Cooperative Control for Uncertain Multiagent Systems via Distributed Output Regulation

The distributed robust output regulation problem for multiagent systems is considered. For heterogeneous uncertain linear systems and a linear exosystem, the controlling aim is to stabilize the closed-loop system and meanwhile let the regulated outputs converge to the origin asymptotically, by the help of local interaction. The communication topology considered is directed acyclic graphs, which means directed graphs without loops. With distributed dynamic state feedback controller and output feedback controller, respectively, the solvability of the problem and the algorithm of controller design are both investigated. The solvability conditions are given in terms of linear matrix inequalities (LMIs). It is shown that, for polytopic uncertainties, the distributed controllers constructed by solving LMIs can satisfy the requirements of output regulation property.


Introduction
Recently, there are amounts of researches on cooperative control for multiagent systems (MASs) because of broad applications.An MAS is a practical model to describe dynamic agents which can exchange information by communication, such as unmanned air vehicles and sensor networks.According to different control objectives, problems of consensus, tracking, formation, flocking and the rest have been widely studied.
Among those cooperative control problems, the consensus problem and the tracking problem share some common characteristics.The consensus problem requires the MAS to reach an agreement by protocols based on local information.In [1,2], the consensus problem is primarily studied, and the basic problem framework is formed.The consensus problem has been investigated for different kinds of agents, such as first-order integrators in [3], second-order integrators in [4,5], linear systems in [6][7][8], and nonlinear systems in [9][10][11][12].Recently, the output consensus problem for heterogeneous systems also attracted researchers.The dynamics and even dimensions of the agents are possibly different, so it is desirable to focus on the synchronization of outputs.In [13], the consensus of a class of second-order integrators with unknown nonlinear dynamics is considered.As for high-order systems, the frequency domain approach is used to discuss the consensus of heterogenous linear systems in [14,15].Uncertain minimum-phase linear MASs are studied in [16], by a low-gain approach.In [7], general uncertain linear MASs are considered, and a sufficient and necessary condition for the solvability of the output consensus problem is proposed.It is admirable that Wieland et al. introduce an important concept of internal model to cooperative control, which is also fundamental in output regulation theory.
For the leader-follower consensus problem, also called consensus tracking problem, it involves one or multiple leaders and several followers.A leader is usually the target to be tracked, or the agent that directly receives the information of the target.Distributed controllers are designed to help all the agents to track one or multiple leaders by cooperation.The tracking problem for MASs has been studied in a lot of papers, such as [17][18][19][20][21][22].Note that for heterogenous MASs, the outputs of all the agents are required to be synchronized in the issue of both consensus problem and tracking problem.To consider the two kinds of problems under a unified framework is one of the motivations to introduce distributed output regulation (DOR) problem.

Mathematical Problems in Engineering
According to [23], output regulation problem involves an exosystem and a regulated output defined by a combination of the measurement output and the output of the exosystem.Controllers are designed to stabilize the closed-loop system and modulate the regulated outputs to the origin.So, it has an attracting performance on solving a tracking problem in the presence of disturbances.The classic output regulation theory cannot directly be applied to MASs nevertheless.Actually the controllers obtained are probably not in a distributed form.So, a framework of DOR for MASs is introduced in [24][25][26].What is mainly different from the classic theory is that the controllers have to be distributed, and only local information is available.In [24], homogenous linear MASs and a directed topology are considered, and dynamic state/output feedback controllers are designed.In [25], heterogenous MASs are challenged, and effective controllers are obtained under the directed acyclic topology.Different from the two works above, the limits on topology are dispelled in [26] by reconstructing the form of controllers.The communication of relative states of controllers replaces the communication of relative outputs.
This paper is basically motivated by [25].In [25], the distributed controllers are robust to uncertain dynamics with sufficiently small uncertainties.However, more analyses based on information of the uncertainties are not involved in [25].What we focus on in our paper is that how to design robust controllers if the uncertainty is structured.We suppose that the uncertainties are in a polytopic form.For the distributed robust output regulation problem, we give sufficient conditions of the solvability in terms of linear matrix inequalities (LMIs) and present an approach of controllers design.
An outline of this paper is as follows.In Section 2, some preliminaries and the problem statement are given.In Section 3, for distributed output regulation problem with polytopic uncertainties, the sufficient conditions of solvability and algorithms of both state and output feedback controllers design are proposed.In Section 4, a practical example is taken to show the control effect of our approach, which is compared with that of the algorithm given in [25].In Section 5, a conclusion is given.
The following notations will be used throughout this paper.R is the set of real numbers.  is the -dimensional identity matrix.0 × is the zero matrix with  rows and  columns, and 0 is the zero matrix with appropriate dimensions.For a symmetric matrix  ∈ R × ,  > 0 which means that  is positive definite.  is the transposition of the matrix .diag block ( 1 , . . .,   ) means a block diagonal matrix with  1 , . . .,   as the diagonal entries.⊗ denotes the Kronecker product.

Problem Statement
Let us begin with notations in graph theory [27].A graph is denoted by G = (V, E), where V = {V 1 , V 2 , . . ., V  } is the set of  nodes and E ⊆ V × V is the set of edges.An edge from node V  to node V  is denoted by (, ) ∈ E. A subset of E{( 1 ,  2 ), ( 2 ,  3 ), . . ., (  ,  +1 )} is called a path from Consider an exosystem with  ∈ R  as the state, whose dynamics can be described as follows: where  ∈ R × is a constant matrix, satisfying the following assumption as that in [23,25].
Assumption 1.  has no eigenvalues with negative real parts.
A MAS consists of  nonidentical dynamic agents which can exchange information among neighborhood.For  = 1, . . ., , the th agent can be expressed by where   ∈ R   ,   ∈ R   ,   ∈ R  , and   ∈ R  are, respectively, the state, the control input, the measurement output, and the regulated output of the th agent. ∈ R × is a certain constant matrix, while those matrices   ∈ R   ×  ,   ∈ R   ×  , and   ∈ R ×  are uncertain matrices represented as where   ,   , and   are known constant matrices, and   are nonnegative constants satisfying that ∑   =1   = 1, for  = 1, . . ., ,  = 1, . . .,   .  ∈ R   × is an arbitrary constant matrix.
The controlling aim is to stabilize the closed-loop system and also to regulate  1 , . . .,   to the origin.For  = 1, . . ., , if   is available to the th agent, the output regulation of the th agent is simple to be achieved by classic output regulation theory.However, only some of the agents can get information of their own regulated outputs, which are called leader nodes.The set of their serial numbers is denoted by L, while other agents utilize the relative output among neighbors to accomplish the output regulation property.To ensure that all the agents can receive the information of the exosystem by local interaction, the communication graph of the MAS satisfies the following assumption.
Assumption 2. Graph G is a directed acyclic graph.And for each nonleader node V  , there exists a leader node V  such that a path from node V  to node V  exists.
In this note, two kinds of distributed dynamic feedback controllers are considered.
(I) Distributed dynamic state feedback controller: where 1 ,  2 ,  1 , and  2 are designed matrices with appropriate dimensions.(II) Distributed dynamic output feedback controller: ,  1 , and  2 are designed matrices with appropriate dimensions.
The variables   () in distributed feedback control laws (4) and ( 6) are measurable relative outputs.For  ∈ L,   () is the regulated output   ().For  ∉ L,   () is the average of relative output errors between th agent and its neighbors.
In the sequel, we rewrite the closed-loop system into a composite form.Let 1  be a column vector with all the elements as 1.   and   , respectively, denote the th row of  and 1  .For  ∈ L, . . .
Problem 3. Distributed output regulation problem: design controllers in the form of ( 4) or ( 6) such that the closed-loop system (8) has the following properties.
(i) It is exponentially stable at the origin with  = 0.

Main Results
In this section, we give two theorems about the solvability of Problem 3 and the approach of controller design.First of all, we introduce the concept of quadratic stability and related lemmas, which will be used later.
Definition 4 (Amato [28]).Consider a parametric uncertain linear system given by where () ∈ R  0 ,  0 ∈  ⊂ R  0 is the vector of uncertain parameters, where  is a hyperbox, and (⋅) is continuous.This system is said to be quadratically stable (QS) in  if and only if there exists a symmetric positive definite matrix  ∈ R  0 × 0 such that for all  0 ∈ ,   ( 0 ) + ( 0 ) < 0.
Lemma 6 (Amato [28]).Assume that the uncertain system is in a polytopic form; that is, where  1 , . . .,   ∈ R  0 × 0 and conv{⋅} represents the convex hull of the following matrices.It is QS if and only if there exists a positive definite symmetric matrix  such that for  = 1, . . ., ,     +   < 0.
Second, we need to recall the concept of internal model and its property.
Definition 7 (Huang [23]).Given any square matrix , a pair of matrices (G 1 , G 2 ) is said to incorporate a -copy internal model of the matrix  if the pair satisfies that where  1 ,  2 , and  3 are arbitrary constant matrices of appropriate dimensions,  is any nonsingular matrix with the same dimension as G 1 , and  1 ,  2 are described as follows: where for  = 1, . . ., ,   is a constant square matrix of dimension   for some integer   and   is a constant column vector of dimension   such that (i)   and   are controllable, (ii) the minimal polynomial of  divides the characteristic polynomial of   .
Lemma 8 (Huang [23]).Under Assumption 1, assume that be exponentially stable, where Â, B, and Ĉ are any matrices with appropriate dimensions.Then, for any matrices Ê and F of appropriate dimensions, the following matrix equations have a unique solution  and .Moreover,  satisfies Ĉ+ F = 0.
Based on these preparations, theorems on solvability of Problem 3 with state/output feedback controllers are given as follows.
Proof.Let   () = (   ,    )  .Then, the subsystems of (8) with controller (4) can be written as follows: for  ∈ L, )  () , (19) and for  ∉ L, According to [29], by relabeling the nodes, a directed acyclic graph could be put into an ordered form.That is to say, for all edges (, ) ∈ E,  >  holds.Notice that matrix   is consequently a block lower triangular matrix and system (8) with  = 0 is asymptotically stable if and only if all the subsystems below are asymptotically stable Consider LMI (17).When it holds for a symmetric positive definite matrix P and a matrix   , let   = P−1  , and let   =     .Pre-and postmultiplied by   , ( 17) is equivalent to the following inequality: According to Lemma 6, when it holds, the subsystems ( 21) are all QS.And according to Lemma 5, for any polytopic uncertainties, the subsystems are exponentially stable.The system (8) with  = 0 is consequently exponentially stable.The condition (i) of Problem 3 has been satisfied.In the following, the error () is proved to converge to zero.
Remark 10.When there is no uncertainty in the system (2),   =  1 ,   =  1 , and   =  1 ,  = 1, . . ., .Then, the conclusion of Theorem 9 still holds if the solvability of LMI ( 17) is replaced by the statement that the pair ( 1 ,  1 ) is stabilizable.In fact, according to [23], from the two statements, (i) the pair ( 1 ,  1 ) is stabilizable, (ii) Assumption 1 holds, and the pair ( 1 ,  2 ) is an internal model of ; it is followed that the pair Then, Problem 3 is solvable by a dynamic output feedback controller (6) with where Proof.For  = 1, . . ., , the subsystems of ( 8) with ( 6) and (31) can be written in such a form: for  ∈ L, and for  ∉ L, Similar to the proof of the previous theorem, the stability of the system (8) with  = 0 is dependent on the following subsystems: Let Then, the subsystem (34) is similar to the following system: where According to Lemma 6, the system (36) is QS if there exists a symmetric positive definite matrix   > 0 such that Suppose that   is a block diagonal matrix, which means that   = ( ) , and   ∈ R   ×  .Then, the inequality above can be rewritten as where Notice that Π 11, < 0 is just the inequality (22), which is equivalent to LMI (17).When Π 11, < 0 holds, the matrices P ,   ,   = P−1  ,   = ( 1 ,  2 ) =   P have already been obtained.Let    1 =  1 ,    2 =  2 , and    3 =  3 .We turn the inequality (39) into LMI (29).This implies that the system (8) with  = 0 is asymptotically stable.
The rest part of the proof is similar to that in Theorem 9. Recalling that ) is Hurwitz and ( 1 ,  2 ) incorporates a -copy internal model of , we can obtain that for any matrices   and   , the following matrix equations have a unique solution   and   .And meanwhile, By calculation, it can be verified that lim  → ∞   () = 0. Consequently, lim  → ∞   () = 0.When Assumption 2 holds, the above statement is equivalent to lim  → ∞ () = 0.This completes the proof.
Remark 12. Consider the robust tracking problem for heterogenous MAS in the presence of disturbances.Suppose that an active leader is described as and there is an environmental disturbance () satisfying The model of the MAS can be given in where   ,   ,   , and    are uncertain matrices with polytopic uncertainties.The control target is to design distributed controllers such that the outputs of followers track the output of the leader.That is to say, for  = 1, . . ., ,   () =   () −   () converges to zero.This problem is just Problem 3 with  = (   ,   )  ,  = diag block (  ,   ),  = (0   ), and  = (−  0).The robust tracking problem for the systems (44), therefore, can be studied under the framework of DOR problem.It can be solved by designing the controller (4) or ( 6) if the conditions in Theorems 9 or 11 are satisfied.
Remark 13.By Theorems 9 and 11, the sufficient condition of the solvability for Problem 3 depends on the dynamics of each agent.For a heterogenous MAS,  LMIs have to be solved to obtain appropriate distributed controllers, which limits scalability.When the agents share some common characteristics, such as nominal parts, the complexity of LMIs will decrease.Specially, if agents are identical in the sense of common nominal parts and common bounds of the uncertainties, the solution to one LMI will construct the required control law.

A Numerical Example
In this section, a numerical example of a heterogenous MAS consisting seven agents is given to show the control effect of controllers obtained by Theorems 9 and 11.A comparison is also given between our controllers and that of [25].In the example, suppose that there is a constant environmental noise and the reference signal (the output of the tracking target) is a sinusoidal signal.Let  = ( 1 ,  2 ,  3 )  , where  1 ,  2 serve as the reference signals and  3 as the exogenous disturbance.Obviously, the differential equations about them are ω 1 () = − 1 (), ω 2 () = − 2 (), and ω 3 () = 0. Therefore, the exosystem can be described by (1) with According to [23], we can immediately obtain a 1-copy internal model as follows: ) . (46) In the example, three kinds of linear agents with 2, 3, and 4 orders are considered.2 or 3 basement matrices are randomly valued for each kind.We present the detailed description of basement matrices in the appendix.
The communication network is described by the directed acyclic graph shown in Figure 1 (29) need to be solved to construct the output feedback controllers.In this situation, we obtain  1 ,  2 and  3 for  = 1, . . ., 7, which are also listed in the appendix due to space limitation.And then the distributed dynamic output feedback controllers are given as (6).For simplicity, we denote this control law as controller   .
On the other hand, we translate the polytopic uncertainty into an equivalent form that consists of a nominal part and an uncertain part.In this case, the nominal part  1 , . . .,  7 ,  1 , . . .,  7 , and  1 , . . .,  7 can be valued as the average of the basement matrices.Then, by the algorithm in [25], for  = 1, . . ., 7, the candidate control law can be chosen as (6) with where For simplicity, this control law obtained by [25] is denoted by controller   .Note that both controllers   and   are robust controllers against the model uncertainty.To compare the effects, two cases of uncertainties are considered in the example.For one case, the model uncertainties denoted by Δ 1 are rather close to the "nominal part" set earlier.For  = 1, . . ., 7,  = 1, . . .,   , the values of   are randomly chosen around 1/  , whose values are shown in the appendix.For another case, the model uncertainties denoted by Δ 2 are valued freely with reasonable coefficients, which are relatively farther from the "nominal part." With the same initial values, the numerical results of the regulated output for each agent are demonstrated in Figure 2 and Figure 3.Our controller   is effective in the presence of either Δ 1 or Δ 2 .However, as for Controller   , when the uncertainty is relatively small, it can help to achieve the output regulation property.While in the case of Δ 2 , the regulated errors diverge rapidly as shown in Figure 3(b).As a consequence of the above, for Problem 3 with polytopic uncertainties, Theorems 9 and 11 help construct distributed dynamic state/output feedback controllers of stronger robustness, at the price of increasing the calculation complexity.

Conclusions
In this paper, we consider the distributed robust output regulation problem for MASs under the topology of directed acyclic graphs.As an extension of the existing results, we focus on a special class of parametric uncertainty.Assume that the dynamics of heterogeneous agents are in a polytopic form, and we study the solvability of the problem and the design of distributed controllers.Both dynamic state feedback controller and dynamic output feedback controller are under consideration.At last, a practical example is presented to validate our results.By local interaction, a team of heterogenous agents is required to achieve a common output, which is generated by an exosystem.The controller generated by our theorem can realize the property of output regulation and the performance of tracking the reference signal and rejecting the disturbances as a special case.

Mathematical Problems in Engineering
The coefficients in case of the uncertainty Δ 1 are as follows:

Figure 1 :
Figure 1: Information flow among exosystem and agents.
the path is called a loop.  denotes the neighbor set { : (, ) ∈ E, 1 ≤  ≤ }, whose cardinality is |  |.A constant matrix   = [  ] ∈ R × is called the adjacency matrix of graph G if   = 1/|  | when (, ) ∈ E and   = 0 when (, ) ∉ E. And a constant matrix  = [  ] =   −   ∈ R × is called the Laplacian matrix of graph G.A graph is called an undirected graph if for all 1 ≤ ,  ≤ ,   =   .Or else, it is called a directed graph.If a directed graph does not contain a loop, it is called a directed acyclic graph.