Sampled-Data State-Feedback Stabilization of Probabilistic Boolean Control Networks: A Control Lyapunov Function Approach

Set stabilization of logical control networks: A minimum node control approach

In network systems, control using minimum nodes or pinning control can be effectively used for stabilization problems to cut down the cost of control. In this paper, we investigate the set stabilization problem of logical control networks. In particular, we study the set stabilization problem of probabilistic Boolean networks (PBNs) and probabilistic Boolean control networks (PBCNs) via controlling minimal nodes. Firstly, an algorithm is given to search for the minimum index set of pinning nodes. Then, based on the analysis of its high computational complexity, we present optimized algorithms with lower computational complexity to ascertain the network control using minimum node sets. Moreover, some sufficient and necessary conditions are proposed to ensure the feasibility and effectiveness of the proposed algorithms. Furthermore, a theorem is presented for PBCNs to devise all state-feedback controllers corresponding to the set of pinning nodes. Finally, two models of gene regulatory networks are considered to show the efficacy of obtained results.

Stability of Sampled-Data Systems With Packet Losses: A Nonuniform Sampling Interval Approach

In this article, inspired by the Halanay inequality, we study stability of sampled-data systems with packet losses by proposing a nonuniform sampling interval approach. First, a sampled-data controller with an exponential gain is put forward to reduce conservatism. We obtain the sufficient condition for linear sampled-data systems to be exponentially stable by extending the famous Halanay inequality to sampled-data systems. The obtained sufficient conditions indicate that the maximal-allowable bound of sampling intervals is determined by the constant terms in the Halanay inequality, and the decay rate is presented in the form of a Lambert function. Compared with some existing results on the stability of sampled-data systems by using the Gronwall-Bellman Lemma, the conservatism induced by the exponential term via the Gronwall-Bellman Lemma can be reduced to some extent. Considering the phenomenon of packet losses, a new lemma is further proposed to generalize the proposed Halanay-like inequality. The results derived by the new lemma permit that there exist some sampling intervals with the upper bound violating the desired condition of the Halanay-like inequality. This permits us to establish exponential stability in significant cases that do not satisfy the Halanay-like inequality needed in the previous results. Finally, the sampled-data local exponential stability is investigated for nonlinear systems with strong nonlinearity.

Pinning Stabilizer Design for Probabilistic Boolean Control Networks via Condensation Digraph

This article investigates the design of pinning controllers for state feedback stabilization of probabilistic Boolean control networks (PBCNs), based on the condensation digraph method. First, two effective algorithms are presented to achieve state feedback stabilization of the considered system from the perspective of condensation digraph. One is to find the desired matrix, and the other is to search for the minimum number of pinned nodes and specific pinned nodes. Then, all the mode-independent pinning controllers can be designed based on the desired matrix and pinned nodes. Several examples are delineated to illustrate the validity of the main results.

A State Space Approach to Decentralized Fault SE-Coprognosability of Partially Observed Discrete Event Systems

The problem of fault prognosis in the context of discrete event systems (DESs) is a crucial subject to study the security and maintenance of cyber-physical systems. In this article, the decentralized fault prognosis of partially observed DESs is analyzed with a universal state-estimate-based protocol. It follows (M,K) as the performance bound of any expected decentralized prognosers, where any fault can be predicted K steps before its occurrence and the fault is guaranteed to occur within M steps once a corresponding fault alarm is issued. To determine whether expected decentralized prognosers exist, the notion of state-estimate-coprognosability (SE-coprognosability) under the case of one fault type is proposed. Compared with existing other kinds of coprognosability, SE-coprognosability is a more generalized concept. Meanwhile, combining the formal method and algebraic state space approach, a novel state estimation algorithm is presented and based on which, the verification of SE-coprognosability is also solved.

Graph-Based Function Perturbation Analysis for Observability of Multivalued Logical Networkss

Observability is a fundamental concept for the synthesis of both linear systems and nonlinear systems. This article devotes to discussing the robustness of observability for multivalued logical networks (MVLNs) subject to function perturbation and establishing a graph-based framework. First, based on the transition graph of undistinguishable pairs of states, a new graph-based criterion is presented for the observability of MVLNs. Second, a candidate set consisting of all suspicious undistinguishable pairs of states is defined, based on the cardinality of which and the graph-based condition, a series of effective criteria are proposed for the robustness of observability subject to function perturbations. Finally, the obtained results are applied to the robust observability analysis of the p53-MDM2 negative feedback regulatory loop.

Set Stabilization of Probabilistic Boolean Control Networks: A Sampled-Data Control Approach

This article investigates the set stabilization of probabilistic Boolean control networks (PBCNs) under sampled-data (SD) state-feedback control within finite and infinite time, respectively. First, the algorithms are, respectively, proposed to find the sampled point set and the largest sampled point control invariant set (SPCIS) of PBCNs by SD state-feedback control. Based on this, a necessary and sufficient criterion is proposed for the global set stabilization of PBCNs by SD state-feedback control within finite time. Moreover, the time-optimal SD state-feedback controller is designed. It is interesting that if the sampled period (SP) is changed, the time of global set stabilization of PBCNs may also change or even the PBCNs cannot achieve set stabilization. Second, a criterion for the global set stabilization of PBCNs by SD state-feedback control within infinite time is obtained. Furthermore, all possible SD state-feedback controllers are obtained by using all the complete families of reachable sets. Finally, three examples are presented to illustrate the effectiveness of the obtained results.