J.W. van der Woude

On the structure at infinity of a structured system

Abstract We develop a graph-theoretic characterization of the generic structure at infinity of the transfer matrix of a structured system. We show that the generic structure at infinity can be determined by means of algorithms from combinatorial optimization based on the max-flow min-cut theorem, and on results concerning minimal-cost flows. As an application of the obtained characterization, we propose a structural version of two well-known disturbance decoupling problems, and we derive graph-theoretic necessary and sufficient conditions for the solvability of each of the two problems.

A graph theoretic characterization for the rank of the transfer matrix of a structured system

In this paper structured systems are considered and the generic rank of the transfer matrix of such systems is introduced. It is shown that this rank equals the maximum number of vertex disjoint paths from the input vertices to the output vertices in the graph that can be associated to the structured system. This maximum number of disjoint paths can be calculated using techniques from combinatorics. As an application a structural version of the well-known almost disturbance decoupling problem is proposed.

Almost non-interacting control by measurement feedback

Consider a linear system Σ that, apart from a control input and a measurement output, has two exogeneous inputs and two exogenous outputs. Controlling such a system by means of a measurement feedback compensator Σc results in a closed loop system with two inputs and two outputs. Hence, the closed loop transfer matrix can be partitioned as a two by two block matrix. The problem addressed in this paper consists of the following. Given Σ and any positive number ge, is it possible to find Σc such that the off-diagonal blocks of the closed transfer matrix, in a suitable norm, are smaller than ϵ? For the solvability of this problem necessary and sufficient conditons will be derived.

On the stability of the pulsewidth-modulated C/spl acute/uk converter

It has been observed that a stabilizing controller for the state-space-average model of a pulsewidth-modulated (PWM) DC-DC converter does not automatically imply a stabilizing controller for the DC-DC converter itself, especially in the case of the C/spl acute/uk converter. By applying multifrequency averaging as a generalization of the state-space averaging technique, the stationary periodic signals of the C/spl acute/uk converter can be obtained by solving a set of linear equations. Using these equations, we are able to analyze stability aspects of the open-loop and closed-loop PWM C/spl acute/uk converter. Simulations are included in open- and closed-loop situations.

On the periodic behavior of PWM DC-DC converters

In this paper, we present a method to compute the periodic behavior of PWM DC-DC converters. The method is based on Fourier series and can be seen as a generalization of the state space averaging techniques that are well-known in the field of DC-DC converters. The method of this paper can be used to compute the periodic behavior of a DC-DC converter up to any degree of accuracy. This can be done in open loop as well as in closed loop situations. In an open loop situation the method only requires the solving a set of linear equations in the Fourier coefficients. It is shown that the periodic behavior in an open loop situation always is stable. In a closed loop situation the method requires the solving of a set of equations consisting of the previously mentioned set of linear equations in the Fourier coefficients, combined with an additional nonlinear equation in the relative pulse width (duty ratio). In both situations the method of this paper can be implemented in an efficient way. The method is illustrated by means of known examples taken from the literature and is shown to be useful in establishing whether or not a periodic behavior in a closed loop situation is stable.

A note on pole placement by static output feedback for single-input systems

Abstract In this note we derive necessary and sufficient conditions for the solvability of the problem of pole placement by static output feedback formulated for single-input systems.

A New Simultaneous Decomposition of a Matrix Quaternity Over an Arbitrary Division Ring with Applications

In this article, we give a new simultaneous decomposition concerning the general matrix quaternity over an arbitrary division ring ℱ. A practical algorithm for the decomposition of the matrix quaternity is also presented. As applications, we establish some necessary and sufficient conditions for the existence of the general solutions to the systems of matrix equations and over ℱ. In addition, we give the expressions of the general solutions to the systems when the solvability conditions are satisfied. Numerical examples are also given to illustrate the results of this article. Moreover, we mention that the findings of this article extend the some known results in the literature.

On the existence of a common solution X to the matrix equations AiXBj=Cij, (i,j)∈Γ

In this paper conditions are derived for the existence of a common solution X to the matrix equations AiXBj=Cij,(i,j)∈Γ, where the matrices Ai,Bj,Cij and X have suitable dimensions and the (i,j)’s are index pairs in some set Γ. The purpose of this paper is to present, for certain specific sets of index pairs Γ, verifiable necessary and sufficient solvability conditions that are stated directly in terms of the matrices and that do not use Kronecker products.

Disturbance decoupling by measurement feedback for structured transfer matrix systems

Structured transfer matrix systems are linear systems given by transfer matrices of which the infinite zero order of each nonzero entry is known, while the associated infinite gains are unknown and assumed mutually independent. In this paper necessary and sufficient conditions are derived for the generic solvability of the well-known disturbance decoupling problem by measurement feedback for structured transfer matrix systems. Generic solvability here means solvability for almost all possible values for the infinite gains of the nonzero transfer matrix entries. The conditions will be stated in terms of a bipartite graph that can be easily associated with a structured transfer matrix system. The advantage of this is that the conditions then can be verified by means of graph theoretic algorithms.

Zero Orders and Dimensions of Some Invariant Subspaces in Linear Structured Systems

AbstractIn this paper we consider linear structured systems which represent a large class of parameter-dependent linear systems and we study invariants for such systems under a large group of transformations including state feedback. In this context we consider the dimension of the maximal output-nulling invariant subspace of a linear structured system, the number and structure of its invariant zeros and its structure at infinity. We give generic characterizations of the invariants in terms of properties of the directed graph that can be naturally associated with a linear structured system.