# Internal Model Approach to Cooperative Robust Output Regulation for Linear Uncertain Time-Delay Multi-Agent Systems

###### Abstract

In this paper, we study the cooperative robust output regulation problem for linear uncertain multi-agent systems with both communication delay and input delay by the distributed internal model approach. The problem includes the leader-following consensus problem of linear multi-agent systems with time-delay as a special case. We first generalize the internal model design method to systems with both communication delay and input delay. Then, under a set of standard assumptions, we have obtained the solution of the problem via both the state feedback control and the output feedback control. In contrast with the existing results, our results apply to general uncertain linear multi-agent systems, accommodate a large class of leader signals, and achieve the asymptotic tracking and disturbance rejection at the same time.

## 1 Introduction

In this paper, we consider the cooperative robust output regulation for linear uncertain time-delay systems of the following form:

(1) |

where , , and are the system state, measurement output, and control input of the subsystem, is the input delay, and is the exogenous signal representing the reference input to be tracked or/and disturbance to be rejected and is assumed to be generated by the exosystem of the form

(2) |

where is a constant matrix.

The regulated output for each subsystem is defined as

(3) |

where .

Let with be the Banach space of continuous functions mapping the interval into endowed with the supremum norm. We assume .

The plant (1) and (2) can be viewed as a multi-agent systems with the exosystem (2) as the leader and the subsystems of (1) as the followers. The communication
topology can be described by a directed graph ^{1}^{1}1See Appendix for a summary of digraph., where is the node set with the node 0 associated with the exosystem (2) and all the other nodes associated with the subsystems (1), and is the edge set. The edge , if and only if the control can access the state and / or the output of subsystem .
If , node is called a neighbor of the node . We use to denote the neighbor set of node with respect to .

Due to the communication constraint and the communication time-delay, we are limited to consider the class of distributed control laws with the communication delay. Mathmetically, such a control law is described as follows:

(4) |

where , , and are linear functions of their arguments, represents the communication delay among the agents. The control law (4) is called a distributed dynamic state feedback control law, and is further called a distributed dynamic output feedback control law if the function is independent of any state variable.

In recent years, the cooperative output regulation problem of multi-agent systems has received extensive attention [15, 16, 17, 20]. The problem is interesting because its formulation includes the leader-following consensus, synchronization or formation as special cases. Like the output regulation problem of a single linear system [1, 3, 4], there are two approaches to handling the cooperative output regulation problem of multi-agent systems. The first one is called feedforward design [15, 16]. This approach makes use of the solution of the regulator equations and a distributed observer to design an appropriate feedforward term to exactly cancel the steady-state tracking error. The second one is called distributed internal model design [17, 20]. This approach employs a distributed internal model to convert the cooperative output regulation problem of an uncertain multi-agent system to a simultaneous eigenvalue assignment problem of a multiple augmented system composed of the given multi-agent system and the distributed internal model. The internal model approach has at least two advantages over the feedforward design approach in that it can tolerate perturbations of the plant parameters, and it does not need to solve the regulator equations.

More recently, the feedforward approach was further extended to the cooperative output regulation problem for exactly known linear multi-agent systems with time-delay [9]. However, since this approach cannot handle the model uncertainties and the control law has to rely on the solution to the regulator equations, we will further develop a distributed internal model approach to deal with the cooperative output regulation problem of uncertain multi-agent systems subject to both input delay and communication delay.

As a special case of the cooperative output regulation problem, the leader-following consensus problem of linear multi-agent systems has been studied in several papers. Some typical references that handle the communication time-delay are [7], [8], [12], [13], [14], [19], [21] and [25]. In particular, in [14], the communication time-delays were considered in the leaderless consensus problem for single-integrator multi-agent systems under undirected and fixed network topology. In [25], the leader-following consensus problem of double integrator multi-agent systems with non-uniform time-varying communication delays was studied under fixed and switching topologies. On the other hand, input delay is also inevitable due to the processing and connecting time for packets arriving at each agent [24]. Cooperative control of multi-agent systems with input delay has been studied in, say, [18], [22], [24] and the references therein. In particular, the reference [24] considered the leaderless consensus problem of high-order linear multi-agent systems with both communication delay and input delay with directed and fixed network.

As mentioned before, the problem formulation of this paper is general enough to include the leader-following consensus problem of general multi-agent systems with both communication delay and input delay as a special case. Moreover, by adopting the distributed internal model approach, our control law is able to handle model uncertainty, and simultaneously achieve asymptotic tracking and disturbance rejection for a large class of signals generated by a linear autonomous system called exosystem.

Technically, this paper is most relevant to [10] and [17]. Specifically, reference [10] studied a special case of this paper with in the system (1). For this case, since there is no communication constraint on the control law (4), we can use the full state feedback control or the full output feedback control to handle the problem. However, in the current case, we have to employ distributed control law which makes the design of our control law much more complicated. On the other hand, reference [17] treated the same problem as this paper for a special case of the system (1) with by a special case of the control law (4) with . However, due to the input delay and communication delay, the proof of the main results of this paper is much more sophisticated than the proof of the main results in [17]. We have to introduce or establish some specific technical lemmas to establish our main results.

The rest of this paper is organized as follows. Section 2 gives the problem formulation and some preliminaries. A general framework is established in Section 3. Section 4 presents our main results. One example is used in Section 5 to illustrate our results. Finally, we close the paper with some concluding remarks in Section 6.

Notation. For , , col. For any matrix , where , is the column of . denotes the Kronecker product of matrices. Let denote the complex plane. For , let denote the real part of .

## 2 Problem formulation and preliminaries

Like in [17], all matrices in (1) can be uncertain. Let , where represent the nominal part of these matrices, and are the perturbations of these matrices. For convenience, we denote the system uncertainties with a vector

Now, we can state our problem as follows:

###### Definition 2.1

Linear cooperative robust output regulation problem: given the system (1), the exosystem (2), and a digraph , design a control law of the form (4) such that the closed-loop system satisfies the properties 2.1 and 2.2 as follows.

###### Property 2.1

The nominal closed-loop system is exponentially stable when .

###### Property 2.2

There exists an open neighborhood of such that, for any and any initial conditions , and , the regulated output .

###### Remark 2.1

It is noted that the problem studied in [17] is a special case of the above problem when both the communication delay and the input delay are zero. The presence of these two delays makes our problem formulation more realistic and, as will be seen later, the handling of the problem more challenging.

For the solvability of the above problem, some assumptions are stated as follows.

###### Assumption 2.1

There exist matrices , , such that , , , .

###### Assumption 2.2

All the eigenvalues of are on the imaginary axis.

###### Assumption 2.3

The matrix pair is stabilizable.

###### Assumption 2.4

The matrix pair is detectable.

###### Assumption 2.5

For all , where denotes the spectrum of ,

(5) |

###### Assumption 2.6

The digraph contains a directed spanning tree with the node as the root.

###### Assumption 2.7

has no eigenvalues with positive real parts.

## 3 A general framework

To construct a specific control law, let and be the weighted adjacent matrix and Laplacian of the digraph , respectively. Let be an nonnegative diagonal matrix whose diagonal element is . Then, we have [7, 15]

where is an column vector whose elements are all and satisfies .

In terms of the elements of , we can define a virtual regulated output for each follower subsystem as follows:

(6) |

Note that the subsystem can access the regulated error if and only if the node is the neighbor of the node .

###### Remark 3.1

In order to make use of the internal model principle to handle the systems with input delay and communication delay, we need to generalize the concept of the minimum p-copy internal model to the following form:

###### Definition 3.1

A pair of matrices is said to be the minimal p-copy internal model of the matrix if the pair takes the following form:

(7) |

where is a constant square matrix whose characteristic polynomial equals the minimal polynomial of , and is a constant column vector such that is controllable.

Having defined the virtual regulated output and introduced the p-copy internal model, we can describe our distributed dynamic state feedback control law as follows:

(8) |

where , , with to be specified later, are constant matrices of appropriate dimensions to be designed later, are defined in (7), and, respectively, our distributed dynamic output feedback control law as follows:

(9) |

where , , , and with to be specified later, , are constant matrices of appropriate dimensions to be designed later and are defined in (7).

Let , , , , , , , , , , , . Then, we define an auxiliary system as follows:

(10) |

Clearly, the matrix pair is the minimal pN-copy internal model of the matrix . Thus, by Definition 3.1, the following system

(11) |

is an internal model of (10). The composition of the auxiliary system (10) and the (11) is called the augmented system of (10) and is put as follows:

(12) |

###### Remark 3.2

It can be seen that the internal model in [10] is a special case of (11) by setting . It is shown in Lemma 1.27 of [6] that if the matrix pair is the minimal p-copy internal model of the matrix , then the following matrix equation

(13) |

has a solution only if . This property is the key for establishing the following result.

The role of an internal model is to convert the output regulation problem of the given plant (10) to the stabilization problem of the augmented system (12). To be more precise, we have the following lemma.

###### Lemma 3.1

Under Assumption 2.2,

(i) suppose a static state feedback control law of the form

(14) |

stabilizes the nominal plant of the augmented system (12). Then, the dynamic state feedback control law of the form

(15) |

solves the robust output regulation problem of the auxiliary system (10).

(ii) suppose a dynamic output feedback control law of the form

(16) |

where , stabilizes the nominal plant of the augmented system (12). Then, the dynamic output feedback control law of the form

(17) |

solves the robust output regulation problem of the auxiliary system (10). By Remark 3.1, under Assumption 2.6, either of the two control laws also solves the cooperative robust output regulation problem of the given plant (1).

Before giving the proof of Lemma 3.1, we still need some remarks. First, under the coordinate transformation , the closed-loop system composed of system (10) and (15) or (17) can be put into the following form:

(18) |

where , , under the dynamic state feedback, , and

and, under the dynamic output feedback, , and

###### Remark 3.3

It can be deduced from Lemma 2.1 of [10], under Assumption 2.2, if the closed-loop system (18) satisfies Property 2.1, then, for each , and any matrix of appropriate dimension, there exists a unique matrix that satisfies the following matrix equation:

(19) |

Moreover, by Lemma 2.2 of [10], under Assumption 2.2, if the controller (15) or (17) renders the closed-loop system (18) Property 2.1, then, the same controller solves the linear robust output regulation problem if and only if, for each , there exists a unique matrix that satisfies the following matrix equations:

(20) | ||||

Now, we will give the proof of Lemma 3.1 as follows.

Proof: Note that the closed-loop system (18) can also be viewed as a composition of the augmented system (12) and a static state feedback control of the form respectively, a dynamic output feedback control law of the form . Thus, the closed-loop system (18) satisfies Property 2.1. By Remark 3.3, under Assumption 2.2, it suffices to prove that the matrix equations (20) have a unique solution under either the static state feedback controller or the dynamic output feedback controller. In fact, by Remark 3.3, the first equation of (20) has one unique solution . Thus, we only need to prove that also satisfies the second equation of (20). We will do so for the static state feedback control case and the dynamic output feedback case, respectively.

Part (i): Let with and expand the first equation of (20) to the following form:

(21) |

where

(22) |

Since the second equation of (21) is in the form (13), by Remark 3.2, . That is, also satisfies the second equation of (20).

Part (ii): Let with , and . Partition to , where with the dimension of . Then, it can be verified that, under the control law , the first equation of (20) can be expanded to the following form:

(23) |

where

(24) |

Since the second equation of (23) is in the form (13), by Remark 3.2, . That is, also satisfies the second equation of (20).

###### Remark 3.4

In order to apply Lemma 3.1 to our problem, it is not enough to show that the nominal part of the augmented system (12) is stabilizable by a static state feedback control law of the form (14) or a dynamic output feedback control law of the form (16). We actually need to show that the nominal part of the augmented system (12) is stabilizable by a distributed static state feedback control law of the form , (or a distributed dynamic output feedback control law of the form ,where , ). As a result, the distributed state feedback control law (8) ( or the distributed output feedback control law (9)) solves the cooperative output regulation problem of the system (1). What makes this stabilization problem much more challenging than the problem in [17] is that the augmented system (12) is subject to both input delay and communication delay. We need to first establish a few lemmas to lay the foundation of our approach.

## 4 Main result

To establish some Lemmas in this section, we need to first cite the following lemma.

###### Lemma 4.1

(Lemma 3.2 in [9]) Consider the system

(25) |

where , are some constant matrices, are arbitrary time-delays, , and is any measurable, essentially bounded function over . Assume that the origin of the unforced system is exponentially stable and . Then, . Moreover, exponentially if exponentially.

###### Lemma 4.2

Proof: Under Assumptions 2.2, 2.3 and 2.5, by Lemma 1.26 of [6], is stabilizable. Moreover, under additional Assumption 2.7, we have that has no eigenvalues with positive real parts. Therefore, there exists a nonsingular matrix such that

(27) |

where all the eigenvalues of the matrix have negative real parts, all the eigenvalues of the matrix are on the imaginary axis and is controllable. Then, system (26) is equivalent to the following system:

(28) |

By Lemma 1 of [24], there exists a matrix , where satisfies

(29) |

and is the positive definite solution of the ARE

(30) |

with some sufficiently small such that, for , the systems are all asymptotically stable.

Let . Then, under the control law , the closed-loop system of (28) is as follows.

(31) |

Since for , subsystem is asymptotically stable, by Lemma 4.1, for subsystem is asymptotically stable. The proof is thus completed with .

###### Lemma 4.3

Consider the system of the form

(32) | ||||

where , , is the minimal p-copy internal model of as defined in (7), and . Then, under Assumptions 2.2, 2.3, 2.5 and 2.7, there exist matrices and , such that under the state feedback control law , system (32) is asymptotically stable if and only if Assumption 2.6 is satisfied.

Proof: (If Part:) Denote the eigenvalues of by . Under Assumption 2.6, by Remark 3.1, for , have positive real parts. Let be the nonsingular matrix such that is in the Jordan form of . Let and . Then, is governed by the following system:

(33) |

where .

Denote with . Partition as , where and and as , where