In this paper we study linear structured systems described by means of system matrices of which only the zero/non-zero structure is known and where the non-zeros are supposed to have independent values. The structure of linear structured systems can be represented by means of various types of graphs, like directed graphs or dynamic graphs. Here we use both type of graphs because they enable us to formulate and study certain controllability properties in a uniform and straightforward way. In this paper we extend the results of a previous paper containing a partial characterisation of the fixed part of the controllable subspace of linear structured systems. This fixed part is defined as the part of the controllable subspace that is independent of the values to the non-zeros, and therefore can be seen as the robust part of the controllable subspace. It turns out that, by considering the generic dimension of the controllable subspace, a characterisation of the fixed part can be obtained. The latter dimension equals the size of the minimal set of nodes in the dynamic graph that separates between the set of input nodes and the set of final state nodes. Computing the supremal of such minimal separating sets, we are capable of characterising the fixed part. In the paper we indicate how this supremal minimal separating set can be obtained insightfully and efficiently using the recursive nature of the dynamic graph. Our results are illustrated by some meaningful examples.@en

This paper addresses difference flatness for structured LTI discrete-time systems. Two forms of necessary and sufficient conditions for an output to be a structural flat output are given. First, a preliminary result algebraically defines a flat output in terms of invariant zeros regardless whether an LTI system is structured or not. Next, the conditions are expressed in terms of graphical conditions to define a structural flat output. Checking for the graphical conditions calls for algorithms that have polynomial-time complexity and that are commonly used for digraphs. The tractability of the conditions is illustrated on several examples.@en

In this paper we study the controllability of interconnected networks that are described by means of structured linear systems with state-like and control variables. We assume that the systems operate in discrete time with the set of integers as the time axis. Further, we assume that the state-like variables for their evolution only depend on recent values of their neighbours with, however, unknown weight factors. These recent values may be one step back in time, but also more steps. This yields a description of the systems by means of matrices containing fixed zeros and free parameters, together with a time lag structure. Knowing the dependency and lag structure, we represent (the structure of the) systems by means of weighted directed graphs and study questions concerning their structural controllability, where the latter has to be defined in an appropriate way, i.e., in behavioural sense. We provide a necessary and sufficient characterization of structural controllability of our systems using a graph representation. The obtained characterization makes use of well-known and efficient algorithms from graph theory. We prove that in this context finding the minimal number of driver (controller) nodes is an NP-hard problem. The concepts and results of the paper are illustrated on academic examples and on a gene regulatory network.@en

In this paper we consider interconnected networks that are described by means of structured linear systems with state and control variables. We represent these systems, whose matrices contain fixed zeros and free parameters, by means of directed graphs and study questions concerning controllability and the controllable subspace. We show in this paper that the controllable subspace can have a part that will be present for almost all values of the free parameters. It actually is a subspace of the controllable subspace and will be referred to as the fixed controllable subspace. The subspace can then be seen as a kind of robustly controllable part of the system. Indeed, it is a subspace in the state space with the generic property that states in it can be steered in an arbitrary way. We derive a characterization of the fixed controllable subspace using the graph representation. The obtained characterization makes use of well-known algorithms from optimization and networks theory. To get some more insight in the components in the fixed part, we also give a representation of the structured linear systems by means of bipartite graphs. Using the Dulmage–Mendelsohn decomposition, we are able to decompose our structured systems in such a way that in some special cases, the fixed controllable subspace can be obtained directly from the decomposition.@en