A natural observer for optimal state estimation in second order linear structural systems

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Abstract

A state observer for mechanical and structural systems is derived in the context of the second order differential equation of motion of linear structural systems. The proposed observer possesses similar characteristics to the Kalman filter in the sense that it minimizes the trace of the state error covariance matrix within the predefined structure of the feedback gain. The main contribution of the paper consists of the fact that the proposed observer can be implemented directly as a modified linear finite element model of the system, subject to collocated corrective forces proportional to the measured response. The proposed algorithm is effectively illustrated in two different types of second order systems; a close-coupled spring–mass–damper multi-degree of freedom system and a plate subject to transverse vibrations.

Highlights

► A natural state observer for linear second order systems is presented. ► The proposed observer minimizes the state error covariance given the structure of the feedback gain. ► The proposed state observer can be implemented directly as a modified finite element model of the system. ► The proposed state observer can account directly for colored excitation without state augmentation.

Introduction

Observers constitute a powerful tool to maximize the amount of information that can be extracted from measured vibration signals in mechanical and structural systems. Specifically, observers allow for the estimation of unmeasured response quantities in a system, given sparsely measured time history response and a model of the system. The vast majority of the existing body of theory regarding observers [7], [8], [14], [16], [17], [21] have been developed almost entirely in the context of first order state-space formulation.

First order state-space formulation is the most general and unified approach to treat estimation problems in diverse engineering and non-engineering applications. In practice, however, to implement standard first order observers in structural and mechanical systems modeled through finite element (FE) models significant pre- and post-processing is necessary. The system matrices need to be assembled from the FE model, the observer gain matrix computed and a simultaneous first order differential equation solver needs to be implemented separately. As has been noted by previous authors [9], [13], it makes the implementation in large scale FE models, typical of large space and civil structures, cumbersome, computationally inefficient and sometimes prohibitive for real-time applications, such as state feedback control [20].

In addition to the implementational issues, theoretical issues have also been raised by some authors related to the consistency of first order observers when applied to second order symmetric systems; especially loss of symmetry, definiteness and the fact that first order observers might yield an estimate of the state that does not correspond to the physical state sought. This was clearly demonstrated by Balas [2]. He concluded that unless certain restrictions are placed on the feedback gain matrix, estimates of the velocity generated by first order observers do not correspond to derivatives of the displacement estimates. Observers that satisfy this requirement are called natural observers because they maintain the second order structure of the underlying system whose state one is trying to estimate.

Implementation and theoretical issues mentioned above have prompted the development of “natural second order observers” for finite and infinite dimensional second order systems [2], [6]. The main objective is to show the conditions under which one can achieve arbitrary pole placement with a natural second order observer given that the system is observable and controllable (the reader is referred to the work of Hughes and Skelton [11], [12] regarding observability and controllability of second order systems). This allows for arbitrarily rapid decay of the state error under unknown initial conditions. The problem of natural second order observer design for finite dimensional systems is very closely related to the problem of eigenvalue assignment in a symmetric quadratic pencil [1], [5], [19]. In this type of problems one is interested in selecting matrices in order to achieve a certain location of the pencil eigenvalues while maintaining a certain internal structure inherited from the boundary conditions and physical characteristics of the underlying system. As it will be shown, the problem treated in this paper has two fundamental differences with respect to traditional design of natural second order observers. The first one is that unmeasured disturbances and measurement noise are considered, therefore it is closer to second order optimal state estimation [9], [13], [18]. The second difference lies in the fact that we not only require that the estimates of velocity correspond to the derivatives of the displacement estimates, but in addition we also require that the resulting observer be realizable as a second order finite element model (a physical system). This last requirement is fundamental in understanding the physical meaning of the Kalman filter in the context of structural dynamics and in improving the computational efficiency of the estimation, specially in the light of the powerful finite element solvers currently available.

The problem of second order state estimation has been previously considered in [9], [13]. The work by Joshi [13] proposes an estimator with guaranteed convergence while minimizing the state error covariance matrix. The proposed estimator requires the solution of two N-by-N coupled matrix algebraic equations, where N is the number of degrees of freedom. The proposed estimator is not natural and therefore cannot be implemented as a finite element model. On the other hand, the work of Hashemi and Laub [9] considers the problem of second order state estimators in the light of possible numerical efficiencies that can be obtained by exploiting the second order nature of the equations, specifically by approximating the second order equation of motion as a finite difference, the structure of the problem is preserved and the Ricatti equations required to deduce the feedback gain were reduced from one 2N-by-2N (as in the Kalman filter) to three N-by-N Ricatti equations. It is argued in [9] that this modification results in significant numerical efficiency without compromising the accuracy. As can be seen, both of the works cited above address important issues related to estimation in second order systems, but they do not deal with the issue of physical realizability.

In addition to the fundamental body of work cited in the above paragraphs, among the most recent and relevant work in application of observers for monitoring and state estimation of large scale structural systems are [3], [4], [10], [23]. The work of Charmichael reported in [3] constitutes one of the first applications of the idea of state estimation to structural engineering. It applies the Kalman filter to estimate the strain and strain rate of concrete due to unmeasured disturbance. In addition, it also treats the problem of parameter estimation using the extended Kalman filter (EKF) in a single degree of freedom system subject to a ground motion. In [23], Waller and Schmidt treat the general topic of observers and their application to structural dynamics is discussed, especially focusing on the effect of pole placement in the estimation accuracy, and the application of observers for damage detection and parameter estimation. In addition, the idea of a reduced order observer is presented, mainly as a means to deal with the application of observers for large models, however, no reference is made to the idea of natural second order observers. In [4] Ching et al. examine a particle filter approach and apply it in reconstructing the nonlinear response of building structures subject to earthquakes. The main contention is that the EKF does not provide accurate estimates when strong nonlinearities are present and the particle filter is proposed as a means to arrive at more accurate estimates.

In [10], Hernandez and Bernal develop an observer to perform state estimation for linear structural systems in the presence of large modeling error. This is suited for many problems in structural engineering where the main source of discrepancy between model estimates in open loop and the measurements arises due to modeling simplifications and parametric errors. The observer is successfully applied in reconstructing the response of a 52-story steel building subject to a seismic ground motion. Recent application of observers and KF in model updating and damage detection can be found in [24], [15], but this is outside of the scope of the present paper.

The main contribution of this paper is the development of an algorithm to implement second order state estimation under similar conditions as the Kalman filter but within the context of a modified finite element model of the system. That is, we derive a natural estimator that is a finite element model, as opposed to an abstract state-space model. As it will be shown in the body of the paper, this has several advantages: It allows the analyst to physically understand the process of optimal estimation in the context of structural dynamics, many times hidden behind the abstract formulae in state-space, second by preserving the algebraic structure of problem it allows for efficient implementation within finite element solvers, and finally, it satisfies the consistency problem described in [2].

Section snippets

Preliminaries

In this paper we restrict our attention to structural systems whose response can be modeled by the following set of ODE:Mq¨(t)+CDq˙(t)+Kq(t)=b2u(t)where M=MT>0RN×N is the mass matrix, CD=CDT>0RN×N is the damping matrix and K=KT>0RN×N is the stiffness matrix. The vector q(t)RNx1 is the displacement vector of the N degrees of freedom, and b2RNxr defines the spatial distribution of the excitation u(t)Rrx1. By defining the state vector xT(t)=[qT(t)q˙T(t)] Eq. (1) can be recast in first order

State observers

In general, the state estimate provided by an observer can be written in first order state-space form asx^˙(t)=Acx^+G(t)(y(t)Cx^)=(AcG(t)C)x^+G(t)y(t)where x^(t) represents the state estimate. As can be seen from the equation above, the observer state estimate is the response of the system subject to excitations consisting of the weighted difference between measured response and model response estimates (Fig. 1). Discrepancy between measured response and model predictions can arise mainly

Structure of natural second order observers

In this section we provide the necessary conditions for second order realizability of any observer of the formx^˙(t)=Acx^+G(y(t)Cx^)=(AcGC)x^+Gy(t)To begin, consider substituting the Ac and C matrices into Eq. (6). Performing the appropriate partitioning we arrive atq^˙(t)q^¨(t)=0IM1KM1CDq^(t)q^˙(t)+G1G2Δy(t)where Δy(t)=y(t)Cx^ is the output discrepancy at time t. To satisfy the top partition it is necessary that G1Δy(t)=0 which means that either G1=0 or Δy(t)N(G1) t. Since in general G

The proposed observer

For the purposes of this paper we will operate under the premise that velocity measurements are available and propose the following observer:Mq^¨(t)+(CD+MG2c2)q^˙(t)+Kq^(t)=MG2q˙m(t)The physical interpretation of Eq. (21) can be appreciated in Fig. 2. Based on Eqs. (1), (5) and (21) the dynamics of the state error are governed byMe^¨(t)+(CD+c2TDβc2)e^˙(t)+Ke^(t)=w(t)+c2TDβv(t)with the matrix Dβ still free to be selected. A close examination of Eq. (22) yields that the matrix Dβ reduces the

Numerical illustration 1

The purpose of this section is to illustrate the obtention of β (Eq. (29)) and the accuracy of the proposed observer in comparison with a standard KF under ideal conditions. The system considered is a 10 degree of freedom close-coupled spring–mass–damper system. The mass matrix of the system is given byM=diag([1212121212])The springs are uniform with stiffness ks=1000 and each damper is proportional to the spring stiffness by a factor of 0.005. The system is grounded at the first mass with a

Numerical illustration 2

In this case the proposed methodology is implemented in a simulated square flat plate. The dimensions of the plate are 1.00 m×1.00 m. The thickness of the plate is 0.01 m, the elastic modulus is 210,000 MPa and the Poisson ratio is 0.30. The plate is simulated with 400 square finite elements of dimension 0.05 m×0.05 m. The boundary conditions are fixed for displacements and rotations at all edges. Each finite element has 12 d.o.f, four corresponding to the vertical displacement of the four corners,

Conclusions

The paper shows that the Kalman filter is not a natural state observer for the second order structural systems considered in this paper and consequently proposes a second order natural state observer that is realizable as a modified finite element model of the system. The proposed state estimate is computed as the dynamic response of a model of the structure with added dashpots at the measured degrees of freedom and subject to corrective forces at the measured degrees of freedom which are

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