Cascade structural model approximation of identified state space models

General black-box system identification techniques such as subspace system identification and FIR/ARX least squares system identification are commonly used to identify multi-input multi-output models from experimental data. However, in many applications there are a priori given structural information. Here the focus is on linear dynamical systems with a cascade structure, and with one input signal and two output signals. Models of such systems are important in e.g. cascade control applications. It is possible to incorporate such a structure in a prediction error method, which, however, is based on rather advanced numerical non-convex optimization techniques to calculate the corresponding structured model estimate. We will instead study how to use model approximation techniques to approximate a general black-box estimate with a structured model. This will avoid the use of numerical optimization and works well with e.g. subspace system identification, which is a standard method in process industry where cascade systems are very common. The problems of cascade structural model approximation and model reduction are rather non-standard, and we will study several new methods. The basic idea is to first find a higher order but structured model approximation using standard H_¿ model matching techniques, and then in a second step use so-called structured balanced model reduction to find lower order structured approximation. Structured balanced model reduction is a rather new approach, with powerful model order selection tools and error bound results. The results of the corresponding two step model approximation approach seem promising, as illustrated by a simple numerical example.