Semi-tensor Product Of Matrices Research Articles

New method for stability of probabilistic Boolean networks with inequality constraints

This paper studies the stability of probabilistic Boolean networks (PBNs) with state inequality constraints by establishing a new algebraic representation. On one hand, combining with the semi-tensor product (STP) of matrices and the law of total expectation, the algebraic representation of PBNs is established. On the other hand, based on the STP of matrices, an algorithm is proposed to determine the solution set of the state inequality constraints. Then, based on the algebraic representation and the inequality constrained solution set, a new form of transition probability matrix is constructed for PBNs. Finally, by the proposed transition probability matrix, new criteria are proposed for the finite-time stability with positive probability of PBNs. Several simulation examples are used to illustrate the effectiveness of the obtained results.

Cluster consensus of finite-field networks with stochastic communication topology

Most of the multi-agent systems (MASs) in real life are with limited storage and communication capabilities; studies on such systems, including consensus problem, however have been mainly based on real number field models. In this article, the cluster consensus problem of MASs with stochastic communication topology over finite-field known as the cluster consensus of finite-field networks (FFNs), is considered. Using the semi-tensor product of matrices, the dynamics of FFNs with stochastic communication topology are equivalently converted into a logical algebraic form, which facilitates further studies. Then a permutation system is introduced. The relationship between cluster consensus of FFNs with stochastic communication topology and the set stability with probability one (SSPO) of the permutation system is revealed. In such an approach, some verifiable criteria can be derived for the cluster consensus of FFNs with stochastic communication topology, without requesting the connectivity of the whole network. Finally, an example is presented to validate the main results.

Analysis and control of demand response in smart grids: An evolutionary game method

As an effective strategy for load management in smart grids, demand response establishes a bidirectional connection between the electricity supplier and users. Based on the networked evolutionary game theory, this paper studies the demand-response issue for a class of smart grids by using the semi-tensor product of matrices. The paper proceeds as follows. (i) Considering the dynamic interactions between the supplier and users, the demand response is modeled as a heterogeneous networked evolutionary game and is expressed as dynamical form by semi-tensor product. (ii) A sufficient and necessary condition is provided to verify the convergence to a fixed point of the considered system. (iii) A feedback controller is designed to ensure the system electricity consumption and price to maintain at a desired level. Finally, an example is presented to illustrate the feasibility of the proposed method.

A Real Method for Solving Octonion Matrix Equation $$AXB=C$$ Based on Semi-tensor Product of Matrices

A Real Method for Solving Octonion Matrix Equation $$AXB=C$$ Based on Semi-tensor Product of Matrices

Construction of Universal Approximators for Multi-Input Single-Output Hierarchical Fuzzy Systems

As an important branch of fuzzy systems, hierarchical fuzzy system (HFS) has a wide range of applications in system science, medical science and engineering. Using the semi-tensor product of matrices, this paper investigates the universal approximation of multi-input single-output HFSs, and develops the construction algorithms of universal approximators. Initially, based on the fuzzy relation matrices of fuzzy logic units with Mamdani-type fuzzy rules, the algebraic formulation of HFSs consisting of fuzzy logic units is proposed. After that, the universal approximation of HFSs is explored via the algebraic formulation, and some effective algorithms are presented to construct the universal approximators of HFSs with respect to different scenarios of approximated functions. Finally, the effectiveness of obtained results is verified by the on-ramp metering of freeway.

Minimal pinning control for set stability of Boolean networks

In this paper, set stability of Boolean networks (BNs) is studied based on semi-tensor product of matrices. First, an index-vector is introduced to verify whether a BN can be set stable from an initial set to a destination set. Then, an algorithm is proposed to obtain the minimal set of pinning nodes to achieve set stability, if a BN is not set stable. Moreover, in order to reduce the computational complexity, a hybrid pinning control technique is proposed, where the designed controllers are not entirely dependent on all the nodes of the networks. Further, the issue of synchronization is discussed by using the proposed hybrid pinning control methods. Finally, simulations are presented to illustrate the effectiveness of the obtained results in this paper.

Minimum-time strategy optimization for networked evolutionary games with bankruptcy mechanism

In this paper, the minimum-time strategy optimization problem for NEGs with bankruptcy mechanism is considered based on semi-tensor product of matrices. Firstly, in order to describe the impact of bankruptcy mechanism on evolution, a novel matrix called controlled state transition matrix is constructed by deleting all invalid states, which will also reduce the computational complexity of subsequent analysis and design. Secondly, by assorting to the redefined matrix, the minimum-time strategy optimization problem for NEGs with bankruptcy mechanism can be solved by designing state feedback control which can guarantee the convergence of states without violating the bankruptcy mechanism. Meanwhile, the verification condition is derived to verify the existence of the state feedback control, and an algorithm for seeking all possible minimum-time state feedback controllers is constructed. Finally, a typical example is illustrated to validate the effectiveness of the proposed method.

A Novel Mathematical Characterization for Switched Linear Systems Based on Automata and Its Stabilizability Analysis

Mathematical characterization of switched systems based on automata is one of the most challenging problems in the context of hybrid dynamical systems. This paper investigates the mathematical representation and stabilization of a class of slowly and fastly switched linear systems with automata in the hybrid coupling framework. To address such an issue, by using the semi-tensor product of matrices, a so-called “automaton-dependent switched linear system” is established to generalize the traditional switched system, which is characterized mathematically in the new form of a hybrid dynamic system consisting of a continuous-time differential equation and a discrete-time difference equation. Moreover, a novel automaton-dependent constrained switching scheme is developed for the stabilization of the proposed model with unstable subsystems. Next, by designing slow mode-dependent average dwell time (S-MDADT), fast mode-dependent average dwell time (F-MDADT), slow cycle-dependent cycle dwell time (S-CDCDT) and fast cycle-dependent cycle dwell time (F-CDCDT) switching laws, a generalized stabilization lemma for automaton-dependent switched nonlinear systems with both stable and unstable modes inside and outside the cycle is derived in the term of digital transformations and multiple Lyapunov functions. Furthermore, based on the stabilizability criteria of nonlinear case, by setting digital transformations and linear matrix inequalities, a new stabilization theorem for automaton-dependent switched linear systems with unstable subsystems is obtained. It should be noted that the proposed scheme provides a more precise estimation of the upper and lower bounds for the dwell time of the system with unstable modes. Finally, a practical converter circuit example and a numerical example are given to show the validity of the designed techniques.

The stability degree analysis of profiles for networked evolutionary games with switching topology

In this paper, we investigate the stability degree of profiles for a class of networked evolutionary games (NEGs) with switching topology using the semi-tensor product of matrices. Firstly, we calculate the algebraic form of NEGs with switching topology, whose strategy updating rule is myopic best response adjustment rule and network updating rule is voting principle. Secondly, the strong stability degree and weak stability degree of a profile are defined, and some necessary and sufficient conditions to detect whether a profile is strong or weak k−degree stable are provided. Furthermore, we propose the state feedback controller algorithms to guarantee that a profile is strong or weak k−degree stable, respectively. Finally, a numerical example is provided to illustrate the effectiveness of our results.

Algebraic Formulation and Application of Multi-Input Single-Output Hierarchical Fuzzy Systems With Correction Factors

Hierarchical fuzzy systems (HFSs) are successfully applied to control systems and computer networks. Particularly, correction factor is introduced to regulate fuzzy rules and alleviate the problems of memory usage and real-time performance of systems. This paper considers the multi-input single-output HFSs with correction factors, and proposes an algebraic framework by using the semi-tensor product of matrices. Firstly, a kind of fuzzy correction factor matrix (FCFM) is constructed to realize the fuzzification of correction factor, based on which, an algebraic form is developed for two-input single-output fuzzy systems. Secondly, the structure diagram of HFSs is decomposed into several fuzzy logic units, which have the form of two-input singleoutput fuzzy systems with FCFM. Based on this operation, the algebraic formulation of serial, parallel and hybrid HFSs with correction factors is proposed. Finally, the effectiveness of HFSs with FCFM is verified by studying the on-ramp metering of freeway.

Multi-sensor fusion particle filtering for Boolean networks with multi-step randomly-delayed measurements

This paper is concerned with developing particle filters for Boolean networks with multi-step randomly-delayed measurements. Based on the semi-tensor product of matrices and dummy selection matrices, a generalized measurement model including multi-step randomly-delayed measurements is introduced. Besides, multi-sensor fusion particle filtering is proposed and two recursive algorithms are designed to minimize the mean-square estimation errors by fusing multi-sensor measurements. Simulation results show the effectiveness of the proposed multi-sensor fusion particle filters. It can be seen from the simulation results that the estimation performance of multi-sensor fusion has been improved compared to that of one sensor.

STP-based verification and synthesis of state opacity for logical finite state machines

Finite-state machines are among the most important models for studying the logical dynamic behavior of cyber–physical systems, and their security and privacy are urgent problems to be solved at present. This paper explores the state-opacity-based privacy verification and synthesis problems for finite-state machines from an algebraic perspective. First, the dynamics of a finite-state machine can be established as an algebraic expression using the semi-tensor product of matrices. Beginning with this novel representation, we propose an algebraic criterion to verify whether the privacy state is opaque to intruders. Subsequently, we investigate the opacity-enhancement synthesis problem from two perspectives. On one hand, we design a feedback supervisory controller by disabling controllable events such that the closed-loop system is opaque with respect to privacy states; on the other hand, we investigate the editing of an observable function simply by assigning specific events in the observation channel instead of changing the transition behavior of the system. We observe that these synthesis problems can be addressed by calculating a system of algebraic equations, and we further develop two algorithms to obtain the controller or function. Finally, an interesting example is presented to demonstrate the validity of the proposed algebraic method. Taken together, these results are conducive to a systematic understanding of privacy enhancement problems.

Asynchronous Control of Stochastic Switched Boolean Control Networks With Piecewise-Homogeneous Dwell Time.

In this article, the l₁-induced performance of the stochastic switched Boolean control network (BCN) is investigated. The switched signal is considered to follow a time-varying probability distribution, the switching of which is considered to have a random dwell time. The asynchronous state feedback control (SFC) is studied to achieve the control objective. This kind of control can avoid the failure of the control due to the inconsistency between the system mode and the control mode, so the results obtained are more general. Using the semitensor product of matrices, the algebraic form of the considered BCN is represented. Under this framework, sufficient conditions are obtained to ensure that the closed-loop system is stochastic stabilized with a prescribed l₁-induced performance level ɣ. Parameters can be solved by inequalities. In addition, when the dwell time converges to infinity, the probability distribution of the switched signal becomes fixed. Necessary and sufficient conditions are presented to ensure the stabilization of the closed system under asynchronous SFC as well as the design of the asynchronous SFC. Then, sufficient condition is obtained for the prescribed l₁-induced performance level. Examples are presented to show the effectiveness of the obtained results.

Modeling, analysis, and dynamics of Bayesian games via matrix-based method

A matrix-based framework for the modeling, analysis and dynamics of Bayesian games are presented using the semi-tensor product of matrices. Static Bayesian games are considered first. A new conversion of Bayesian games is proposed, which is called an action-type conversion. Matrix expressions are obtained for Harsanyi, Selten, and action-type conversions, respectively. Certain properties are obtained, including two kinds of Bayesian Nash equilibria. Then the verification of Bayesian potential games is considered, which is proved to test the solvability of corresponding linear equations equivalently. Finally, the dynamics of evolutionary Bayesian games are considered. Two learning rules for Bayesian potential games are proposed, which are type-based myopic best response adjustment and logit response rule, respectively. Markovian dynamic equations are obtained for the proposed strategy updating rules and convergence is proved.

Stability Analysis of Probabilistic Boolean Networks With Switching Discrete Probability Distribution

This article analyzes the stability of probabilistic Boolean networks (PBNs) with switching discrete probability distribution (DPD). First, the dynamics of PBNs with switching DPD is converted to an algebraic form by using the semitensor product of matrices. Second, based on the algebraic form, two essential concepts, that is, expectation of number of trajectories (ENT) and state expectation matrix, are proposed. Third, some criteria are presented for the finite-time stability and asymptotical stability of PBNs with switching DPD by resorting to the properties of ENT. Finally, two illustrative examples are worked out to show the effectiveness of the obtained new results. Computational complexity is still challenging for this article, which limits the application of ENT-based framework to large-size PBNs with switching DPD.

Stabilization of Delayed Boolean Networks Using Constrained State Pinning Control

Pinning control is applied to ensure the stabilization of Boolean networks (BNs) with time delay parameter. The time delay parameter in this article follows an independent identical distribution. Using semi-tensor product of matrices, the considered BN with time delay is converted to a high dimensional switching BN and the switching signal is the time delay signal. Different from general switching BN, the structure matrices of all subsystems are independent with each other, structure matrices of all subsystems for the high dimensional switching BN depend on the original BN structure matrix. Pinning control is designed to guarantee the global stochastic stability of the considered system. Furthermore, the global stochastic stability of BN with time delay parameter is proved to be equivalent to the stability of BN without time delay parameter, which greatly reduces the computational complexity and simplifies the control design. For the considered BN with finite cycles, constrained pinning control is applied with limitation that only one state of each undesired cycle is under control. With this constraint, the minimal number of pinning control nodes is further investigated. Some algorithms are presented to obtain the new structure matrix, with which, pinning control can be solved by the obtained methods. Both numerical examples and biological example illustrate the effectiveness of the results.

Observability Decomposition of Boolean Control Networks

This article investigates the observability decomposition of Boolean control networks (BCNs) under the algebraic frame based on the semitensor product of matrices. First, the definition for the observability decomposition of BCNs is proposed, which is consistent with the observability decomposition of traditional linear control systems. Then, by using the vertex partition method, a necessary and sufficient vertex partition condition for the observability decomposition of BCNs is proposed. Next, an algorithm is designed for the analysis and realization of observability decomposition. Finally, a biological example and a counterexample are analyzed to illustrate the obtained results.

Necessary and Sufficient Conditions for Security of Hidden Markov Boolean Control Networks Under Shifting Attacks

In this paper, output feedback control (OFC) stabilization of hidden Markov Boolean control networks (HMBCNs) is studied. Using semi-tensor product of matrices, the OFC problems to be solved are presented in algebraic form. All feasible OFC gains have been characterized. A special kind of attack on the HMBCNs, named shifting attack, is studied. Such kind of attack does not change the control policy via attacking the value of the output measurements, but via passing false information to the controller after receiving the output measurements. Necessary and sufficient conditions are presented for the security of the considered HMBCNs under shifting attacks. Besides, for those systems under attack, some constraints are considered. Necessary and sufficient conditions for the security of HMBCNs with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\gamma$</tex-math></inline-formula> -measurements protected are also presented. Moreover, the smallest number of measurements to be protected is further elaborated. Finally, a simulation example based on a biological system is used to illustrate the effectiveness of the obtained results.

Mean-square stability of discrete-time switched systems under modeled random switching

This paper studies the mean-square stability of discrete-time linear switched systems with random switching, where the switching signal is the output of a logic dynamical switching model driven by an independent and identically distributed (i.i.d.) process. This class of switching is referred to as the modeled random switching. By regarding the switching model as a part of the system, a combined switched system with i.i.d. switching is obtained, which is of a hybrid nature, that is, the augmented state vector has both logic and continuous components. The semi-tensor product of matrices and the vector representation of logic are applied to merge the logic and the continuous components of the state. The equivalence between the mean-square stability of the original switched system and that of the merged switched system under i.i.d. switching is proved. By deriving the dynamics of the column-stacking form of the covariance matrix, necessary and sufficient conditions of mean-square stability for discrete-time linear switched systems with modeled random switching are obtained. An illustrative example is provided to demonstrate the usage of the proposed results.

Agent Transformation of Bayesian Games

A new transformation that converts a Bayesian game to a so-called <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ex-ante</i> agent game (a normal-form game) is proposed. Differently from the existing transformation proposed by R. Selten that changes a Bayesian game to an <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">interim</i> agent game (also a pure normal-form game), we prove that the new transformation preserves potentiality. In addition, there is a nonpotential Bayesian game whose <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ex-ante</i> agent game is potential. We also prove that there is one-to-one correspondence between pure Bayesian Nash equilibria (BNE) of Bayesian games (if one exists) and pure Nash equilibria (NE) of the resulting <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ex-ante</i> agent games. Then, we provide a sufficient and necessary condition for a Bayesian game to have an <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ex-ante</i> agent potential game. By using these results, one can transform pure BNE seeking in Bayesian games to pure NE seeking in their <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ex-ante</i> agent games [by using the potential functions of the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ex-ante</i> agent games (if one exists)], where previously pure BNE seeking in Bayesian games by using potential functions can only be done in Bayesian potential games (BPGs). Particularly, we prove for two-player games that BPGs are exactly the Bayesian games having <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ex-ante</i> agent potential games. Furthermore, by using the semi-tensor product of matrices, a potential equation for finite Bayesian games is developed. Based on the potential equation, algorithms for verifying potentiality and for searching pure BNE in finite Bayesian games are designed. Finally, the results are applied to a routing problem with incomplete information.