We provide a necessary and sufficient graph-theoretical characterization of quotient fixes modes occurring in parametric decentralized control systems. Specifically, we introduce the notion of structurally quotient fixed modes (SQFMs) that generically captures the quotient fixed modes and only depends on the system’s structure. Additionally, we provide an efficient polynomial-time algorithm for the verification of this graph-theoretical condition. We show that this algorithm can be parallelized, and linear-time computational complexity approximation algorithms can be considered to attain a sub-optimal solution. Lastly, we discuss the implications of the actuation–sensing–communication capabilities and the systems’ interconnections on the existence of SQFM.

我们提供了在参数分散控制系统中出现的商定模式的必要和充分的图论表征。具体来说，我们介绍了结构商固定模式（SQFM）的概念，该概念通常捕获商固定模式，并且仅取决于系统的结构。此外，我们提供了一种有效的多项式时间算法来验证这种图论条件。我们证明了该算法可以并行化，并且可以考虑使用线性时间计算复杂度近似算法来获得次优解。最后，我们讨论了驱动，传感，通信功能以及系统互连对SQFM的影响。