Design of sensor networks for instantaneous inversion of modally reduced order models in structural dynamics
Introduction
The dynamic forces acting on a structure and the corresponding system states are of great importance to many engineering applications. Often, however, the dynamic forces and resulting system states can hardly be obtained by direct measurements, e.g. for wind loads, and have to be determined indirectly from dynamic measurements of the system response using system inversion techniques.
Originally, force identification and state estimation problems were treated separately. Force identification problems were initially solved off-line in a deterministic setting. Many methods were proposed, most of them based on the inversion of the frequency response function [1], [2], [3] or making use of a time domain approach [4], [5], [6], [7]. Several state estimation algorithms have been proposed for linear as well as for non-linear systems [8], [9], [10], [11]. A recursive deterministic method was presented by Klinkov and Fritzen [12], estimating both the input and system states from a set of output measurements. Currently, the attention is shifted to the development of recursive combined deterministic-stochastic approaches [13], [14]. These methods do not only account for measurement errors, but also for modeling errors and additional unknown vibration sources. An algorithm was proposed by Gillijns and De Moor, where the input estimation is performed prior to the state estimation step [15]. The algorithm was introduced in structural dynamics by Lourens et al. [16], extending the algorithm for use with reduced-order models. A similar approach was proposed by Niu et al. [17]. Alternatively, the dynamic forces and system states can be jointly estimated using a classical Kalman filter, hereby including the unknown forces in an augmented state vector [18].
This paper focuses on instantaneous system inversion, i.e. inversion without any time delay, covering the majority of inversion algorithms applied in structural dynamics. The invertibility of a system in general depends on three conditions. Firstly, the dynamic forces and/or the corresponding states must be identifiable from the given set of response measurements. Secondly, the system inversion algorithm must be stable, such that small perturbations of the data do not give rise to unbounded errors on the identified quantities. Thirdly, the estimates obtained must be uniquely defined by the measurement data. In the literature, the main requirements on the system description for instantaneous invertibility are extensively documented for the general case of linear systems [19], [20], [21]. For the specific case of linear modally reduced order models, which are often used in structural dynamics, the general conditions can be directly translated into a number of requirements on the sensor network, i.e. sensor types, sensor locations, and number of sensors.
The outline of this paper is as follows. In Section 2, the problem of system inversion is outlined. Next, in 3 Identifiability conditions, 4 Stability conditions, 5 Uniqueness conditions, the requirements on the sensor network are derived, starting from the general conditions for system invertibility, as given in the literature. Section 6 discusses the practical implementation of the requirements for a case study, where a sensor network for a footbridge is designed that allows for the identification of multiple forces on the bridge deck. Finally, in Section 7, the work is summarized.
Section snippets
Problem formulation
In structural dynamics, first principles models, e.g. finite element (FE) models, are widely used. In many cases, modally reduced order models are applied, constructed from a limited number of structural modes. When proportional damping is assumed, the continuous-time decoupled equations of motion for modally reduced order models are given bywhere is the vector of modal coordinates, with being the number of modes taken into account in the model. The
Identifiability conditions
System identifiability requires that the measured output contains information on the quantities that are estimated, i.e. system inputs and/or system states. This condition can be subdivided into two separate requirements, the observability requirement and the direct invertibility requirement, which are both discussed next.
Stability conditions
Stability of system inversion is mainly an issue for time domain inversion algorithms. The stability of the system inversion depends on the poles of the inverse system and therefore on the transmission zeros of the original system [21].
Consider the system described by Eqs. (2), (5). A number is called a finite transmission zero of the system if [26]If λj is a finite transmission zero of the system, there exist vectors and such that
Uniqueness conditions
The uniqueness of the identified forces and/or system states depends on the system transmission zeros, as stated in the following corollary. Corollary 5.1 The input of a system with at least one finite transmission zero cannot be uniquely reconstructed.
This follows directly from the definition of a system transmission zero: there exists an input and an initial state for which the system output is zero (Eq. (21)). As a consequence, the input cannot be uniquely reconstructed from the measured output. When the
Implementation for a footbridge model
In this section, the conditions as derived above are implemented in the design of a sensor network for a footbridge that allows for the identification of multiple forces on the bridge deck. The footbridge, depicted in Fig. 5, is located in Ninove (Belgium) where it crosses the Dender river. It is a two-span cable-stayed steel bridge with a main and secondary span of 36 m and 22.5 m, respectively.
Conclusions
In this paper, general conditions for the invertibility of linear system models have been presented for the specific case of modally reduced order models. The conditions apply to any instantaneous input estimation or joint input-state estimation algorithm and ensure the identifiability, stability, and uniqueness of the identified quantities. It is shown that the general invertibility conditions can be reformulated in terms of the modal characteristics of the structure. The practical
Acknowledgments
The research presented in this paper has been performed within the framework of the project G.0738.11 “Inverse identification of wind loads on structures”, funded by the Research Foundation Flanders (FWO), Belgium. Their financial support is gratefully acknowledged. The authors are all members of the KU Leuven – BOF PFV/10/002 OPTEC – Optimization in Engineering Center. E. Reynders is a Postdoctoral Fellow of the Research Foundation Flanders (FWO), Flanders.
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