arXiv subjects

Daizhan Cheng

Publications and source records attributed to Daizhan Cheng.

At least 37 records · Page 2Linked to original sources

arXiv2023

Transition System Representation of Boolean Control Networks

First, the topological structure of a transition system is studied. Then, two types of transition system (TS) representations of Boolean networks (BNs) and Boolean control networks (BCNs) are investigated. The first kind of representation is state-based, which converts a BCN into a TS with either distinct control or non-distinct control. The second representation is output-based, which is also called the simulation of the original BCN. Some applications are also studied.

eess.SY↗

arXiv2023

Aggregated (Bi-)Simulation of Finite Valued Networks

The paper provides a method to approximate a large-scale finite-valued network by a smaller model called the aggregated simulation, which is a combination of aggregation and (bi-)simulation. First, the algebraic state space representation (ASSR) of a transition system is presented. Under output equivalence, the quotient system is obtained, which is called the simulation of the original transition system. The ASSR of the quotient system is obtained. The aggregated (bi-)simulation is execueted in several steps: a large scale finite-valued network is firstly aggregated into several blocks, each of which is considered as a network where the in-degree nodes and out-degree nodes are considered as the block inputs and block outputs respectively. Then the dynamics of each block is converted into its quotient system, called its simulation. Then the overall network can be approximated by the quotient systems of each blocks, which is called the aggregated simulation. If the simulation of a block is a bi-simulation, the approximation becomes a lossless transformation. Otherwise, the quotient system is only a (non-deterministic) transition system, and it can be replaced by a probabilistic networks. Aggregated simulation can reduce the dimension of the original network, while a tradeoff between computation complexity and approximation error need to be decided.

eess.SY↗

arXiv2023

Semi-Tensor Product of Hypermatrices with Application to Compound Hypermatrices

The semi-tensor product (STP) of matrices is extended to the STP of hypermatrices. Some basic properties of the STP of matrices are extended to the STP of hypermatrices. The hyperdeterminant of hypersquares is introduced. Some algebraic and geometric structures of matrices are extended to hypermatrices. Then the compound hypermatrix is proposed. The STP of hypermatrix is used to compound hypermatrix. Basic properties are proved to be available for compound hypermatrix.

eess.SY↗

arXiv2022

From Dimension-Free Manifolds to Dimension-varying Control Systems

Starting from the vector multipliers, the inner product, norm, distance, as well as addition of two vectors of different dimensions are proposed, which makes the spaces into a topological vector space, called the Euclidean space of different dimension (ESDD). An equivalence is obtained via distance. As a quotient space of ESDDs w.r.t. equivalence, the dimension-free Euclidean spaces (DFESs) and dimension-free manifolds (DFMs) are obtained, which have bundled vector spaces as its tangent space at each point. Using the natural projection from a ESDD to a DFES, a fiber bundle structure is obtained, which has ESDD as its total space and DFES as its base space. Classical objects in differential geometry, such as smooth functions, (co-)vector fields, tensor fields, etc., have been extended to the case of DFMs with the help of projections among different dimensional Euclidean spaces. Then the dimension-varying dynamic systems (DVDSs) and dimension-varying control systems (DVCSs) are presented, which have DFM as their state space. The realization, which is a lifting of DVDSs or DVCSs from DFMs into ESDDs, and the projection of DVDSs or DVCSs from ESDDs onto DFMs are investigated.

eess.SY↗

arXiv2022

Invariant and Dual Invariant Subspaces of $k$-valued Networks

Consider a $k$-valued network. Two kinds of (control) invariant subspaces, called state and dual invariant subspaces, are proposed, which are subspaces of state space and dual space respectively. Algorithms are presented to verify whether a dual subspace is a dual or dual control invariant subspace. The bearing space of $k$-valued (control) networks is introduced. Using the structure of bearing space, the universal invariant subspace is introduced, which is independent of the dynamics of particular networks. Finally, the relationship between state invariant subspace and dual invariant subspace of a network is investigated. A duality property shows that if a dual subspace is invariant then its perpendicular state subspace is also invariant and vice versa.

eess.SY↗

arXiv2022

Dimension-Free Matrix Spaces

Based on various types of semi-tensor products of matrices, the corresponding equivalences of matrices are proposed. Then the corresponding vector space structures are obtained as the quotient spaces under equivalences, which are called the dimension-free Matrix spaces (DFESs). Certain structures and properties are investigated. Finaly, the Lie bracket structure of general linear algebra is extended to DfMSs to make them Lie algebras, called dimension-free general linear algebra (DFGLA). Inspire of the fact that the DFGLAs are of infinite dimension, they have most properties of finite dimensional Lie algebras, whicl are studied in the paper.

math.OC↗

arXiv2022

A Remark on Evolution Equation of Stochastic Logical Dynamic Systems

Modelling is an essential procedure in analyzing and controlling a given logical dynamic system (LDS). It has been proved that deterministic LDS can be modeled as a linear-like system using algebraic state space representation. However, due to the inherently non-linear, it is difficult to obtain the algebraic expression of a stochastic LDS. This paper provides a unified framework for transition analysis of LDSs with deterministic and stochastic dynamics. First, modelling of LDS with deterministic dynamics is reviewed. Then modeling of LDS with stochastic dynamics is considered, and non-equivalence between subsystems and global system is proposed. Next, the reason for the non-equivalence is provided. Finally, consistency condition is presented for independent model and conditional independent model.

math.OC↗

arXiv2021

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 semi-tensor product (STP) of matrices and the algebraic state-space representation (ASSR) of Boolean networks, this paper reveals that in addition to this explicit order, there is 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, hidden order provides a global picture to describe the evolution of the overall network. It is our conjecture that the order of networks is mainly determined by the dual attractors via their corresponding hidden orders. The previously obtained results about Boolean networks are further extended to the k-valued case.

eess.SY↗

arXiv2021

On Networks Over Finite Rings

A (control) network over a finite ring is proposed. Using semi-tensor product (STP) of matrices, a set of algebraic equations are provided to verify whether a finite set with two binary operators is a ring. It is then shown that the STP-based technique developed for logical (control) networks are applicable to (control) networks over finite rings. The sub-(control) network over an ideal of the bearing ring is revealed. Then the product ring is proposed and the (control) network over product ring is investigated. As a key result, the decomposition theorem, called the Decomposition Principle (DP), is proposed, which shows a (control) network over a product ring is decomposable into sub-(control)networks over each factor rings, which makes the (control) properties of a network can be revealed by its factor sub-networks over factor rings. Using DP, the control problems of a control network are investigated via its sub control networks. Particularly, using DP, the control of linear networks over product rings is discussed in detail. Finally, the representation theorem is presented, which shows that each (control) network over a finite set can be expressed as a (control) network over a product ring.

eess.SY↗

arXiv2021

A Formula for Designing Zero-Determinant Strategies

A formula is presented for designing zero-determinant(ZD) strategies of general finite games, which have $n$ players and players can have different numbers of strategies. To this end, using semi-tensor product (STP) of matrices, the profile evolutionary equation for repeated finite games is obtained. Starting from it, the ZD strategies are developed for general finite games, based on the same technique proposed by Press and Dyson \cite{pre12}. A formula is obtain to design ZD strategies for any player directly, ignoring the original ZD design process. Necessary and sufficient condition is obtained to ensure the effectiveness of the designed ZD strategies. As a consequence, it is also clear that player $i$ is able to unilaterally design $|S_i|-1$ dominating linear relations about the expected payoffs of all players. Finally, the fictitious opponent player is proposed for networked evolutionary networks (NEGs). Using it, the ZD-strategies are applied to NEGs. It is surprising that an individual in a network may use ZD strategies to conflict the whole rest network.

cs.GT↗

arXiv2021

A Comprehensive Survey on STP Approach to Finite Games

Nowadays the semi-tensor product (STP) approach to finite games has become a promising new direction. This paper provides a comprehensive survey on this prosperous field. After a brief introduction for STP and finite (networked) games, a description for the principle and fundamental technique of STP approach to finite games is presented. Then several problems and recent results about theory and applications of finite games via STP are presented. A brief comment about the potential use of STP to artificial intelligence is also proposed.

cs.GT↗

arXiv2021

Matrix Expression of Bayesian Game

A matrix-based framework for Bayesian games is presented, using semi-tensor product of matrices. Static Bayesian games are considered first. Matrix expression of Bayesian games is proposed. Three kinds of conversions, which convert Bayesian games to complete information games are investigated, certain properties are obtained, including two kinds of Bayesian-Nash equilibriums. Finally, dynamic Bayesian games are considered. Markoven dynamic equations are obtained for some strategy updating rules.

math.OC↗

arXiv2021

Weighted and Near Weighted Potential Games with Application to Game Theoretic Control

An algorithm is proposed to verify whether a finite game is a weighted potential game (WPG) without pre-knowledge on its weights. Then the algorithm is also applied to find the closest WPG for a given finite game. The concept and criterion of near weighted potential games (NWPGs) are given, indicating the evolutionary equivalence between a finite game and its closest WPG. Based on this, a method is proposed for replacing PGs by NWPGs in game theoretic control, which improved the applicability of PG-based optimization.

cs.GT↗

arXiv2021

On Perfect Hypercomplex Algebra

The set of associative and commutative hypercomplex numbers, called the perfect hypercomplex algebra (PHA) is investigated. Necessary and sufficient conditions for an algebra to be a PHA via semi-tensor product(STP) of matrices are reviewed. The zero set is defined for non-invertible hypercomplex numbers in a given PHA, and a characteristic function is proposed for calculating zero set. Then PHA of different dimensions are considered. First, $2$-dimensional PHAs are considered as examples to calculate their zero sets etc. Second, all the $3$-dimensional PHAs are obtained and the corresponding zero sets are investigated. Third, $4$-dimensional or even higher dimensional PHAs are also considered. Finally, matrices over pre-assigned PHA, called perfect hypercomplex matrices (PHMs) are considered. Their properties are also investigated.

math.RA↗

arXiv2021

On Identification of Boolean Control Networks

A new analytical framework consisting of two phenomena: single sample and multiple samples, is proposed to deal with the identification problem of Boolean control networks (BCNs) systematically and comprehensively. Under this framework, the existing works on identification can be categorized as special cases of these two phenomena. Several effective criteria for determining the identifiability and the corresponding identification algorithms are proposed. Three important results are derived: (1) If a BN is observable, it is uniquely identifiable; (2) If a BCN is O1-observable, it is uniquely identifiable, where O1-observability is the most general form of the existing observability terms; (3) A BN or BCN may be identifiable, but not observable. In addition, remarks present some challenging future research and contain a preliminary attempt about how to identify unobservable systems.

eess.SY↗

arXiv2021

Invariant Subspace Approach to Boolean (Control) Networks

A logical function can be used to characterizing a property of a state of Boolean network (BN), which is considered as an aggregation of states. To illustrate the dynamics of a set of logical functions, which characterize our concerned properties of a BN, the invariant subspace containing the set of logical functions is proposed, and its properties are investigated. Then the invariant subspace of Boolean control network (BCN) is also proposed. The dynamics of invariant subspace of BCN is also invariant. Finally, using outputs as the set of logical functions, the minimum realization of BCN is proposed, which provides a possible solution to overcome the computational complexity of large scale BNs/BCNs.

eess.SY↗

arXiv2020

State Feedback Stabilization of Generic Logic Systems via Ledley Antecedence Solution

In this paper, the application of Ledley antecedence solutions in designing state feedback stabilizers of generic logic systems has been proposed. To make the method feasible, two modifications are made to the original Ledley antecedence solution theory: (i) the preassigned logical functions have been extended from being a set of equations to an admissible set; (ii) the domain of arguments has been extended from the whole state space to a restricted subset. In the proposed method, state feedback controls are considered as a set of extended Ledley antecedence solutions for a designed iterative admissible sets over their corresponding restricted subsets. Based on this, an algorithm has been proposed to verify the solvability, and simultaneously to provide all possible state feedback stabilizers when the problem is solvable. All stabilizers are optimal, which stabilize the logic systems from any initial state to the destination state/state set in the shortest time. The method is firstly demonstrated on Boolean control networks to achieve point stabilization. Then, with some minor modifications, the proposed method is also proven to be applicable to set stabilization problems. Finally, it is shown that in $k$-valued and mix-valued logical systems, the proposed method remains effective.

math.LO↗

arXiv2020

Self-Triggered Scheduling for Boolean Control Networks

It has been shown that self-triggered control has the ability to reduce computational loads and deal with the cases with constrained resources by properly setting up the rules for updating the system control when necessary. In this paper, self-triggered stabilization of Boolean control networks (BCNs), including deterministic BCNs, probabilistic BCNs and Markovian switching BCNs, is first investigated via semi-tensor product of matrices and Lyapunov theory of Boolean networks. The self-triggered mechanism with the aim to determine when the controller should be updated is given based on the decrease of the corresponding Lyapunov functions between two successive sampling times. We show that the self-triggered controllers can be chosen as the conventional controllers without sampling, and also can be optimally constructed based on the triggering conditions.

eess.SY↗