## 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).

Many recent works (cai2017adaptive; wu2017adaptive; klotz2014asymptotic) solved the leader-following synchronization problem by assuming that some/all followers know the leader’s parameters. In contrast, solving knowledge-based leader-following synchronization problem is much challenging, since no followers can get access to the leader’s knowledge (i.e., parameters) and only part of them can sense the leader’s state or output signals. There are generally two approaches to handle the uncertain leader. One approach is the so-called distributed adaptive internal model design approach (su2013cooperative). Then, under the assumption that the leader has a unitary relative degree and strongly minimum phase properties, the synchronization problem can be converted to a global adaptive robust stabilization problem. Another is the adaptive distributed observer design approach, which was studied recently in wanghuang2018 and wangmeng2020, where each follower node maintains an observer, estimating the state and/or learning the parameters of the leader node in a distributed manner, and then a decentralized controller is designed for each follower node to drive it to track the output of its observer. Some early tries of the observer design can also be found in modares2016optimal, where the convergence analysis of the estimated parameters is missing. Very recently, baldi2020distributed also studied the knowledge-based leader-following synchronization problem and proposed an adaptive distributed observer to estimate the state and learn the knowledge of a linear leader. They pointed out that it is a starting point to extend the approach to more general cases. Although much attention has been paid to uncertain linear leader dynamics, insufficient investigation has been reported for knowledge-based leader-following networks with the nonlinear dynamics, to the best of the author’s knowledge.

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.Motivated by the above-mentioned statements, in this paper, we consider a more practical scenario of knowledge-based leader-following synchronization problems, where the leader node has nonlinear dynamics and is not known by any followers. We aim to design a fully distributed control law. We first establish a learning-based fully distributed observer for a class of nonlinear leader systems whose parameters are not known precisely. Under some standard assumptions, this learning-based distributed observer can estimate and pass the leader’s state to each follower through local interaction without knowing the leader’s parameters. Under an additional assumption that the leader’s regressor matrix is persistently exciting, this distributed observer can also asymptotically adaptively learn the leader’s parameters. Our observer design removes the bounded Jacobian matrices assumption or the quadratic condition on the coupled nonlinear systems (wangslotine2004; wang2006theoretical; zhou2006adaptive). Moreover, the global information of the communication graph has been completely removed by using an adaptive technique, different from the one proposed in li2014designing, which is not applicable in our case of nonlinear leader dynamics. As an application, 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 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

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

(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

(2a) | ||||

(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(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

(4a) | ||||

(4b) | ||||

(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)

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

(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

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,

## 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 ,

(6a) | ||||

(6b) | ||||

(6c) | ||||

(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 ,

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 ,

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

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

(7a) | ||||

(7b) | ||||

(7c) |

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

(8) |

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

(9a) | ||||

(9b) |

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

###### Lemma 0

Proof: 1): Define

(11) |

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

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

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

where we consider as a function of via the relationship

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

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

Then, for , we will have

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

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

It is noted that

where , and

for any and , which further implies

Then, for any , we have

Thus

Then, by letting , we have

for some positive constant .

###### Theorem 5

Proof: Consider the Lyapunov function candidate

(16) |

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

(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

A sufficient condition to make the above inequality hold is

(18) |

Therefore, for any sufficiently smooth positive function with the form

(19) |

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

(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

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

To show (14), differentiating gives,

(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

Besides, for ,

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

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

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

(22a) | ||||

(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 (

6) can adaptively learn the true values of the leader’s parameters.###### 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

(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

For any , both and are periodic functions with periodic ,

From (23) we have

Then, It can be verified that