Algebraic State Space Representation Research Articles

Hidden Order of Boolean Networks.

It is a common belief that the order of a Boolean network is mainly determined by its attractors, including fixed points and cycles. Using the semi-tensor product (STP) of matrices and the algebraic state-space representation (ASSR) of the Boolean networks, this article reveals that in addition to this explicit order, there is a certain implicit or hidden order, which is determined by the fixed points and limit cycles of their dual networks. The structure and certain properties of dual networks are investigated. Instead of a trajectory, which describes the evolution of a state, the hidden order provides a global horizon to describe the evolution of the overall network. We conjecture that the order of networks is mainly determined by the dual attractors via their corresponding hidden orders. Then these results about the Boolean networks are further extended to the k -valued case.

Total-activity conservation of probabilistic Boolean networks and its applications

Conservation law is a basic property of biological networks, which exhibits the total-activity conservation in Boolean networks (BNs). This article analyzes the total-activity conservation of probabilistic Boolean networks (PBNs) based on the algebraic state space representation approach. The concept of total-activity conservation is proposed for PBNs, and a vector-based criterion is presented to verify the total-activity conservation of PBNs. Moreover, the impact of deterministic function perturbation on the total-activity conservation of PBNs is analyzed. Finally, as an application of total-activity conservation for PBNs, the robust total-activity conservation of BNs subject to stochastic function perturbation is explored.

Transition Analysis of Stochastic Logical Control Networks

Transition analysis is essential for analyzing and regulating logical control networks (LCNs). The algebraic state-space representation method, which relies on the semi- tensor product of matrices, has been shown to allow deterministic LCNs to be represented as linear- like systems. However, due to the inherent uncertainty, it is hard to obtain the algebraic expression of a stochastic LCN. A unified paradigm for the transition analysis of LCNs with stochastic and deterministic dynamics is provided in this research. First, the algebraic expression of LCN with deterministic dynamics is reviewed. Second, the algebraic expression of LCN with stochastic dynamics is considered, where the non-equivalence between the dispersed form and the integrated form is proposed. Then the reason for the non-equivalence is provided. After that, a consistency condition is presented to bridge the gap between the independent model and the conditionally independent model. Lastly, we specifically point out that probabilistic LCN satisfies the consistency criteria, allowing one to calculate the probabilistic LCN transition matrix by using a power-reducing operator.

Cluster Synchronization of Finite-Field Networks.

This technical paper investigates the cluster synchronization of finite-field networks (FFNs) based on the algebraic state space representation. By resorting to the semi-tensor product of matrices, the cluster synchronization of an FFN can be completely converted into the set stability of state trajectory. Then, a necessary and sufficient condition for the cluster synchronization of FFNs is obtained based on the invariant subset and one-step transition matrix. In particular, the obtained results are applicable to check the leader-follower synchronization and group consensus of FFNs. Finally, a numerical example is given to illustrate the feasibility of the obtained results.

Total-Activity Conservation Analysis and Design of Boolean Networks.

The law of conservation of mass, represented in Boolean networks (BNs) as total-activity conservation, is one of the typical properties of biological networks. This article analyzes the total-activity conservation of BNs based on the algebraic state-space representation (ASSR) approach. First, the total-activity-conservative matrix is defined and a matrix-based criterion is proposed to verify the total-activity conservation of BNs. Meanwhile, when function perturbation is considered, robust total-activity conservation is investigated. Second, by means of the pseudo-Boolean function generated by total-activity conservation, a constructive design procedure of the Boolean dynamics is given to achieve the total-activity conservation. Third, the total-activity conservation of switched Boolean networks (SBNs) under arbitrary switching signal is studied, which together with network aggregation achieves the total-activity conservation of large-scale BNs.

Structural Observability of Boolean Control Networks via Topology-Based Approach

This brief aims to investigate the structural observability of Boolean control networks, followed by the recent research stream, with available topology structure but unknown node dynamics. To begin with, two kinds of structural observability are defined along with two systems. Then, by adopting the dependency graphs, several necessary and sufficient conditions for structurally arbitrary-input and single-input observability are derived, respectively. Instead of employing the algebraic state space representation framework, this graphic-theoretical approach takes full advantage of node-to-node interaction, thus decreasing the time complexity to a great extent.

Event‐triggered control for sampled‐data set stabilization of switched delayed boolean control networks

This paper considers the event‐triggered control design for the uniform sampled‐data set stabilization of switched delayed Boolean control networks (SDBCNs). First, using the algebraic state space representation method, SDBCNs are converted into the equivalent algebraic form. Second, using the algebraic form, the uniform sampled‐data reachable sets are constructed, based on which, a necessary and sufficient condition is obtained for the uniform sampled‐data set stabilization of SDBCNs. Finally, the event‐triggered mechanism is presented, and a sufficient condition is proposed to design the time‐variant state feedback event‐triggered controller for the uniform sampled‐data set stabilization of SDBCNs.

Finite-time sampled-data set stabilisation of delayed probabilistic Boolean control networks

In this paper, the finite-time constrained set stabilisation of delayed probabilistic Boolean control networks (DPBCNs) is considered via sampled-data control (SDC). The main mathematical tool is semi-tensor product of matrices, which is used to convert the dynamics of DPBCNs into an equivalent algebraic state space representation (ASSR). Based on the ASSR of DPBCNs, the constrained sampled-data transition probability matrix is constructed. Then, new criteria are proposed for the finite-time constrained reachability of DPBCNs under uniform SDC. By virtue of finite-time constrained reachability, a procedure is established to design state feedback sampled-data stabilisers for DPBCNs. The obtained results are applied to the ‘minimal’ model of lactose operon in E. coli.

Finite‐time observability of probabilistic Boolean control networks

AbstractIn this paper, the finite‐time observability (FO) of probabilistic Boolean control networks (PBCNs) based on set reachability and parallel extension is investigated. Under algebraic state space representation of PBCNs, finite‐time set reachability (FSR), finite‐time single input sequence set reachability (FSSSR), finite‐time arbitrary input sequence set reachability (FASSR), as well as finite‐time output feedback set reachability (FOSR) are addressed, where some efficient criteria are proposed. In the second part of this paper, several types of FO are provided, which can be converted to the corresponding FSR based on the parallel extension technique. Consequently, the obtained results for FSR are utilized to deal with FO. Finally, two examples including a biological network are given to demonstrate the theoretical results.

Controllability and stabilization of periodic switched Boolean control networks with application to asynchronous updating

This paper investigates the periodic switching point controllability and stabilization of periodic switched Boolean control networks (PSBCNs), and applies the obtained results to the stabilization of deterministic asynchronous Boolean control networks (DABCNs). Firstly, using the algebraic state space representation of PSBCNs, a kind of periodic switching point controllability matrix is constructed, based on which, a necessary and sufficient condition is presented for the periodic switching point reachability and controllability of PSBCNs. Secondly, using the reachable set of PSBCNs, a constructive procedure is proposed to design time-variant state feedback controllers for the periodic switching point stabilization of PSBCNs. Finally, by converting the dynamics of DABCNs into the form of PSBCNs, the time-variant state feedback stabilization problem of DABCNs is solved.

Constrained set controllability of logical control networks with state constraints and its applications

State constraints are very common in actual networked systems. The main purpose of this article is to investigate the constrained set controllability of k-valued logical control networks (KVLCNs), the states of which are within a finite set. At first, the dynamics of KVLCNs with state constraints (SC-KVLCNs) can be equivalently written as an bilinear system by applying algebraic state space representation approach. After that, a method is presented to construct a kind of constrained set controllability matrix, based on which, a criterion is put forward to solve the constrained set controllability problem of KVLCNs. In addition, the constrained set controllability is applied to the partial stabilization, output tracking, output synchronization and observability of SC-KVLCNs. Finally, to implement the main results of this article, three examples are given.

Minimum-Time and Minimum-Triggering Observability of Stochastic Boolean Networks

This article investigates the observability of Markovian jump Boolean networks (MJBNs) via algebraic state space representation approach. A necessary and sufficient criterion in the form of linear programming is derived for the asymptotic observability in distribution of MJBNs, and several conditions are obtained for the finite-time observability based on the properties of nilpotent matrices. Subsequently, in order to minimize the time consumption, a maximum principle is established to address the minimum-time observability problem. With regard to the event-triggered output feedback observability, an efficient procedure is developed to minimize the number of triggering events. Finally, three numerical examples are employed to demonstrate the effectiveness of theoretical results.

State Feedback Stabilization of Large-Scale Logical Control Networks via Network Aggregation

Finding a computationally tractable method to solve the feedback stabilization problem of large-scale logical control networks (LCNs) is a challenging issue. This article combines network aggregation and algebraic state-space representation (ASSR) to solve the state feedback stabilization problem of large-scale LCNs. First, the whole network of large-scale LCNs is divided into several small subnetworks via network aggregation. Second, the mode-dependent state feedback gain is explored for the stabilization of switched LCNs based on the ASSR method. Third, using the mode-dependent state feedback gain in each subnetwork, an effective result is proposed to design the state feedback gain for the stabilization of large-scale LCNs. Finally, two illustrative examples are given to demonstrate the effectiveness of the obtained new results.

Robust stability of switched delayed logical networks with all unstable modes

Logical network is a special kind of positive systems with finite state space and logical operations. This paper analyzes the robust stability of switched delayed logical networks (SDLNs) with all unstable modes. The main mathematical tool is the semi-tensor product of matrices, which is used to convert the dynamics of SDLNs with all unstable modes into an equivalent algebraic state space representation. Following the algebraic state space representation framework, the concept of robustly stable parts is proposed, based on which, a new criterion is established for the robust stability of SDLNs with all unstable modes. Finally, two examples are given for the verification of main results.

Impulsive control design for output tracking of probabilistic Boolean control networks

This study investigates the impulsive control for the output tracking of probabilistic Boolean control networks (PBCNs) by the algebraic state-space representation method. Firstly, a novel system with augmented control is established by regarding the impulsive time as a special kind of control, which is equivalent to the considered impulsive PBCN (IPBCN). Secondly, based on the largest control invariant subset, a necessary and sufficient condition is proposed to design impulsive control for the output tracking problem of IPBCNs. Moreover, a constructive method is put forward to design the state feedback controller and state-dependent impulsive time sequence. Finally, an example is given to demonstrate the effectiveness of the obtained results.

Finite-Time Consensus of Finite Field Networks With Stochastic Time Delays

This brief aims to solve the finite-time consensus problem of finite field networks (FFNs) with stochastic time delays. An algebraic state space representation (ASSR) approach is introduced for the investigation of this problem. Firstly, the concept of finite-time consensus with probability one is defined for FFNs with two kinds of stochastic time delays. Secondly, the dynamics of FFNs with time delays is converted into an ASSR, based on which, the dynamics of FFNs with probabilistic time delay is converted into a linear discrete-time stochastic system. It is shown that the finite-time consensus of FFNs with probabilistic time delay is equivalent to the finite-time set stability of the obtained system. Thirdly, a verifiable criterion is put forward for the finite-time consensus of FFNs with probabilistic time delay. Finally, using the dummy matrix, the dynamics of FFNs with Markov jump time delay is also converted into a linear discrete-time stochastic system, and the corresponding criterion is proposed for the finite-time consensus of FFNs with Markov jump time delay.

Finite-Time Controllability and Set Controllability of Impulsive Probabilistic Boolean Control Networks

This paper addresses the finite-time controllability and set controllability of impulsive probabilistic Boolean control networks (IPBCNs). Firstly, using the algebraic state space representation (ASSR) method, the transition probability matrix of IPBCNs is established. Secondly, a kind of finite step reachability matrix with probability one is constructed, based on which, several effective criteria are proposed for the finite-time controllability with probability one of IPBCNs. Thirdly, a necessary and sufficient condition is presented for the finite-time set controllability with probability one of IPBCNs by constructing the set controllability probability distribution vector. Finally, the obtained results are extended to switching topology case.

Finite-Time Time-Variant Feedback Stabilization of Logical Control Networks With Markov Jump Disturbances

This brief investigates the finite-time time-variant state feedback stabilization of k-valued logical control networks (KVLCNs) with Markov jump disturbances via the algebraic state space representation (ASSR) method. Firstly, an equivalent Markov jump system is established for the considered KVLCNs with Markov jump disturbances by using ASSR. Secondly, a sequence of reachable sets is defined, based on which, a control design procedure is established for the time-variant state feedback stabilization of KVLCNs with Markov jump disturbances. Finally, the obtained result is applied to lac operon in the bacterium Escherichia coli.

Output Tracking Control Design of Switched Boolean Control Networks

In this letter, the output tracking control design of switched Boolean control networks (SBCNs) is investigated via state feedback and output feedback control. The algebraic state-space representation method which resorts to the semi-tensor product (STP) of matrices is utilized; necessary and sufficient conditions for the solvability of the output tracking control problem are presented. A constructive procedure is given to obtain all possible switching signal-dependent state feedback and output feedback controllers under arbitrary switching signals such that the output of SBCNs tracks a time-varying reference trajectory. Finally, a SBCN model of the lactose regulation in the Escherichia Coli bacteria is considered to illustrate the effectiveness of the proposed results.

Function Perturbation Impact on Feedback Stabilization of Boolean Control Networks.

Function perturbation analysis of Boolean networks is an important topic in the study of gene regulation due to gene mutation or immeasurable variables. This brief studies the function perturbation impact on feedback stabilization of Boolean control networks (BCNs) by using the algebraic state space representation approach. First, the state feedback stabilization control design of BCNs is recalled and the function perturbation problem is formulated. Second, given a state feedback stabilizer, it is robust to the considered function perturbation if one of the following three cases happens: 1) the block where the function perturbation occurs is different from the block which is affected by the state feedback stabilizer (Case 1); 2) when Case 1 does not happen, the perturbed column converges to the equilibrium faster than the original column (Case 2); 3) when Cases 1 and 2 do not happen, the perturbed column does not belong to the reachable set of the original column (Case 3). Third, when the perturbed column belongs to the reachable set of the original column, a constructive procedure is proposed to modify the given state feedback stabilizer to be robust to the function perturbation. Finally, the obtained new results are applied to the function perturbation analysis of lactose operon in Escherichia coli. The main novelty of this brief is to develop a new theoretical framework for the robustness of feedback controllers of BCNs with respect to function perturbation, which is not solved in the existing literature.