TY - JOUR

T1 - A dynamic graph characterisation of the fixed part of the controllable subspace of a linear structured system

AU - van der Woude, Jacob

AU - Commault, Christian

AU - Boukhobza, Taha

N1 - Accepted author manuscript

PY - 2019

Y1 - 2019

N2 - 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.

AB - 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.

KW - Controllable subspace

KW - Graph theory

KW - Linear structured systems

KW - Maximal linkings

KW - Minimal separators

KW - Robust part

UR - http://www.scopus.com/inward/record.url?scp=85066405929&partnerID=8YFLogxK

U2 - 10.1016/j.sysconle.2019.05.002

DO - 10.1016/j.sysconle.2019.05.002

M3 - Article

AN - SCOPUS:85066405929

VL - 129

SP - 17

EP - 25

JO - Systems & Control Letters

JF - Systems & Control Letters

SN - 0167-6911

ER -