Hierarchical subspace identification of directed acyclic graphs

Abstract

In this paper, a hierarchical PO-MOESP subspace identification algorithm for directed acyclic graphs (DAGs) is presented. The state of every node and the structure of the graph are assumed to be unknown. The method computes a hierarchical partition of the DAG by using projection matrices typically used in subspace identification methods applied in cascade. Then, the column space of the observability matrix of every node is sequentially estimated by projecting away the input-output data from the previous levels. A past input–output instrumental variable approach is adopted to deal with the noise. The topology of multilevel DAGs is revealed by dedicated projections applied on every level. Moreover, we provide a brief study of a more general class of DAGs that can be accurately represented by multilevel DAGs of reduced interconnection structure. Finally, three simulation examples are provided to show the effectiveness of the proposed methodology.

Keywords:

• subspace-identification
• PO-MOESP
• directed acyclic graphs
• RQ factorisation
• structure detection

Notes

1. In this case we assume that G = (V, E) does not consider self interconnections (cycles). This means that the set E does not contain the pair i → i.

2. Without loss of generality we assume that the dimensions of the signals for every node are the same, however, all the results can be extended for vector signals of different sizes.

3. In this case the matrices of the form , Φ^{(1, 2)}_s, W^{(1, 2)}_{s, 2}, P^{(1, 2)}_{s, 2} are combinations of the parameters of the system and will not be defined explicitly for brevity.

4. The block matrices F^{(q_m, q_ℓ)}_s, P^{(q_m, q_ℓ)}_s and W^{(q_m, q_ℓ)}_s contain combinations of the parameters of the system.

5. We also require that the matrix is full rank. This condition can be derived similarly as done in Verhaegen and Verdult (2007).

6. In this case, the matrix depends on the parameters of the system and depends on both the parameters of the system and input–output data.

Additional information

Funding

This research is supported by the NWO Veni [grant number 11930] “Reconfigurable Floating Wind Farms” and by CONICYT National Committee for Science and Technology, Government of Chile.