一些真實的物理系統，如電化學領域等，較適合使用分數階微分方程來建立模型，能夠比整數階微分方程得到更精準的系統描述。此外，分數階建模還有一有用特性，即能以簡單低階模型對高階系統提供良好的描述。分數階系統的應用需要大量運算，近年因為電腦運算速度的快速進步，分數階系統逐漸成為重要的研究主題。
本論文提出利用繼電器回饋對未知系統進行閉環鑑別，僅需對系統進行一次響應試驗，即可結合輸出入量測和頻率響應原理來獲得系統的分數一階時延模型。在分數階模型已知的情況下，可使用直接合成法進行控制器設計，然而所得的控制器形式相當複雜，不利於實際應用。為解決此問題，本文提出利用最小平方法進行PIλDμ控制器的全域最佳參數求解，可將複雜控制器轉為簡單的PID形式控制器，其中頻率範圍的選取以及頻率權重的調整，對PIλDμ控制器的設計有重要影響。
最後本論文結合兩者提出自動調諧方法，先利用繼電器回饋鑑別出分數一階時延模型，然後根據模型設計適合的PIλDμ控制器。模擬研究顯示，本自動調諧方法對於各種整數階和分數階系統，皆能鑑別出有效的分數階模型，並且所得之PIλDμ控制器具有良好的控制性能。

Some real physical systems, such as electrochemistry, are more suitable to use fractional order differential equations to build models, which can obtain more accurate system descriptions than integer order differential equations. In addition, fractional order modeling has a useful feature, that is, it can describe high-order systems well with simple low-order models. The application of fractional order systems requires considerable calculations. In recent years, due to the rapid advancement of computer computing speed, fractional order systems have gradually become an important research topic.
This thesis proposes the use of relay feedback to identify unknown systems in closed-loop operation. This requires only a single response test on the system, and then the input-output measurements and frequency response theory can be combined to find the fractional first order plus time delay (FFOPTD) model of the system. When the fractional order model is known, the direct synthesis method can be used to design a controller. However, the resulting controller form is quite complicated, which is not feasible in practical applications. To resolve this problem, the thesis proposes to use the least square method to find the globally optimal solution of PIλDμ controller parameters. This can transform the complex controller into a simple controller of PID form. The selection of the frequency range and the adjustment of the frequency weights are important for the design of PIλDμ controller.
Finally, this thesis combines the two methods to present an auto-tuning method. First, the FFOPTD model is identified using relay feedback, and then a suitable PIλDμ controller is designed based on the identified model. Simulation study shows that the proposed auto-tuning method can effectively identify the fractional order models of various integer order or fractional order systems, and the resulting PIλDμ controller possesses good control performance.

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