Control Of Logical Dynamical Systems And Its Applications On Decision-Making Problems Via Semi-Tensor Product Of Matrices Approach

Utilizing the semi-tensor product of matrices as the mathematical tool,the algebraic state-space expression of logical dynamical systems as the solving approach,this dissertation investigates solutions of several theoretical and practical issues of intelligent inference and intelligent decision-making.This dissertation first studies two basic theoretical issues on controlling logical dynamical systems and then turns to the applications on two categories of decision-making problems.This dissertation follows the rational decision-making stages proposed by the famous pioneer in artificial intelligence:Prof.Herbert Alexander Simon,using the semi-tensor product of matrices approach,models and investigates some representative problems in the information retrieval,strategy design,decision choice,and strategy implementation procedures.Several theoretical results are obtained.Considering that the development of logical dynamical systems and the applications on decision-making problems are at the primary stage,with the help of the semi-tensor product,more explorations are meaningful.The main contents are as follows:·The output tracking(or output regulation)problem of Boolean control networks is considered,where the tracking object is the output of a reference system.The concept of control attractor is proposed and an easily computable formula is obtained to calculate the set of control attractors.Then it is proved that the maximum control invariant subset within a preassigned subset is the set of control attractors and the states that can be driven to the attractors within finite steps.An auxiliary system,which combines the original Boolean control networks and the reference systems,is constructed.Combining the control attractors of the auxiliary system with the set controllability method,an easily verifiable necessary and sufficient condition for the solvability of the output tracking problems is obtained.For the solvable problems,open-loop and closed-loop control designs are presented,which can realize the tracking within the minimum time.·Observability and reconstructibility of Boolean control networks are investigated.By constructing an auxiliary system,the control of the state pairs of the original Boolean control network is converted into the control of the auxiliary system.Utilizing the set controllability technique and the concept of the control attractor sets of Boolean control networks,the set controllability tasks of the auxiliary Boolean control networks,which are equivalent to the observability or reconstructibility problems,are proposed and solved.·Controllability of incomplete logical dynamical systems is considered.The necessary and sufficient condition for controllability is obtained.By analyzing the avoiding pairs and the forbidden states,the wolf-sheep-cabbage and the missionaries-cannibals problems are converted into controllability of logical dynamical systems.Then the corresponding formulas and the optimal solutions are provided.·The conventional pairing problem(CPP)is revisited and extended to a networked form,which is called the networked pairing problem(NPP).First,we give the definition and modeling of NPPs.The existence of stable arrangements in NPPs is proved.Moreover,the properties of arrangements are also acquired.Second,an algorithm is proposed to verify whether an arrangement is stable,after which stability criteria of NPPs are obtained,by virtual of the algebraic state-space representations.Next,event-triggered control is introduced to ensure the global stability of NPPs,based on which the stabilization result is obtained.·The incomplete-profile games(IPGs),that is,the games with infeasible pro-files,are studied.Using the semi-tensor product of matrices,the structures of IPGs,and the dynamics of evolutionary IPGs are constructed.A necessary and sufficient condition is obtained to assure an IPG being potential.Moreover,an algorithm is proposed to find the feasible set of an IPG,which guarantees the game to be potential.To make the method more practically applicable,we consider near potential IPGs,which are mimics of near potential games.To this end,the decomposition of an IPG is also presented.Finally,for IPGs with several same size feasible sets,an algorithm is given to choose a feasible set that has the best potential approximation.·The profile-dynamic based fictitious play(PDBFP)is proposed as an improvement of the fictitious play.The model and the updating rule of PDBFP are presented via the semi-tensor product.Some convergence properties of the best response PDBFP are obtained,including the absorption of Nash equilibria.Examples are also provided with simulating results to show the advantages of PDBFP.