Jacob van der Woude

Survey Generic properties and control of linear structured systems: a survey

In this survey paper, we consider linear structured systems in state space form, where a linear system is structured when each entry of its matrices, like A,B,C and D, is either a fixed zero or a free parameter. The location of the fixed zeros in these matrices constitutes the structure of the system. Indeed a lot of man-made physical systems which admit a linear model are structured. A structured system is representative of a class of linear systems in the usual sense. It is of interest to investigate properties of structured systems which are true for almost any value of the free parameters, therefore also called generic properties. Interestingly, a lot of classical properties of linear systems can be studied in terms of genericity. Moreover, these generic properties can, in general, be checked by means of directed graphs that can be associated to a structured system in a natural way. We review here a number of results concerning generic properties of structured systems expressed in graph theoretic terms. By properties we mean here system-specific properties like controllability, the finite and infinite zero structure, and so on, as well as, solvability issues of certain classical control problems like disturbance rejection, input-output decoupling, and so on. In this paper, we do not try to be exhaustive but instead, by a selection of results, we would like to motivate the reader to appreciate what we consider as a wonderful modelling and analysis tool. We emphasize the fact that this modelling technique allows us to get a number of important results based on poor information on the system only. Moreover, the graph theoretic conditions are intuitive and are easy to check by hand for small systems and by means of well-known polynomially bounded combinatorial techniques for larger systems.

Conditions for the structural existence of an eigenvalue of a bipartite (min, max, +)-system

In this paper we consider bipartite (min, max, +)-systems. We present conditions for the structural existence of an eigenvalue and corresponding eigenvector for such systems, where both the eigenvalue and eigenvector are supposed to be finite. The conditions are stated in terms of the system matrices that describe a bipartite (min, max, +)-system. Structural in the previous means that not so much the numerical values of the finite entries in the system matrices are important, rather than their locations within these matrices. The conditions presented in this paper can be seen as an extension towards bipartite (min, max, +)-systems of known conditions for the structural existence of an eigenvalue of a (max, +)-system involving the (ir)reducibility of the associated system matrix. Although developed for bipartite (min, max, +)-systems, the conditions for the structural existence of an eigenvalue also can directly be applied to general (min, max, +)-systems when given in the so-called conjunctive or disjunctive normal form.

Power Algorithms for (\max,+) - and Bipartite (\min,\max,+) -Systems

In this paper we consider(max,+)-systems and bipartite (min,max,+)-systems.We present so-called power algorithms that under some mild conditionson the structure of the systems determine eigenvalues and correspondingeigenvectors in an iterative way. We present simple proofs forour algorithms and we illustrate our algorithms by means of someexamples also clarifying the difference with existing power algorithms.

Generic Properties and Control of Linear Structured Systems

Abstract In this paper we deal with generic properties and control of linear systems taking into account prior structural knowledge available from physical considerations. We consider linear structured systems in state space form, where a linear system is structured when each entry of its matrices, like A, B, C and D , is either a fixed zero or a free parameter. The location of the fixed zeros in these matrices constitutes the structure of the system. Indeed, most physical systems which admit a linear model are structured. A structured system is representative of a class of linear systems in the usual sense. It is of interest to investigate properties of structured systems which are true for almost any value of the free parameters, therefore also called generic properties. Interestingly a lot of classical properties of linear systems can be studied in terms of genericity . Moreover, these generic properties can in general be checked by means of directed graphs that can be associated to a structured system in a natural way. We review here a number of results concerning generic properties of structured systems expressed in graph theoretic terms. By properties we mean here system specific properties like controllability, the finite and infinite zero structure, and so on, as well as, solvability issues of certain classical control problems like disturbance rejection, input-output decoupling, and so on. In this paper we do not try to be exhaustive but instead, by a selection of results, we would like to motivate the reader to appreciate what we consider a wonderful modeling and analysis tool. We emphasize the fact that this modeling technique allows us to get a number of important results based on poor information on the system only. Moreover the graph theoretic conditions are intuitive and are easy to check by hand for small systems, and by means of well-known combinatorial techniques for larger systems.

A Characterization of the Eigenvalue of a General (Min,Max, +)-System

In this paperwe consider general (min,max,+)-systems for whichthe eigenvalue exists. The major contribution of this paper isa characterization of the eigenvalue of such systems in termsof the convergence of a specific iteration. Conditions for theeigenvalue to exist are also given.In this paper we consider general (min,max,+)-systems for which the eigenvalue exists. The major contribution of this paper is a characterization of the eigenvalue of such systems in terms of the convergence of a specific iteration. Conditions for the eigenvalue to exist are also given.

Currents' Physical Components (CPC) in the time-domain: Single-phase systems

This paper presents a time-domain derivation of the Currents Physical Components (CPC) power theory using standard results from mathematical systems theory. Consequently, piece-wise continuous currents and voltages, such as square waves and sawtooth signals, can naturally be taken into account without approximating them by a finite number of harmonics. The results are illustrated using two electrical circuits that have been used in the literature to reveal the need for a proper power decomposition.

Asymptotic growth rate of stochastic max-plus systems that with a positive probability have a sunflower-like support

In this paper the asymptotic growth rate of stochastic max-plus linear systems is studied. Special attention is paid to systems whose system matrix with a positive probability is supported by a basic sunflower graph, i.e., a graph that contains precisely one circuit, which has length one (a self-loop), and in which each node has precisely one predecessor. It is shown that for such systems all state components have the same asymptotic growth rate. The result is illustrated by means of an example. Also two generalizations will be briefly presented

Stability aspects of a multifrequency model of a PWM converter

In this paper the stability of a multifrequency model of a PWM converter is investigated. A multifrequency model is a model based on Fourier series that contains as a special case the so-called state space average model. In contrast to a state space average model a multifrequency model may also include so-called higher order harmonics, where the zeroth order harmonic corresponds to the (moving) average. This paper focuses on a specific PWM converter, namely a ( uk converter, and it is proved that a multifrequency model of a ( uk converter with fixed duty ratio is asymptotically stable. This result generalizes the known corresponding result for a state space average model of a ( uk converter with fixed duty ratio. Taking all the harmonics into account the result also illustrates the well-known fact that a ( uk converter with a fixed duty ratio and a finite switching frequency is asymptotically stable in the following sense. If the signals in a ( uk converter do not correspond with a periodic behaviour, they will however do so in the limit, i.e. as time goes to infinity the signals will become periodic, and this limiting periodic behaviour is unique. Although the paper mainly deals with the stability issues for a ( uk converter, it is possible to use the ideas of the paper to derive similar results for other types of PWM converters.

An ergodic theorem for stochastic max-plus linear systems

Abstract In this paper the asymptotic growth rate of stochastic max-plus linear systems is studied. For non-random irreducible max-plus linear systems the existence of this growth rate is well known. For stochastic systems conditions for the existence of the growth rate will be developed. The conditions are mild and natural, and weaker than the conditions used in literature until now. The interpretation of the conditions is that for the asymptotic growth rate to exist the stochastic max-plus linear system should be dominated from below by a non-random irreducible max-plus linear system. The latter condition need not to be true always, but should at least hold with some positive probability. The investigations in this paper are motivated by means of some examples.

A simplex-like method to compute the eigenvalue of an irreducible (max,+)-system

Abstract In this paper, we present an alternative method to compute the eigenvalue of an irreducible ( max ,+) -system. The method resembles the well-known simplex method in linear programming in the sense that the eigenvalue and a corresponding eigenvector are obtained by going along the boundary of a polygon-like set, while increasing the number of equalities in some ( max ,+) -algebraic eigenvalue–eigenvector expression, until only equalities are left over. The latter is unlike the normal linear programming approach where, going along the boundary of a polygon-like set, a linear functional is optimized.