基于Tustin变换的分数阶微分算子近似离散化

Discrete Approximation of Fractional-order Differentiator Based on Tustin Transform

分数阶微分算子的离散化是分数阶控制器数字化实现的关键。对基于Tustin变换的分数阶微分算子直接离散化方法进行了研究和比较。概述了分数阶微积分及其离散化,介绍了用于Tustin算子展开的幂级数展开法、连分式展开法和Muir递归展开法;并给出了展开方法的算法表达式。定义了误差指标函数,举例比较了以上三种分数阶微分算子离散化方法的优缺点。仿真比较表明:连分式展开法在较宽频带内对分数阶微分算子具有最好的近似特性,但计算复杂度大;幂级数展开法和Muir递归展开法近似效果相当,但前者具有较大计算效率优势。在分数阶数字控制器实现过程中应根据具体情况选择合适的分数阶算子离散化方法。

The discretization of fractional-order differentiator is the key step in digital implementation of a frac- tional-order controller. The Tustin transform based direct discretization methods for fractional-order differentiator is focused on. Firstly, fractional order differintegral and its discretization are reviewed briefly. Secondly, power series expansion, continued fractional expansion and Muir-recursion expansion are presented and the algorithm expressions are introduced. The error function is defined and an illustrative example is given to compare the advantages and dis- advantages of the discretization methods. The simulations and comparisons show that continued fractional expansion has better properties in a wide frequency band for fractional-order differentiator, while the computational complexity is high. Power series expansion and Muir-recursion expansion have similar properties and the former has the advan- tage of computational efficiency. It is necessary to select suitable discretization method in the implementation of digital fractional order controller.