A survey on observability of Boolean control networks

Observability is a fundamental property of a partially observed dynamical system, which means whether one can use an input sequence and the corresponding output sequence to determine the initial state. Observability provides bases for many related problems, such as state estimation, identification, disturbance decoupling, controller synthesis, etc. Until now, fundamental improvement has been obtained in observability of Boolean control networks (BCNs) mainly based on two methods-Edward F. Moore's partition and our observability graph (or their equivalent representations found later based on the semitensor product (STP) of matrices (where the STP was proposed by Daizhan Cheng)), including necessary and sufficient conditions for different types of observability, extensions to probabilistic Boolean networks (PBNs) and singular BCNs, even to nondeterministic finite-transition systems (NFTSs); and the development (with the help of the STP of matrices) in related topics, such as computation of smallest invariant dual subspaces of BNs containing a set of Boolean functions, multiple-experiment observability verification/decomposition in BCNs, disturbance decoupling in BCNs, etc. This paper provides a thorough survey for these topics. The contents of the paper are guided by the above two methods. First, we show that Moore's partition-based method closely relates the following problems: computation of smallest invariant dual subspaces of BNs, multiple-experiment observability verification/decomposition in BCNs, and disturbance decoupling in BCNs. However, this method does not apply to other types of observability or nondeterministic systems. Second, we show that based on our observability graph, four different types of observability have been verified in BCNs, verification results have also been extended to PBNs, singular BCNs, and NFTSs. In addition, Moore's partition also shows similarities between BCNs and linear time-invariant (LTI) control systems, e.g., smallest invariant dual subspaces of BNs containing a set of Boolean functions in BCNs vs unobservable subspaces of LTI control systems, the forms of quotient systems based on observability decomposition in both types of systems. However, there are essential differences between the two types of systems, e.g., "all plausible definitions of observability in LTI control systems turn out to be equivalent" (by Walter M. Wonham 1985), but there exist nonequivalent definitions of observability in BCNs; the quotient system based on observability decomposition always exists in an LTI control system, while a quotient system based on multiple-experiment observability decomposition does not always exist in a BCN.