Understanding Stochastic Subspace Identification

The data driven Stochastic Subspace Identification techniques is considered to be the most powerful class of the known identification techniques for natural input modal analysis in the time domain. However, the techniques involves several steps of “mysterious mathematics” that is difficult to follow and to understand for people with a classical background in structural dynamics. Also the connection to the classical correlation driven time domain techniques is not well established. The purpose of this paper is to explain the different steps in the SSI techniques of importance for modal identification and to show that most of the elements in the identification techniques have simple counterparts in the classical time domain techniques. Introduction Stochastic Subspace Identification (SSI) modal estimation algorithms have been around for more than a decade by now. The real break-through of the SSI algorithms happened in 1996 with the publishing of the book by van Overschee and De Moor [1]. A set of MATLAB files were distributed along with this book and the readers could easily convince themselves that the SSI algorithms really were a strong and efficient tool for natural input modal analysis. Because of the immediate acceptance of the effectiveness of the algorithms the mathematical framework described in the book where accepted as a de facto standard for SSI algorithms. However, the mathematical framework is not going well together with normal engineering understanding. The reason is that the framework is covering both deterministic as well as stochastic estimation algorithms. To establish this kind of general framework more general mathematical concepts has to be introduced. Many mechanical engineers have not been trained to address problems with unknown loads enabling them to get used to concepts of stochastic theory, while many civil engineers have been trained to do so to be able to deal with natural loads like wind, waves and traffic, but on the other hand, civil engineers are not used to deterministic thinking. The book of van Overschee and De Moor [1] embraces both engineering worlds and as a result the general formulation presents a mathematics that is difficult to digest for both engineering traditions. It is the view point of the present authors, that going back to a more traditional basis of understanding for addressing the response of structural systems due to natural input (ambient loading) makes things more easy to understand. In this paper, we will look at the SSI technique from a civil engineering (stochastic) point of view. We will present the most fundamental steps of the SSI algorithms based on the use of stochastic theory for Gaussian distributed stochastic processes, where everything is completely described by the correlation functions in time domain or by the spectral densities in frequency domain. Most modal people still like to think about vibrations in continuous time, and thus the discrete time formulations used in SSI are not generally accepted. Therefore a short introduction is given to discrete time models and it is shown how simple it is to introduce the description of free responses in discrete time. In the SSI technique it seem mysterious to many people why the response data is gathered together in a Block Hankel matrix, orders of magnitude larger than the original amount of data. Therefore the structure of the Block Hankel matrix is related to traditional covariance estimation, and it is shown how the subsequent so-called Projection of this Hankel matrix onto itself can be explained in terms of covariances and thus results in a set of free responses for the system. Then finally it is explained how the physics can be estimated by performing a singular value decomposition of the projection matrix. It is avoided to get into discussions about how to estimate the statistical part of the general model. Normally when introduced to the SSI technique, people will start looking at the innovation state space formulation involving mysterious Kalman states and a Kalman gain matrix that has nothing to do with the physics. This makes most engineers with a normal background in dynamics fall of the train. In this formulation, the general model is bypassed, however the mysterious Kalman states are introduced and explained as the states for the free responses estimated by the projection. Thus, this is an invitation to the people that were disappointed in the first place to get back on the track, take a another ride with the SSI train to discover that most of what you will see you can recognize as generalized procedures well established in classical modal analysis. The discrete time formulation We consider the stochastic response from a system as a function of time