Downloads

Su, Y., Cai, H., & Huang, J. The Cooperative Output Regulation by the Distributed Observer Approach. International Journal of Network Dynamics and Intelligence. 2022, 1(1), 20–35. doi: https://doi.org/10.53941/ijndi0101003

Survey/review study

The Cooperative Output Regulation by the Distributed Observer Approach

Youfeng Su 1, He Cai 2, and Jie Huang 3,*

1 College of Computer and Data Science, Fuzhou University, Fuzhou 350116, China

2 School of Automation Science and Engineering, South China University of Technology, Guangzhou 510641, China

3 Department of Mechanical and Automation Engineering, The Chinese University of Hong Kong, Hong Kong, China

* Correspondence: jhuang@mae.cuhk.edu.hk

 

 

Received: 4 November 2022

Accepted: 13 December 2022

Published: 22 December 2022

 

Abstract: The cooperative output regulation problem (CORP) is an extension of the leader-following consensus problem of multi-agent systems (MASs), and has been studied by two approaches, namely, the distributed observer (DO) approach and the distributed internal model (DIM) approach. The two approaches are, respectively, the extensions of the classical feedforward control approach and the classical internal model approach (for a single system) to the MASs. This paper overviews the CORP by the DO approach with the emphasis on linear MASs. After formulating the CORP, we present the evolution process of three types of DOs and the corresponding solutions to the CORP. Furthermore, some variants and extensions of the DO approach are also briefly surveyed for completeness.

Keywords:

cooperative output regulation distributed observer multi-agent systems

1. Introduction

The output regulation problem (ORP) has been one of the central control problems in the control community for over a half century [1-3]. The ORP aims to design a feedback control law for a given plant such that the output of the plant asymptotically track a class of reference inputs in the presence of a class of disturbances while ensuring the internal stability of the closed-loop system. Both reference inputs and disturbances are generated by the so-called exosystem. The ORP was first studied for linear systems with both the reference input and disturbance being step functions [4-5], and then extended to various general cases in, for instance, [6-12]. Two approaches, namely, the feedforward control approach and the internal model (IM) approach have been developed, where the former makes use of the solution of the regulator equations to obtain the precise feedforward control quantity to cancel the tracking error incurred by the exogenous signal, while the latter converts the ORP of the given plant to a stabilization problem of the augmented system composed of the given plant and a well designed IM. An advantage of the IM approach is that it is able to deal with systems with parametric uncertainties.

Since 2000, the cooperative control of MASs has become a mainstream control problem following the publications of a few celebrated papers, say, [13-16]. The cooperative control of MASs was first studied for simple linear MASs where each agent is a single integrator [15-19], double integrator [20], or harmonic oscillator [21]. The consensus of general linear homogenous MASs was treated in [22] and the consensus problem of some weak nonlinear MASs was also investigated in [23] for Lipschitz nonlinear systems, in [24] for Euler-Lagrange systems, and in [25] for rigid-body systems. Since around 2010, the research on the cooperative control has focused on more complex MASs featuring strong nonlinearities, time-delays, large uncertainties, and jointly connected switching networks, and a more challenging objective (such as simultaneous tracking and disturbance rejection) has been attracting the attention.

The CORP is an extension of the leader-following consensus problem of MASs in the sense that it handles simultaneously the asymptotic tracking and disturbance rejection problems for heterogeneous and possibly uncertain MASs. The CORP is also a generalization of the conventional ORP for a single plant to a MAS with a leader by viewing the leader system as the exosystem and the follower system as the controlled plant. Generalizing the feedforward control approach and the IM approach for a single plant to their distributed counterparts, respectively, leads to the so-called DO approach and the so-called DIM approach. This paper will mainly focus on the DO approach.

A DO is a dynamic compensator aimed to estimate certain information of the leader system over a communication network and transmit the estimated information to the follower systems. Cascading a purely decentralized control law with DO gives a distributed control law. The DO was first developed in [26] over static networks for the purpose of solving the CORP for linear heterogeneous MASs, and was later generalized to jointly connected switching networks in [27]. Since 2011, the DO has experienced three phases of development. To be specific, the original DO as given in [26-27] only estimates and transmits the leader's signal to followers by assuming every follower knows the dynamics of the leader [26-29]. In practice, the dynamics of the leader may not be known by every follower. Thus, in the second phase, the capability of the original DO was enhanced to be able to estimate and transmit both the leader's signal and the dynamics of the leader. This enhanced DO only assumes that the leader's children know the dynamics of the leader and is thus fully distributed. Since this enhanced DO is able to estimate the leader's system matrix, it is called an adaptive distributed observer (ADO) for a known leader system [30, 31]. In a more realistic setting, the leaders' dynamics may contain uncertain parameters. In this case, none of the followers know the leader's exact dynamics. Thus, the third phase of DO needs to be able to estimate the leader's unknown parameters. Such a dynamic compensator is called ADO for an uncertain leader. The ADO for an uncertain leader was first studied in [32], which did not consider the convergence of the estimated unknown parameters of the leader to their real values. In [33], an ADO was further developed for an uncertain leader system and the sufficient condition was also provided on the convergence of the estimated unknown parameters to the actual parameters. The COPR subject to an uncertain leader system was further studied in [34]. More aspects of the CORP by the DO approach can be found in [35-40]. A comprehensive treatment on the CORP by the DO approach can be found in [41].

The rest of this paper is organized as: Section 2 formulates CORP of MASs. Section 3 overviews the three types of DOs. After reviewing the conventional feedforward control for solving linear ORP of one single control system in Section 4, we further summarize, in Section 5, the solvability conditions of the CORP by all three types of DOs. In Section 6, we present some variants and extensions of the DO as well as the applications of the DO to some other cooperative control problems such as the consensus problem of Euler-Lagrange systems, attitude synchronization problem of spacecraft systems, rendesvous/flocking problem, and formation problem. The CORP can also be handled by the DIM approach and the integrated approach that combines the DO approach and the DIM approach. In Section 7, we close this paper by giving a brief overview of the DIM approach and the integrated approach.

2. A Formulation of CORP

In this section, we give a general formation of the CORP for linear MASs. An MAS consists of a group of individual agents. A general representation for a continuous-time linear MAS having  agents can be described as:

where , and  are the state, control input, measurement output, and performance output of the  agent of (1), respectively,  and  are the exogenous signal and the uncertain parametric vector, respectively. All nine matrices in (1) are smooth in .

The exogenous signal  is typically produced by the following system:

where , and  are the state, measurement output, reference output, and unknown constant vector, respectively,  and  are smooth in , and  is known and constant.

Systems (1) and (2) together are treated as an MAS of  agents. Since (2) generates reference signals for the performance outputs of (1) to track, we call (2) and (1) as the leader system and the follower system, respectively. If the dynamics of (1) does not contain , then (1) is called a leaderless MAS of  agents.

If (1) contains no uncertain parametric vectors, we can simplify it to:

where all nine matrices are known. If (2) is known exactly, we can also simplify it to:

where , and  are all known.

The information exchange among the agents of a leader-follower MAS defined by (2) and (1) is described by a switching graph 1 (See [41, Section 1.2] for a detailed description of a switching graph.), where  is the node set, and  is the edge set. The node  is associated with the leader, and, for , the node  is associated with the  follower. For , the edge  if and only if, at the time instant , the control  can make use of the measurement output .

For , let  be the entries of the adjacent matrix of . Then, we consider the following control law:

where  and  are smooth,  consists of the signals and /or the parameters of the leader system. Since the control law (5) satisfies the communication constraints, we call (5) a distributed control law.

The error output (or regulated output)  of the  follower is defined as:

where , and . If , and  are all known, (6) takes the following simplified form:

For simplicity, this paper focuses on the known follower system (3) without uncertain parametric vectors, while the leader system can be either known as in (4) or uncertain as in (2). The CORP is formulated as follows:

Problem 1. Given the systems (3), (4) (or (2)), and the graph  (or a static communication graph ), find a distributed control law of the form (5) such that the closed-loop system satisfies the following properties:

• Property 1: The origin of the closed-loop system with  set to zero is asymptotically stable.

• Property 2: For any initial condition, the solution of the closed-loop system satisfies 

3. The Purely Decentralized Feedforward Control

Let us first assume that the state  can be used by the control input  of each follower. Then, Problem 1 can be handled by the classical feedforward control method under some standard assumptions as follows:

Assumption 1: The matrix pairs , are stabilizable.

Assumption 2: The matrix pairs , are detectable.

Assumption 3: There exist matrix pairs , that satisfy the following matrix equations:

Remark 1: Assumption 1 guarantees that Property 1 can be achieved by the state feedback control. Assumption 2 together with Assumption 1 guarantees that Property 1 can be achieved by the measurement output feedback control. Equations (8) are called the regulator equations, and the solvability of equations (8) is the necessary condition of the solvability of Problem 1 [11].

Two classes of control laws are considered.

1. Purely Decentralized Full Information Static State Feedback:

where  are constant with appropriate dimensions.

2. Purely Decentralized MeasurementOutput Feedback plus Feedforward:

where  are constant with appropriate dimensions.

When N=1, the full information control law (9) was originally given in [11], where it is shown that, under Assumptions 1 and 3, the classical ORP can be solved by (9) if  are chosen such that  are Hurwitz, and  with  given in (8). The solvability of the classical ORP via the measurement output feedback plus feedforward control law (10) was considered in [29], where  can be chosen the same as those in (9), and under additional Assumption 2,

with  being such that  are Hurwitz. Moreover, when  is unmeasurable for any follower, and

one can also find a measurement output feedback control law by estimating  as well. Such a case can be found in [3, Section 1.2].

4. Three Types of the Distributed Observers

This section overviews three types of DOs, which can be viewed as a concentrate of [41, Chapter 4].

Given any switching communication graph  with  nodes, define a compensator of the following form: for ,

where  is some globally defined smooth function. If, for any initial condition, the solution of systems (4) and (12) satisfies, for , then (12) is called a distributed observer (DO) for (4).

To construct a DO for (4), we list the following assumptions.

Assumption 4: There exists a positive number  such that, for all , and , the union graph  contains a spanning tree with node  as the root.

To introduce the next assumption, let  be a subgraph of  where  and  is obtained from  by removing all edges incident to node .

Assumption 5:  is undirected for any .

When the communication graph is static, we simplify the notation  and  by  and , respectively. In this case, Assumption 4 is simplified to the following one.

Assumption 6:  contains a spanning tree with node  as the root.

Various assumptions for the leader system are as follows:

Assumption 7:  is detectable.

Assumption 8: The real parts of all the eigenvalues of the matrix  are non-positive.

Assumption 9: The matrix  is marginally stable.

4.1. Distributed Observer for a Known Leader System

In this subsection, let us allow the matrices  and  of the leader system (4) to be known by every follower for all .

Consider the following DO candidate:

where , and  is to be designed.

Let , and .2(For  column vectors .) Then,

System (14) is called the error system of (13). Clearly, (13) is a DO for (4) if and only if the origin of (14) is asymptotically/exponentially stable.

If the leader's state is available, i.e., . Then, letting  where  is a positive real number, (13) is simplified to the following DO:

and the error system (14) becomes:

In what follows, we call (13) and (15) the state-based DO and output-based DO, respectively.

The state-based DO was first given in [26] over static networks where it was shown that, under Assumptions 6, for sufficiently large , the origin of (16) is exponentially stable. This result was later extended to jointly-connected switching networks in [27] where it was shown that, under Assumptions 4 and 8, for any , the origin of (14) is exponentially stable.

Under Assumption 7, let  be the unique solution of

Then, under Assumptions 6 and 7, with  and ,3 (Given any matrix  and  denote the minimum and maximum real parts of eigenvalues of , respectively.) the error system (14) with  is exponentially stable [41, Theorem 3.1], which directly leads to the output-based DO over static networks. Under Assumptions 4, 5, 7, and 9, it was shown in [42] using the generalized Barbalat's lemma that (16) is asymptotically stable, which further leads to the output-based DO over jointly-connected switching networks as can be found in [29].

In some applications, the asymptotic stability of system (14) may not be enough. One needs system (14) to be exponentially stable. By using the generalized Krasovskii-LaSalle theorem together with the scaling invariant property [43], one can further show that, under Assumptions 4, 5, 7, and 9, the origin of (14) is exponentially stable, where  for any  with  satisfying (9). By adopting the generalized Krasovskii–LaSalle theorem for switched time-varying systems [44], one can further show that the exponential stability is uniform w.r.t. a group of switching signals.

Remark 2: There are some other variants of this type of DOs. For example, [45] considered the case where each agent only has access to part of the output of the leader system. [38, 46] studied the local observer together with global distributed observer. [47] considered the finite-time convergent DO for a special leader system in chained integrator form. [48] treated leader systems with bounded inputs. DOs with self-tuning gain assignment can be found in [49-50], and the gain assignment for nonovershooting performance was proposed in [51].

4.2. Adaptive Distributed Observer for a Known Leader System

The DO (13) is not fully distributed in the sense that the system matrices  and  are used by every follower. To obtain a fully distributed DO, in this subsection, we further introduce the so-called adaptive distributed observer (ADO) for leader system (4), which not only estimates the state of the leader, but also the three matrices , and . Two scenarios are considered corresponding to the two scenarios in the previous subsection.

The output-based ADO candidate is designed as follows:

The compensator (18) is further called an output-based ADO, if, for , and =0 all exponentially.

When , i.e., , there is no need to estimate . Then, we obtain the state-based ADO candidate in the following form:

The compensator (19) is further called a state-based ADO if, for , and  all exponentially.

Remark 3: In many cases, the role of the matrices  and  is to select the elements of . In these cases, we can assume  and  are known by every follower, and hence, there is no need to estimate  and . Consequently, the output-based ADO (18) can be simplified to:

and the state-based ADO (19) can be simplified to:

The state-based ADO (21) was first proposed in [30] over switching networks under Assumptions 4, 5, and the assumption that the system matrix  is neutrally stable. Assumption 5 was removed in [52], and the neutral stability assumption on  was relaxed to Assumption 8 in [53]. The case for static networks was established in [31] under Assumption 6. The study of the general form (19) over both switching and static networks can be found in [41, Section 4.4]. Technically, the error dynamics for this class of DOs is a perturbed system with the nominal system in the form (16). If the communication graph  is switching, the state-based ADO (19) candidate is indeed an ADO if Assumptions 4 and 8 are satisfied and . If the graph  is static and Assumption 6 holds, then the state-based ADO (19) candidate is indeed an ADO with .

A special type of the output-based ADO was given in [54], and a more general form of (18) can be found in [41, Section 4.4]. Technically, the error dynamics for this class of DOs is a perturbed system with the nominal system in the form (14). If the communication graph  is switching, the output-based ADO (18) candidate is indeed an output-based ADO if Assumptions 4, 5, and 9 are satisfied and . If the graph  is static and Assumption 6 holds, then the output-based ADO (19) candidate is indeed an output-based ADO with .

Remark 4: Other ADO variants, for example, an ADO with self-turning gains was proposed in [55].

4.3. Adaptive Distributed Observer for an Uncertain Leader System

Both DO and ADO assume that the leader's dynamics are known by at least some followers. In this subsection, it is assumed that the leader's dynamics contains some uncertain parameters, and hence none of followers knows the exact dynamics of the leader system. For simplicity, let us consider a special case of uncertain leader system (2) as follows:

where  satisfies the following assumption:

Assumption 10: For all , the matrix  is neutrally stable and nonsingular.

In fact, a neutrally stable matrix allows a simple zero eigenvalue. As the zero eigenvalue is known and hence can be dealt with by the DO, we exclude this case by requiring that the matrix  be nonsingular in Assumption 10. Thus, without loss of generality, we can assume that

Let the mapping  be such that, for any ,

Then, for , an ADO candidate, which is independent of the unknown vector , is proposed as follows:

where . The dynamic compensator (24) is further called ADO for the leader (22) if .

The ADO for an unknown leader system was first studied in [32]. However, [32] did not consider the convergence of the estimated unknown parameters of the system matrix of the leader to their actual values. The same problem was further pursued in [34] with an ADO (24). It is shown in [34] that, under Assumptions 6 and 10, (24) is valid for any  if the communication graph  is static and the subgraph  is undirected. It is also shown that, as long as the leader signal is persistently exciting, the estimated unknown parameters of the system matrix of the leader will asymptotically tend to their actual values, that is, .

Remark 5: Extensions of the ADO for an uncertain leader over directed acyclic static graphs and directed acyclic switching graphs were given in [56] and [57], respectively. Other results dealing with the uncertain leader can be found in [58-59].

5. The Cooperative Output Regulation via the Distributed Observer Approach

We are now ready to consider the solvability of the CORP by the DO approach. This section can be viewed as a concentrate of [41, Chapter 9].

Notice that the purely decentralized control laws (9) and (10) can be unified into the following form:

where  are some constant matrices.

Let the following system

with a smooth function  in its argument, denote some DO of the leader presented in Section 4. Now, replacing  in (25) by  gives the following cascade-connected control law:

which is indeed a distributed control law since, at each time , for any  makes use of  if and only if the leader is a neighbor of the th follower. Moreover, it is shown in [29] that the distributed control law (27) solves the CORP as long as the purely decentralized control law (25) solves the ORP.

5.1. The Distributed Observer Based Approach for a Known Leader System

Let (26) be given by the DO (13), and let (25) be given by control laws (9) and (10), respectively. Then, we obtain two distributed control laws for solving the CORP (Problem 1) as follows.

1. Distributed full information control law:

2. Distributed measurement output feedback plus feedforward control law:

where  are the same as those in (9) and (10), and  is given in (13).

Problem 1 was first formally formulated in [26] for the leader system with , and solved over static networks using distributed control law (28), and then solved over jointly connected switching networks in [27] using distributed control laws (28) and (29), respectively. The solution to the CORP with general output matrix  was given in [29] by using the distributed control laws (28) and (29), respectively. It is noted that the solvability conditions are the combinations of the solvability conditions of the ORP (e.g. Assumptions 1 - 3) and those for establishing the DO (14) or its special case (15) (e.g. Assumptions 4 - 9). The same problem was also studied in [28] via the dynamic measurement output feedback design by combining the DO and the Luenberger observer.

5.2. The Adaptive Distributed Observer Based Approach for a Known Leader System

Employing ADO (18) enables us to solve the CORP without requiring the control of every follower knows the the leader's system matrix . Since this design will result in a closed-loop system whose origin is not an equilibrium point when  is set to zero, we modify Problem 1 to the following one.

Problem 2: Given systems (3), (4) (or (2)), and the graph , find a distributed control law of the form (5) such that the closed-loop system satisfies the following properties.

• Property 3: The solution of the closed-loop system exists and is uniformly bounded over  for uniformly bounded .

• Property 4: The solution of the closed-loop system satisfies 

Clearly, if the closed-loop system satisfies Property 1, it also satisfies Property 3.

Since the solution of the regulator equations (8) associated with every follower system relies on , the solution of (8) cannot be obtained without knowing . To overcome this difficulty, an iterative approach for obtaining the solution of (8) was reported in [31]. To summarize the approach in [31], let , be a sequence such that  exponentially, and let  be defined as follows:

Then, under Assumption 3, for any initial condition, the time-varying compensator

where  and , generates a uniformly bounded solution  such that

where 4 (For any column vector  for some positive integers  and , where, for ). Moreover, if the exponential convergent rate of  is at least , then there exists  such that, for any , the exponential convergent rate of  is at least .

Consequently, one can find the approximation of  in (9) and (10) by . To save the notation, let us adopt the simpler ADO where  and assume  is known by every follower. In this case, with the ADO (21), two types of distributed control laws corresponding to (9) and (10) for solving the CORP (Problem 2) can be synthesized as follows.

3. Distributed full information control law:

4. Distributed measurement output feedback plus feedforward control law:

where  are the same as those in (9) and (10),  are given in (21),  are defined in (30).

Problem 2 over static networks was first formally formulated in [31] for the leader system with  and solved using the distributed control law (32). The detailed study over both switching and static networks can be found in [41, Section 8.3].

5.3. The Adaptive Distributed Observer Based Approach for an uncertain Leader System

We now consider the uncertain leader system described in (22), which entails the ADO (24).

The regulator equations (8) in this case can be re-written as

where the solution may depend also on the uncertain parametric vector , which is denoted by . Like the previous subsection, let the sequence , be generated by the ADO (24), and let  be defined as follows:

Whenever  exponentially, under Assumption 3, for any initial condition, the time-varying compensator

where  and  are the same as those in (31), generates a uniformly bounded solution  such that

where . Consequently,  in (9) and (10) can be approximated by . Therefore, as reported in [34], with the ADO (24), two types of the distributed control laws corresponding to (9) and (10) for solving the CORP (Problem 2) can be synthesized as follows.

5. Distributed full information control law:

6. Distributed measurement output feedback plus feedforward control law:

where  are the same as those in (9) and (10),  are given in (24),  are defined in (36).

6. Other Variants and Extensions

This section briefly overviews some other variants and extensions of the CORP by the DO approach.

6.1. Other Types of Systems

The CORP of discrete-time linear MASs can be found in [60-62] based on various discrete-time distributed observers [62-65].

For the CORP of some other types of MASs, please refer to [66-70] for time-delay MASs, [71-73] for singular MASs, and [74-75] for switched MAS, and [76-77] for multiple parabolic PDE systems.

6.2. Cooperative Containment Control Problem

The cooperative containment control problem arises when there are multiple leaders. This problem can also be dealt with by the DO approach, where the states of the DO are steered to some weighted average of the leaders' states. Some representative publications can be found in, for example, [78-82].

6.3. Cooperative Synchronization Problem

While the CORP can be viewed as the extension of the leader-following consensus problem, the cooperative synchronization problem can be viewed as an extension of the leaderless consensus problem. This line of research can be found in a number of papers, for example, in [83-85].

6.4. Performance of the Distributed Observer

Extensive investigations on the performance of the DO were conducted including the input saturation design [86-87], the quantized feedback design [88], the fault tolerant control design [89], the indirect adaptive approach [90], and the  performance design [91].

6.5. Other Applications of the Distributed Observer

In addition to the CORP for linear MASs, the DO is also applicable to some other problems including some nonlinear systems. For example, the leader-following consensus problem of multiple Euler-Lagrange systems over jointly connected switching communication networks by the state-based DO was first studied in [92]. The same problem was further considered in [30] by a state-based ADO for the case where the leader system is neutrally stable. The neutral stability assumption in [30] was relaxed in [53] to allow the matrix  to have eigenvalues with non-positive real parts. The uncertain leader system was considered in [33, 93]. Investigation on time delay issues for information exchange over the communication network can be found in [94] and [95]. References [53, 96] and [97] studied the rejection of external disturbances for Euler-Lagrange systems by the DO approach. The DO approach has also been applied to solve the leader-following attitude consensus problem for rigid body systems. For static communication networks, state feedback, output feedback, and adaptive control schemes were proposed in [98-99], and [100], respectively. These results were further extended in [101-104] to switching communication networks. Formation control of multiple rigid body systems was investigated in [105]. Rendezvous and flocking problems were reported in [106-107].

6.6. Other Issues

To implement the DO in digital platforms, various sampled-data feedback control and event-triggered feedback control approaches have been studied. The sampled-data distributed observer based approach was investigated in [108, 109], and various event-triggered DO-based approaches and DO-based integrated control approaches can be found in [110-121].

When the system model is unknown but the feedback data can be freely used, the reinforcement learning has been shown to be a useful tool for obtaining proper approximations for the solution of the regulator equations as well as the gain matrices in the DO and the distributed controller [122-129].

Security is one of the central issues for cyber-physical systems. Various DO-based distributed resilient control (robust against DOS attacks or Byzantine attacks) can be found in [130-137]. Private protect DO based controllers can be found in [138].

7. Conclusions

This paper has given an overview on the CORP of MASs by the DO approach. Since the DO approach needs to make use of the solution of the regulator equations to synthesize the control law, it cannot handle the model uncertainties by itself. On the other hand, like the classical IM approach, the DIM approach is also able to deal with the model uncertainties.

The distributed -copy IM design was first studied in [139] over static and acyclic communication networks. This static and acyclic assumption was removed in [140] when agents have the same nominal dynamics. Some other attempts on this topic can be found in [141-144]. The distributed canonical IM design for SISO linear minimum phase systems with identical relative degree was treated in [143], and was extended to other uncertain linear MASs [145-146]. An advantage of the DIM approach is that it is able to deal with the global CORP of various uncertain nonlinear systems such as the output feedback systems with unity relative degree in [147], with higher relative degree in [148] and the strict feedback systems in [149]. The case where the exosystem is linear and contains uncertain parameters was studied for multiple nonlinear systems in output feedback form with unity relative degree in [150], and high relative degree in [151]. Other studies involving DIM control can be found in, for instance, [152-154].

Nevertheless, the DIM approach can only handle every time connected graphs. By integrating the DO approach and the DIM approach, one obtains a so-called integrated control approach which is capable of handling the nonlinearity and uncertainty of various systems over jointly connected switching networks.

Studies on integrating the DO approach with the -copy IM can be found in [49, 155-159], which can deal with linear MASs with sufficiently small uncertain parametric vectors. Studies on integrating the DO approach with the canonical IM can be found in [160-162], which can deal with various linear/nonlinear MASs with arbitrary large uncertain parametric vectors.

Author Contributions: Youfeng Su and Jie Huang: conceptualization; Youfeng Su and He Cai: investigation; Youfeng Su and Jie Huang: writing — original draft preparation; Youfeng Su, He Cai and Jie Huang: writing —review and editing; Jie Huang: supervision; Youfeng Su, He Cai and Jie Huang: funding acquisition. All authors have read and agreed to the published version of the manuscript.

Funding: This work is supported by the National Natural Science Foundation of China (62173092, 62173149, and 61973260), the Hong Kong Research Grants Council (14201621), and the Natural Science Foundation of Guangdong Province, China (2022A1515011262)

Conflicts of Interest: The authors declare no conflict of interest.

Acknowledgments: In this section, you can acknowledge any support given which is not covered by the author contribution or funding sections. This may include administrative and technical support, or donations in kind (e.g., materials used for experiments).

References

  1. Wonham WM (1985) Linear multivariable control: A geometric approach, a geometric approach, 3rd edn. Springer, New York. Available online:https://link.springer.com/book/10.1007/978-1-4684-0068-7(accessed on 11 November 2022)
  2. Knoblich HW, Isidori A, Flokerzi D (1993) Topics in control theory. Birkhauser, Boston. Available online:https://link.springer.com/book/10.1007/978-3-0348-8566-9(accessed on 11 November 2022) DOI: https://doi.org/10.1007/978-3-0348-8566-9
  3. Huang J (2004) Nonlinear output regulation: theory and applications. SIAM, Philadelphia. Available online:https://epubs.siam.org/doi/book/10.1137/1.9780898718683?mobileUi=0(accessed on 11 November 2022) DOI: https://doi.org/10.1137/1.9780898718683
  4. Johnson CD, Accommodation of external disturbances in linear regulator and servomechanism problem. IEEE Trans Autom Control, 1971, 16(6): 535−544. DOI: https://doi.org/10.1109/TAC.1971.1099830
  5. Smith HW, Davison EJ, Design of industrial regulators: integral feedback and feedforward control. Proc IEEE, 1972, 199(8): 1210−1216. DOI: https://doi.org/10.1049/piee.1972.0233
  6. Cheng L, Pearson JB, Frequency-domain synthesis of multivariable linear regulators. IEEE Trans Autom Control, 1978, 23(1): 3−15. DOI: https://doi.org/10.1109/TAC.1978.1101694
  7. Davison EJ, The output control of linear time-invariant multivariable systems with unmeasurable arbitrary disturbance. IEEE Trans Autom Control, 1972, 17(5): 621−630. DOI: https://doi.org/10.1109/TAC.1972.1100084
  8. Davison EJ, A generalization of the output control of linear time-invariant multivariable systems with unmeasurable arbitrary disturbances. IEEE Trans Autom Control, 1975, 20(6): 788−792. DOI: https://doi.org/10.1109/TAC.1975.1101104
  9. Davison EJ, The robust control of a servomechanism problem for linear time-invariant multivariable systems. IEEE Trans Autom Control, 1976, 21(1): 25−34. DOI: https://doi.org/10.1109/TAC.1976.1101137
  10. Francis BA, Wonham WM, The internal model principle of control theory. Automatica, 1976, 12(5): 457−465. DOI: https://doi.org/10.1016/0005-1098(76)90006-6
  11. Francis BA, The linear multivariable regulator problem. SIAM J Control Optim, 1977, 15(3): 486−505. DOI: https://doi.org/10.1137/0315033
  12. Wonham WM, Pearson JB, Regulation and internal stabilization in linear multivariable systems. SIAM J Control Optim, 1974, 12(1): 5−18. DOI: https://doi.org/10.1137/0312002
  13. Ogren P, Egerstedt M, Hu X, A control Lyapunov function approach to multiagent coordination. IEEE Trans Robot Autom, 2002, 18(5): 847−851. DOI: https://doi.org/10.1109/TRA.2002.804500
  14. Fax JA, Murray RM, Information flow and cooperative control of vehicle formations. IEEE Trans Autom Control, 2004, 49(9): 1465−1476. DOI: https://doi.org/10.1109/TAC.2004.834433
  15. Jadbabaie A, Lin J, Morse AS, Coordination of groups of mobile autonomous agents using nearest neighbor rules. IEEE Trans Autom Control, 2003, 48(6): 988−1001. DOI: https://doi.org/10.1109/TAC.2003.812781
  16. Olfati-Saber R, Murray RM, Consensus problems in networks of agents with switching topology and time-delays. IEEE Trans Autom Control, 2004, 49(9): 1520−1533. DOI: https://doi.org/10.1109/TAC.2004.834113
  17. Lin Z (2005) Coupled dynamic systems: from structure towards stability and stabilizability, PhD dissertation, University of Toronto, Toronto, Canada. Available online:https://www.semanticscholar.org/paper/Coupled-Dynamic-Systems%3A-From-Structure-Towards-Lin-Francis/6af6b4a0b4140704b008d78ba37fba93501caee0(accessed on 11 November 2022)
  18. Moreau L (2004) Stability of continuous-time distributed consensus algorithms. In: Proceedings of the 41st IEEE conference on decision and control, pp 3998–4003. Available online: https://ieeexplore.ieee.org/document/1429377(accessed on 11 November 2022)
  19. Ren W, Beard RW, Consensus seeking in multiagent systems under dynamically changing interaction topologies. IEEE Trans Autom Control, 2005, 50(5): 655−661. DOI: https://doi.org/10.1109/TAC.2005.846556
  20. Hu J, Hong Y, Leader-following coordination of multi-agent systems with coupling time delays. Phys A Stat Mech Appl, 2007, 374(2): 853−863. DOI: https://doi.org/10.1016/j.physa.2006.08.015
  21. Ren W, Synchronization of coupled harmonic oscillators with local interaction. Automatica, 2008, 44(2): 3195−3200. DOI: https://doi.org/10.1016/j.automatica.2008.05.027
  22. Tuna SE (2008) LQR-based coupling gain for synchronization of linear systems. arxiv.org/abs/0801.3390
  23. Song Q, Cao J, Yu W, Second-order leader-following consensus of nonlinear multiagents via pinning control. Syst Control Lett, 2010, 59(9): 553−562. DOI: https://doi.org/10.1016/j.sysconle.2010.06.016
  24. Nuño E, Ortega R, Basañez L, Hill D, Synchronization of networks of nonidentical Euler-Lagrange systems with uncertain parameters and communication delays. IEEE Trans Autom Control, 2011, 56(4): 935−941. DOI: https://doi.org/10.1109/TAC.2010.2103415
  25. Bai H, Arcak M, Wen JT, Rigid body attitude coordination without inertial frame information. Automatica, 2008, 44(12): 3170−3175. DOI: https://doi.org/10.1016/j.automatica.2008.05.018
  26. Su Y, Huang J, Cooperative output regulation of linear multi-agent systems. IEEE Trans Autom Control, 2012, 57(4): 1062−1066. DOI: https://doi.org/10.1109/TAC.2011.2169618
  27. Su Y, Huang J, Cooperative output regulation with application to multi-agent consensus under switching network. IEEE Trans Syst Man Cybern Part B Cybern, 2012, 42(3): 864−875. DOI: https://doi.org/10.1109/TSMCB.2011.2179981
  28. Su Y, Huang J, Cooperative output regulation of linear multi-agent systems by output feedback. Syst Control Lett, 2012, 61(12): 1248−1253. DOI: https://doi.org/10.1016/j.sysconle.2012.09.005
  29. Huang J, Certainty equivalence, separation principle, and cooperative output regulation of multi-agent systems by the distributed observer approach. Control Complex Syst: Theory and Appl, 2016, 14: 421−449. DOI: https://doi.org/10.1016/B978-0-12-805246-4.00014-8
  30. Cai H, Huang J, The leader-following consensus for multiple uncertain Euler-Lagrange systems with an adaptive distributed observer. IEEE Trans Autom Control, 2016, 61(10): 3152−3157. DOI: https://doi.org/10.1109/TAC.2015.2504728
  31. Cai H, Lewis FL, Hu G, Huang J, The adaptive distributed observer approach to the cooperative output regulation of linear multi-agent systems. Automatica, 2017, 75: 299−305. DOI: https://doi.org/10.1016/j.automatica.2016.09.038
  32. Modares H, Nageshrao SP, Lopes GAD, Babuska R, Lewis FL, Optimal model-free output synchronization of heterogeneous systems using off-policy reinforcement learning. Automatica, 2016, 71: 334−341. DOI: https://doi.org/10.1016/j.automatica.2016.05.017
  33. Wang S, Huang J, Adaptive leader-following consensus for multiple Euler-Lagrange systems with an uncertain leader system. IEEE Trans Neural Netw Learn Syst, 2019, 30(7): 2188−2196. DOI: https://doi.org/10.1109/TNNLS.2018.2878463
  34. Wang S, Huang J, Cooperative output regulation of linear multi-agent systems subject to an uncertain leader system. Int J Control, 2019, 94(4): 952−960. DOI: https://doi.org/10.1080/00207179.2019.1625445
  35. Grip HF, Saberi A, Stoorvogel AA, Synchronization in networks of minimum-phase, non-introspective agents without exchange of controller states: homogeneous, heterogeneous, and nonlinear. Automatica, 2015, 54: 246−255. DOI: https://doi.org/10.1016/j.automatica.2015.01.011
  36. Meng Z, Yang T, Dimarogonasa DV, Johansson KH, Coordinated output regulation of heterogeneous linear systems under switching topologies. Automatica, 2015, 53: 362−368. DOI: https://doi.org/10.1016/j.automatica.2015.01.009
  37. Yaghmaie FA, Lewis FL, Su R, Output regulation of linear heterogeneous multi-agent systems via output and state feedback. Automatica, 2016, 67: 157−164. DOI: https://doi.org/10.1016/j.automatica.2016.01.040
  38. Seyboth GS, Ren W, Allgöwer F, Cooperative control of linear multi-agent systems via distributed output regulation and transient synchronization. Automatica, 2016, 68: 132−139. DOI: https://doi.org/10.1016/j.automatica.2016.01.068
  39. Yan F, Gu G, Chen X, A new approach to cooperative output regulation for heterogeneous multi-agent systems. SIAM J Control Optim, 2018, 56(3): 2074−2094. DOI: https://doi.org/10.1137/17M1121068
  40. Li X, Soh YC, Xie L, Lewis FL, Cooperative output regulation of heterogeneous linear multi-agent networks via H∞ performance allocation. IEEE Trans Autom Control, 2019, 64(2): 683−696.
  41. Cai H, Su Y, Huang J (2022) Cooperative control of multi-agent systems: distributed-observer and internal-model-approaches. Springer Nature Switzerland AG, Switzerland DOI: https://doi.org/10.1007/978-3-030-98377-2
  42. Su Y, Huang J, Stability of a class of linear switching systems with applications to two consensus problems. IEEE Trans Autom Control, 2012, 57(6): 1420−1430. DOI: https://doi.org/10.1109/TAC.2011.2176391
  43. Lee TC, Tan Y, Mareels I, Analyzing the stability of switched systems using common zeroing-output systems. IEEE Trans Autom Control, 2017, 62(10): 5138−5153. DOI: https://doi.org/10.1109/TAC.2017.2679069
  44. Su Y, Lee TC, Output feedback synthesis of multi-agent systems with jointly connected switching networks: a separation principle approach. IEEE Trans Autom Control, 2022, 67(2): 941−948. DOI: https://doi.org/10.1109/TAC.2021.3077352
  45. Liu T, Huang J, An output-based distributed observer and its application to the cooperative linear output regulation problem. Control Theory Technology, 2019, 17(1): 62−72. DOI: https://doi.org/10.1007/s11768-019-8172-z
  46. Abdessameud A, Tayebi A, Distributed output regulation of heterogeneous linear multi-agent systems with communication constraints. Automatica, 2018, 91: 152−158. DOI: https://doi.org/10.1016/j.automatica.2018.01.020
  47. Zuo Z, Defoort M, Tian B, Ding Z, Distributed consensus observer for multiagent systems with high-order integrator dynamics. IEEE Trans Autom Control, 2020, 65(4): 1771−1778. DOI: https://doi.org/10.1109/TAC.2019.2936555
  48. Hua Y, Dong X, Hu G, Li Q, Ren Z, Distributed time-varying output formation tracking for heterogeneous linear multiagent systems with a nonautonomous leader of unknown input. IEEE Trans Autom Control, 2019, 64(10): 4292−4299. DOI: https://doi.org/10.1109/TAC.2019.2893978
  49. Li Z, Chen MZQ, Ding Z, Distributed adaptive controllers for cooperative output regulation of heterogeneous agents over directed graphs. Automatica, 2016, 68: 179−183. DOI: https://doi.org/10.1016/j.automatica.2016.01.076
  50. Dong Y, Chen J, Huang J, A self-tuning adaptive distributed observer approach to the cooperative output regulation problem for networked multi-agent systems. Int J Control, 2019, 92(8): 1796−1804. DOI: https://doi.org/10.1080/00207179.2017.1411610
  51. Schmid R, Aghbolagh HD, Nonovershooting cooperative output regulation of linear multiagent systems by dynamic output feedback. IEEE Trans Control Netw Syst, 2019, 6(2): 526−536. DOI: https://doi.org/10.1109/TCNS.2018.2846180
  52. Liu W, Huang J, Adaptive leader-following consensus for a class of higher-order nonlinear multi-agent systems with directed switching networks. Automatica, 2017, 79: 84−92. DOI: https://doi.org/10.1016/j.automatica.2017.02.010
  53. Liu T, Huang J, Leader-following consensus with disturbance rejection for uncertain Euler-Lagrange systems over switching networks. Int J Robust Nonlinear Control, 2019, 29(18): 6638−6656. DOI: https://doi.org/10.1002/rnc.4741
  54. Cai H, Huang J, Output based adaptive distributed output observer for leader–follower multiagent systems. Automatica, 2021, 125: 109413. DOI: https://doi.org/10.1016/j.automatica.2020.109413
  55. Dong Y, Xu S, Hu X, Coordinated control with multiple dynamic leaders for uncertain Lagrangian systems via self-tuning adaptive distributed observer. Int J Robust Nonlinear Control, 2017, 27(16): 2708−2721. DOI: https://doi.org/10.1002/rnc.3706
  56. Wang S, Huang J, Adaptive distributed observer for an uncertain leader with an unknown output over directed acyclic graphs. Int J Control., 2022, 94(12): 3424−3432. DOI: https://doi.org/10.1080/00207179.2020.1766117
  57. He C, Huang J, Adaptive distributed observer for an uncertain leader over acyclic switching digraphs. Int J Robust Nonlinear Control, 2022, 32(2): 873−899. DOI: https://doi.org/10.1002/rnc.5864
  58. Wu Y, Lu R, Shi P, Su H, Wu ZG, Adaptive output synchronization of heterogeneous network with an uncertain leader. Automatica, 2017, 76: 183−192. DOI: https://doi.org/10.1016/j.automatica.2016.10.020
  59. Wang S, Meng X, Adaptive consensus and parameter estimation of multiagent systems with an uncertain leader. IEEE Trans Autom Control, 2021, 66(9): 4393−4400. DOI: https://doi.org/10.1109/TAC.2020.3046215
  60. Huang J, The cooperative output regulation problem of discrete-time linear multi-agent systems by the adaptive distributed observer. IEEE Trans Autom Control, 2017, 62(4): 1979−1984. DOI: https://doi.org/10.1109/TAC.2016.2592802
  61. Liu T, Huang J, Adaptive cooperative output regulation of discrete-time linear multiagent systems by a distributed feedback control law. IEEE Trans Autom Control, 2018, 63(12): 4383−4390. DOI: https://doi.org/10.1109/TAC.2018.2823266
  62. Liu T, Huang J, Discrete-time distributed observers over jointly connected switching networks and an application. IEEE Trans Autom Control, 2021, 66(4): 1918−1924. DOI: https://doi.org/10.1109/TAC.2020.2998933
  63. Su Y, Huang J, Two consensus problems for discrete-time multi-agent systems with switching network topology. Automatica, 2012, 48(9): 1988−1997. DOI: https://doi.org/10.1016/j.automatica.2012.03.029
  64. Huang J, Leader-following consensus for a class of discrete-time multi-agent systems under directed switching networks. IEEE Trans on Autom Control, 2017, 62(8): 4086−4092. DOI: https://doi.org/10.1109/TAC.2016.2622695
  65. Lee TC, Xia W, Su Y, Huang J, Exponential consensus of discrete-time systems based on a novel Krasovskii–LaSalle theorem under directed switching networks. Automatica, 2018, 97: 189−199. DOI: https://doi.org/10.1016/j.automatica.2018.07.022
  66. Lu M, Huang J, Cooperative output regulation problem for linear time-delay multi-agent systems under switching network. Neurocomputing, 2016, 190: 132−139. DOI: https://doi.org/10.1016/j.neucom.2016.01.025
  67. Yan Y, Huang J, Cooperative output regulation of discrete-time linear time-delay multiagent systems. IET Control Theory Appl, 2016, 10(16): 2019−2026. DOI: https://doi.org/10.1049/iet-cta.2016.0106
  68. Yan Y, Huang J, Cooperative output regulation of discrete-time linear time-delay multiagent systems under switching network. Neurocomputing, 2017, 241: 108−114. DOI: https://doi.org/10.1016/j.neucom.2017.02.022
  69. Yang J, Yu H, Chen T, Cooperative output regulation with asynchronous transmissions and time-varying delays. IEEE Trans Autom Control, 2022, 67(3): 1438−1445. DOI: https://doi.org/10.1109/TAC.2021.3077347
  70. Luo S, Xu X, Liu L, Feng G (online) Leader-following consensus of heterogeneous linear multiagent systems with communication time-delays via adaptive distributed observers. IEEE Trans Cybern. https://doi.org/10.1109/TCYB.2021.3115124 DOI: https://doi.org/10.1109/TCYB.2021.3115124
  71. Ma Q, Xu S, Lewis FL, Cooperative output regulation of singular heterogeneous multiagent systems. IEEE Trans Cybern, 2016, 46(6): 1471−1475. DOI: https://doi.org/10.1109/TCYB.2015.2436971
  72. Wang S, Huang J, Cooperative output regulation of singular multi-agent systems under switching network by standard reduction. IEEE Trans Circuits Syst I-Regul Papers, 2018, 65(4): 1377−1385. DOI: https://doi.org/10.1109/TCSI.2017.2755686
  73. Liu X, Xie Y, Li F, Gui W, Cooperative output regulation of singular multi-agent systems under adaptive distributed protocol and general entirety method. Syst Control Lett, 2022, 138: 1−10. DOI: https://doi.org/10.1016/j.sysconle.2020.104628
  74. Xue M, Tang Y, Ren W, Qian F, Practical output synchronization for asynchronously switched multi-agent systems with adaption to fast-switching perturbations. Automatica, 2020, 116: 1−12. DOI: https://doi.org/10.1016/j.automatica.2020.108917
  75. Ma Y, Zhao J, Distributed adaptive integral-type event-triggered cooperative output regulation of switched multiagent systems by agent-dependent switching with dwell time. Int J Robust Nonlinear Control, 2020, 30: 2550−2569. DOI: https://doi.org/10.1002/rnc.4892
  76. Deutscher J, Cooperative output regulation for a network of parabolic systems with varying parameters. Automatica, 2021, 125: 109446. DOI: https://doi.org/10.1016/j.automatica.2020.109446
  77. Deutscher J, Robust cooperative output regulation for a network of parabolic PDE systems. IEEE Trans Autom Control, 2022, 67(1): 451−459. DOI: https://doi.org/10.1109/TAC.2021.3079352
  78. Liang H, Zhou Y, Ma H, Zhou Q, Adaptive distributed observer approach for cooperative containment control of nonidentical networks. IEEE Trans Syst Man Cybern: Syst, 2019, 49(2): 299−307. DOI: https://doi.org/10.1109/TSMC.2018.2791513
  79. Haghshenas H, Badamchizadeh MA, Baradarannia M, Containment control of heterogeneous linear multi-agent systems. Automatica, 2015, 54: 210−216. DOI: https://doi.org/10.1016/j.automatica.2015.02.002
  80. Lui DG, Petrillo A, Santini S, An optimal distributed PID-like control for the output containment and leader-following of heterogeneous high-order multi-agent systems. Inf Sci, 2020, 541: 166−184. DOI: https://doi.org/10.1016/j.ins.2020.06.049
  81. Wang Q, Dong X, Wen G, Lv J, Ren Z (online) Practical output containment of heterogeneous nonlinear multiagent systems under external disturbances. IEEE Trans Cybern. https://doi.org/10.1109/TCYB.2022.3175769 DOI: https://doi.org/10.1109/TCYB.2022.3175769
  82. Zhou P and Chen BM, Formation-containment control of Euler–Lagrange systems of leaders with bounded unknown inputs. IEEE Trans Cybern., 2022, 52(7): 6342−6353. DOI: https://doi.org/10.1109/TCYB.2020.3034931
  83. Kim H, Shim H, Seo JH, Output consensus of heterogeneous uncertain linear multi-agent systems. IEEE Trans Autom Control, 2011, 56(1): 200−206. DOI: https://doi.org/10.1109/TAC.2010.2088710
  84. Wieland P, Sepulchre R, Allgöwer F, An internal model principle is necessary and sufficient for linear output synchronization. Automatica, 2011, 47(5): 1068−1074. DOI: https://doi.org/10.1016/j.automatica.2011.01.081
  85. Zhu L, Chen Z, Middleton RH, A general framework for robust output synchronization of heterogeneous nonlinear networked systems. IEEE Trans Autom Control, 2016, 61(8): 2092−2107. DOI: https://doi.org/10.1109/TAC.2015.2492141
  86. Shi L, Li Y, Lin Z, Semi-global leader-following output consensus of heterogeneous multi-agent systems with input saturation. Int J Robust Nonlinear Control, 2018, 28: 4916−4930. DOI: https://doi.org/10.1002/rnc.4291
  87. Zhou P, Chen BM, Semi-global leader-following output consensus of heterogeneous systems with all agents subject to input saturation. Int J Robust Nonlinear Control, 2022, 32: 4648−664. DOI: https://doi.org/10.1002/rnc.6045
  88. Ma J, Yu X, Liu L, Ji HB, Feng G, Global cooperative output regulation of linear multiagent systems with limited bandwidth. IEEE Trans Control Netw Syst, 2022, 9(2): 1017−1028. DOI: https://doi.org/10.1109/TCNS.2021.3128500
  89. Song G, Song P, Lim CP, Distributed fault-tolerant cooperative output regulation for multiagent networks via fixed-time observer and adaptive control. IEEE Trans Control Netw Syst, 2022, 9(2): 845−855. DOI: https://doi.org/10.1109/TCNS.2021.3124915
  90. Baldi S, Azzollini IA, Ioannou PA, A distributed indirect adaptive approach to cooperative tracking in networks of uncertain single-input single-output systems. IEEE Trans Autom Control, 2021, 66(10): 4844−4851. DOI: https://doi.org/10.1109/TAC.2020.3038742
  91. Yaghmaie FA, Movric KH, Lewis FL, Su R, Sebek M, H∞-output regulation of linear heterogeneous multiagent systems over switching graphs. Int J Robust Nonlinear Control, 2018, 28: 3852−3870. DOI: https://doi.org/10.1002/rnc.4108
  92. Cai H, Huang J, Leader-following consensus of multiple uncertain Euler-Lagrange systems under switching network topology. Int J Gen Syst, 2014, 43(3-4): 294−304. DOI: https://doi.org/10.1080/03081079.2014.883714
  93. Lu M, Liu L, Leader-following consensus for multiple uncertain Euler-Lagrange systems with unknown dynamic leader. IEEE Trans Autom Control., 2019, 64(10): 4167−4173. DOI: https://doi.org/10.1109/TAC.2019.2892384
  94. Lu M, Liu L, Leader-following consensus of multiple uncertain Euler-Lagrange systems subject to communication delays and switching networks. IEEE Trans Autom Control., 2018, 63(8): 2604−2611. DOI: https://doi.org/10.1109/TAC.2017.2771318
  95. Lu M, Liu L, Robust synchronization control of switched networked Euler-Lagrange systems. IEEE Trans Cybern, 2022, 52(7): 6834−6842. DOI: https://doi.org/10.1109/TCYB.2020.3031009
  96. Feng Z, Hu G, Ren W, Dixon W, Mei J, Distributed coordination of multiple unknown Euler-Lagrange systems. IEEE Trans Control Netw Syst, 2018, 5(1): 55−66. DOI: https://doi.org/10.1109/TCNS.2016.2578039
  97. Wang T, Huang J (2021) Leader-following consensus of multiple uncertain Euler-Lagrange systems subject to unknown disturbances over switching networks. In: Proceedings of the 40th Chinese control conference DOI: https://doi.org/10.23919/CCC52363.2021.9550025
  98. Cai H, Huang J, The leader-following attitude control of multiple rigid spacecraft systems. Automatica, 2014, 50: 1109−1115. DOI: https://doi.org/10.1016/j.automatica.2014.01.003
  99. Cai H, Huang J, Leader-following attitude consensus of multiple rigid body systems by attitude feedback control. Automatica, 2016, 69: 87−92. DOI: https://doi.org/10.1016/j.automatica.2016.02.010
  100. Cai H, Huang J, Leader-following adaptive consensus of multiple uncertain rigid spacecraft systems. Sci China Inf Sci, 2016, 59: 1−13. DOI: https://doi.org/10.1007/s11432-015-5442-3
  101. Liu T, Huang J, Leader-following attitude consensus of multiple rigid body systems subject to jointly connected switching networks. Automatica, 2018, 92: 63−71. DOI: https://doi.org/10.1016/j.automatica.2018.02.012
  102. Wang T, Huang J, Leader-following adaptive consensus of multiple uncertain rigid body systems over jointly connected networks. Unmanned Syst, 2020, 8(2): 85−93. DOI: https://doi.org/10.1142/S2301385020500065
  103. Wang T, Huang J, Consensus of multiple spacecraft systems over switching networks by attitude feedback. IEEE Trans Aerosp Electron Syst, 2020, 56(3): 2018. DOI: https://doi.org/10.1109/TAES.2019.2940306
  104. Lu M, Liu L, Leader-following attitude consensus of multiple rigid spacecraft systems under switching networks. IEEE Trans Autom Control., 2020, 65(2): 839−845. DOI: https://doi.org/10.1109/TAC.2019.2920074
  105. Wang T, Huang J, Time-varying formation control with attitude synchronization of multiple rigid body systems. Int J Robust Nonlinear Control, 2022, 32(1): 181−204. DOI: https://doi.org/10.1002/rnc.5825
  106. Dong Y, J. Huang, Leader-following rendezvous with connectivity preservation of a class of multi-agent systems. Automatica, 2013, 49: 1386−1391. DOI: https://doi.org/10.1016/j.automatica.2013.02.024
  107. Dong Y, J, Huang. Flocking with connectivity preservation of multiple double integrator systems subject to external disturbances by a distributed control law. Automatica, 2015, 55: 197−203. DOI: https://doi.org/10.1016/j.automatica.2015.03.006
  108. Liu W, Huang J, Sampled-data cooperative output regulation of linear multi-agent systems. Int J Robust Nonlinear Control, 2021, 31(10): 4805−4822. DOI: https://doi.org/10.1002/rnc.5503
  109. Zheng S, Shi P, Agarwal RK, Lim CP, Periodic event-triggered output regulation for linear multi-agent systems. Automatica, 2020, 122: 109223. DOI: https://doi.org/10.1016/j.automatica.2020.109223
  110. Liu W, Huang J, Event-triggered global robust output regulation for a class of nonlinear Systems. IEEE Trans Autom Control, 2017, 27(11): 5923−5930. DOI: https://doi.org/10.1109/TAC.2017.2700384
  111. Liu W, Huang J, Event-triggered cooperative robust practical output regulation for a class of linear multi-agent systems. Automatica, 2017, 85: 158−164. DOI: https://doi.org/10.1016/j.automatica.2017.07.049
  112. Liu W, Huang J, Robust practical output regulation for a class of uncertain linear minimum-phase systems by output-based event-triggered control. Int J Robust Nonlinear Control, 2017, 27(18): 4574−4590. DOI: https://doi.org/10.1002/rnc.3815
  113. Liu W, Huang J, Cooperative global robust output regulation for a class of nonlinear multi-agent systems by distributed event-triggered control. Automatica, 2018, 93: 138−148. DOI: https://doi.org/10.1016/j.automatica.2018.03.062
  114. Liu W, Huang J, Event-triggered cooperative global robust practical output regulation for second-order uncertain nonlinear multi-agent. IEEE Trans Neural Netw Learn Syst, 2018, 29(11): 5486−5498. DOI: https://doi.org/10.1109/TNNLS.2018.2803142
  115. Liu W, Huang J, Global robust practical output regulation for nonlinear systems in output feedback form by output-based event-triggered control. Int J Robust Nonlinear Control, 2019, 29(6): 2007−2025. DOI: https://doi.org/10.1002/rnc.4472
  116. Su Y, Xu L, Wang X, Xu D, Event-based cooperative global practical output regulation of multi-agent systems with nonlinear leader. Automatica, 2019, 107: 600−604. DOI: https://doi.org/10.1016/j.automatica.2019.06.008
  117. Liang D, Huang J, Robust output regulation of linear systems by event-triggered dynamic output feedback control. IEEE Trans Autom Control, 2021, 66(5): 2415−2422. DOI: https://doi.org/10.1109/TAC.2020.3010772
  118. Qian Y, Liu L, Feng G, Cooperative output regulation of linear multiagent systems: an event-triggered adaptive distributed observer approach. IEEE Trans Autom Control, 2021, 66(2): 833−840. DOI: https://doi.org/10.1109/TAC.2020.2985947
  119. Cheng B, Li Z, Wang X, Cooperative output regulation of heterogeneous multi-agent systems with adaptive edge-event-triggered strategies. IEEE Trans Circuits Syst Ⅱ Express Briefs, 2020, 67(10): 2199−2203. DOI: https://doi.org/10.1109/TCSII.2019.2953930
  120. Deng C, Wen C, Huang J, Zhang X, Zou Y, Distributed observer-based cooperative control approach for uncertain nonlinear MASs under event-triggered communication. IEEE Trans Autom Control, 2022, 67(5): 2669−2676. DOI: https://doi.org/10.1109/TAC.2021.3090739
  121. Zhang H, Chen J, Wang Z, Fu C, Song S, Distributed event-triggered control for cooperative output regulation of multiagent systems with an online estimation algorithm. IEEE Trans Cybern, 2022, 52(3): 1911−1923. DOI: https://doi.org/10.1109/TCYB.2020.2991761
  122. Gao W, Jiang Z, Learning-based adaptive optimal output regulation of linear and nonlinear systems: an overview. Control Theory Technol, 2022, 20: 1−9. DOI: https://doi.org/10.1007/s11768-022-00081-3
  123. Kiumarsi B, Lewis FL, Output synchronization of heterogeneous discrete-time systems: A model-free optimal approach. Automatica, 2017, 84: 86−94. DOI: https://doi.org/10.1016/j.automatica.2017.07.004
  124. Modares H, Lewis FL, Kang W, Davoudi A, Optimal synchronization of heterogeneous nonlinear systems with unknown dynamics. IEEE Trans Autom Control, 2018, 63(1): 117−131. DOI: https://doi.org/10.1109/TAC.2017.2713339
  125. Gao W, Jiang Z, Lewis FL, Wang Y, Leader-to-formation stability of multi-agent systems: An adaptive optimal control approach. IEEE Trans Autom Control, 2018, 63(10): 3581−3587. DOI: https://doi.org/10.1109/TAC.2018.2799526
  126. Jiang Y, Fan J, Gao W, Chai T, Lewis, F L, Cooperative adaptive optimal output regulation of nonlinear discrete-time multi-agent systems. Automatica, 2020, 121: 109149. DOI: https://doi.org/10.1016/j.automatica.2020.109149
  127. Chen C, Frank L, Xie K, Xie S, Liu Y, Off-policy learning for adaptive optimal output synchronization of heterogeneous multi-agent systems. Automatica, 2020, 119: 109081. DOI: https://doi.org/10.1016/j.automatica.2020.109081
  128. Yang Y, Modares H, Wunsch DC, Yin Y, Leader-follower output synchronization of linear heterogeneous systems with active leader using reinforcement learning. IEEE Trans Neural Netw Learn Syst, 2018, 29(6): 2139−2153. DOI: https://doi.org/10.1109/TNNLS.2018.2803059
  129. Gao W, Mynuddin M, Wunsch DC, Jiang ZP (online) Reinforcement learning-based cooperative optimal output regulation via distributed adaptive internal model. IEEE Trans Neural Netw Learn Syst. https://doi.org/10.1109/TNNLS.2021.3069728 DOI: https://doi.org/10.1109/TNNLS.2021.3069728
  130. Deng C, Wen C, Distributed resilient observer-based fault-tolerant control for heterogeneous multiagent systems under actuator faults and DoS attacks. IEEE Trans Control Netw Syst, 2020, 7(3): 1308−1318. DOI: https://doi.org/10.1109/TCNS.2020.2972601
  131. Feng Z, Hu G, Secure cooperative event-triggered control of linear multiagent systems under DoS attacks. IEEE Trans Control Syst Technol, 2020, 28(3): 741−752. DOI: https://doi.org/10.1109/TCST.2019.2892032
  132. Xu Y, Fang M, Pan YJ, Shi K, Wu ZG, Event-triggered output synchronization for nonhomogeneous agent systems with periodic denial-of-service attacks. Int J Robust Nonlinear Control, 2021, 31(6): 1851−1865. DOI: https://doi.org/10.1002/rnc.5223
  133. Yan J, Deng C, Wen C, Resilient output regulation in heterogeneous networked systems under Byzantine agents. Automatica, 2021, 133: 109872. DOI: https://doi.org/10.1016/j.automatica.2021.109872
  134. Deng C, Wen C, MAS-based distributed resilient control for a class of cyber-physical systems with communication delays under DoS attacks. IEEE Trans Cybern, 2021, 51(5): 2147−2358. DOI: https://doi.org/10.1109/TCYB.2020.2972686
  135. Deng C, Zhang D, Feng G, Resilient practical cooperative output regulation for MASs with unknown switching exosystem dynamics under DoS attacks. Automatica, 2022, 139: 110172. DOI: https://doi.org/10.1016/j.automatica.2022.110172
  136. Zhang D, Deng C, Feng G (online) Resilient cooperative output regulation for nonlinear multi-agent systems under DoS attacks. IEEE Trans Autom Control. https://doi.org/10.1109/TAC.2022.3184388 DOI: https://doi.org/10.1109/TAC.2022.3184388
  137. Du S, Xu W, Qiao J, Daniel WC (online) Resilient output synchronization of heterogeneous multiagent systems with DoS attacks under distributed event-/self-triggered control. IEEE Trans Neural Netw Learn Syst. https://doi.org/10.1109/TNNLS.2021.3105006 DOI: https://doi.org/10.1109/TNNLS.2021.3105006
  138. Liu XK, Zhang JF, Wang J, Differentially private consensus algorithm for continuous-time heterogeneous multi-agent systems. Automatica, 2020, 122: 109283. DOI: https://doi.org/10.1016/j.automatica.2020.109283
  139. Wang X, Hong Y, Huang J, Jiang ZP, A distributed control approach to a robust output regulation problem for multi-agent linear systems. IEEE Trans Autom Control, 2010, 55(12): 2891−2895. DOI: https://doi.org/10.1109/TAC.2010.2076250
  140. Su Y, Hong Y, Huang J, A general result on the robust cooperative output regulation for linear uncertain multi-agent systems. IEEE Trans Autom Control, 2013, 58(5): 1275−1279. DOI: https://doi.org/10.1109/TAC.2012.2229837
  141. Huang C, Ye X, Cooperative output regulation of heterogeneous multi-agent systems: an H∞ criterion. IEEE Trans Autom Control, 2013, 59(1): 267−273. DOI: https://doi.org/10.1109/TAC.2013.2272133
  142. Koru AT, Sarsilmaz SB, Yucelen T, Johnson EN, Cooperative output regulation of heterogeneous multiagent systems: a global distributed control synthesis approach. IEEE Trans Autom Control, 2021, 66(9): 4289−4296. DOI: https://doi.org/10.1109/TAC.2020.3032496
  143. Su Y, Huang J, Cooperative robust output regulation of a class of heterogeneous linear uncertain multi-agent systems. Int J Robust Nonlinear Control, 2014, 24(17): 2819−2839. DOI: https://doi.org/10.1002/rnc.3027
  144. Hong Y, Wang X, Jiang Z, Distributed output regulation of leader-follower multi-agent systems. Int J Robust Nonlinear Control, 2013, 23(1): 48−66. DOI: https://doi.org/10.1002/rnc.1814
  145. Zhang Y, Su Y, Wang X, Distributed adaptive output feedback control for multi-agent systems with unknown dynamics. IEEE Trans Autom Control, 2021, 66(3): 1367−1374. DOI: https://doi.org/10.1109/TAC.2020.2994293
  146. Zhang Y, Su Y, Cooperative output regulation for linear uncertain MIMO multi-agent systems by output feedback. Sci China Inf Sci, 2018, 61(9): 092206. DOI: https://doi.org/10.1007/s11432-017-9255-2
  147. Dong Y, Huang J, Cooperative global robust output regulation for nonlinear multi-agent systems in output feedback form. J Dyn Syst Meas Control-Trans ASME, 2014, 136(3): 031001. DOI: https://doi.org/10.1115/1.4026135
  148. Dong Y, Huang J, Cooperative global output regulation for a class of nonlinear multiagent systems. IEEE Trans Autom Control, 2014, 59(5): 1348−354. DOI: https://doi.org/10.1109/TAC.2013.2292723
  149. Su Y, Huang J, Cooperative global output regulation for nonlinear uncertain multi-agent systems in lower triangular form. IEEE Trans Autom Control, 2015, 60(9): 2378−2389. DOI: https://doi.org/10.1109/TAC.2015.2400713
  150. Su Y, Huang J, Cooperative global output regulation for a class of nonlinear uncertain multi-agent systems with unknown leader. Syst Control Lett, 2013, 62(6): 461−467. DOI: https://doi.org/10.1016/j.sysconle.2013.02.013
  151. Wang X, Su Y, Xu D, Nonlinear output-feedback tracking in multiagent systems with an unknown leader and directed communication. Sci China Inf Sci, 2021, 64: 222202. DOI: https://doi.org/10.1007/s11432-020-3108-6
  152. Ding Z, Consensus output regulation of a class of heterogeneous nonlinear systems. IEEE Trans Autom Control, 2013, 58(10): 2648−2653. DOI: https://doi.org/10.1109/TAC.2013.2255973
  153. Isidori A, Marconi L, Casadei G, Robust output synchronization of a network of heterogeneous nonlinear agents via nonlinear regulation theory. IEEE Trans Autom Control, 2014, 59(10): 2680−2691. DOI: https://doi.org/10.1109/TAC.2014.2326213
  154. Priscoli FD, Isidori A, Marconi L, Pietrabissa A, Leader-following coordination of nonlinear agents under time-varying communication topologies. IEEE Trans Control Netw Syst, 2015, 2(4): 393−405. DOI: https://doi.org/10.1109/TCNS.2015.2426752
  155. Wang L, Wen C, Guo F, Cai H, Su H, Robust cooperative output regulation of uncertain linear multi-agent systems not detectable by regulated output. Automatica, 2019, 101: 309−317. DOI: https://doi.org/10.1016/j.automatica.2018.12.018
  156. Kawamura S, Cai K, Kishida M, Distributed output regulation of heterogeneous uncertain linear agents. Automatica, 2020, 119: 109094. DOI: https://doi.org/10.1016/j.automatica.2020.109094
  157. Basu H, Yoon SY, Robust cooperative output regulation under exosystem detectability constraint. Int J Control, 2018, 93(5): 1102−1114. DOI: https://doi.org/10.1080/00207179.2018.1493537
  158. Wang L, Wen CY, Liu ZT, Su HY, Cai JP, Robust cooperative output regulation of heterogeneous uncertain linear multiagent systems with time-varying communication topologies. IEEE Trans Autom Control, 2020, 65(10): 4340. DOI: https://doi.org/10.1109/TAC.2019.2954349
  159. Bi C, Xu X, Liu L, Feng G, Robust cooperative output regulation of heterogeneous uncertain linear multiagent systems with unbounded distributed transmission delays. IEEE Trans Autom Control, 2022, 67(3): 1371−1383. DOI: https://doi.org/10.1109/TAC.2021.3069718
  160. Li R, Leader-following output synchronization for a class of uncertain nonlinear multi-agent systems under uniformly connected network. Asian J. Control, 2016, 18(1): 1−11.
  161. Liu W, Huang J, Cooperative global robust output regulation for nonlinear output feedback multi-agent systems under directed switching networks. IEEE Trans Autom Control, 2017, 62(12): 6339−6352. DOI: https://doi.org/10.1109/TAC.2017.2712747
  162. Liu T, Huang J, Cooperative robust output regulation for a class of nonlinear multi-agent systems subject to a nonlinear leader system. Automatica, 2019, 108: 108501. DOI: https://doi.org/10.1016/j.automatica.2019.108501