In this paper, an algebraic and robust fractional order differentiator is designed for a class of fractional order linear systems with an arbitrary differentiation order in $[0,2]$. It is designed to estimate the fractional derivative of the pseudo-state with an arbitrary differentiation order as well as the one of the output. In particular, it can also estimate the pseudo-state. Different from our previous works, the considered system no longer relies on the matching conditions, which makes the system model be more general. First, the considered system is transformed into a fractional differential equation from the pseudo-state space representation. Second, based on the obtained equation, a series of equations are constructed by applying different fractional derivative operators. Then, the fractional order modulating functions method is introduced to recursively give algebraic integral formulas for a set of fractional derivatives of the output and a set of fractional derivative initial values. These formulas are used to non-asymptotically and robustly estimate the fractional derivatives of the pseudo-state and the output in discrete noisy cases. Third, the required modulating functions are designed. After giving the associated estimation algorithm, numerical simulation results are finally given to illustrate the accuracy and robustness of the proposed method.

本文针对一类具有任意微分阶的分数阶线性系统设计了代数且鲁棒的分数阶微分器。 $[0，2]$。它用于估计具有任意微分阶数和输出之一的伪状态的分数导数。特别地，它也可以估计伪状态。与我们以前的工作不同，所考虑的系统不再依赖于匹配条件，这使得系统模型更加通用。首先，将所考虑的系统从伪状态空间表示转换为分数阶微分方程。其次，基于所获得的方程，通过应用不同的分数阶导数运算符来构造一系列方程。然后，引入分数阶调制函数方法以递归给出输出的一组分数导数和一组分数导数初始值的代数积分公式。这些公式用于非渐近和鲁棒地估计在离散噪声情况下伪状态和输出的分数导数。第三，设计所需的调制功能。在给出了相关的估计算法之后，最后给出了数值仿真结果以说明该方法的准确性和鲁棒性。