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This paper introduces structural identification using expectation maximization (STRIDE), a novel application of the expectation maximization (EM) algorithm and approach for output-only modal identification. The EM algorithm can be used to estimate the maximum likelihood parameters of a state-space model. In this context, the state-space model represents the equation of motion for a linear dynamic system. STRIDE is an iterative procedure that uses Kalman filtering and Rauch-Tung-Striebel (RTS) smoothing equations to produce estimates of the unobserved states; these calculations are based on the observed data and prior estimates of the state-space parameters. With this information, the conditional likelihood of the model is maximized and the state-space parameters are updated at each iteration. Once an iteration meets user-prescribed convergence criterion, the algorithm ends-yielding maximum likelihood estimates (MLE) for the state-space model parameters. The modal properties of the structure are then extracted from these MLE. The performance of STRIDE is compared in detail with eigenvalue realization algorithm-natural excitation technique (ERA-NExT) and eigenvalue realization algorithm-observer Kalman filter identification of output-only systems (ERA-OKID-OO) identification algorithms in the analyses of ambient vibration data from the Northampton Street Bridge and Golden Gate Bridge, both collected using a dense wireless sensor network. A computational comparison shows that STRIDE provides a successful identification at a significantly lower model order than ERA-NExT, ERA-OKID-OO, or auto-regressive (AR), simultaneously requiring fewer cumulative floating point operations than ERA-OKID-OO in both applications.

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There are many occasions in structural health monitoring (SHM) on which collected data sets contain missing observations. Such instances may occur as a result of failed communications or packet losses in a wireless sensor network or as a result of sensing and sampling methods-for example, mobile sensing. By implementing modified expectation and maximization steps, structural identification using expectation maximization (STRIDE) is capable of processing data in these circumstances and is the first modal identification technique to formally accept data with missing observations. This paper presents the STRIDE algorithm, a statistical perspective of missing data, and new STRIDE equations that account for missing observations. Expectation step (E-step) equations are given explicitly for both partially observed time steps and those not fully observed. The maximization step (M-step) provides state-space parameter updates in terms of available observations and missing-data state-variable statistics. This paper also discusses the performance and convergence behavior of STRIDE with missing data. Finally, two applications are presented to exemplify common use in network reliability and mobile sensing, both using data collected at the Golden Gate Bridge. This paper demonstrates that sensor network data containing a significant amount of missing observations can be used to achieve a comprehensive modal identification. A successful real-world identification with simulated mobile sensors quantifies the preservation of spatial information, establishing the benefits of this type of network and emphasizing a line of inquiry for future SHM implementations.

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This article uses the formulation of the structural identification using expectation maximization (STRIDE) algorithm for compatibility with the trun-

Historically, structural health monitoring (SHM) has relied on fixed sensors, which remain at specific locations in a structural system throughout data collection. This paper introduces state-space approaches for processing data from sensor networks with time-variant configurations, for which a novel truncated physical model (TPM) is proposed. The state-space model is a popular representation of the second-order equation of motion for a multidegree of freedom (MDOF) system in first-order matrix form based on field measurements and system states. In this mathematical model, a spatially dense observation space on the physical structure dictates an equivalently large modeling space, i.e., more total sensing nodes require a more complex dynamic model. Furthermore, such sensing nodes are expected to coincide with state variable DOF. Thus, the model complexity of the underlying dynamic linear model depends on the spatial resolution of the sensors during data acquisition. As sensor network technologies evolve and with increased use of innovative sensing techniques in practice, it is desirable to decouple the size of the dynamic system model from the spatial grid applied through measurement. This paper defines a new data class called dynamic sensor network (DSN) data, for efficiently storing sensor measurements from a very dense spatial grid (very many sensing nodes). Three exact mathematical models are developed to relate observed DSN data to the underlying structural system. Candidate models are compared from a computational perspective and a truncated physical model (TPM) is presented as an efficient technique to process DSN data while reducing the size of the state variable. The role of basis functions in the approximation of mode shape regression is also established. Two examples are provided to demonstrate new applications of DSN that would otherwise be computationally prohibitive: high-resolution mobile sensing and BIGDATA processing.