Minimal Control Nodes for Strong Structural Observability of Discrete-Time Iterative Systems: Explicit Formulas and Polynomial-Time Algorithms

In this article, we initiate an important class of vertex-marked directed graphs, referred to as the
strong structurally observable graphs
(SSOGs), to be the fundamental structural architectures ensuring the observability of several prevalent types of discrete-time iteration systems. This article also answers the minimal control node problem of SSOGs by carrying out two explicit formulas and the corresponding polynomial-time algorithms. Primitively, by exploring the strong structural observability of Boolean networks, an underlying class of SSOGs is proposed as an extended counterpart of network structures of observable conjunctive Boolean networks, while we call for particular attention to network structures instead of node dynamics. With such a class of SSOGs, we justify that a general class of discrete-time iterative systems can also be well addressed. In our formulation, the robustness against the uncertainties in nodal dynamics is achieved by merely exploiting the information on network structures. Two minimal synthesis strategies are, respectively, established to render a directed graph strongly structurally observable in a polynomial amount of time via placing a minimal number of sensors or controlling the nodes. Notably, apart from explicitly calculating the minimal set of control nodes, the structural insights are also leveraged to account for the close relations between the observability of considered iteration systems and observed-path-compatible paths and cycles in the network structures. For certain special yet worthy cases, the SSOGs are specialized in the strong structural observability of finite-field networks and derive the minimum pinned node theorem of Boolean networks with a biological case study as an efficient demonstration.