Due to high computational load of ideal realization of fractional order elements, fractional order transfer functions are commonly implemented via integer-order, limited-band approximate models. An important side effect of such a non-ideal fractional order controller function realization for control applications is that the approximate fractional order models may deteriorate practical performance of optimal control tuning methods. Two major concerns come out for approximate realization in fractional-order control. These are stability preservation and model response matching properties. This study revisits four fundamental fractional order approximation methods, which are Oustaloup's method, CFE method, Matsuda's method and SBL fitting method, and considers stability preservation, time and frequency response matching performances. The study firstly presents a detailed review of Oustaloup's method, CFE method, Matsuda's method. Then, a modified version of SBL fitting method is presented. The stability preservation properties of approximation methods are investigated according to critical root placements of corresponding approximation method. Stability issue is highly significant for control applications. For this reason, a detailed analysis and comparision of stability preservation properties of these four approximation methods are investigated. Moreover, approximate implementations of an optimally tuned FOPID controller function are performed according to these four methods and compared for closed loop control of a large time delay system. Findings of this study indicate a fact that approximate models can considerably influence practical performance of optimally tuned FOPID control systems and ignorance of limitations of approximation methods in optimal tuning solutions can significantly affect real world performances.

由于理想的分数阶元素实现的计算量很大，因此分数阶传递函数通常是通过整数阶有限带近似模型来实现的。对于控制应用而言，这种非理想的分数阶控制器功能实现的重要副作用是，近似的分数阶模型可能会降低最佳控制调整方法的实际性能。在分数阶控制中实现近似有两个主要问题。这些是稳定性保持和模型响应匹配属性。本文研究了四种基本的分数阶逼近方法，分别是Oustaloup方法，CFE方法，Matsuda方法和SBL拟合方法，并考虑了稳定性保持，时间和频率响应匹配性能。研究首先对Oustaloup方法，CFE方法，Matsuda方法进行了详细的综述。然后，提出了SBL拟合方法的改进版本。根据相应逼近方法的关键根位置，研究逼近方法的稳定性保持性质。稳定性问题对于控制应用而言非常重要。因此，对这四种近似方法的稳定性保持特性进行了详细的分析和比较。此外，根据这四种方法执行了最佳调谐FOPID控制器功能的近似实现，并针对大时延系统的闭环控制进行了比较。