Stability and stabilization of Boolean networks

The stability of Boolean networks and the stabilization of Boolean control networks are investigated. Using semi‐tensor product of matrices and the matrix expression of logic, the dynamics of a Boolean (control) network can be converted to a discrete time linear (bilinear) dynamics, called the algebraic form of the Boolean (control) network. Then the stability can be revealed by analyzing the transition matrix of the corresponding discrete time system. Main results consist of two parts: (i) Using logic coordinate transformation, the known sufficient condition based on incidence matrix has been improved. It can also be used in stabilizer design. (ii) Based on algebraic form, necessary and sufficient conditions for stability and stabilization, respectively, are obtained. Copyright © 2010 John Wiley & Sons, Ltd.     

Minimum‐Time State Feedback Stabilization of Constrained Boolean Control Networks

Ministry in 2005, and the National Prize of Natural Science of China in 2008. Currently, he is Associate Editor IMA Journal of Math Control and Inform., and a Technical Committee member of IFAC (TC2.3). This paper investigates the state feedback stabilization of Boolean control networks (BCNs) with state and input constraints by using the semi‐tensor product of matrices. Firstly, a kind of constrained input‐state incidence matrix is proposed for constrained BCNs, which contains all the reachability information of the constrained systems. Secondly, based on the constrained input‐state incidence matrix, a necessary and sufficient condition is presented for the stabilization of constrained BCNs via state feedback. Thirdly, a general procedure is proposed to design all possible minimum‐time state feedback stabilizers. The study of an illustrative example shows that the new results obtained are effective in analyzing the stabilization of constrained BCNs.     

Nonlinear systems possessing linear symmetry

This paper tackles linear symmetries of control systems. Precisely, the symmetry of affine nonlinear systems under the action of a sub‐group of general linear group GL(n,?). First of all, the structure of state space (briefly, ss) symmetry group and its Lie algebra for a given system is investigated. Secondly, the structure of systems, which are ss‐symmetric under rotations, is revealed. Thirdly, a complete classification of ss‐symmetric planar systems is presented. It is shown that for planar systems there are only four classes of systems which are ss‐symmetric with respect to four linear groups. Fourthly, a set of algebraic equations are presented, whose solutions provide the Lie algebra of the largest connected ss‐symmetry group. Finally, some controllability properties of systems with ss‐symmetry group are studied. As an auxiliary tool for computation, the concept and some properties of semi‐tensor product of matrices are included. Copyright © 2006 John Wiley & Sons, Ltd.     

Synchronization of Interconnected Multi‐valued Logical Networks

This paper studies the synchronization of interconnected k‐valued logical networks, as well as that of interconnected higher order k‐valued logical networks, based on their algebraic representations. For interconnected k‐valued logical networks, one necessary and sufficient criterion of synchronization is established via the definition of synchronization and another equivalent condition is given by combining synchronization with fixed points and cycles. Furthermore, with some special properties of semi‐tensor product, the third necessary and sufficient condition for synchronization of interconnected k‐valued logical networks is raised in terms of transition matrices. Then, the limit set method can be used to analyze the synchronization of interconnected higher order k‐valued logical networks. Finally, three examples, including the Boolean model for interconnected biochemical oscillators in the cell cycle, are presented to illustrate effectiveness of the results.     

Observability conditions of Boolean control networks

Observability is a basic, yet challenging, issue when studying Boolean control networks (BCNs). Recently, a criterion for observability of controllable BCNs was proposed by using the algebraic representation of logical dynamics based on the technique of the semi‐tensor product of matrices. In this paper, we present new necessary and sufficient conditions guaranteeing observability of a BCN without preassuming its controllability. The conditions are hence more general. Some examples are worked out to illustrate the obtained results. Copyright © 2013 John Wiley & Sons, Ltd.     

L2 stability,H∞ control of switched homogeneous nonlinear systems and their semi‐tensor product of matrices representation

This paper investigates the problems of L₂ stability and H_∞ control of switched homogeneous nonlinear systems. This paper gives that if a switched homogeneous nonlinear system with disturbance is internally homogeneously asymptotically stable, then it has a finite L₂ gain, and if a switched homogeneous nonlinear system with disturbance and controls is homogeneously stabilizable under the zero disturbance condition, its H_∞ control problem is solvable under some mild conditions, and a homogeneous solution is given. Then, via the semi‐tensor product of matrices method, the aforementioned obtained results are transformed to linear‐like forms, and the aforementioned obtained Hamilton–Jacobi–Isaacs inequality is transformed to a linear‐like matrix inequality, which makes it feasible to compute the solutions by computer. Copyright © 2012 John Wiley & Sons, Ltd.