In this article, we deal with the study of the determinantal assignment problem (DAP) when the parameters of the compensator are not entirely free, but some of them are fixed. The problem is reduced to a restricted form of an exterior algebra problem (decomposability of multivectors), which is referred to as partial decomposability problem. We study this problem and in case that this problem has no solution, we examine the problem of approximate partial decomposability. We treat the problem of exact or partial decomposability into a vector and a multivector of lower dimension. If this procedure is repeated then this results in an approximation of the initial multivector into a decomposable vector. The approximation of a vector by an optimal decomposable multivector is a nonlinear procedure and has been solved completely using the power method. The method developed in this article, although it produces a suboptimal solution, can be used alternatively for the solution of DAP or the approximate DAP, as a shorter and easier approach, because it is based on known tools as the singular value decomposition. We apply these results to treat the restricted approximate decomposability problem, which leads to approximate solutions to the pole placement and zero assignment problems.

中文翻译：

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部分固定结构行列式分配问题


在本文中，我们研究补偿器参数不完全自由，但其中一些参数是固定的情况下的行列式分配问题（DAP）。该问题被简化为外代数问题（多向量的可分解性）的限制形式，称为部分可分解性问题。我们研究这个问题，如果这个问题无解，我们检查近似部分可分解性问题。我们将精确或部分可分解性问题处理为一个向量和一个较低维度的多重向量。如果重复此过程，则将导致初始多向量近似为可分解向量。用最优可分解多向量来逼近向量是一个非线性过程，并且已使用幂法完全求解。本文开发的方法虽然产生了次优解，但可以替代地用于 DAP 或近似 DAP 的解，作为一种更短且更容易的方法，因为它基于已知的奇异值分解工具。我们应用这些结果来处理受限近似可分解性问题，从而得出极点配置和零分配问题的近似解。