10. Robust Identification of Stable MIMO Modal State Space Models

As systems become more and more complex, representing them with differential equations or transfer functions becomes cumbersome and even more so if the number of inputs and outputs are growing. Using a state space model representation largely alleviates this problem as it provides a convenient and compact way to model and analyze MIMO systems.

In various engineering applications, e.g., structural coupling, virtual sensing, time-domain simulation, and control, state space models have been proven to be a useful and practical system representation. Different “direct” state space model identification algorithms exist in literature, e.g., prediction error methods and time-domain/frequency-domain subspace identification methods. Generally, subspace identification algorithms consist of geometrical projections to estimate the states of the unknown system or to estimate the extended observability matrix from which the state matrices are retrieved. In such methods, the model order is traditionally determined by inspecting and truncating the singular values of the projection (hence the name “subspace” methods). When applied to real, sometimes noisy, measurement data, it is not obvious to decide on the model order. Another issue with the subspace identification techniques and in particular with the frequency-domain variants is that the stability of the poles is not always guaranteed.

In this paper, a modal-based approach to derive a robust and stable state space model starting from a set of measured frequency response functions (FRFs) will be presented. The approach makes use of the Polymax modal parameter estimator to derive a modal model which can be further improved by the iterative MLMM algorithm when needed. The clear stabilization diagram obtained using Polymax can be used as a valid tool to determine the order of the system under test. The obtained modal model is then transformed to an equivalent state space model. The lower and the upper residual terms, which are used to model the out-of-band modes in the modal model, are transformed to the so-called residual compensation modes allowing a straightforward conversion from a modal model to a state space model. Since the modal model contains only stable poles, the stability of the identified state space model is guaranteed. The presented approach will be validated using industrial datasets and compared with the well-known “N4SID” implementations of time- and frequency-domain subspace identification methods.