Finite-state systems have applications in systems biology, formal verification and synthesis problems of infinite-state (hybrid) systems, etc. As deterministic finite-state systems, logical control networks (LCNs) consist of a finite number of nodes which can be in a finite number of states and update their states. In this paper, we investigate the synthesis problem for controllability and observability of LCNs by state feedback under the semitensor product framework. We show that state feedback can never enforce controllability of an LCN, but sometimes can enforce its observability. We prove that for an LCN $Sig$ and another LCN $Sig$ obtained by feeding a state-feedback controller into $Sig$, (1) if $Sig$ is controllable, then $Sig$ can be either controllable or not; (2) if $Sig$ is not controllable, then $Sig$ is not controllable either; (3) if $Sig$ is observable, then $Sig$ can be either observable or not; (4) if $Sig$ is not observable, $Sig$ can also be observable or not. We also prove that if an unobservable LCN can be synthesized to be observable by state feedback, then it can also be synthesized to be observable by closed-loop state feedback (i.e., state feedback without any input). Furthermore, we give an upper bound for the number of closed-loop state-feedback controllers that are needed to verify whether an unobservable LCN can be synthesized to be observable by state feedback.
Download