We describe a method for analyzing the structure of a system of nonlinear integro-differential–algebraic equations (IDAEs) that generalizes the $\Sigma$-method for the structural analysis of differential–algebraic equations. The method is based on the sparsity pattern of the IDAE and the $\nu$-smoothing property of a Volterra integral operator. It determines which equations and how many times they need to be differentiated to determine the index, and it reveals the hidden constraints and compatibility conditions in order to prove the existence of a solution. The success of the $\Sigma$-method is indicated by the non-singularity of a certain Jacobian matrix. Although it is likely the $\Sigma$-method can be directly applied with success to many problems of practical interest, it can fail on some solvable IDAEs. Accordingly, we also present two techniques for addressing these failures.

我们描述了一种用于分析非线性积分-微分-代数方程组（IDAE）的系统结构的方法，该方法可将 $\Sigma$-方法用于微分-代数方程的结构分析。该方法基于IDAE的稀疏模式和$\nu$-Volterra积分算子的平滑特性。它确定哪些方程式以及需要微分多少次才能确定索引，并且揭示了隐藏的约束条件和兼容性条件，以证明解决方案的存在。成功的$\Sigma$-方法由某个雅可比矩阵的非奇异性表示。虽然很可能$\Sigma$-方法可以成功地直接应用于许多具有实际意义的问题，在某些可解决的IDAE上可能会失败。因此，我们还提出了两种解决这些故障的技术。