=∫t+Tt1a˙v1(s)ds=0, ∫t+Tt v1(s)v2(s)(1−v21(s))ds=0.

For any , both and are periodic functions with periodic ,

 ∫t+Ttv21(s)ds> 0, ∫t+Ttv22(s)ds> 0, ∫t+Ttv21(s)v22(s)(1−v21(s))2ds> 0.

From (23) we have

 ∂p(v,ω)∂ω= [v2000−v1(1−v21)v2]=ϕ(v).

Then, It can be verified that