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This paper investigates the leader-based distributed optimal control problem of discrete-time linear multi-agent systems (MASs) on directed communication topologies. In particular, the communication topology under consideration consists of only one directed spanning tree. A distributed consensus control protocol depending on the information between agents and their neighbors is designed to guarantee the consensus of MASs. In addition, the optimization of energy cost performance can be obtained using the proposed protocol. Subsequently, a numerical example is provided to demonstrate the effectiveness of the presented protocol.

Inspired by biological motion in nature, the cooperative motion of multi-agent systems (MASs) has been studied extensively in the past decade (

In practice, it is necessary to investigate the control problem of multi-agent systems in discrete time with most computer systems being discrete structures. In the study by

In practical applications, the energy cost performance of the designed protocols should be considered carefully, especially for the systems with low loadability, for example, autonomous underwater vehicles and spacecrafts. In the study by _{
∞
} optimal tracking problem of a class of physically interconnected large-scale systems with a strict feedback form and saturated actuators into the equivalent control problem of MASs; meanwhile, a feedback control algorithm is designed to learn the optimal control input of the system. In the study by

However, to the authors’ best knowledge, there are very few studies focusing on the optimal control of discrete-time MASs only containing a directed spanning tree. In this study, the leader-based distributed optimal control problem of discrete-time linear MASs on directed communication topologies is investigated. A distributed discrete-time consensus protocol based on the directed graph is designed, and it is proved that the optimization of energy cost performance can be satisfied with the presented consensus protocol. Furthermore, the optimal solution can be obtained by solving the algebraic Riccati equation (ARE), and the design of the protocol presented in this study does not require global communication topology information and relies on only the agent dynamics and relative states of neighboring agents, which means that every agent manages its protocol in a fully distributed way.

_{
n
} is an identity matrix of _{
N
} is a _{
m×n
} denotes a zero matrix of order ^{−1} and ^{T} are the inverse matrix and transpose matrix of

A digraph _{
j
}, _{
i
}) is included in the set _{
i
} to _{
j
}. A path from _{
i
} to _{
j
} is made up of a set of edges (_{
i
}, _{
l1}), … (_{ln}, _{
j
}). A graph is supposed to be connected if a path from _{
i
} to _{
j
} for all pairs of (_{
i
}, _{
j
}) existed. An adjacency matrix _{
ii
} = 0, and _{
ij
} = 1, _{
ij
} = −_{
ij
} for

Considering a group of

the leader agent’s index is defined as 0, and the leader agent’s and the follower agent’s index are defined as 1, … ,

(Matrix Inversion Lemma (^{
N×N
}, ^{
N×M
} and the general matrices ^{
N×N
}, ^{
N×M
} holds. Then, the inverse of the matrix(

In this section, a distributed optimal controller is designed to solve the consensus of the system in

Since

Let

A distributed optimal controller is developed as follows.

The error system can be obtained by taking the difference of

Inspired by reference given by

In ^{
T
} (

Next, the protocol presented in

For the given matrices ^{
T
} > 0 and ^{
T
} > 0, the cost function

i) Optimization of Cost Function

(i–i) Necessity

The optimal control input can be solved from the equation as follows.

Let λ(_{
N
} ⊗

Let ^{T} > 0. Since

According to

Considering the costate variable _{
N
} ⊗

Let ^{T} > 0 holds true, then we have

It is to be noted that if

(i–ii)

Considering the following equation:

Based on

Since

Let

Let

Hence, it indicates that the optimal performance index

(ii)

Based on the expressions of

It is inferred from _{
k→∞
}

As a consequence, the conditions in

Based on _{
i
}(_{
i
}(

The topology considered in this study is a structure containing only one directed spanning tree, which means that the agent can only obtain the information of a single neighbor, and we prove the effectiveness of the proposed distributed optimal controller under the abovementioned conditions. In fact, the proposed controller is also suitable for the case with the general case, such as reference given by

In this section, a numerical example is provided to demonstrate the effectiveness of the proposed controller.

Considering a network with seven agents, the communication topology is described by

Communication topology among seven agents.

Let _{3}, and the coupling strength _{0} (0) = [0.2–0.2 0.3]^{T}, _{1} (0) = [0.1 0.2 0.2]^{T}, _{2} (0) = [−0.15–0.1 0.1]^{T}, _{3} (0) = [0.3 0.2 0.1]^{T}, _{4} (0) = [−0.2 0.2–1.1]^{T}, _{5} (0) = [1.3 0.1–0.1]^{T}, and _{6} (0) = [1.0 0.5 1.5]^{T}. Then, the trajectories of the state norm and tracking error norm are shown in

State norm of seven agents.

State tracking error norm.

It can be seen from

Control input _{
i
} of six agents.

Moreover, the trajectories of energy cost performance

Trajectories of energy cost performance.

In this study, the leader-based distributed optimal control of discrete-time linear MASs only containing a directed spanning tree has been investigated. A distributed optimal consensus control protocol is presented to guarantee that multiple followers can successfully track the leader. It can be proved that the proposed protocol can ensure the optimization of the energy performance index with the optimal gain parameters which can be realized by solving the ARE. Moreover, the design of the protocol presented in this study is independent with the global information of topologies, which indicates that every agent manages its protocol in a fully distributed way. Finally, a numerical example which illustrates the effectiveness of the designed protocol is reported.

The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author.

GH is responsible for the simulation and the writing of this manuscript. ZZ is responsible for the design idea of this study. WY is responsible for the revision of this manuscript.

This study was supported in part by the National Key R&D Program of China 2019YFB1310303, in part by the Key R&D program of Shaanxi Province, 2021GY-289, in part by the National Natural Science Foundation of China under Grant U21B2047, Grant U1813225, Grant 61733014, and Grant 51979228.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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