Decoupling And Decomposition Problems Of Boolean Control Networks

Boolean control networks(BCNs)are discrete dynamic control systems composed of Boolean variables and Boolean functions.BCNs have been widely used in many fields,including cell differentiation,immune response,biological evolution,neural network,gene regulation and so forth.The results of this dissertation are obtained in the theoretical framework of algebraic representation of BCNs based on semi-tensor product of matrices.By using the algebraic representation of BCNs,this dissertation introduces the vertex-colored state transition graph of BCNs.Based on the vertex-colored state transition graph,the decoupling problem and system decomposition problem of BCNs are studied by using the vertex partition method.The disturbance decoupling problem,input-output decoupling problem,observability decomposition problem and cascade decomposition problem of BCNs are completely solved.Compared with the existing results,by using the vertex partition method,necessary and sufficient conditions for the solvability of these problems are obtained,which reveal the essence of these problems concisely and intuitively.The main contributions of this dissertation are as follows:1.For the disturbance decoupling problem of BCNs,a new concept called original disturbance decoupling is proposed.Compared with the existing concept of disturbance decoupling,this concept does not depend on system decomposition and is more consistent with the traditional concept of disturbance decoupling for linear control systems.The equivalent algebraic conditions for the realization of the original disturbance decoupling are obtained.Then,by considering the vertex-colored state transition graph of BCNs and using the vertex partition method,necessary and sufficient partition conditions for original disturbance decoupling and disturbance decoupling are obtained.These partition conditions intuitively reveal the relationship between the two disturbance decoupling concepts,which also shows the essence of disturbance decoupling.In addition,an algorithm is designed to search for these partition conditions.2.For the input-output decoupling problem of BCNs,by considering the vertexcolored state transition graph of BCNs and using the vertex partition method,necessary and sufficient partition conditions for two definitions of input-output decoupling are obtained.Based on the partition conditions,the essential relationship between the two definitions of input-output decoupling is revealed and the essence of input-output decoupling also is revealed for the first time.Besides,the method to verify these partition conditions is given.3.For the observability decomposition of BCNs,the definition for observability decomposition is proposed for the first time.By using the vertex partition method,a necessary and sufficient partition condition for the observability decomposition of BCNs is derived.Besides,an algorithm is designed to search for the partition condition,and the observability decomposition can be constructively realized by the partition condition.4.For the cascading decomposition problem of BCNs,the concept of nested perfect equal vertex partition is proposed.Based on the nested perfect equal vertex partition,necessary and sufficient partition conditions for the solvability of two types of cascading decomposition problems are obtained.Given the proposed partition conditions,the logical coordinate transformations can be constructively obtained to realize the corresponding cascading decomposition forms.