Jun-e Feng

Stabilization of Markovian Systems via Probability Rate Synthesis and Output Feedback

This technical note is concerned with the stabilization problem of Markovian jump linear systems via designing switching probability rate matrices and static output-feedback gains. A novel necessary and sufficient condition is established to characterize the switching probability rate matrices that guarantee the mean square stability of Markovian jump linear systems. Based on this, a necessary and sufficient condition is provided for the existence of desired controller gains and probability rate matrices. Extensions to the polytopic uncertain case are also provided. All the conditions are formulated in terms of linear matrix inequalities with some equality constraints, which can be solved by two modified cone complementarity linearization algorithms. Examples are given to show the effectiveness of the proposed method.

Internal positivity preserved model reduction

This article studies model reduction of continuous-time stable positive linear systems under the Hankel norm, H ∞ norm and H 2 norm performance. The reduced-order systems preserve the stability as well as the positivity of the original systems. This is achieved by developing new necessary and sufficient conditions of the model reduction performances in which the Lyapunov matrices are decoupled with the system matrices. In this way, the positivity constraints in the reduced-order model can be imposed in a natural way. As the model reduction performances are expressed in linear matrix inequalities with equality constraints, the desired reduced-order positive models can be obtained by using the cone complementarity linearisation iterative algorithm. A numerical example is presented to illustrate the effectiveness of the given methods.

l1-gain analysis and model reduction problem for Boolean control networks

In this paper, the weighted l1-gain analysis and l1 model reduction problem for Boolean control networks are proposed and investigated via semi-tensor product method. First, the input energy and output energy are described by pseudo-Boolean functions, based on which the definition of the weighted l1-gain is introduced. By constructing a co-positive Lyapunov function, a sufficient condition is established to ensure that a Boolean control network is not only internally asymptotically stable, but also has an l1-gain no more than a given scalar. Along this line, by virtue of the properties of semi-tensor product, the l1 model reduction problem of a Boolean control network is defined and converted to the l1-gain problem of another Boolean control network with more nodes. A sufficient condition for the l1 model reduction problem is then derived immediately, and an algorithm is presented to compute the matrices in the reduced order model. Finally, two examples, including the Boolean model for biochemical oscillators in the cell cycle, are displayed to show the feasibility of the theoretical results.

Mix-valued logic-based formation control

The formation control (FC) problem is investigated via a mix-valued logic-based approach. First, a trajectory-tracking algorithm of mix-valued logic control networks is proposed. Then, a new formulation of FC problems is established and a feedback control is designed to solve FC problems. The mathematical description of partial-formation control (PFC) problems is then designed as a structure of logical networks. An interesting practical example of PFC is also presented and discussed in detail.

On solutions to the matrix equations XB−AX=CY and XB−AX^=CY

Abstract The solution of the generalized Sylvester real matrix equation XB − AX = CY is important in stability analysis and controller design in linear systems. This paper presents an explicit solution to the generalized Sylvester real matrix equation XB − AX = CY . Based on the derived explicit solution to the considered generalized Sylvester real matrix equation, a new approach is provided for obtaining the solutions to the generalized Sylvester quaternion j-conjugate matrix equation XB − A X ^ = CY using the real representation of a quaternion matrix. The closed form solution is established in an explicit form for this generalized Sylvester quaternion j-conjugate matrix equation. The existing complex representation method requires the coefficient matrix A to be a block diagonal matrix over complex field, while it is any suitable dimension quaternion matrix in the present paper. Therefore, we generalize the existing results.

Further Results on Disturbance Decoupling of Mix-Valued Logical Networks

This note investigates the disturbance decoupling problems (DDPs) of mix-valued logical control networks (MLCNs) via semi-tensor product method. By using prime factor decomposition, a unique Y-friendly subspace of mix-valued logical networks (MLNs) is derived. A necessary and sufficient condition is obtained for the solvability of DDPs. An algorithm is proposed to find all disturbance decoupling controllers. This approach is more operational than the existing results in which the Y-friendly subspace is not unique. Finally, an illustrative numerical example is given to show the effectiveness of the proposed method.

On solutions of the matrix equation AX=B with respect to semi-tensor product

Abstract This paper studies the solutions of the matrix equation AX = B with respect to semi-tensor product. Firstly, the matrix–vector equation AX = B with semi-tensor product is discussed. Compatible conditions are established for the matrices, and a necessary and sufficient condition for the solvability of the matrix–vector equation is proposed. In addition, concrete solving methods are provided. Based on this, the solvability of the matrix equation AX = B with semi-tensor product is studied, and several examples are presented to illustrate the efficiency of the results.

Estimates of the spectral condition number

In this article, new upper and lower bounds for the spectral condition number are obtained. These bounds are constructed based on the Frobenius norm of some matrices related to the given matrix and its inverse. Hence, unlike some existing bounds, these new bounds are smooth functions with respect to the elements in the matrix. It is theoretically established that the new bounds are also sandwiched by the true value of the spectral condition number and its estimates using the Frobenius norms. Moreover, the bounds give the exact value of the spectral condition number when the matrix is unitary or of order less than 3. The new upper bound provided, via statistical numerical comparison, is shown to be the best when compared with existing results.

Comments on “Disturbance Decoupling of Boolean Control Networks”

A systematic method of disturbance decoupling for Boolean control networks has been developed in paper . However, the purpose of this note is to show that the necessary and sufficient condition for the existence of disturbance decoupling controllers needs a mild modification, which is provided.

Consensus and r-consensus problems for singular systems

In this paper, the problem of consensus for continuous time singular systems of multi-agent networks is considered. The definition of r-consensus is introduced for singular systems of multi-agent networks. Firstly, linear systems with algebraic constraints are considered, and the corresponding results about consensus and average-consensus are derived. Then r-consensus and consensus problems of singular systems are investigated. Sufficient conditions of r-consensus and consensus are obtained, respectively. Finally, an illustrative example is given to show the effectiveness of the proposed method.