## 2 Theoretical Basis of Time-Varying Target Formation  
This section introduces the concept of graph theory and the connotation of affine transformation and defines the nominal formation configuration, as well as our control target, which lays the foundation for the subsequent design of time-varying target configuration.  
In the formation control of multi-agent, the agents cannot communicate with each other directly, and the communication between them needs to be established through the network topology. Graph theory is a very useful tool for studying cooperation systems. As a result, a formation is usually modeled by a graph, with vertices corresponding to agents and edges corresponding to the specified distance between the agents (Kal  
Given a set of points \(\{p_{i}\}_{i=1}^{n}\) in \(R^{d}\), the affine span of these points (called \(S\)) can be expressed as  
\[S=\{\sum_{i=1}^{n}a_{i}p_{i}:a_{i}\in R\,\sum_{i=1}^{n}a_{i}=1\} \tag{1}\]  
Let the configuration matrix \(P\in R^{n\times d}\) and an augmented matrix \(\overline{P}\in R^{n\times(d+1)}\) be  
\[P(p)=\left[\begin{array}{c}P_{1}^{T}\\ \vdots\\ p_{n}^{T}\end{array}\right],\overline{P}(p)=\left[\begin{array}{c}p_{1}^{T} &1\\ \vdots&\vdots\\ p_{n}^{T}&1\end{array}\right]=[P(p),1_{n}] \tag{2}\]  
where \(1_{n}\triangleq[1,\cdots,1]^{T}\in R^{n}\). By definition, if and only if the rows of \(P(p)\) are linearly independent, \(\{p_{i}\}_{i=1}^{n}\) are affinely independent. Because \(P(p)\) has \(d+1\) columns, there exist at most \(d+1\) points that are affinely independent in \(R^{d}\).  
To achieve the formation control, a leader-follower multi-agent is needed to track time-varying goal configurations, as described below.  
Definition 1 (Target formation): The time-varying configuration of the target formation has the form of  
\[p^{*}(t)=[I_{n}\otimes A(t)]r+1_{n}\otimes b(t) \tag{3}\]  
where \(r=[r_{1}^{T},\cdots,r_{n}^{T}]^{T}=[r_{1}^{T},r_{f}^{T}]^{T}\in R^{dn}\) is a constant configuration, and \(A(t)\in R^{d\times d}\), \(b(t)\in R^{d}\) are continuous of \(t\). The desired position of the agent \(i\in v\) in the target formation is  
\[p^{*}(t)=A(t)r+b(t) \tag{4}\]  
In the above equation, \(r\) is the typical geometric configuration that needs to be maintained during the formation process and \((G,r)\) is the nominal formation. In fact, the target configuration is an affine transformation of the nominal configuration that varies with time. There is a one-to-one correspondence between the positions of the different leaders and the affine transformations \((A,b)\). As a consequence, the affine transformation of the entire formation can be achieved by controlling the position of the leader. The objective of this paper is to control the multi-agent tracking the time-varying target formation and satisfy \(p(t)\to p^{*}(t)\) during every split time.  
In this article, there is no independent control law for leaders because we assume that leaders can be well controlled by humans or some intelligent program. It is known that the leader's position corresponds to the expected value in the target configuration, that is \(p_{l}(t)=p_{l}^{*}(t)\) for all \(t\). Thus, our goal of control is to steer the followers to achieve \(p_{f}(t)\to p_{f}^{*}(t)\) at different time intervals.  
Design of Time-Varying Target Formation Control for Multi-Agent Systems Based on the Improved Hungarian Algorithm