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*International Journal of Network Dynamics and Intelligence*. 2022,

*1*(1), 20–35. doi: https://doi.org/10.53941/ijndi0101003

This work is licensed under a Creative Commons Attribution 4.0 International License.

*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

- 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)
- 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)
- 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)
- Johnson CD, Accommodation of external disturbances in linear regulator and servomechanism problem. IEEE Trans Autom Control, 1971, 16(6): 535−544.
- Smith HW, Davison EJ, Design of industrial regulators: integral feedback and feedforward control. Proc IEEE, 1972, 199(8): 1210−1216.
- Cheng L, Pearson JB, Frequency-domain synthesis of multivariable linear regulators. IEEE Trans Autom Control, 1978, 23(1): 3−15.
- Davison EJ, The output control of linear time-invariant multivariable systems with unmeasurable arbitrary disturbance. IEEE Trans Autom Control, 1972, 17(5): 621−630.
- 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.
- Davison EJ, The robust control of a servomechanism problem for linear time-invariant multivariable systems. IEEE Trans Autom Control, 1976, 21(1): 25−34.
- Francis BA, Wonham WM, The internal model principle of control theory. Automatica, 1976, 12(5): 457−465.
- Francis BA, The linear multivariable regulator problem. SIAM J Control Optim, 1977, 15(3): 486−505.
- Wonham WM, Pearson JB, Regulation and internal stabilization in linear multivariable systems. SIAM J Control Optim, 1974, 12(1): 5−18.
- Ogren P, Egerstedt M, Hu X, A control Lyapunov function approach to multiagent coordination. IEEE Trans Robot Autom, 2002, 18(5): 847−851.
- Fax JA, Murray RM, Information flow and cooperative control of vehicle formations. IEEE Trans Autom Control, 2004, 49(9): 1465−1476.
- 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.
- 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.
- 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)
- 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)
- Ren W, Beard RW, Consensus seeking in multiagent systems under dynamically changing interaction topologies. IEEE Trans Autom Control, 2005, 50(5): 655−661.
- 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.
- Ren W, Synchronization of coupled harmonic oscillators with local interaction. Automatica, 2008, 44(2): 3195−3200.
- Tuna SE (2008) LQR-based coupling gain for synchronization of linear systems. arxiv.org/abs/0801.3390
- 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.
- 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.
- Bai H, Arcak M, Wen JT, Rigid body attitude coordination without inertial frame information. Automatica, 2008, 44(12): 3170−3175.
- Su Y, Huang J, Cooperative output regulation of linear multi-agent systems. IEEE Trans Autom Control, 2012, 57(4): 1062−1066.
- 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.
- Su Y, Huang J, Cooperative output regulation of linear multi-agent systems by output feedback. Syst Control Lett, 2012, 61(12): 1248−1253.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- Meng Z, Yang T, Dimarogonasa DV, Johansson KH, Coordinated output regulation of heterogeneous linear systems under switching topologies. Automatica, 2015, 53: 362−368.
- Yaghmaie FA, Lewis FL, Su R, Output regulation of linear heterogeneous multi-agent systems via output and state feedback. Automatica, 2016, 67: 157−164.
- 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.
- 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.
- 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.
- Cai H, Su Y, Huang J (2022) Cooperative control of multi-agent systems: distributed-observer and internal-model-approaches. Springer Nature Switzerland AG, Switzerland
- 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.
- 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.
- 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.
- 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.
- Abdessameud A, Tayebi A, Distributed output regulation of heterogeneous linear multi-agent systems with communication constraints. Automatica, 2018, 91: 152−158.
- 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.
- 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.
- Li Z, Chen MZQ, Ding Z, Distributed adaptive controllers for cooperative output regulation of heterogeneous agents over directed graphs. Automatica, 2016, 68: 179−183.
- 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.
- 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.
- 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.
- 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.
- Cai H, Huang J, Output based adaptive distributed output observer for leader–follower multiagent systems. Automatica, 2021, 125: 109413.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- Su Y, Huang J, Two consensus problems for discrete-time multi-agent systems with switching network topology. Automatica, 2012, 48(9): 1988−1997.
- 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.
- 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.
- Lu M, Huang J, Cooperative output regulation problem for linear time-delay multi-agent systems under switching network. Neurocomputing, 2016, 190: 132−139.
- Yan Y, Huang J, Cooperative output regulation of discrete-time linear time-delay multiagent systems. IET Control Theory Appl, 2016, 10(16): 2019−2026.
- Yan Y, Huang J, Cooperative output regulation of discrete-time linear time-delay multiagent systems under switching network. Neurocomputing, 2017, 241: 108−114.
- 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.
- 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
- Ma Q, Xu S, Lewis FL, Cooperative output regulation of singular heterogeneous multiagent systems. IEEE Trans Cybern, 2016, 46(6): 1471−1475.
- 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.
- 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.
- 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.
- 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.
- Deutscher J, Cooperative output regulation for a network of parabolic systems with varying parameters. Automatica, 2021, 125: 109446.
- Deutscher J, Robust cooperative output regulation for a network of parabolic PDE systems. IEEE Trans Autom Control, 2022, 67(1): 451−459.
- 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.
- Haghshenas H, Badamchizadeh MA, Baradarannia M, Containment control of heterogeneous linear multi-agent systems. Automatica, 2015, 54: 210−216.
- 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.
- 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
- 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.
- Kim H, Shim H, Seo JH, Output consensus of heterogeneous uncertain linear multi-agent systems. IEEE Trans Autom Control, 2011, 56(1): 200−206.
- Wieland P, Sepulchre R, Allgöwer F, An internal model principle is necessary and sufficient for linear output synchronization. Automatica, 2011, 47(5): 1068−1074.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- Lu M, Liu L, Robust synchronization control of switched networked Euler-Lagrange systems. IEEE Trans Cybern, 2022, 52(7): 6834−6842.
- 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.
- 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
- Cai H, Huang J, The leader-following attitude control of multiple rigid spacecraft systems. Automatica, 2014, 50: 1109−1115.
- Cai H, Huang J, Leader-following attitude consensus of multiple rigid body systems by attitude feedback control. Automatica, 2016, 69: 87−92.
- Cai H, Huang J, Leader-following adaptive consensus of multiple uncertain rigid spacecraft systems. Sci China Inf Sci, 2016, 59: 1−13.
- Liu T, Huang J, Leader-following attitude consensus of multiple rigid body systems subject to jointly connected switching networks. Automatica, 2018, 92: 63−71.
- 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.
- Wang T, Huang J, Consensus of multiple spacecraft systems over switching networks by attitude feedback. IEEE Trans Aerosp Electron Syst, 2020, 56(3): 2018.
- 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.
- 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.
- Dong Y, J. Huang, Leader-following rendezvous with connectivity preservation of a class of multi-agent systems. Automatica, 2013, 49: 1386−1391.
- 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.
- Liu W, Huang J, Sampled-data cooperative output regulation of linear multi-agent systems. Int J Robust Nonlinear Control, 2021, 31(10): 4805−4822.
- Zheng S, Shi P, Agarwal RK, Lim CP, Periodic event-triggered output regulation for linear multi-agent systems. Automatica, 2020, 122: 109223.
- 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.
- Liu W, Huang J, Event-triggered cooperative robust practical output regulation for a class of linear multi-agent systems. Automatica, 2017, 85: 158−164.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- Gao W, Jiang Z, Learning-based adaptive optimal output regulation of linear and nonlinear systems: an overview. Control Theory Technol, 2022, 20: 1−9.
- Kiumarsi B, Lewis FL, Output synchronization of heterogeneous discrete-time systems: A model-free optimal approach. Automatica, 2017, 84: 86−94.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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
- 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.
- 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.
- 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.
- Yan J, Deng C, Wen C, Resilient output regulation in heterogeneous networked systems under Byzantine agents. Automatica, 2021, 133: 109872.
- 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.
- 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.
- 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
- 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
- Liu XK, Zhang JF, Wang J, Differentially private consensus algorithm for continuous-time heterogeneous multi-agent systems. Automatica, 2020, 122: 109283.
- 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.
- 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.
- Huang C, Ye X, Cooperative output regulation of heterogeneous multi-agent systems: an H∞ criterion. IEEE Trans Autom Control, 2013, 59(1): 267−273.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- Dong Y, Huang J, Cooperative global output regulation for a class of nonlinear multiagent systems. IEEE Trans Autom Control, 2014, 59(5): 1348−354.
- 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.
- 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.
- 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.
- Ding Z, Consensus output regulation of a class of heterogeneous nonlinear systems. IEEE Trans Autom Control, 2013, 58(10): 2648−2653.
- 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.
- 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.
- 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.
- Kawamura S, Cai K, Kishida M, Distributed output regulation of heterogeneous uncertain linear agents. Automatica, 2020, 119: 109094.
- Basu H, Yoon SY, Robust cooperative output regulation under exosystem detectability constraint. Int J Control, 2018, 93(5): 1102−1114.
- 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.
- 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.
- 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.
- 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.
- 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.