本論文針對個體網絡的運動可持續性問題提出不同的解決方案並進行比較。在特定條件下，利用一群個體來描繪符合美學的運動軌跡，其可視為類分散控制，每個個體均能自我控制與修正。

次擺線圖案的需求為網絡的邊際穩定。其主要分為兩部分，第一部分是僅使用局部交互來呈現運動軌跡，第二則是確保網絡的邊際穩定性。第一部分之成果已陳述於先前的研究，而本篇論文則著重於如何保持邊際的穩定性。目前針對該議題之相關研究缺乏了對系統穩健性的探討，故本篇論文主要針對此部分進行研究。

保持邊際穩定性的需求在控制工程中並不常見。為了解決這個問題，本篇論文應用了三種方法：模型參考自適應控制、積分滑動模式控制和干擾觀測器控制。結果證明這些控制方法對於可能發生的各種不確定性與干擾是穩健的。
然而，這些方法的限制在於每個個體模型都被視為單質點，不適用於任何類型的機器人。本篇論文主要分為三部分，第一部分為前人研究與數學模式之文獻回顧，第二部分講述解決問題的方法與其相關描述和功能，最後一部分為呈現模擬的結果及其方案間之比較。

本論文結果分析表明，三個控制器均能夠完成目標，而每個控制器都具有其優點和缺點。根據模擬結果，可以得出結論為任何控制器的使用均各有利弊。然而，若目的為保持邊際穩定性，則所有控制器均滿足要求。

This dissertation presents different solutions and compare them for a motion sustainability of a network of agents. Under specific condition, a group of agents is used to depict aesthetic patterns. It is a decentralized-control-like where each agent is able to control and correct itself.

In the formation control field, a common trait is to keep a fixed-distance between agents, and thus even if the whole fleet of agents is moving. Fixed-formation maintenance is widely studied and posses a profound background. However in some scenarios; may they be military, observation, rescuing or cleaning, it can be more useful to have a time-varying-formation. The research focusing on this challenge leaded to depiction of aesthetically pleasant patterns, called trochoid.

To draw trochoid patterns, a requirement of the network of agents is to be marginally stable. As this feature is necessary. Such patterns are depicted only if the overall network is marginally stable, and vice versa, thus it rises a problem regarding the stability. From there, the motivation is double. First the pattern is drawn, second the marginal stability is maintained. Marginal stability of the overall network is a criteria needed to generate specific patterns. This marginal stability relies on the values of the gains of each agents. If any of agent's gain is uncertain the formation is not maintained. Hence, by using robustification of the gains and make them tend to a specific value, the marginal stability of the overall network is ensured.

Moreover, if the system becomes unstable, then there exist a chance of collision between agents. The collision of two agents in the motion is one of the motivation because, in this kind of formation, awareness of a risk of collision is key. If a collision is possible, then a switch to a reactive control law might be performed. This switch can be regarded as equivalent to a step disturbance in the behavior law. Hence, by ensuring the robustness of the gains, the formation will be ensured, and agents will prevent from collision.

The desire of keeping the marginal stability is atypical in control engineering. To tackle the problem, three methods were applied: model reference adaptive control, integral sliding mode control, and disturbance observer control. It is demonstrated that these methods are robust against the diverse uncertainties / disturbances that could occur. The limitations of the methods rely on the fact that every agents is modeled as a single integrator and thus were not applied on any kind of robot. The dissertation is divided essentially in three parts: a history and literature review with mathematics reviews, a second part talking about the methods to solve the problem with their description and functioning, and a last part concerning the simulations with results and comparisons.

Analysis of the responses shows that the three controllers are able to complete the objective, each of them having benefits and drawbacks that are exposed. Based on these results, we conclude that the use of any of the controller relies on the situation one have at hands. However if the aim is just to maintain marginal stability, all controllers satisfy the requirement.

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