A dual Kalman filter approach for state estimation via output-only acceleration measurements

https://doi.org/10.1016/j.ymssp.2015.02.001Get rights and content

Highlights

  • A novel dual Kalman filter is proposed for output-only state and input estimation.

  • Sparse pure acceleration measurements and a model of the structure are used.

  • The unobservability issues attributed to augmented Kalman filter are resolved.

  • The low frequency drift is properly tackled in an online and autonomous fashion.

  • The efficiency of the proposed method is verified by numerical simulations.

Abstract

A dual implementation of the Kalman filter is proposed for estimating the unknown input and states of a linear state-space model by using sparse noisy acceleration measurements. The successive structure of the suggested filter prevents numerical issues attributed to un-observability and rank deficiency of the augmented formulation of the problem. Furthermore, it is shown that the proposed methodology furnishes a tool to avoid the so-called drift in the estimated input and displacements commonly encountered by existing joint input and state estimation filters. It is shown that, by fine-tuning the regulatory parameters of the proposed technique, reasonable estimates of displacements and velocities of structures can be accomplished.

Introduction

This paper contributes to the problem of state estimation in the entire body of the metallic structures that undergo vibrations due to unknown input forces during their operational life, aiming at prediction of fatigue damage identification. The idea of using the estimated response of the structures for fatigue damage identification was first suggested by Papadimitriou et al. [1]; where a technique was introduced that uses the Kalman filter for estimating power spectral densities of the strain in the body of the structure thereby predicting the remaining fatigue life. To estimate the fatigue damage, a time history of the strains in the hotspot points of the structure is required. To estimate the strain in a point of interest, the displacement field around that point is needed; therefore, a reliable state estimate could lead to a reliable fatigue damage identification.

The subject of estimation of the states of a partially observed dynamic system in an stochastic frame has been studied by many scientists and there are well developed algorithms to manage both linear (e.g. the Kalman filter [2]) and nonlinear (e.g. the particle filter [3], the unscented Kalman filter [4]) state-space models. Dealing with structural systems, the states of the system are displacements and velocities of the response of the system at some points, namely degrees-of-freedom (DOF) on the structure. In practical cases, it is difficult or sometimes impossible to measure displacements and velocities of the system, hence when a knowledge of the displacements and velocities is required, a state estimation algorithm could be used to provide estimates of the whole state of the system. The Bayesian filters that exist in the literature, take advantage of the correlation between the observable part of the state of the system and the hidden part, and furnish an estimate of the whole state with the associated uncertainty of it. Within the context of the structural dynamics, over the continuous time the equations of motion, namely the dynamic model of the system define an intrinsic correlation between the states of the system. Dealing with discrete models, the kinematic relation between the accelerations, velocities and displacements of the system enters the dynamic model of the system and together with kinetics shape the discrete state-space model.

Within such a context, Ching and Beck [5] estimated the unknown states of a structure using incomplete output data from a structure excited by uncertain dynamic loading, to estimate the likelihood of any particular unobserved response of the structure exceeding a prescribed threshold. Hernandez [6] proposed an observer that possesses similar characteristics to the Kalman filter in the sense that it minimizes the trace of the state error covariance matrix. The main notion behind the algorithm is that the proposed observer can be implemented as a modified linear finite element model of the system, subject to collocated corrective forces proportional to the measured response. It has been used to estimate the number of threshold crossings in the bending moment history of a simulated tall vertical structure subject to turbulent wind load and fatigue damage [7]. The methodology has further been experimentally validated via a laboratory test [8]; where the measured stress at the locations of interest was compared to estimates obtained by well-established estimation methods such as Luenberger observers and the Kalman filter relying on the using a limited number of velocity measurements. Smyth and Wu [9] proposed a multi-rate Kalman filter for the online fusion of measured displacement and acceleration data sampled at different rates. The filter is designed to circumvent problems related to the integration of accelerometer or the differentiation of displacement data in situations where both these response quantities are collocated and available in different sampling rates. Gao and Lu [10] used the Kalman filter in combination with an ARX model for damage detection in linear structural systems where only accelerations at some degrees of freedom are measured. Reynders and De Roeck used the linear Kalman filter as part of the subspace identification technique for modal analysis [11] to directly estimate states from measured data without knowledge of the system model. The linear Kalman filter is also used by Bernal [12] in a damage detection scheme for linear systems based on the hypothesis test of whiteness of the innovations, taking advantage of the fact that correlations emerge when either the properties of the system or the process noise deviate from the values that the filter is formulated for. In very recent work, Bernal and Ussia [13] have proposed a sequential deconvolution method for input reconstruction in linear time invariant systems, and have presented a detailed study of the necessary conditions for identifying the input force as well as the stability conditions associated with a segmented implementation of the proposed method. Through theoretical derivations, it is shown that when the number of inputs is less than or equal to the number of observations the non-collocated inputs can be identified by performing deconvolution. Further, to facilitate continuous input reconstruction over time, a sequential implementation of the deconvolution method is proposed by authors. The method is an interesting alternative for the case where the aim is input reconstruction in the event where the input load locations are known and additionally the inputs to be reconstructed are fewer than the monitored outputs. In such a case one may solve for the inputs and then recast the problem into a standard Kalman filtering framework which would take care of the process and measurement noise involved, as well as the uncertain initial conditions. Instead, the method developed herein proposes a parallel carrying out of these tasks in one compact formulation.

For nonlinear state estimation and parameter identification in civil engineering, the extended Kalman filter (EKF) has been the de facto standard in the past mainly due to its ease of implementation, robustness and suitability for real-time applications. In recent years, however, many alternative techniques have been proposed. In a first extension for alleviating the issues that arise through linearization in the EKF, Julier and Uhlmann [4] have proposed the unscented Kalman filter (UKF), in which the evolution of the statistics of the state of the system is performed through a sampling scheme. It has been shown by Mariani and Ghisi [14] that at the price of a higher computational burden the UKF outperforms the EKF dealing with nonlinear parameter identification problems. To mitigate the issues pertinent to high computational costs, Eftekhar Azam et al. [15] proposed a parallel implementation of the UKF. Ching et al. [16] compared the performance of the EKF with that of the particle filter (PF) applied to identification of system matrices of a linear multi storey shear building. Chatzi and Smyth [17] and Eftekhar Azam et al. [18] compared the performance of the UKF and PF applied to identification of the parameters of nonlinear constitutive models. One of the advantages of the PF in comparison to the EKF is that it is applicable to highly nonlinear systems with non-Gaussian uncertainties. In turn, one major drawback of the particle filters is that when dealing with high dimensional state-space models, the computational burden of the generic particle filters increase exponentially. To alleviate this issue, recently techniques have been developed that improve the ensemble of the so-called particles. Chatzi and Smyth [19] have used evolutionary particle filters for structural health monitoring where, the use of the standard PF is combined with mutation operators to enhance the particles. Eftekhar Azam and Mariani [20] have used a hybrid extended Kalman particle filter for online damage detection of linear and nonlinear multi storey shear type buildings. In the abovementioned works, the process and observation noise covariances are always assumed as known parameters of the problem; however, in practice the covariances of the noise parameters should be appropriately estimated to ensure that an optimal prediction is furnished by the filters [21]. Moreover in practical implementations in online monitoring, the possible outliers contained in the observation process can have detrimental effects on the estimates including instability of the identified parameters. In treating this issue and for enabling online and continuous monitoring, Mu and Yuen have proposed a novel robust outlier resistant EKF for online parameter identification of linear time variant structural systems [22].

Although the joint state and parameter identification task is a subject frequently addressed in recent years, the joint identification of state and input information is a topic less treated so far in the literature. Structural systems are inherently characterized by uncertainty, relating to measurement errors, sensor noise, inefficacy of the numerical models and lack of a priori knowledge on the system and loading conditions. It this paper, the latter source of uncertainty, i.e. the lack of information regarding the input to the system is the core of the study. In practice, one common approach is to assume the unknown input as a zero mean white Gaussian process and make use of the aforementioned Bayesian techniques for state estimation; however, in many cases this assumption is violated and therefore it may lead to major adverse effects on the accuracy of the estimations. To address this issue, a number of optimal filtering techniques in the presence of unknown input have been proposed. In a pioneering work, Kitanidis developed an unbiased minimum-variance recursive filter for input and state estimation of linear systems without direct transmission; his algorithm did not make any a-priori assumption on the input [23]. The latter filter is not globally optimal in the mean square error sense. Hsieh has proposed a new formulation of the Kitanidis filter which is more convenient for practical applications [24]. Gillijns and De Moor proposed a new filter for joint input and state estimation for linear systems without direct transmission [25]. Their filter is globally optimal in the minimum-variance unbiased sense. Later Gillijns and De Moor developed a new formulation of the aforementioned filter which included a direct transmission term in its structure [26].

In more recent years, Lourens et al. [27] have proposed an extension of the method developed in [26] to cope with the numerical instabilities that arise when the number of sensors surpasses the order of the model, i.e. when a large number of sensors is used in combination with a reduced-order model assembled from a relatively small number of modes. The modified algorithm was used to predict and estimate the input force and accelerations of a simulated steel beam, a laboratory test beam and a large scale steel bridge. It was reported that, although the algorithm provides a reasonable prediction of the accelerations, the input force estimates are affected by spurious low frequency components that must be filtered out in this case. It is worth noting, that in dealing with the joint state and parameter estimation, Chatzi and Fuggini [28] have proposed a technique to cope with the issues related to the spurious low frequency components in the displacement estimates by introducing artificial displacement measurements into the observation vector. Lourens et al. [29] have proposed an augmented Kalman filter (AKF) for unknown force identification in structural systems, and concluded that the AKF is prone to numerical instabilities due to un-observability issues of the augmented system matrix.

In this paper, a dual implementation of the Kalman filter is proposed to estimate the unknown input and states of a linear state-space model; however, the input estimation itself is a secondary goal compared to state estimation, as the objective is to estimate the fatigue damage accumulation. It is assumed that a limited number of noisy acceleration measurements are available. The successive structure of the suggested filter prevents numerical problems attributed to un-observability and rank deficiency of the AKF. Additionally, it is shown that the expert guess on the covariance of the unknown input provides a tool for avoiding the so-called drift effect in the estimated input force and displacements. The drift is linked to the integral nature of these quantities in the presence of acceleration information. The effectiveness and performance of the proposed method is ascertained via numerical analysis carried out on a shear model of a building as well as the numerical model of the Pirelli Tower [30], [31], a land mark skyscraper located at Milan, Italy. It is concluded that, by fine-tuning the covariance of the fictitious process noise of the unknown input, a reasonable estimate of the state, useful for fatigue damage estimation could be accomplished.

The paper starts with a section devoted to a brief formulation of the state-space equations for linear dynamical systems. The next section introduces the dual scheme by use of the Kalman filter for estimation of both the unknown input and state of linear state-space models and is followed by a section on the numerical comparison of the dual Kalman filter, the augmented Kalman filter [29] and the filter proposed by Gillijn and De Moor (GDF) [26]. The paper is concluded by a section on the numerical investigation of the performance of the proposed algorithm when applied to both input and state estimation of the Pirelli Tower, located in Milan, Italy.

Section snippets

Mathematical formulation of the problem

A linear structural dynamics problem is typically formulated using the following continuous time second order differential equation:Mu¨(t)+Cu̇(t)+Ku(t)=f(t)=Spp(t)where u(t)n denotes the displacement vector and K, C and Mn×n stand for the stiffness, damping and mass matrix, respectively. f(t)n is the excitation force, which herein is presented as a superposition of time histories p(t)m that are influencing some degrees-of-freedom on the structure as indicated via the influence matrix Sp

Dual Kalman filter for input and state estimation

Consider the following discrete time state-space equation:ζk+1=Aζk+Bpk+vkζdk=Gζk+Jpk+wkwhere vkζ is the process noise assumed, zero-mean, white with covariance Qζ, and wkis the zero mean, white, measurement noise of covariance R. The problem at hand is to estimate the unknown input pk and the hidden or partially observed state ζk of the system using the noisy observations dk in an online fashion. In doing so, a dual implementation of the Kalman filter is proposed in this section. The proposed

Simulated example

To assess the performance of the proposed algorithm, an 8 DOF shear building (see Fig. 1) with system properties introduced in [36] is adopted, where the value of the mass of each floor is assumed to be 625 tones, and the inter-storey stiffness of each floor is equal to 109 kgf/m. Additionally, the modal damping ratio of each mode is assumed to be 2%.

Throughout the numerical analysis section, it is assumed that only accelerations of the response of the structure at the storey levels are

Application to a field inspired test case: the Pirelli tower

As a case study, we investigate the capability of AKF, GDF and DKF in state estimation using unknown input by considering the Pirelli Tower in Milan, shown in Fig. 18. The building features 39 stories, and its total height is about 130 m. The plan dimensions of the standard floor are approximately 70×20 m2. The structural components of the building are entirely made of cast-in-place reinforced concrete. In particular, the lateral load resisting system is comprised of the four triangular cores,

Conclusions

In this study, a dual implementation of the Kalman filter, namely the DKF, has been proposed to estimate the full states of a linear state-space model with unknown inputs. It is assumed that a limited number of noisy acceleration measurements are available. This data together with the known physical model of the system are incorporated into the DKF for accomplishing this objective. It has been demonstrated that the successive structure of the suggested filter prevents numerical problems

Acknowledgments

This research has been implemented under the “ARISTEIA” Action of the “Operational Program Education and Lifelong Learning” and was co-funded by the European Social Fund (ESF) and Greek National Resources. Authors are indebted to Dr. Gianluca Barbella and Prof. Federico Perotti, who provided the numerical model of the Pirelli Tower.

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